Eindhoven University of Technology MASTER Inverse problems in Black-Scholes … · Abstract The Black-Scholes equation for option pricing is a famous model in nancial mathematics

Eindhoven University of Technology

MASTER

Inverse problems in Black-Scholesoption price forecasting with locally reconstructed volatility

Wolf J

Award date2015

Link to publication

DisclaimerThis document contains a student thesis (bachelors or masters) as authored by a student at Eindhoven University of Technology Studenttheses are made available in the TUe repository upon obtaining the required degree The grade received is not published on the documentas presented in the repository The required complexity or quality of research of student theses may vary by program and the requiredminimum study period may vary in duration

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors andor other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights

bull Users may download and print one copy of any publication from the public portal for the purpose of private study or research bull You may not further distribute the material or use it for any profit-making activity or commercial gain

Inverse Problems inBlack-Scholes OptionPrice Forecasting withLocally Reconstructed

Volatility

Masterrsquos Thesis

Jan-Philipp Wolf

Department of Mathematics and Computer ScienceCentre for Analysis Scientific computing and Applications

SupervisorsProf dr Sorin Pop TU Eindhoven (NL) University of Bergen (NO)

Dr Michiel Hochstenbach TU Eindhoven (NL)Drir Jan ten Thije Boonkkamp TU Eindhoven (NL)Prof dr Christian Rohde Universitat Stuttgart (DE)

SimTech No 6

Eindhoven 30th November 2015

Abstract

The Black-Scholes equation for option pricing is a famous model in financial mathematics Givena strike price and expiration date it tries to determine at what price an option should trade todayLots of research has been carried out in determining the validity of its suppositions implementa-tion details and extensions to accommodate for more realistic assumptions Noting the equivalenceof the Black-Scholes equation to the heat transfer equation reversing the time in this equationleads to an ill-posed problemFirst we present a way to treat this problem mathematically and numerically to forecast optionprices via the solution of a time inverse parabolic problem based on the work of Klibanov [29]This model introduces new boundary and initial conditions and leaves behind traditional notionslike the strike price and expiration date of an option contract We try to identify the most efficientapproach for the numerical solution of the problemSecondly we aim to improve prediction quality by incorporating more realistic market volatilitydata In fact obtaining volatility implied by todayrsquos option price requires the solution of anothernonlinear inverse problem Depending on the assumptions we impose on the dependence of volat-ility on maturity and strike price the complexity of problems and methods used to solve themvaries greatly Due to its computational simplicity we follow a linearization approach presentedby Bouchaev amp Isakov [11 21] The reconstructed volatility surface resulting from this methodcould then be used in the forecasting model and results be compared to the volatility functionused in Klibanov [29] However note that this is merely an experiment because convergence ofthe approximate solution to the exact has not been proved yet in this caseNext we discuss a trading strategy based on the result of the time inverse model and back testthis strategy with real market data We assert that trading only makes sense if the bid ask spreadof the optionrsquos underlying is large enough and see this conjecture confirmed when back testingFinally we are also interested in the prediction quality for different boundary conditions and theimpact of decision criteria on the strategyrsquos performance

Inverse Problems in Black-Scholes Option Price Forecasting with Locally ReconstructedVolatility

Preface

rdquoA gift is pure when it is given from the heart to the right person at the right time and at theright place and when we expect nothing in returnrdquo

The Bhagavad Gita

After five years of study across multiple disciplines and at two universities there are quite a fewpeople I feel indebted to Let me start off by thanking my parents and my sister for their relentlessand subtle support all the way through my academic life I appreciate that they allowed me tofollow my interests and never interfered with any of my plans

Secondly I would like to thank prof Rohde in Stuttgart and prof Pop in Eindhoven for super-vising a self initiated project that did not fully match their field of expertise However prof Poprsquosoversight allowed me to work with just the right specialists at TUe I want to express my thank-fulness to dr Michiel Hochstenbach for his help concerning inverse problems and to dr Jan tenThije Boonkkamp for his insights on numerical aspects Furthermore I express my gratitude toMikhail Klibanov from the University of North Carolina at Charlotte for answering my questionsregarding his paper

Thirdly I have to express my special gratitude to all my fellow students in Stuttgart who I re-member for their kind helpfulness and willingness to try to understand a problem by its root Ilargely enjoyed studying in groups and I will cherish these hours for the years to come During mystay in Eindhoven I have met some of the most amazing people from all around the world Thankyou all for making this period of my life one of the most memorable experiences imaginable andbroadening my horizon manifoldFinally thanks to Joos Buijs and Thijs Nugteren for providing the LATEXtemplate for this thesis

Jan-Philipp WolfEindhoven October 2015

Contents

Contents vii

List of Figures ix

List of Tables xi

1 Introduction 1

2 Preliminaries 521 Black-Scholes model (BSM) 522 On implied volatility 723 Model shortcomings and criticism 8

3 Time Inverse Parabolic Problem Option Price Forecasting 931 In analogy to the heat equation 932 Option price forecasting Time inverse BSM 1133 Solution approaches 12

331 Error estimation 12332 Tikhonov regularization 13

34 Convergence results Carleman estimates 13341 Carleman estimate 14342 Stability estimates and Regularization 15343 Discretization 17344 LSQR 21345 Minimization via Conjugate gradient method 22346 Multifrontal Sparse QR factorization 22

35 Numerical Experiments 23351 The backward-time problem Crank-Nicolson 23352 Verification test time inverse heat solution 23353 Assessment How to include boundary conditions 24354 Bid-Ask spread requirements 24355 Choice of regularization parameter 26356 Numerical instability of gradient method and lsqr algorithm 28357 Alternative Boundary conditions 28

4 Parameter Identification Implied Volatility as an Inverse Problem 3141 Ill-posedness of the inverse problem and literature overview 3242 Linearization approach by Isakov 32

421 Continuous time formula Dupire 32422 Linearization at constant volatility 33423 Uniqueness and Stability 34424 Numerical solution of the forward problem 35425 Numerical tests reconstruction of given volatility surfaces 36

CONTENTS

426 Towards a real world scenario Smooth interpolation of market prices andremarks 37

5 Putting everything into action A large-scale real market data study 3951 Option exchanges amp BATS 3952 Data gathering 3953 Data filtering 3954 Trading strategy 4055 Backtesting 40

551 Restriction on options with significant bid-ask spread A Comparison 41552 Modification Negativity of extrapolated ask price 41553 Impact of boundary conditions and decision criterion 42554 Influence of volatility coefficient 42555 Test on most actively traded options 42556 Alternative one-day trading strategy 42557 Forecasting with locally reconstructed implied volatility Future work 44

6 Conclusions 45

Bibliography 47

viii Inverse Problems in Black-Scholes Option Price Forecasting with Locally ReconstructedVolatility

List of Figures

11 First known printed forward contract for a share in the Dutch West India Companyworth 500 Flemish pounds at 1785 with a maturity of one year This print datesback to 1629 2

12 Payoff and profit for a call option on the left respectively put option onthe right with strike price K = 1 3

13 Payoff and profit for a short call option on the left respectively short putoption on the right with strike price K = 1 4

31 Heat equation solved forward and backward 1032 Price of an option with strike price K = 50$ T = 512 years in dependence of spot

price S Verification of analytical solution as calculated by Matlab for call andput vs Crank-Nicolson scheme for call and put 24

33 Heat equation solved backward with noise level δ = 10minus3 δ = 10minus2 andδ = 10minus1 and true initial distribution ψ(x) for different terminal times Te 25

34 Solution of price forecasting for set boundaries vs free boundaries com-pared to forward solution of the problem as obtained by Crank-Nicolson method

2535 L-Curve in double logarithmic scale for options with different bid ask spreads of

the underlying large spread ∆ = 26 sa+sb2 vs tight spread ∆ = 012 sa+sb

2for regularization parameter λ isin (10minus8 1) 26

36 GCV function G(λ) for the full range of regularization parameters λ isin (10minus8 1) onthe left and zoomed in around the minimum on the right 27

37 Solution as obtained using different methods reference solution Matlabrsquos back-

slash (Multifrontal QR) Matlabrsquos LSQR and backslash solution of the

gradient system Regularization parameter was set to λ = 01 2938 Price using quadratic polynomial +5 minus5 regression +5

minus5 mean +5 minus5 weighted average +5 minus5 29

41 Implied volatility surface for MSFT options Data retrieved on 05072015 31

42 Reconstructed volatility surface compared to prescribed volatility surface 3743 Price of put option for different strikes on MSFT stock with maturity date

17062016 as observed on 14092015 with smoothing spline 38

List of Tables

51 Comparison between samples from all options versus samples from options withsignificant bid-ask spread 42

52 Effects of incorporating the modified trading rule mentioned in Section 552 4253 Impact of different combinations of boundary conditions and decision criteria as

presented in Section 343 4354 Experiment Linear regression function as volatility 4355 Experiment Most actively traded options 4356 Experiment One-day trading strategy 43

Chapter 1

Introduction

Prediction is very difficult especially if it is about the futureMarkus M Ronner 1918Correct attribution is hard especially for the pastDoug Arnold 2010

Since this thesis deals with the prediction of option prices we give a short overview over theclasses of market prediction techniques and their relation to the term market efficiency Addition-ally we outline the history of option trading to better understand the usefulness of this financialdevice to the world economy Finally modern day terminology and technical details are introducedand instructive examples serve to understand the mechanics of options trading the leverage effectas well as some of the risks involvedFirst of all the question whether the prediction of market movement is possible at all hinges onthe question of market efficiency A market is deemed efficient if all relevant information is alwaysincorporated into the current share prices The efficient-market hypothesis then states that itis consequently impossible for anybody to outwit the market and stock prices behave accordingto the closely related random walk hypothesis which states that stock prices are governed by arandom walk and are therefore unpredictable However the joint hypothesis problem refers to thefact that testing markets for efficiency is impossible because any such test would have to involvean equilibrium asset pricing model which by definition is not able to capture anomalous behaviorAny evidence for anomalous market behavior may therefore be the result of market inefficiency oran insufficient model In short there is no way to tell where anomalies originate from if equilibriumasset pricing models are used and therefore it is impossible to prove market efficiency In factthere are now three forms of market efficiency in existence Namely weak-form semi-strong-formand strong-form market efficiency They vary in the degree to which information is incorporatedinto current share prices and the speed in which they are incorporated For more information onthese variations of market efficiency see eg Jones [23] Behavioral finance or economics tries toexplain market inefficiencies by analyzing prevalently human factors like irrational decision mak-ing mental emotional filters and the likeStock market prediction has existed as long as modern day trading exists The most importantaccounts are made by Joseph de la Vega [47] who describes aspects of technical analysis in Dutchmarkets as early as in the 17th century There are three broad categories of prediction methodsFundamental analysis technical analysis and alternative methods Whereas fundamental tries tomake predictions based on an analysis of the stock issuing company technical analysis is concernedwith finding patterns in past prices using eg time series analysis Statistical techniques as variousmoving average methods and the use of candlesticks are widely applied as well The third andmost recent development concerns machine learning ie making predictions using feed forwardneural networksThis project is a challenging mathematical problem with a link to large real world data sets andapplication to but not limited to finance For future works there is a connection of the math-ematical field of inverse problems to machine learning to be explored However this project isbased on a recent publication dealing with option price forecasting in connection with a classicinverse problem by Klibanov [29] The beauty of mathematics is that the methods used in thistopic are directly transferable to other inverse problems ranging from inverse scattering Au [4]electrocardiography Sarikaya [41] precision measurement in chip production Soulan [44]

The previous section dealt with stock market prediction whereas this thesis is about optionprice prediction Let us clarify the terminology and connection between stocks and options now

CHAPTER 1 INTRODUCTION

Figure 11 First known printed forward contract for a share in the Dutch West India Companyworth 500 Flemish pounds at 1785 with a maturity of one year This print dates back to 1629

An option is the right but not the obligation to buy or sell an agreed upon amount of stock at acertain point of time in the future at a predetermined price The price paid for this right is calledthe option premium When one obtains the option to buy stock this is referred to as a call optionwhereas when one obtains the right to sell stock this is referred to as a put option Since thevalue of an option depends on the underlying stock options are derivatives ie they derive theirvalue from another financial product Options are not restricted to stocks but are also availableon commodities interest rates foreign currencies and bondsTo better understand the (original) purpose of this special instruments let us have a look at thehistory of forward contracts and options Forward contracts are closely linked to options with theonly difference being that the buying respectively selling is not optional but mandatory Forwardtrading of commodities as grain or salt had already appeared in medieval times Indeed one ofthe first option buyers was the Greek philosopher Thales of Miletus who bought the right to rentolive presses in anticipation of a larger than usual olive harvest When this harvest turned out tobe larger than usual indeed he rented out the spaces for a higher price thus making him one ofthe first option exercisers Saunders [42] Even though forward contracts have also been found onMesopotamian clay tablets written in cuneiform scripture dating back as far as 1750 BC thereare few records on how this form of financial instrument was traded among business men and howprices were set Insightful records are only available from the 17th century with Joseph Pensode la Vegarsquos Confusion de confusiones [47] from 1688 being one of the most important resourcesespecially for the situation in the Netherlands Around the year 1500 is the time when with therise of a system of periodical fairs trading in options became common for people who were notdirectly involved in trading the underlying commodities Venice became a city of regular fairs Thefirst permanent markets were to be found in Antwerp Belgium with foreign merchants settlingin the Scheldt port after 1500 Goetzmann [16] Another milestone are the Dutch voyages to Asiastarting around 1590 which resulted in the emergence of companies like VOC Dutch East andWest Company as well as in a spike in futures trading One example of how such a contract lookedlike at that time can be seen in Fig 11 Furthermore the public opinion regarding derivativeshad changed from a negative view influenced by the Roman laws under which it was regardedas taking chances and therefore equated to betting to a more liberal one where regulation was

2 Inverse Problems in Black-Scholes Option Price Forecasting with Locally ReconstructedVolatility

0 05 1 15 2

s price of underlying

Pcpayoff

0 05 1 15 2

s price of underlyingPppayoff

Figure 12 Payoff and profit for a call option on the left respectively put option on theright with strike price K = 1

demanded rather than outright bans For more information on the history of derivative tradingwe refer to Goetzmann [16] from which lots of inputs for this section have been taken As statedin Haug [20] there were active option exchanges in London Paris and New York as well as otherEuropean markets in the late 1800 and early 1900 Furthermore there was a Foreign ExchangeOption Market from 1917 to 1921 as reported by Mixon [35] There is proof that traders usedsophisticated techniques to price options in the American equity option in 1870 (see Kairys [24])or on the Johannesburg Stock Exchange in the early 20th century (see Moore [36]) With theinstallation of the first transatlantic telegraph cable in 1858 by Cyrus West Field and his AtlanticTelegraph Company arbitrage trading emerged From Haug [20] rdquoAlthough American securitieshad been purchased in considerable volume abroad after 1800 the lack of quick communicationplaced a definite limit on the amount of active trading in securities between London and NewYorkmarkets (see Weinstein 1931) Furthermore one extant source Nelson (1904) speaks volumesan option trader and arbitrageur SA Nelson published a book The ABC of Options and Arbit-rage based on his observations around the turn of the twentieth century The author states thatup to 500 messages per hour and typically 2000 - 3000 messages per day were sent between theLondon and the New York market through the cable companies Each message was transmittedover the wire system in less than a minute In a heuristic method that was repeated in DynamicHedging by one of the authors (Taleb 1997) Nelson describe in a theory-free way many rigor-ously clinical aspects of his arbitrage business the cost of shipping shares the cost of insuringshares interest expenses the possibilities to switch shares directly between someone being longsecurities in New York and short in London and in this way saving shipping and insurance costsas well as many more similar tricksrdquo

Nowadays options on stocks commodities foreign currencies bonds and interest rates aretraded on exchanges all around the world Along with options on different kinds of underlyingthere are also different styles of options The most common option is the European style optionEuropean options can only be executed on the day of maturity On the contrary American optionscan be executed on or before the day of maturity These two most common types of options arealso called vanilla options Furthermore there are binary options where the payoff is either afixed amount or nothing and all kinds of exotic options which could involve foreign exchangerates as in quanto options depend on multiple indexes as with basket options or depend on thevalue of the underlying not only at maturity but also on multiple dates before maturity (Asianlookback barrier digital spread options etc) Note that these options are usually traded over

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

s price of underlyingP

payoff

Figure 13 Payoff and profit for a short call option on the left respectively short putoption on the right with strike price K = 1

the counter (OTC) To get a feel for the significance of this kind of trading note that in 2013the Chicago Board Options Exchange (CBOE) which is the biggest American option exchangereported a trading volume of 561503190262$ To better understand the link between stock priceand option price and to understand how options work take a look at the payoff diagrams for acall and put option with strike price K = 1$ in Fig 12 The payoff in the left plot of a calloption is Pc = max (sminusK 0) = (sminusK)+ The difference between payoff and profit is the optionpremium ie the amount the buyer pays to obtain the right to exercise the option Converselythere is a payoff from a put option (plot on the right in Fig 12) once the underlying price s atmaturity is lower than the strike price and one profits from this option once this differences isgreater than the premium payed to the put option seller The profit from a put option is thereforePp = max (K minus s 0) = (K minus s)+ From these quantities another vital relation can be derivedThe equality Pc minus Pp = (s minus K)+ minus (K minus s)+ = s minus K is called the put-call parity providing adirect link between the price of a put and the price of the corresponding call option Furthermorethis means that only either the call or the put has to be priced Next consider the case of writingcall and put options In the case of a short call this means that one sells the right to buy sharesrespective in the case of a short put the right to sell shares to a second party The payoff andprofit diagrams for these kinds of options are presented in Fig 13 Note that short call optionsare especially risky since there is no limit on possible losses Finally it is worth mentioning thatleverage can be created through options Assume you have $ 1000 to invest Instead of buying 10shares of company ABC at $ 100 you could buy five option contracts at eg 200 a piece allowingyou to control 500 shares instead of 10 Note that one option contract is usually written on 100shares This example also reveals the risk associated with option trading

Chapter 2

Preliminaries

The underlying model for all the considerations to follow is the option pricing model of FischerBlack and Myron Scholes [8] from 1973 Therefore we will explain the modelrsquos assumptions andsimplifications derive the model equation and a closed form solution for European style options 1Furthermore the modelrsquos limitations will be analyzed and the reader will be made aware of possiblemodifications improving the accuracy of results In fact one of the most delicate quantities in thismodel is the volatility coefficient σ Historical data reveals that constant volatility as assumedin the standard Black-Scholes model can not reproduce real world market data For this reasonstrategies to better model volatility will be presented Finally we deal with the criticism BSMhas received and we outline the similarities with older pricing options which have been used sincethe 19th century (see eg Haug amp Taleb [20])

21 Black-Scholes model (BSM)

A short derivation of the Black-Scholes equation is given The aim is to calculate a fair price of anEuropean option given the underlying does not pay dividends A European call option is the rightto buy an asset at a specified strike price K in time T from today In this case the option writer isobliged to sell the asset The price paid for the right to execute the deal by the buyer respectivelythe money received by the option issuer is called the premium In the case of a put option thebuyer of the option has the right to sell the asset at price K in time T whereas the option issueris obliged to buy the asset In 1973 Fischer Black and Merton Scholes [8] proposed their famousmodel on how to price European options with riskless interest rate r where the underlying doesnot pay dividends The basic idea is to buy and short sell a certain amount of options at the sametime to eliminate risk This is a list of assumptions on both assets and the market

1 The risk-free rate of return is constant in time

2 The underlying (ie the stock) does not pay dividends

3 The logarithmic returns of the stock follow an infinitesimal geometric Brownian motion withconstant drift and volatility

4 The market allows for no arbitrage

5 Fractional buy and sell positions on the stock are possible at all times

6 Fractional lending and borrowing at the riskless rate r is possible at all times

7 Frictionless market there are no transaction fees or costs

We assume the price of an optionrsquos underlying s follows a geometric Brownian motion

s= microdt+ σdW (21)

This signifies that the uncertainty is coming from the term Brownian motionW (t) In a geometricBrownian motion the logarithm of the quantity (in this case s) follows a Brownian motion Heremicro is the expected return of the stock and σ is related to the standard deviation and is called

1European style the option can only be exercised on the expiration date

CHAPTER 2 PRELIMINARIES

volatility of the log returns in percent per year Since dW (t) is a Wiener process the followingproperties hold

E[dW ] = 0 E[dW 2] = dt (22)

To understand the relation between the volatility σ and the variance of ds consider

E[ds] = E[smicrodt+ σsdW ] = E[smicrodt] + E[σsdW ] = smicrodt+ σsE[dW ] = smicrodt (23)

V ar[ds] = E[ds2]minus E[ds]2 = E[(smicrodt+ σsdW )2]minus (smicrodt)2 = σ2s2E[dW 2] (24)

Since the standard deviation is the square-root of the variance the volatility σ is proportional

toradic

V ar[ds]

sradicdt

We now want to find a fair price u(s t) for an option in dependence on the time t

and value of the underlying s where we know the price at maturity u(s T ) Since s follows astochastic process we introduce Itorsquos lemma for two variables

Lemma 1 Assume Xt is an Ito drift diffusion process satisfying

dXt = microtdt+ σtdBt (25)

where Bt is a Brownian motion Also assume that f isin C2 Then in the limit dtrarr 0 we have

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

For a proof see eg Karatzas [25]

We derive and apply Itorsquos lemma to derive the Black-Scholes equation The Taylor expansionof u(t s) is

du =partf

parttdt+

partsds+

part2f

partx2ds2 +O(dt2 ds3) (27)

Recalling equation (21) we substitute s (microdt+ σdW ) for ds

(partu

partt+partu

)(smicrodt+ sσdW ) +

part2u

parts2(s2

(micro2dt2 + 2microdtσdW + σ2dW 2

))+O(dt3 dW 3)

(28)Forming the limit dtrarr 0 and considering that dB2 is O(dt) we obtain

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

We now consider a portfolio containing one short option and partuparts shares of the optionrsquos underlying

Let the value of this so-called delta hedge portfolio at time t be denoted by D(t) = minusu+ partuparts s Let

∆t gt 0 be a time discretization and consider the portfoliorsquos change in value over the time interval[t t+∆t]

∆D = minus∆u+partu

partsδs (210)

Discretizing equations (21) and (29) and inserting them into (210) gives

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

where the Brownian motion dW has vanished Taking the riskless interest rate r into account wehave

∆t= rD (212)

Inserting (211) and D = minusu+ partuparts s into this equation and transforming the discretized operators

into continuous ones we conclude with the famous Black-Scholes PDE

partt+σ2

2s2part2u

parts2+ rs

partsminus ru = 0 (213)

The model is completed by boundary conditions

u(s = 0 t) = 0 forallt u(s t) rarr s for srarr infin (214)

and terminal conditionu(s T ) = maxsminusK 0 (215)

In the case of an European option (an option that can only be exercised at the maturity date)Black and Scholes derived an analytical solution to the PDE This can be done by transformingthe equation into a heat diffusion equation

Lemma 2 The respective explicit formulas for the value of a call C(t s) and put P (t s) are

C(t s) = sN(d1)minusK exp(minusr(T minus t)N(d2)) (216)

P (t s) = K exp(minusr(T minus t))N(minusd2)minus SN(minusd1) (217)

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

d2 = d1 minus σradicT minus t (219)

with the cumulative standard distribution function

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

This means that for every pair (KT ) there is exactly one fair option price u

22 On implied volatility

The most delicate parameter in the Black-Scholes model is the volatility coefficient σ In financialterminology volatility is defined as the standard deviation of the instrumentrsquos yearly logarithmicreturns To understand the concept of logarithmic returns let us first define the simple annualreturn or simple annual rate of return of an investment with value V0 at the start of the year andV1 at the end of the year as

r =V1 minus V0V0

Furthermore the relation between the return R over a period of time t can be converted to anannualized rate or return r by the following relationship

1 +R = (1 + r)t (222)

The continuously compounded return or logarithmic return is then given by

Rl = lnV1V0 (223)

with the logarithmic rate of return over a period t given by

rl =Rl

t (224)

One of the reasons for the usage of logarithmic returns in financial modeling is related to thedefinition of the cumulative returns ri (i isin 0 1 2 k) over k + 1 periods of time ie

Rc = (1 + r0) middot (1 + r1) (1 + rk)minus 1 (225)

which can be represented as a simple sum if the returns ri are logarithmic

ksumi=0

ri (226)

Therefore volatility is a measure for the magnitude of uncertainty regarding an instrumentrsquos(logarithmic) return One distinguishes between two types of volatility namely actual and impliedvolatility Actual volatility is the volatility inferred from historical data with the most recent databeing todayrsquos data One also speaks of actual historical volatility if the last available data datesfrom earlier than today and actual future volatility for the volatility of an instrument endingat a point of time in the future On the other hand there is an implied volatility analogonfor each of these types Implied volatility is the volatility for which a financial model (such asthe Black-Scholes model) returns the actual observed market price The volatility is thereforerdquoimpliedrdquo by the modelRecall that the volatility coefficient in BSM is a constant In reality one observes a differentscenario though depending on which market is considered different strike prices will result indifferent volatility values Usually options with very low or very high strike prices will implyhigher volatilities than near the money options This phenomenon is called the smile effectAdditionally volatility does not seem to only depend on strike price but also on term structureie maturity dates Taking the dependences on strike K and maturity T into account one willend up with an implied volatility surface serving as a representation of the mapping

σ(KT ) R2 rarr R (227)

A vast body of research is available on how to retrieve this volatility information from eg the(modified) Black-Scholes model Some of the methods will be presented others will be mentionedand references will be given in Chapter 4

23 Model shortcomings and criticism

Haug amp Taleb [20] also criticize that Black amp Scholes use an academic and theoretical argument toderive a formula that has been in practitionerrsquos hands for a long time More specifically they cri-ticize that the model is based on dynamic hedging meaning buying and selling fractional amountsof securities at arbitrary times In reality this is not possible due to technical restrictions Fur-thermore they argue that traders do not do valuation based on unknown probability distributionsbut price options such that they are compatible with put-call parity and other instruments in themarket As such the degrees of freedom are heavily restricted and pricing models like Black-Scholesare unnecessary Further points of critique concern some of the main assumptions of the modelFor example the fact that the increments of the price of the underlying are modeled as normallydistributed does not make the model robust to large unexpected changes in the market regimeUltimately that is what caused the decline of Scholesrsquo and Robert C Mertonrsquos LTCM hedge fundwhich relied on dynamic hedging strategies Losses totaled $46 billion within less than 4 monthsin 1998 and resulted in a bailout initiated by the United States Federal Reserve see Edwards [15]Research also exists on modifying the Black-Scholes framework to accommodate for and studythe effects of transaction costs Amster [3] dividend payments and on incorporating non-constantvolatilitySince we are not interested in pricing options directly but in obtaining a trend into which directionan option might be trading in the future we think that it is still important to investigate whetherBlack-Scholesrsquo model can be used for such purposes

Chapter 3

Time Inverse Parabolic ProblemOption Price Forecasting

The Black-Scholes model is a way of deriving a fair price of an option today given a futureexpiration date T The main idea of Klibanov [29] is to reverse the time t and calculate ieforecast a future option price given todayrsquos market data First we show why this inverse problem isill-posed in the sense of Hadamard using the analogy to the heat equation as an accessible exampleNext the full model including new boundary and initial conditions is presented Thereafter wepresent the most common numerical solution approaches and reproduce convergence results whichare based on Carleman estimates for one such class of procedures from Klibanov [29 27] Theregularization technique treating this ill-posed problem is the famous Tikhonov regularizationapproach for the solution of ill-posed inverse problems Some remarks on the role of data errorsand the inevitability of having estimates on the input data error for the successful constructionof regularization algorithms are made Finally some numerical experiments are carried out todetermine optimal regularization parameter influence of different boundary criteria and preferablenumerical solution techniques

31 In analogy to the heat equation

Recalling that the Black-Scholes is of the parabolic type let us consider the heat equation for amoment In its simplest form the heat equation can be written as

ut minus uxx = 0 in Ω sub Rtimes (0infin) (31)

u(x 0) = f(x) for x isin Ω (32)

u(middot t) = 0 on partΩ (33)

For simplicity let us consider the one-dimensional case and let the domain be bounded such thatx isin Ω = (0 π) and let the terminal time T gt 0 be chosen such that t isin (0 T ) sub (0infin) It iswell known that there exists a unique solution to this initial boundary value problem which canbe represented by Fourier series

Lemma 3 The solution of the one-dimensional heat equation (31) with initial and boundaryconditions (32)-(33) is given by the Fourier series

u(x t) =

infinsumk=1

fk sin(kx) exp(minusk2t) (34)

with the Fourier coefficients fk k = 1 2 infin given by

int π

f(x) sin(kx)dx (35)

Now consider the time inverse problem

ut + uxx = 0 in Ω sub Rtimes (0infin) (36)

CHAPTER 3 TIME INVERSE PARABOLIC PROBLEM OPTION PRICE FORECASTING

minus06 minus04 minus02 0 02 04 06

minus1000

Figure 31 Heat equation solved forward and backward

Again consider the one-dimensional case with the bounded domain Ω = (0 π) and t isin (0 T ) sub(0infin)

Lemma 4 The solution of the time inverse heat equation (36) with initial and boundary condi-tions (37))-(38) restricted to one space dimension is given by

u(x t) =

infinsumk=1

fk sin (kx) exp (k2t) (39)

This representation is instructive in understanding why the inverse time problem is ill-posedObserve that for the solution (39) to exist the squares of the Fourier coefficients fk with respectto k have to decay exponentially Another typical problem in the context of inverse problems isthe instability of the solution to small fluctuations in eg initial data Consider the norm of theu(x t)

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

and remember that the Fourier coefficients are obtained via the convolution with the initial dataie f(x) in our case This means that small deviations in fk can result in large variations ofthe solution u(t x) From (310) we can also deduce that the problem becomes more unstableas time t proceeds To confirm these propositions and to get an impression for what this meansin terms of numerical aspects we solve the heat equation forwards and backwards using a naiveexplicit solution scheme and plot the results in Fig 31 For the purpose of demonstration we letΩ = (x t)|x isin (minus05 05) t isin (0 1) and let

f(x) =

100 |x| lt 1

10 else(311)

The result after t = 10 is plotted in Fig 31 on the left Applying the same scheme with timereversed leads to oscillations and a non realistic result as seen in Fig 31 on the right afteronly 10 time steps The discretization parameters dt dx used fulfill the CFL condition dt

dx2 =10minus4

( 160 )

2 = 036 lt 12 All numerical examples to appear from hereon have been executed with Matlab

R2015aThus this problem belongs to a special class of inverse problems namely ill-posed inverse

problems According to Jacques Salomon Hadamard [18] a problem is well-posed if

1 A solution exits

2 The solution is unique

3 The solutions is depends continuously on the initial data

A problem is ill-posed if any of these three stipulations does not hold

The time reverse heat equation is a well-known and classic example for an inverse ill-posedCauchy problem For an overview of methods applied to tackle such problems see eg Seidman [43]Useful information on data errors error estimation and the impossibility to construct solutionalgorithms without any knowledge on the error level of the input data can be extracted fromYagola [48] Alternative techniques as in eg Marie [34] try to solve well-posed approximationsto the ill-posed problem rather than to solve the ill-posed problem

32 Option price forecasting Time inverse BSM

And now remains that we findout the cause of this effectWilliam Shakespeare Hamlet

We use the model presented in Klibanov [29] which entails the Black-Scholes equation along withspecial initial and boundary conditions as follows

Lu =partu

partt+σ(t)2

2s2part2u

parts2= 0 on QT (312)

u(s 0) = f(s) for s isin (sa sb) (313)

u(sb t) = ub(t) u(sa t) = ua(t) for t isin (0 T ) (314)

where the domain is given as

QT = (s t)|s isin (sb sa) t isin (0 T ) (315)

and with

f(s) =ub minus uasb minus sa

s+uasb minus ubsasb minus sa

Note that f(s) is simply a linear interpolation between the optionrsquos bid price ub and ask price uaIn the general context it is not necessary for f to be linear though As we will see later forecastingis done for a rather short time horizon This is why the boundary conditions in (314) namelyub(t) and ua(t) are the extension of the quadratic interpolation polynomial of preceding bid andask prices in the time interval (minusT 0) The same goes for the volatility coefficients σ(t) which isthe extrapolation of the quadratic interpolation polynomial for the volatility in (minusT 0) Let theSobolev space Hk(QT ) be defined as follows

Definition Let 1 le p ltinfin and k isin Z+ Suppose u isin Lp(QT ) and the weak derivatives partαu existfor any multi-index α with |α| le k such that

partαu isin Lp(QT ) |α| le k (317)

We then say that u isinW kp and W kp is called Sobolev space For p = 2 we define Hk =W k2

For our case letH21 be similarly defined as the space of functions u isin L2(QT ) with partsu partssu parttu isinL2(QT )Deviations from the function

F (s t) =ub(t)minus ua(t)

sb minus sas+

ua(t)sb minus ub(t)sasb minus sa

isin H2(QT ) (318)

will be penalized and these relations hold

F (s 0) = f(s) F (sb t) = ub(t) F (sa t) = ua(t) (319)

The fact that deviations of the solution u from the function F are penalized is already a regu-larization technique In the spirit of Tikhonov regularization a solution that solves the originalproblem as well as possible but at the same time fulfills certain restrictions is searched for Re-member for example the noncontinuous dependence of the norm of u(x t) in (310) on the Fouriercoefficients In that case one could for example ask for an approximation to u(x t) with boundednorm Something similar is done here we want that equation (312) is fulfilled in the integralsense while at the same time we demand that u is close to an interpolation of option prices F This gives rise to the Tikhonov functional

Jλ(u) =

int sa

(Lu)2dtdx+ λ uminus FH2(QT ) (320)

where the regularization parameter λ isin R determines how much deviations of u from F arepenalized as well as how smooth the solution isIn the following section the discretization of this model the choice of the regularization parameterλ and alternative regularization techniques are discussed Furthermore the convergence prooffor this Tikhonov functional is replicated from Klibanov [29] and some remarks on the iterativesolver for the resulting system of linear equations are made Another important problem is theway in which the model error can be estimated and the question how this error level relates to thechoice of the regularization parameter λ The chapter is concluded with a handful of numericalexperiments in which the need for higher order Tikhonov regularization is debated and concreterecommendations for the mesh size and regularization parameter λ for this problem are made

33 Solution approaches

There are several techniques that are regularly applied to these types of inverse problems Wewill have a closer look at a method called Tikhonov regularization which will be used for thisproblem However the general problem is sketched first and thereafter we define what a solutionof an ill-posed problem is in connection with perturbations in the input data

331 Error estimation

Since we ultimately want to make statements on the convergence properties of regularized solutionsof ill-posed problems to the exact solution of the problem an interesting question to address iswhether or not it is possible to estimate errors and if so what a priori information is necessary todo so A precise and general survey on data errors and error estimation is given by Yagola [48] Ingeneral a rdquostable regularization method does not guarantee the possibility of an error estimationor a comparison of convergence ratesrdquo [48] Also it is essential to include all a priori informationin the mathematical problem formulation before trying to solve it Let us start by answering thequestion whether it is possible to solve an ill-posed problem without knowledge of the data errorConsider the operator equation

Ax = u x isin X u isin U (321)

Here the unknown is x XU are Hilbert spaces and the operator A X rarr U is bounded andlinear Recalling Hadamardrsquos requirements for well-posed problems we assert that stability meansthat if instead of the exact data (A u) we are given admissible data (Ah uδ) with ||AminusAh|| le h||u minus uδ|| le δ then the approximate solution converges to the exact one as h and δ approachzero The problems of existence and uniqueness of the problem are usually dealt with by taking ageneralized solution of the problem In the case of non-uniqueness one usually takes the solutionwhich has the smaller norm or smaller distance from a fixed element that arises from a prioriknowledge The main problem however is that the search for such a generalized solution can still

be unstable with respect to errors of A and u This brings us to the definition of solution of anill-posed problem Tikhonov [45] in his seminal paper on incorrectly formulated problems andhis regularization method states that to solve an ill-posed problem there has to be a mappingR(h δAh uδ) which

bull maps a regularized solution xhδ = R(h δAh uδ) to the data of the problem with h ge 0δ ge 0 Ah bounded and linear map uδ isin U

bull and guarantees convergence of xhδ rarr x as h δ rarr 0

Note that not all ill-posed problems allow for the construction of such a regularizing algorithmand are therefore termed rdquononregularizable ill-posed problemsrdquo One interesting result is the factthat it is impossible to construct stable regularizing algorithms without the knowledge of the errorlevel (h δ) see Yagola [48] or Bakushinskii [7] for a proof

332 Tikhonov regularization

Apart from giving this definition of the solution to an ill-posed problem Tikhonov also proposed aregularization algorithm The method searches for a minimum xλ = xλ(h δ) isin X of the Tikhonovfunctional

Tλ(x) = ||Ahxminus uδ||2 + λ||x||2 (322)

selecting the regularization parameter λ such that convergence of the approximate solution xhδ tox is ensured We do not give any more general information on Tikhonov functionals here becauseconvergence properties of the Tikhonov-like functional for the case of our inverse problem (320)will be proved in Section 34 For more details on the general case see eg Tikhonov [45 46]

34 Convergence results Carleman estimates

He was a genius My older friends in Uppsala used to tell me about the wonderful years theyhad had when Carleman was there He was the most active speaker in the Uppsala MathematicalSociety and a well-trained gymnast When people left the seminar crossing the Fyris River hewalked on his hands on the railing of the bridge Bo Kjellberg 1995

To prove that the Tikhonov approach guarantees the existence of at most one solution to theinverse problem and to establish convergence rates for the numerical minimization of the Tikhonovfunctional Carleman estimates play a key role Specifically the Carleman estimate allows one toobtain a logarithmic stability estimate (Theorem 1) which in turn guarantees uniqueness of thesolution to the parabolic problem and is needed to establish convergence rates of the Tikhonovminimizer to a solution of this problem The following theorems from Klibanov [27] are presentedfor a general parabolic problem of the form

ut + Lparu = f in QT = Ωtimes (0 T ) u isin H2(QT ) (323)

u(s 0) = g(s) (324)

u(s t) = p(s t) on partΩtimes (0 T ) (325)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

aij(s t)uij A = partt + Lpar (327)

where uj =partupartsj

aij isin C1(QT ) bj b0 isin B(QT ) Here B(QT ) is the space of bounded functions and

L0par is the principal part of the parabolic operator for which the following ellipticity conditionholds there exist two constants micro2 ge micro1 gt 0 such that

micro1|η|2 lensum

aij(s t)ηiηj le micro2|η|2 foralls isin QT forallη isin Rn (328)

Even though it sounds more natural to assume u isin H21(QT ) in (323) the extra smoothnessis needed to prove stability and convergence results later on and is specifically mentioned in eg(337) This way problem (312) becomes a special case in which f = 0 g(s) = f(s) with f(s) asdefined in (313) and p(s t) = ub(t) respectively p(s t) = ua(t) as in (314) Let the lower orderterms of (326) be bounded by a constant M gt 0

bjC(QT ) leM j = 0 n (329)

341 Carleman estimate

Since the following estimate is the key to proving convergence of the Tikhonov functional to aunique solution of (323)-(325) we introduce the concept of Carleman estimates For an intro-ductory overview of the application of Carleman estimates for coefficient inverse problems seeTimonov [31] and Klibanov [27] for the application to ill-posed Cauchy problems Let the con-stant k gt 0 be a number which will be chosen later For λ gt 1 the Carleman Weight Functionϕλ(t) in our case is defined as

ϕλ(t) = (k + t)minusλ t gt 0 (330)

The combination of the following two lemmata which originate from Klibanov [30] will yield theCarleman estimate which will be used in the following proof

Lemma 5 There exists a number λ0 gt 1 and constant C gt 0 both depending on micro1 micro2 andmaxij aijC1(QT ) such that for all λ ge λ0 and for all u isin C21(QT ) the following estimate holds

for all (s t) isin QT

(minusut minus L0paru)uϕ2λ ge micro1|nablau|2ϕ2

λ minus λu2ϕ2λ + divU1 +

partt(minusu2

2ϕ2λ) (331)

|U1| le C|u||nablau|ϕ2λ (332)

Lemma 6 Additionally for λ0 C from Lemma 5 for all λ ge λ0 and for all u isin C21(QT ) thefollowing estimate holds for all (s t) isin QT

(ut + L0paru)2ϕ2

λ ge minusC|nablau|2ϕ2λ + λ(k + t)minus2u2ϕ2

λ+ (333)

(λ(k + t)minus1u2ϕ2

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

|U2| le C|ut||nablau|ϕ2λ (334)

The combination of the preceding two lemmata yields the Carleman estimate we are looking for

Lemma 7 (Carleman estimate) There exists a number a0 isin (0 1) depending on micro1 micro2 andmaxij aijC1(QT ) M such that if k + T lt a0 there exists a number λ1 depending on micro1 micro2

constants CC1 gt 0 and θ depending on micro1 micro2 and maxij aijC1(QT ) M such that for all λ gt λ1

and for all u isin C21(QT ) the following estimate holds for all (s t) isin QT

(ut + L0paru)2ϕ2

λ ge 2

3θmicro1|nablau|2ϕ2

λ + Cλu2ϕ2λ + divU (335)

(Cλ(k + t)minus1(1minus C1(k + t))u2ϕ2

λ minus Cϕ2λ

nsumij=1

aijuiuj

|U | le C(|ut|+ |u|)|nablau|ϕ2λ (336)

Proof The proof for this lemma is only outlined here

342 Stability estimates and Regularization

Consider the family of functions uδ with δ isin (0 1) satisfyingintQT

(parttuδ + Lparuδ)2dsdt le Nδ2 uδ isin H2(QT ) (337)

uδ(middot 0)L2(Ω) leradicNδ (338)

uδ = 0 on partΩtimes (0 T ) (339)

where N gt 0 is a constant independent of δ

Theorem 1 Let uδ satisfy conditions (337)-(339) Let T fulfill T gt a0

2 with a0 from Lemma 7Then there exists a number δ0 = δ0(Lpar) isin (0 1) and a number C10 gt 0 such that the logarithmicstability estimate holds for all uδ(s T )

uδ(middot T )L2(Ω) leC10radicln(δminus1)

(1 + uδH2(QT )) (340)

For every ε isin (0 T2) let β = β(ε) be defined as

β = β(ε) = minus ln(1minus εT )

2 ln(1minus ε(2T ))isin (0

2) (341)

Then also this Holder estimate holds

nablauδL2(QTminusε)+ uδL2(QTminusε)

le C10(1 + uδH2(QT ))δβ (342)

Theorem 2 Uniqueness of the solution to problem (323)-(325) is a direct consequence of The-orem 1

With these estimates at hand the final step is to prove the existence of a solution to the minimiz-ation problem arising with the Tikhonov regularization approach and to find an estimate for theconvergence rate in such a case Assume that there exists a function F isin H2(QT ) satisfying

F (s 0) = g(s) F = p(s t) on partΩtimes (0 T ) (343)

The Tikhonov functional corresponding to the (inverse) problem (323) is

JλLpar(u) = ut + Lparu2L2(QT ) + λ uminus F2H2(QT ) u isin H2(QT ) (344)

subject to initial and boundary conditions (324)minus (325) (345)

Since we are dealing with convergence analysis assume that there exists an exact solution ulowast isinH2(QT ) of (323)-(325)

g minus glowastL2(Ω) le δ f minus flowastL2(QT ) F minus F lowastH2(QT ) le δ (346)

The number δ can be interpreted as the error level in the input data In the case of the forecastingproblem the intervals s and t are defined on are small and it is assumed that approximating theexact F lowast linearly in s and t by F (s t) as defined in (319) results in a statement of the form (346)with δ gt 0 sufficiently small

Theorem 3 (Existence) If there exists a function F isin H2(QT ) satisfying (343) then for everyλ gt 0 there exists a unique minimizer uλ isin H2(QT ) of the functional (344) and the followingestimate holds

uλH2(QT ) leC11radicλ(FH2(QT ) + fL2(QT )) (347)

Proof Define H20 (QT ) = v isin H2(QT ) v|Γ = partnv = 0 Letting v = u minus F one can argue

that the Tikhonov functional is a bounded linear functional on the space H20 (QT ) The aim is to

minimize the following functional subject to boundary conditions (324) and (325)

Jλ(v) = ||Av + (AF minus f)||2L2(QT ) + λ||v||H2(QT ) v isin H20 (QT ) (348)

Assume the minimizer is given by vλ = uλ minus F isin H20 (QT ) and note that due to the variational

principle it should satisfy

(Avλ Ah) + λ[vλ h] = (Ah f minusAF ) forallh isin H20 (QT ) (349)

where (middot middot) denotes the scalar product in L2(QT ) and [middot middot] denotes the scalar product in H2(QT )Define

v hλ = (AvAh) + λ[v h] forallv h isin H20 (QT ) (350)

The expression v h is a new scalar product on H20 (QT ) and its corresponding norm fulfills

radicλ||v||H2(QT ) le vλ le C11||v||H2(QT ) (351)

The norms vH2(QT ) and ||v||H2(QT ) are equivalent and we can rewrite

vλ hλ = (Ah f minusAF ) forallh isin H20 (QT ) (352)

This yields|(Ah f minusAF )| le C(||F ||H2(QT ) + ||f ||L2(QT ))hλ (353)

This shows that the right hand side of (352) is a bounded linear functional on H20 (QT ) Rieszrsquo

theorem then implies a unique solution of the minimization problem The Carleman estimate isnot used in this proof

Theorem 4 (Convergence rate) Assume (346) holds and let λ = λ(δ) = δ2α with α isin (0 1]Also let T lt a02 with a0 isin (0 1) let ε β and let the Carleman estimate be defined as beforeThen there exists a sufficiently small number δ0 = δ0(Lpar a0 QT ) isin (0 1) and constant C12 gt 0

such that for all δ isin (0 δ1α0 ) the following convergence rates hold true

∥∥uλ(δ)(middot T )minus ulowast(middot T )∥∥L2(Ω)

le C12radicln(δminus1)

(1 + ulowastH2(QT )) (354)∥∥nablauλ(δ) minusnablaulowast∥∥L2(QTminusε)

+∥∥uλ(δ) minus ulowast

∥∥L2(QTminusε)

le C12(1 + ulowastH2(QT ))δαβ (355)

Note that the stronger Holder estimate (355) only holds for times close to T whereas the logar-ithmic estimate (354) is valid on the whole of QT

Proof Let vlowast = ulowast minus F lowast so vlowast isin H20 (QT ) and Av

lowast = flowast minusAF lowast Consider

(Avlowast Ah) + γ[vlowast h] = (Ah flowast minusAF lowast) + λ[vlowast h] forallh isin H20 (QT ) (356)

Subtraction of 349 and the definitions vλ = vlowast minus vλ f = flowast minus f F = F lowast minus F yield

(Avλ Ah) + λ[vλ h] = (Ah f minusAF ) + λ[vlowast h]forallh isin H20 (QT ) (357)

The test function is set to h equiv vλ and thus

||Avλ||2L2(QT ) + λ||vλ||H2(QT ) = (Avλ f minusAF ) + λ[vlowast vλ] (358)

Application of Cauchy-Schwarz inequality gives

||Avλ||2L2(QT )+λ||vλ||2H2(QT ) le

2||Avλ||2L2(QT )+

2||fminusAF ||2L2(QT )+

2(||vlowast||H2(QT )+||vλ||H2(QT ))

(359)Using the a priori information (346) this means

||Avλ||2L2(QT ) + λ||vλ||2H2(QT ) le C12δ2 + λ||vlowast||2H2(QT ) (360)

Noting that λ = δ2α with α isin (0 1] we deduce δ2 le λ and this is why (360) implies

||vλ||H2(QT ) le C12(1 + ||vlowast||H2(QT )) (361)

||Avλ||2H2(QT ) le C12(1 + ||vlowast||2H2(QT ))δ2α (362)

For convenience denote wλ = vλ(1+ ||vlowast||H2(QT )) Combining (361) and the Holder estimate from

Theorem 1 imply ||wλ||H2(QT ) le C12δαβ and

vλ||H1(QT+ε) le C12(1 + ||vlowast||H2(QT ))δαβ forallδ isin (0 δ0) (364)

Applying the triangle inequality onto vλ = (ulowastminusu)+ (F lowastminusF ) and using (346) once again yields

||vλ||H1(QT+ε) ge ||uλ minus ulowast||H1(QT+ε) minus ||F lowast minus F ||H1(QT+ε) ge ||uλ minus ulowast||H1(QT+ε) minus δ (365)

Putting the last two inequalities together and arguing that δαβ gt δ since β δ isin (0 1) and α isin (0 1]we get the final result presented in the theorem above

343 Discretization

The domain is given by QT = (sb sa)times (0 T ) with T = 2τ where τ isin (0 14 ) is time in years Thelimitation on τ is due to the restriction on T in Theorem 1 Furthermore sb lt sa always holds Inour case τ = 1255 represents one day The domain is discretized in tuples of the discretizationsof s and t The stock prices s isin (sb sa) are discretized into M equidistant points si = sb + i middot dsi = 0 M Respectively time t isin (0 T ) is split into N equidistant points such that ti = i middot dti = 0 N For all computations N =M = 100 Let unm denote the option price correspondingto a stock price sm at time tn ie unm asymp u(sm tn) The values unm are formatted into a vectorof this form

u = [u11u12 u1Mminus1u1Mu21u22 u2Mminus1u2M uNminus11uNminus12 uNminus1Mminus1uNminus1M

uN1uN2 uNMminus1uNM ]T isin RNtimesM (366)

The discretized version of the Tikhonov functional (320) can then be written as

Jα(u) = dtds

NMminusMsumj=M+1

(uj+M minus ujminusM

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

2s(j minus 1 (mod M) + 1)2

uj+1 minus 2uj + ujminus1

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

2dt(uj+M minus ujminusM minus Fj+M + FjminusM )

2ds(uj+1 minus ujminus1 minus Fj+1 + Fjminus1)

dt2(uj+M minus 2uj + ujminusM minus Fj+M + 2Fj minus FjminusM )

ds2(uj+1 minus 2uj + ujminus1 minus Fj+1 + 2Fj minus Fjminus1)

For the discretization of the Sobolev norm second order finite differences are used to approx-imate the derivatives For the first derivatives a central differencing scheme is used such that thetime derivative at the m-th price sm is given by

nablatum = Dtlowastum +O(dt) (368)

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

To assure correct approximations of the derivative on the boundaries first order accurate forwarddifferences are used on the boundaries Consequently the full set of derivatives is given by

nablatunm = Dtunm +O(dt) (370)

Dt = Dtlowast otimes IM (371)

and is only first-order accurate in total

Definition The Kronecker product otimes is defined as follows Let A isin Rmtimesn and B isin Rptimesq thenthe Kronecker product AotimesB isin Rmptimesnq is the block matrix defined as

AotimesB =

a11B a1nB

am1B amnB

The procedure for the derivative with respect to s is analogous

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Ds = IN otimesDslowast (374)

For the second derivatives the following representation is obtained

∆ttum = Dttlowastum +O(dt) (375)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

Again note that in order to obtain reasonable approximations of the derivative on the boundary thefirst and last row ofDttlowast are modified when compared to standard Toeplitz finite difference matrixThe choice of the stencil ( 12 minus1 12 ) is in accordance with a first order accurate approximation ofthe second derivative on the boundary To see this consider the following two Taylor series

u2 = u1 + dtpartu1partt

+ dt2part2u1partt2

+O(dt3) (377)

u3 = u1 + 2dtpartu1partt

+ 4dt2part2u1partt2

+O(dt3) (378)

Addition of (378)-2middot(377) gives

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Consequently the full set of derivatives is given by

∆ttunm = Dttunm +O(dt) (380)

withDtt = Dttlowast otimes IM (381)

Similarly for the second derivative with respect to s one obtains

∆ssun = Dsslowastun +O(ds) (382)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

The derivation of the boundary conditions is completely analogous to the one for the Dttlowast matrixand the full set of derivatives is given by

∆ssunm = Dssunm +O(ds) (384)

withDss = Dsslowast otimes IN (385)

Also define the Sobolev matrix D

D = [I Dt Ds Dtt Dss]T (386)

The resulting approximation Jα of the Tikhonov functional Jα is then given by

Jα = dsdt Au22+αdsdt(uminus F22+Dt(uminus F )22+Ds(uminus F )22+Dtt(uminus F )22+Dss(uminus F )22

(387)where A = Dt+RDss with R isin RMNtimesMN being a diagonal matrix with elements corresponding

to the factor σ(t)2

Incorporation of initial and boundary conditions

The boundary conditions (314) and initial condition (313) have to be considered as well Sincethe initial and boundary conditions are prescribed they do not constitute degrees of freedom inthe problem of minimizing the Tikhonov functional (367) but they still have to influence thesolution in the right way To achieve this the right hands side of Au = 0 is modified as followsLet ubd isin RMN be the vector containing the discrete boundary values and initial values in the theentries corresponding to the boundaries and zeros otherwise In order to eliminate the prescribednodes from system Au = 0 we subtract minusb = Aubd which yields the new system Au = minusAubdThe rows and columns corresponding to the free nodes can then easily be extracted using an indexvector The resulting matrices are named A D u Considering that there are 2N +M set valuesthe minimization problem reduces slightly in dimension from NM to NM minus (2N +M) unknownsand from 5NM to 5NM minus 5(2N + M) equations This results in the following minimizationproblem

∥∥∥∥[ AradicαD

]uminus

[bradicαDF

]∥∥∥∥2

where A = Dt + RDss is the realization of the discretization of the first term in (367) and thesecond set of equations in (388) corresponds to the discretization of the Sobolev norm in (367)There are different ways to solve this minimization problem namely applying the LSQR algorithmfor the solution of Sparse Linear Equations and Least Square Problems as published in Paige [37]or solving the linear system resulting from the first order optimality condition on the gradient ofJ or applying a sparse QR solver to the linear system (390) All methods are briefly described inthe next sections and one of them is chosen based on performance thereafter

Alternative method

Another way of incorporating the initial and boundary conditions is to see them as free vari-ables and introduce a penalty term in (344) penalizing deviations from the prescribed initial andboundary values This modified Tikhonov functional is given by

Jλ = ||ut + Lparu||2L2(QT ) + λ||uminus F ||2H2(QT ) + γ||uminus ubd||2L2(QT ) (389)

However note that there neither is proof for convergence of this functional towards the exactsolution nor results on how to choose the parameters α γ We mention this to point out thatthere are certain slightly different formulations of the problem which might vary in the quality ofsolution and might possess more or less favourable numerical properties Furthermore it servesas a (numerical) test to determine the influence of the (incorporation of) boundary conditions onthe overall characteristics of the solution The minimization problem to be solved transforms into

∥∥∥∥∥∥ AαDγIF

uminus

0radicαDFγubd

∥∥∥∥∥∥2

where IF isin RNM is a diagonal matrix containing ones in the entries corresponding to initial andboundary values

344 LSQR

LSQR is an iterative method for the solution of sparse (asymmetric) linear equations and sparseleast square problems ie problems of the form Ax = b or min Axminus b2 with A being large andsparsely populated Published in 1982 by Paige amp Saunders [37] it is based on the bidiagonalizationprocedure of Golub and Kahan The algorithm is analytically equivalent to the standard conjugategradient method but achieves better numerical resultsWe aim for a solution of the system Bx = b with B isin Rmtimesm symmetric and positive definiteBriefly the method is an orthogonal projection onto the Krylov space

Km(r0 B) = spanr0 Br0 B2r0 Bmr0 (391)

where r0 = Bx0 minus b is the initial residual The Lanczos algorithm can be used to compute anorthonormal basis of the Krylov space Km In the process it generates a sequence of vectors viand scalars αi βi i gt 0 such that B is reduced to tridiagonal form

Lanczos process

β1v1 = bfor i = 1 2 do

wi = Bvi minus βiviminus1

αi = vTi wi

βi+1vi+1 = wi minus αiviend for

where v0 = 0 and all βi ge 0 are chosen such that vi = 1 After k steps we have

BVk = VkTk + βk+1vk+1eTk (392)

where Tk = tridiag(βi αi βi+1) and Vk = [v1 v2 vk] Multiplying equation (392) by anarbitrary vector yk we obtain

BVkyk = VkTkyk + βk+1vk+1y (393)

where y is the last entry of yk By definition Vk(β1e1) = b and we define

Tkyk = β1e1 (394)

xk = Vkyk

Combining this with (392) we have Bxk = b + yβk+1vk+1 up to working precision Fromhere it is possible to construct iterative methods that solve Bx = b The main difference is theway in which yk is being eliminated from (394) For conjugate gradients one uses the Choleskyfactorization Tk = LkDkL

Tk for example

Let us now turn our attention to the damped least-squares problem of the form

∥∥∥∥[AλI]xminus

]∥∥∥∥2

with λ isin R The solution to this system also satisfies[I AAT minusλ2I

](396)

where r = bminusAx Applying the Lanczos algorithm to this system is the same as using a bidiagon-alization procedure from Golub amp Kahan [17] Furthermore the relations (394) can after 2k + 1iterations be permuted to[

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

where Bk isin Rk+1timesk is a lower bidiagonal matrix One can see that yk is the solution of anotherleast-squares problem

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

which in turn can efficiently be solved using orthogonal transformations

345 Minimization via Conjugate gradient method

Another way of minimizing the functional (367) with respect to the values of uλ at the grid pointsis to form the partial derivatives of the discrete version of the functional with respect to the entriesof uλ isin RMN The discrete analogon of (344) is (367)

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

one could obtain the gradient of the Tikhonov functional by forming all partial derivatives partJλ

partuj

This can be expressed in matrix form as follows

nablaJλ = GuminusHF (3100)

where u F isin RMN contains the entries of the gradient that are independent of uλ and GH isinRMNtimesMN is symmetric and positive definite (SPD) The first order optimality condition leads tothe solution of the following linear system of equations

GuminusHF = 0 (3101)

One possible method for the solution of such linear systems with G being SPD is the conjugategradient method (CG) It is also possible to apply direct sparse solvers to the system Alternativelyan equivalent and much easier derivation of this linear system is obtained by considering thediscrete version of the Tikhonov functional in the form

Jλ(u) = ||Auminusb||22+λ[||IuminusIF ||22+||DtuminusDtF ||22+||DsuminusDsF ||22+||DttuminusDttF ||22+||DssuminusDssF ||22

]and applying the gradient operator to obtain

nablaJλ(u) =ATAuminusAT b+ λIT I[uminus F ] + λDTt Dt[uminus F ] + λDT

s Ds[uminus F ]

+ λDTttDtt[uminus F ] + λDT

ssDss[uminus F ]

=[ATA+ IT I +DTt Dt +DT

s Ds +DTttDtt +DT

ssDss]uminusAT b

minus λ[IT I +DTt Dt +DT

s Ds +DTttDtt +DT

ssDss]F (3102)

where A = Dt +RDss is the discrete differential operator as defined before

346 Multifrontal Sparse QR factorization

An important way to efficiently solve large (sparse) linear system and least-squares problemsdirectly is to factorize matrices into the form A = QR where Q is orthogonal and R is uppertriangular Such a transformation is called QR factorization The first QR factorization meth-ods transformed A one row or column at a time and did not reach theoretical performance dueto irregular access of memory An efficient alternative is an approach which involves symbolicanalysis preordering to eliminate singletons and makes use of different forms of parallelism Themost relevant details for this multifrontal multithreaded QR factorization as implemented fromMatlab R2009a upwards are extracted from Davis [13]The procedure starts with the elimination of singletons which are columns j with a single nonzeroentry aij gt τ where τ is a threshold value or no entry Singletons are permuted to the left and

row i is permuted to the top Column j and row i are then removed from the matrix This isdone for all singletons leading to a matrix [

R11 R12

](3103)

where R11 is upper triangular with diagonal entries larger than τ The remaining matrix A22

is then ordered to reduce fill using Matlabrsquos rdquoCOLAMDrdquo method which orders ATA withoutexplicitly forming it Next rdquoCOLMODrdquo symbolically finds a LLT factorization of the orderedmatrix ATA by only analyzing A itself For this purpose the elimination tree of ATA and thenumber of nonzero entries in each row of R are computed The information contained in theelimination tree is important for the symbolic analysis as well as the numerical factorization Thenext step is the identification of supernodes which are groups of m adjacent rows with (nearly)identical nonzero pattern Each supernode results in one frontal matrix which is a dense subset ofA on which the actual QR factorization is performed The information about supernodes is storedin the frontal matrix tree which gives rise to the frontal matrix after ordering determination ofthe frontal matrix sizes and a corresponding assembly strategy A Householder transformation isthen applied to the frontal matrix Potential ways of exploiting parallelisms are the parallel QRfactorization of parts of the frontal matrix and the QR factorization of a single frontal matrix canbe accelerated by using shared-memory multi-core capabilities of the rdquoLAPACKrdquo QR factorizationpackage

35 Numerical Experiments

In order to extract suitable model and discretization parameters and to validate the model somenumerical tests are carried out These tests are based on the performance of the regularizationalgorithm tested against the downward in time solution of the standard Black-Scholes equationwith suitable final and boundary data Thus the numerical solution of the Black-Scholes backwardsin time is shortly outlined in the first section Thereafter the algorithm is tested for the inversetime heat problem We determine the best way to solve the least-squares problem numerically andintroduce potential alternative boundary conditions The choice of the regularization parameterwill also be dealt with

351 The backward-time problem Crank-Nicolson

The Black-Scholes equation is solved backwards in time which is the forward problem to ourinverse problem to assess the quality of the regularization procedure described above The Crank-Nicolson method will be applied to numerically solve (213) Since the interest rate is assumed tobe zero we obtain the following discretization For all reference forward solution of the problem weuse N = 400 data points in each dimension To verify that the implementation works results forone example are compared to Matlabrsquos implementation of the Black-Scholes price called rdquoblspricerdquoIn Fig 32 one can see a plot of call and put as calculated using the Crank-Nicolsonscheme for an option with expiration T = 512 strike K = 50$ interest rate r = 0 and volatilityσ = 05

352 Verification test time inverse heat solution

To validate the algorithm for a slightly different problem the 1D heat equation is solved backwardin time Let the initial boundary value problem be given by

partt+ αd

part2T

partx2= 0 with x isin (0 π) t isin (0 Te) (3104)

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

Figure 32 Price of an option with strike price K = 50$ T = 512 years in dependence of spotprice S Verification of analytical solution as calculated by Matlab for call and put vsCrank-Nicolson scheme for call and put

where αd is the thermal diffusivity with units [m2

s ] and is set to αd equiv 1 The initial temperaturedistribution is prescribed by ψ(x) = sin(x) The equation is then first solved forwards in timeusing the Crank-Nicolson scheme as mentioned before to obtain the temperature distributionat the final times Te = 005 01 02 04s These distributions are then contaminated with arandom signal with noise level δ = 10minus1 10minus2 10minus3 in order to prevent committing inversecrime The resulting distribution is used as the terminal condition for the algorithm solving theinverse problem up until time t = 0 Finally the reconstructed initial distribution is comparedto the true initial condition ψ(x) in Fig 33 For this test the regularization parameters λ ischosen heuristically and is set to λ = 025 for error level δ = 10minus2 This way the constant α from(354) can be obtained In our test case we obtain α = 015 and we thus choose the regularizationparameter according to λ(δ) = δ03 In accordance with expectations one can see that the qualityof the solution decreases with an increase in noise level and or for larger end times Te Notethat stable solutions are only guaranteed for Te isin (0 12 ) because of the restriction Te lt a02 inTheorem 354 Indeed increasing Te to 1

2 in our numerical example leads to unusable results Theminimization problem (390) is solved using Matlabrsquos sparse QR solver

353 Assessment How to include boundary conditions

As discussed in Section 343 we consider two ways of implementing the initial (313) and boundary(314) conditions In Fig 34 the results are compared One can observe that with a certain choiceof parameters λ and γ the results are similar for both methods However the fixed boundary

method gives better results compared to the alternative boundary method which makes

sense since the forward problem was solved using the same boundary conditions In thefollowing numerical computations the boundary conditions will be directly prescribed

354 Bid-Ask spread requirements

When dealing with real market data we assess that bid-ask spreads of an optionrsquos underlying arevery tight The bid-ask spread ∆ is defined as ∆ = saminussb We observe a typical bid-ask spread of∆ = 005 sa+sb

2 The resulting problem is that the dynamics on the domain Q2τ are not sufficientto extract any additional information when compared to the interpolation of option prices betweenua(t) and ub(t) as defined in (314) Thus we exclude options with spreads ∆ lt 05 sa+sb

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

Figure 33 Heat equation solved backward with noise level δ = 10minus3 δ = 10minus2 andδ = 10minus1 and true initial distribution ψ(x) for different terminal times Te

01 middot 10minus2

2 middot 10minus23 middot 10minus2

time t

spot price s

option

priceu(st)

Figure 34 Solution of price forecasting for set boundaries vs free boundaries compared

to forward solution of the problem as obtained by Crank-Nicolson method

10minus8 10minus7 10minus6 10minus5 10minus4 10minus3 10minus2 10minus1 100 101 102 103 104

||Auminus f ||2

||uminusF|| H

Figure 35 L-Curve in double logarithmic scale for options with different bid ask spreads ofthe underlying large spread ∆ = 26 sa+sb

2 vs tight spread ∆ = 012 sa+sb2 for

regularization parameter λ isin (10minus8 1)

the following considerations

355 Choice of regularization parameter

The magnitude of the regularization parameter λ greatly influences the solutionrsquos behavior Thereare two classes of parameter selection algorithms methods that rely on information on the noiselevel in the input data and such methods that do not and are therefore called noise free parameterchoice rules However in the deterministic case there is always a noise such that for any regular-ization method with error free parameter choice the regularized solution will not converge to thetrue solution This result is also referred to as Bakushinksiirsquos veto [7]

L-curve

The L-curve criterion is a popular error free parameter choice method It is a graphical toolwhich for all valid regularization parameters plots the discrete smoothing norm (in our case thediscrete analogon of ||uminus F ||H2) versus the corresponding residual norm ||Auminus f ||2 In the caseof Tikhonov regularization the L-curve is continuous because one can choose the regularizationparameter from an interval of real number values The L-curve criterion visualizes the tradeoffbetween minimization of the two norms mentioned In a lot of cases the plot in double logarithmicscale of these two norms against each other has the shape of the letter L One then aims to seekthe point where the curvature is maximal being the point corresponding to the corner of the letterL Let us calculate the norms for regularization parameters λ isin (10minus8 1) for one randomly pickedoption and plot the results in double logarithmic scale In Fig 35 one can see that the resultingplot does not resemble the letter L However increasing the difference between the bid andask price of the underlying ndashsb and sa respectivelyndashone can observe that the resulting L-curveactually resembles the letter L Since the bid-ask spreads we are dealing with are typically smallerand since it is difficult to numerically determine this point in this case we move on to the nextpossible parameter selection criterion

Generalized Cross Validation (GCV)

Another error free parameter selection method is the Generalized Cross Validation method (GCV)The idea is to minimize the predictive mean-square error ||Auλminus flowast||2 where flowast is the exact data

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Figure 36 GCV function G(λ) for the full range of regularization parameters λ isin (10minus8 1) onthe left and zoomed in around the minimum on the right

as defined in (346) The GCV function works with the given data f and is defined as

G(λ) =||Auλ minus f ||22

(Tr(I minusAA))2 (3107)

where A = (ATA + λDTD)minus1AT is the regularized inverse Using generalized singular valuedecomposition the matrices A and D can be decomposed as

A = UCXminus1 D = V SXminus1 (3108)

where U V are orthonormal matrices X is nonsingular with ATA orthogonal columns and CSare diagonal matrices with diagonal elements being ordered non-decreasingly Then consider therelations

A(ATA+ λDTD)minus1AT = UCXminus1(XminusT (C2 + λS2)Xminus1)minus1XminusTCTUT (3109)

Tr(I minusA(ATA+ λDTD)minus1AT ) = tr(I)minus tr(C2(C2 + λS2)minus1) (3110)

where we used the fact that U is orthogonal and the relation Tr(ABC) = Tr(CAB) Remarkson the general form Tikhonov form and parameter selection criteria in this case can be foundin Chung [12] The λ0 value corresponding to the minimum of the GCV function G(λ0) is theoptimal parameter then Fig 36 shows one such plot for a random option Observe that theminimum found does not represent the best parameter when compared to the reference forwardsolution uCS and is also hard to determine numerically since machine precision level is reachedA similar scenario is found on page 198 in Hansen [19] where the occasional failure of the methodis mentioned and the GCV function around the minimum is very flat Thus we refrain fromusing the GCV method for our problem and continue with a heuristic estimation of the optimalparameter in the next section

Heuristics Parameter selection by forward solution fitting

Since it seems to be hard to find a regularization parameter in our case we introduce a heuristicapproach now Let the numerical solution of the forward problem as obtained by Crank-Nicolsonscheme be uCS The idea is to pick 100 different options and choose the average regularizationparameter which minimizes the difference between the regularized solution and the forward solu-tion Let the mean-square error between these two solutions be defined as e2 = ||uλminusuCS ||2 The

resulting regularization parameter is λ = 5 middot 10minus3 which is in good agreement with the parameterused by Klibanov [29]

More Heuristics Multiparameter Regularization

We observed that the reconstruction quality is sometimes higher and the range of admissibleregularization parameters larger when the different terms in the (discretized) Sobolev norm ||uminusF ||H2 are assigned different weights More specifically this means that there is not only oneregularization parameter λ but five of them One way to do this is to remove the natural scalingstemming from the finite difference discretization of the Sobolev operator such that

||uminusF ||H2 = λ||uminusF ||2+λ2dt||Dt(uminusF )||2+λ2ds||Ds(uminusF )||2+λdt2||Dtt(uminusF )||2+λds2||Dss(uminusF )||2(3111)Note that by doing so we essentially make the problem a multiparameter Tikhonov regularizationproblem and also leave the framework of the convergence proof presented before For more inform-ation on multiparameter Tikhonov regularization and possible parameter selection criteria in thiscase see among few available papers on this topic for example Ito [22] Reichel [38] or Chung [12]Ito [22] provides further relevant references in section 1 of his paper Rerunning the analysis fromthe preceding section we retrieve as the regularization parameter λ = 075 for our problem Sincethis is only an experimental observation and since the convergence results from Section 34 do nothold for the case of multiple parameters we will not actually apply this approach when forecastingprices in Chapter 5

356 Numerical instability of gradient method and lsqr algorithm

Solving the minimization problem using Matlabrsquos implementation of the LSQR algorithm fromSection 344 and the gradient method discussed in Section 345 led to more unstable resultswhen compared to the multifrontal QR solution from Section 346 The matrix product ATA isactually calculated in the gradient formulation leading to considerable round-off error because thecondition number of ATA is high which explain its unsatisfactory performance In Fig 37 theresults using different solvers are plotted

357 Alternative Boundary conditions

Since the boundary conditions chosen by Klibanov [29] are very sensitive to price changes and havea strong influence on the trading strategy (see Section 54) As an insightful thought experimentconsider for example an option trading at constant 095$ for two days and then making a 5jump to 1$ on the third day ie prices p = [1 1 105] corresponding to the times t = [minus2τminusτ 0]Quadratic interpolation and evaluation of the resulting polynomial until time 2τ imply an anti-cipated price of p2τ = 13 However compare this to the result for a 5 decrease at the thirdday yielding p2τ = 07 This corresponding quadratic functions are plotted in Fig 38Therefore alternative choices of the boundary values (314) are proposed The aim is to still usethe past observations for ub ua at τ isin minus2τminusτ 0 but to reduce the strong dynamics impliedby the quadratic polynomial in (314)

Linear Regression

As a first alternative we propose to use the linear regression function as boundary values As onecan see in Fig 38 the dynamics are damped compared to the quadratic polynomial while stillexhibiting a tendency to increase decrease

Constant Average

One might also argue that a single jump in price does not justify modeling future prices with alasting trend In this case we suggest to use constant boundary conditions using the mean value

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

time t underlying price s

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

Figure 37 Solution as obtained using different methods reference solution Matlabrsquos back-

slash (Multifrontal QR) Matlabrsquos LSQR and backslash solution of the gradient system

Regularization parameter was set to λ = 01

minus8 minus6 minus4 minus2 0 2 4 6 8

middot10minus3

t time

option

priceu(t)

Figure 38 Price using quadratic polynomial +5 minus5 regression +5 minus5 mean +5 minus5 weighted average +5 minus5

of the past observations as the constant boundary condition The corresponding plots are also there in Fig 38

Constant Weighted Average

Similarly the more recent observations could be given more weight when calculating the meanvalue One possibility is to assign the weights ω = [ 16

36 ] to the prices p resulting in the weighted

averagepw = ω1p1 + ω2p2 + ω3p3 = 10250975 (3112)

Chapter 4

Parameter Identification ImpliedVolatility as an Inverse Problem

As already pointed out in the introduction the volatility implied by the Black-Scholes equation isobserved to depend on both the underlying price s and the time t In reality one can observe thatoptions whose strike is either far out of or far into the money have a greater implied volatility whilevolatility tends to decrease the more the time to expiry This dependence can be seen in so-calledvolatility surface plots in which implied volatility is plotted against the strike price K and timeto expiry T For example consider Fig 41 in which the volatility data for 288 call options onMicrosoft (MSFT) shares on 05072015 were plotted According to Rubinstein [40] this effect hasonly been observed in stock options after the financial breakdown of 1987 whereas it was alreadyknown for foreign currency options by then As a result many practitioners and researchers triedto extend the Black-Scholes model to incorporate the volatility coefficients dependence on stockprice and time This class of models is therefore called local volatility models and the Black-Scholesin the extended form becomes

partt+

2s2σ(s t)2

part2u

parts2+ smicro

The volatility coefficient is assumed to satisfy 0 lt m lt σ(s t) lt M ltinfin and belong to the Holderspace Cλ(ωlowast) 0 lt λ lt 1 for some interval ωlowast sub (0infin) Furthermore there is still a final conditionimposed at the expiry date T

u(s T ) = (sminusK)+ = max(0 sminusK) 0 lt s (42)

The inverse problem of option pricing can then be formulated as determining the volatility σ forgiven values

u(slowast tlowastKT ) = ulowast(KT ) K isin ωlowast T isin Ilowast = T1 T2 (43)

where slowast is the price of the underlying at time tlowast and is assumed to be contained in ωlowast ie slowast isin ωlowastThe main problem with coefficient inverse problems (CIP) is that the cost functionals associatedwith the Tikhonov formulation of the problem are non-convex This means that the cost functional

Time to expiry T

Strike K

latilit

Figure 41 Implied volatility surface for MSFT options Data retrieved on 05072015

CHAPTER 4 PARAMETER IDENTIFICATION IMPLIED VOLATILITY AS AN INVERSEPROBLEM

potentially has many ravines and one should roughly know the location of the minimum to finda global minimum One way to obtain globally convergent numerical algorithms is to constructstrictly convex Tikhonov-like cost functionals using Carleman Weight Functions (CWF) For sometheory on this topic see for example Klibanov [28] Timonov [31] In the following the non-linearcoefficient inverse problem (41)-(43) will be linearized to obtain estimates of the volatility σ(s t)However if one wants to use this s-dependent volatility coefficient in the forecasting problem fromsection 32 the convergence estimates of section 34 do not hold anymore The challenge is toconstruct a modified Tikhonov-like functional such that convergence can be proved in this caseInspiration for this task can be drawn from eg Klibanov [26 28] or Timonov [31] In any casethe algorithm from section 32 will be tested with the data obtained from the solution of the CIPin a numerical experiment

41 Ill-posedness of the inverse problem and literature over-view

In the past 35 years many approaches have been made to reconstruct the volatility surface given adiscrete set of option prices Avellaneda [5] applies dynamic programming to minimize a relativeentropy functional Bouchaev [9] approximates the option price as the sum of the Black-Scholesprice and a correction for the non-constant volatility stemming from the parametrix expansion ofthe fundamental solution whereas Lagnado and Osher [32] minimize a Tikhonov like functionalto solve the inverse problem For more complete surveys on the topic see eg Zubelli [49] orReisinger [39] Since the solution of the nonlinear inverse option pricing model (nonlinear inthe sense of the volatility depending nonlinearly on K and T ) is unstable therefore numericallychallenging and regularization techniques hard to apply on sparse real market data we turn toa linearization approach suggested by Isakov [11 21] which is numerically more robust to sparsedata and better suited to the analysis of a large data set in terms of computational complexity

42 Linearization approach by Isakov

Based on the aforementioned papers by Isakov [11 21] the volatility is being linearized at twodifferent times T1 T2 The local volatility is assumed to be given by

2σ(s t)2 =

2σ20 + flowast0 (s) + tflowast1 (s) (44)

where flowast0 flowast1 isin C0(ω) are small perturbations of σ0 which in turn is the constant implied volatility

defined as the solution of the Black-Scholes equation with constant volatility coefficient Calcu-lation of σ0 is straightforward and can be done by applying a Newton fix point algorithm to theBlack-Scholes equation Alternative ways as eg rational approximation Li [33] also exist Out-side of the interval ωlowast we set flowast0 = flowast1 = 0 This conjecture is acceptable if the volatility is notchanging fast with respect to stock price and slow with respect to timeSince the analysis of the problem is analytically and numerically easier in terms of the strikeprice K and maturity date T the Black-Scholes equation is formulated in terms of these variablesyielding the Dupire equation This dual equation is derived in the next section

421 Continuous time formula Dupire

Dupire [14] derived a dual equation to the Black-Scholes in 1993 which is not defined in terms ofs and t but describes the option price in dependence of strike K and expiry T The result stemsfrom the question whether under certain conditions the Black-Scholes model can be modified toaccommodate for non-constant volatility while keeping the model complete in the sense of beingable to hedge the option with the underlying asset The question was whether it is possible to

find a risk-neutral diffusion process of the form

S= r(t)dt+ σ(S t)dW (45)

given arbitrage-free prices u(KT ) The relation

u(KT ) =

int infin

max (S minusK 0)φT (S)dS =

int infin

(S minusK)φT (S)dS (46)

results in

φT (K) =part2u

partK2(KT ) (47)

after differentiating twice with respect to K where φT is the risk-neutral density function of thespot at time T The question is whether there exists a unique diffusion process dx = a(x t)dt +b(x t)dW generating these densities φT Assuming for the moment that the interest rate is zeroconsider dx = b(x t)dW Using the Fokker-Planck equation and some calculation it is shown inDupire [14] that the following relation holds

b2(KT )

part2u

partK2=partu

partT(48)

for European options From this b can be obtained because part2upartK2 and partu

partT are known from observablemarket data One only has to think of suitable interpolation or discrete differentiation proceduresto obtain these quantities Comparing this to the diffusion process (45) one identifies the volatility

σ = b(St)S The derivation for non-zero interest rate is slightly more complicated and results in

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

partK+ (r minus micro)u = 0 (49)

422 Linearization at constant volatility

Equation (49) and final condition (42) are transformed by

y = ln(K

slowast) τ = T minus tlowast (410)

a(y τ) = σ(slowastey τ + tlowast) U(y τ) = u(slowastey τ + tlowast) (411)

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

party+ (r minus micro)u = 0 (412)

u(y 0) = slowast(1minus ey)+ y isin R(413)

The additional market data become

u(y τj) = uj(y) y isin ω j = 1 2 (414)

where ω is the transformed interval ωlowast and uj(y) = ulowast(slowastey Tj) Also note that

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

withf0(y) = flowast0 (s

lowastey) + tlowastflowast1 (slowastey) f1(y) = flowast1 (s

lowastey) (416)

The idea is to construct the solution u as u = V0 + V + v where v vanishes quadratically withrespect to (f0 f1) and V0 solves (412) with a = σ0 and V is the solution to

partτminus 1

part2V

party2+ (

2σ20 + micro)

party+ (r minus micro)V = α0(f0(y) + τf1(y)) (417)

V (y 0) = 0 y isin R (418)

α0(y τ) =slowastradic2πτσ0

eminus y2

2τσ20+cy+dτ

d = minus 1

2+ micro)2 + microminus r (420)

andV (y τj) = Vj(y) = uj(y)minus V0(y τj) j = 1 2 y isin ω (421)

This linearization is in accordance with the theory presented in eg Bouchaev [10] for the inverseoption pricing model Further substitution of V = ecy+dτW yields

partτminus 1

part2W

party2= α(f0(y) + τf1(y)) 0 lt τ lt τlowast y isin R (422)

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

and the final dataW (y τj) =Wj(y) = eminuscyminusdτjVj(y) (425)

In one last simplification α is replaced by the y independent function

α1(τ) =slowastradic2πτσ0

and the domain R is restricted to a finite interval Ω sup ω

423 Uniqueness and Stability

Theorem 5 Let ω = (minusb b) and f0 = f1 = 0 outside ω Furthermore demand

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

τ1τ2(τ2 minus τ1)2σ20

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

This guarantees the uniqueness of the solution (f0 f1) isin Linfin(ω) times Linfin(ω) of the inverse optionpricing problem (422)-(426)

Proof The proof will only be sketched here For the complete proof see Isakov [21] Onestarts by proving the uniqueness of a solution to the approximate inverse option pricing problem(422)-(426) by applying a Fourier transform to (422) solving the generated ordinary differentialequation and using a Taylor expansion of the exponential function to prove boundedness of thesystem of integral equations Using the definition of Sobolev norms via the Fourier transform thestability and uniqueness is proved

Equivalence to the following system of Fredholm equationsintω

K0(x y τ1)f0(y) +K1(x y τ1)f1(y)dy =W1(x) x isin ω (429)intω

K0(x y τ2)f0(y) +K1(x y τ2)f1(y)dy =W2(x) x isin ω (430)

is then shown where

K0(x y τ) = Slowast

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

K1(x y τ) = Slowast(

radicτradic2σ0

(|y| minus |xminus y|)eminus (|xminusy|+|y|)2

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

with Slowast = slowast

radicπ

and Wj(x) = eminuscxminusdτjVj(x) with Vj(x) = uj(x) minus V0(x τj) as defined before

This means that it suffices to show the uniqueness and stability of the solution to this system ofFredholm equations to prove uniqueness of the linearized option pricing problem

424 Numerical solution of the forward problem

This section deals with the numerical solution of the forward problem

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

The forward problem is solved using an implicit respective combination of explicit and implicitmethods Consider the Dupire equation (412) which is being discretized using the so-calledθ-method For simplicity we set r = micro = 0 Time derivatives are discretized using first order finitedifferences whereas derivatives with respect to the strike price are second order accurate finitedifferences The θ-method requires a θ-weighted average of explicit and implicit discretization tobe formed Inserting these averages into (435) yields

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

unj+1 minus 2unj + unjminus1

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

unj+1 minus unjminus1

Defining σ = a(yτ)2δt2δx2 and ρ = minusa(yτ)2δt

4lowastδx we rearrange equation (435)

un+1j + 2σθun+1

j minus σθun+1j+1 minus ρθun+1

j+1 minus σθun+1jminus1 + ρθun+1

jminus1 = (435)

unj minus (1minus θ)σ2unj + (1minus θ)σunj+1 + (1minus θ)ρunj+1 + σ(1minus θ)unjminus1 minus ρ(1minus θ)unjminus1

j = 2 N minus 1

We prescribe zero Dirichlet boundary conditions such that Un1 = Un

N = 0 which yields

j = 1 minusσθun+12 minus ρθun+1

2 = (1minus θ)σun2 + (1minus θ)ρun2 (436)

j = N minusσθun+1Nminus1 + ρθun+1

Nminus1 = (1minus θ)σunNminus1 minus (1minus θ)ρunNminus1 (437)

In matrix notation we have

0 minusσθ minus ρθ 0 0 middot middot middot 0minusσθ + ρθ 1 + 2σθ minusσθ minus ρθ 0 middot middot middot 0

0 minusσθ + ρθ 1 + 2σθ minusσθ minus ρθ middot middot middot 0

0 middot middot middot 0 minusσθ + ρθ 1 + 2σθ minusσθ minus ρθ0 middot middot middot middot middot middot 0 minusσθ + ρθ 0

0 (1minus θ)(σ + ρ) 0 0 middot middot middot 0(1minus θ)(σ minus ρ) 1minus 2σ(1minus θ) (1minus θ)(σ + ρ) 0 middot middot middot 0

0 (1minus θ)(σ minus ρ) 1minus 2σ(1minus θ) (1minus θ)(σ + ρ) middot middot middot 0

0 middot middot middot 0 (1minus θ)(σ minus ρ) 1minus 2σ(1minus θ) (1minus θ)(σ + ρ)0 middot middot middot middot middot middot 0 (1minus θ)(σ minus ρ) 0

Aun+1 = Bun (440)

For all computations we chose θ = 1

425 Numerical tests reconstruction of given volatility surfaces

In order to test the proposed algorithm the Dupire equation (49) is solved with a prescribed

volatility function a(y τ) =σ20

2 + f(y) to obtain the values uj(y) Then the system of Fredholmequations (429) (430) is solved and the reconstructed volatility perturbation a(y τ) = σ0

2 +f0(y) + τf1(y) is compared to the given volatility a(y τ) We make use of the fact that

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

( |xminus y|+ |y|σ0

radic2τ

) (441)

where erfc is the complementary error function The integrals are calculated using a compositemidpoint rule For more information on the numerics of (ill-posed) Fredholm integral equationsof the first kind see eg Baker [6] We discretize the domain ω = (minusb b) with N grid points Thisway (429) (430) becomes

Nsumk=1

dyK0(x yk τ1)f0(yk) +K1(x yk τ1)f1(yk) =W1(x) x isin ω (442)

Nsumk=1

If we let the collection of yk k = 1 2 N coincide with the points xj j = 1 2 N thisresults in a square block matrix K isin R2Ntimes2N consisting of the square matrices K10K11K20K21

with entries

K10jk = dyK0(xj yk τ1) K11jk = dyK1(xj yk τ1) (444)

such that

[K10 K11

K20 K21

] (446)

This way the system of equations (429) (430) can be written

Kf = W (447)

The unknown volatility perturbation f(y) is then obtained by solving the linear system (447) as

f = Kminus1W (448)

where f = [f0f1]T W = [W1W2]

T isin R2N and where the middot quantities are the approximationsto their respective smooth counterparts For example the entries of f0 isin RN are f0j asymp f0(xj)j = 1 N

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Figure 42 Reconstructed volatility surface compared to prescribed volatility surface

Numerical example

We try to recover a given volatility surface function a(y τ) = 12σ

20 + 1

2y2 Let s = 20 σ0 = 1

2 y isin (minus1 15) The equation (412) is solved backwards with volatility σ0 and a(y τ) to obtainVj(y) and thereafter the Fredholm system (447) is solved The results are presented in Fig 42The quality of the reconstruction is not satisfactory and the reason for this behaviour are not clearat the moment

426 Towards a real world scenario Smooth interpolation of marketprices and remarks

Since uj j = 1 2 has to be at least twice differentiable on ω in order for f to be continuous the realmarket prices u(τj y) j = 1 2 which are only sparsely available ie for certain combinations ofmaturity T and strike priceK have to be smoothly approximated and or interpolated For exampleone could use smoothing splines to achieve this Given n data (xi yi) with x1 lt x2 lt x3 lt middot middot middot lt xnsuch that yi = micro(xi) The smoothing spline approximation micro of micro is the minimizer of

nsumi=1

(yi minus micro(xi))2 + η

int xn

(d2micro(x)dx2

dx (449)

Here η gt 0 is a smoothing parameter regulating the trade-off between a good data fit and asmooth approximation In fact for η rarr 0 the solution will converge towards the interpolatingspline whereas for η rarr infin the solution will get smoother and converge towards the linear leastsquares fit A small test on some prices of put options on the MSFT stock suggests that thismethod is suitable for this purpose The result which can be seen in Fig 43 has been generatedusing Matlabrsquos Curve Fitting Toolbox and the suggested smoothing parameter in this case isη = 03738In the context of the inverse problem of section 32 one could take the corresponding optionrsquos strikeprice and maturity date and reconstruct the volatility as follows For the choice of Ilowast = T1 T2there are two possibilities either take the options maturity as T1 and take the next biggest contractduration as T2 or take T2 as the optionrsquos maturity and T1 as the next smallest maturity Regardingthe interval ω it should contain todayrsquos stock price slowast One approach would be to take all strikesin a range of plusmn30 around the stock price slowast This choice reflects the observed reconstructionquality of the volatility which deteriorates for options that are deep in or out of the money

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Figure 43 Price of put option for different strikes on MSFT stock with maturity date17062016 as observed on 14092015 with smoothing spline

Chapter 5

Putting everything into action Alarge-scale real market data study

51 Option exchanges amp BATS

The data was collected via Yahoo Finance httpfinanceyahoocomoptions which retrievesits options data from the BATS exchange which had a market share in the American optionsmarket of 99 in March 2015 [1] The biggest American options exchange is the Chicago Boardof Options (CBOE) with a option trading volume of more than $561 billion in 2013 [2] Regulartrading hours on the American exchanges are generally from 0930 ET to 1615 ET There are alsoextended trading hours from 0030 ET until 0930 ET and from 1615 ET until 0915 ET +1d Withinthe European market Deutsche Borse owned Eurex Exchange is the most important exchange

52 Data gathering

For information on how to use the Yahoo Finance API visit httpwwwjarloocomyahoo_

finance To retrieve the option data we used the Remote Data Access functionality of the PythonData Analysis Library pandas httpwwwpandaspydataorg Quite some effort went intoimplementing this multi-threaded data grabber incorporating delays to synchronize real-time stockquotes with delayed option quotes and creating a MongoDB httpwwwmongodborg database tostore the collected data Data was collected in one hour intervals starting from 0930 ET duringregular trading hours This means that 7 data sets were collected each day starting from 1st July2015 until 4th August 2015 The information is stored in CSV format and one such data set hasa file size of around 40MB A list of 3746 symbols for which option data was collected can bedownloaded from https000tinyuploadcomfile_id=08865665819161381532

53 Data filtering

We collect data in the following formatStrike Expiry Type Symbol LastBid Ask Chg PctChg VolOpen Int IV Root IsNonstandard UnderlyingUnderlying Price Quote Time Underlying Symbol Underlying Bid Underlying Ask

In order to obtain meaningful results the raw data has to be filtered first Only options withsufficient trade volume were taken into account Next for backtesting reasons option data thatwere not available for five consecutive days as well as zero entries are removed from the data setand a mapping of dates to consecutive trading days with results being stored in a vector calledTV is appended Option with identical last price or volume on two consecutive days were alsofiltered because this is interpreted as a sign of inactivity Additionally a vector containing thetime to expiry in years named TTE is also appended Eventually the datasets (per option) whichare imported to Matlab contain the following dataStrike Bid Ask Implied volatility Underlying Bid Underlying Ask TTE TV

Assume there are nt days for which data is available for option XYZ The dataset XY Z for optionXYZ will then be of dimension XY Z isin R8timesnt For the following computations only every tenth

CHAPTER 5 PUTTING EVERYTHING INTO ACTION A LARGE-SCALE REALMARKET DATA STUDY

entry of the database was considered to keep calculation times manageable on a consumer classcomputer

54 Trading strategy

Remember that this model does not distinguish between call and put options since this informationis usually incorporated by means of the boundary conditions However we solve the (inverse time)Black-Scholes equation on the new interval (sb sa) For trading decisions the calculated price of theoption is evaluated at the midpoint between the ask and bid price of the underlying sm = sb+sa

2 Since the bid-ask spread is narrow trades are only executed if our calculated price uα(sm τ)surmounts the so-called decision criterion d(t) by an amount of b = 002$ at least Since we arecalculating uα(sm t) for t isin (0 2τ) the trading strategy is a two time unit strategy This meansthat depending on the prices uα(sm τ) uα(sm 2τ) we execute one of the strategies listed belowRemember that τ can be one minute one hour one day or every fraction of 255 days between 0and 14

1 If uα(sm τ) gt d(τ) + b and uα(sm 2τ) gt d(2τ) + b buy two contracts of the option todayie at t = 0 Then sell one contract at t = τ and the second one at t = 2τ Apply thestrategy after two time units ie at t = 2τ again

2 If uα(sm τ) gt d(τ) + b but uα(sm 2τ) le d(2τ) + b buy one contract today ie at t = 0Sell the contract after one time unit ie at t = τ and apply the strategy on that day again

3 If uα(sm τ) le d(τ) + b but uα(sm 2τ) gt d(2τ) + b buy one contract today ie at t = 0and sell it after two time units ie at t = 2τ

4 If uα(sm τ) le d(τ) + b and uα(sm 2τ) le d(2τ) + b do not buy anything and apply thestrategy after time τ has passed

For now we use the extrapolated ask price ua(t) as the decision criterion ie d(t) = ua(t) Wemention this because we will choose different combinations of boundary conditions and decisioncriteria in later sections

55 Backtesting

bull Options are usually traded by means of option contracts which is written on 100 shares ofthe underlying

bull Trading is possible irrespective of the account balance rarr money can be borrowed at all timesand without costs

bull Transaction costs are not considered but one might argue that the buffer b accounts forthem

bull Interest rates are not taken into account ie r equiv 0

bull Data is required to be available for 5 consecutive days in order to back test In a real worldapplication of the algorithm more decisions would be made

Since our trading strategy only incorporates trades with holding periods of one and two day(s)we store profits respectively losses in the matrices X1 and X2 For nt days of observable datawe have nt minus 2 trading opportunities for one day holding periods and nt minus 2 minus 1 for two dayholdings This is due to the fact that we need two days of history data to apply our strategy Ifno options are considered the matrices will have the following dimensions X1 isin Rnotimesntminus2 andX2 isin Rnotimesntminus2minus1 For implementation reasons we append a zero column vector z = 0 isin Rnotimes1 toX2 to make its dimension coincide with the dimension of X1 Ie X2 is redefined as X2 = [X2 z]

To clarify matters assume there is a list of options labeled 1 2 j no Now entry X1(i j)contains the change in price of the ith option from the jth day to the j + 1th day of observationie X1(i j) = (ub(j+1)+ua(j+1))2minusua(j) Note that the ask price on the day of buying ua(j)is quoted whereas the midpoint between bid and ask price on the day of selling is quoted underthe assumption that we are able to trade within the bid ask range on average Next an indicatormatrix I isin Rnotimesntminus2 is defined Whenever the algorithm suggests a strategy for a certain optionthe corresponding number will be written into I at the corresponding position The labeling isas follows strategy 1 = 1 strategy 2 = 2 strategy 3 = 3 no trade = 0 In a next step thisindicator matrix is split intro three separate matrices of the same dimension such that I1 containsthe non zero entries of I which equal one and I2 I3 the non zero entries which equal 2 and 3respectively These indicator matrices allow us to pick the dailytwo day profitloss from X1 andX2 for the days on which the algorithm suggested trading This is a comfortable way to calculateprofit per option or per strategy Consider for example the profits (or losses) generated by strategy1 Since this strategy includes two holding periods the total profit is a combination of entries inX1 and X2 More precisely let X1d1 and X2d1 be the entries of X1 and X2 respectively whosepositions coincide with the non zero entries of I1 The subscript 1d1 encodes a holding period ofone day (1d) for strategy 1 (1) For example one obtains the profit per option for strategy 1 bysumming along the second dimension of X1 The account balance as plotted in the figures acrossthe following sections is the sum of all optionsrsquo profitlosses for all strategies combined such thatbalance with initial balance b0 on the ith trading day is

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d1(a b) +Xi1d2(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

551 Restriction on options with significant bid-ask spread A Com-parison

As witnessed and explained in the previous section the information content for options whoseunderlyings have very small bid-ask spreads is very small To mitigate this we restrict our analysisto those options with significant bid-ask spreads and investigate possible potential for profit amongthis data subset Let the bid-ask spread of the underlying be defined as ∆ = sa minus sb Such adata subset contains only options for which the relative spread ∆

(sa+sb)2gt ε where ε is the

relative spread threshold We demand a minimum relative spread of ε = 06 for an option to beconsidered rdquoan option with significant bid ask spreadrdquo The testing procedure is as follows Thedataset containing all options is referred to as dataset 1 The dataset containing the options withsignificant bid ask spread is referred to as dataset 2 We take 40 random samples out of each ofthe datasets such that the number of decisions made by the algorithm is roughly the same Thenwe simulate the trading strategy on each of the datasetsrsquo samples and calculate mean values ofcharacteristic quantities q which we denote by emptyq The initial account balance is b0 = 100$ andit is assumed that money can be borrowed at no costs at any time A comparison of back testingresults can be seen in Table 51 The algorithm triggers fewer trades for dataset 2 than for dataset1 This results in significantly higher end balance emptyb(nt minus 2) 9493$ vs 8868$ Note howeverthat the ratio between profitable and losing trades is still the same

552 Modification Negativity of extrapolated ask price

In case the extension of the quadratic polynomial ua(t) onto (0 2τ) is negative at any pointt isin (0 2τ) no trade is recommended The reason therefore is that the extrapolated ask price is aboundary condition for the inverse problem and even though the solution of such a problem couldstill contain information about the evolution of the price when compared to the decision criterionua we do feel like it is not reasonable to allow for negative option prices The results whichcan be seen in Table 52 suggest improved results for both datasets Especially the increase inperformance for dataset 1 is remarkable when compared with the data from Table 51 (∆emptyb(nt minus2) = +693$) Therefore all the computations to follow respect this modified trading rule

Measure Dataset 1 Dataset 2emptyb(nt minus 2) 8868$ 9493$empty decisions made 114 112empty profitable trades 13 8empty losing trades 40 27

Table 51 Comparison between samples from all options versus samples from options with signi-ficant bid-ask spread

Table 52 Effects of incorporating the modified trading rule mentioned in Section 552

553 Impact of boundary conditions and decision criterion

From the boundary conditions from Section 343 we chose to test the regression boundary condi-tions and back test it Furthermore we also investigate the effects of using different combinationsof boundary conditions and decision criteria The abbreviations rdquobcrdquo and rdquodcrdquo stand for theboundary conditions and decision criterion used respectively In this context rdquoquadrdquo refers tothe quadratic polynomial and rdquoregrdquo refers to the case where regression is applied on to the re-spective data for use as boundary condition or decision criterion See Table 53 for the results Animportant observation is that the regression boundary conditions and regression as the decisioncriterion lead to way lower number of trades It might be interesting to further investigate thebehavior of this strategy for longer observation horizon

554 Influence of volatility coefficient

Another important quantity is the volatility coefficient σ(t) The quadratic extrapolation men-tioned in Section 32 is compared with a linear regression function obtained from the volatilityobservations at times minus2τ τ 0 The impact can be seen in Table 54 Fewer trades are recom-mended when compared to the cases with the volatility which is quadratic in time (see Table 52)

555 Test on most actively traded options

One expects the prediction quality to be better for actively traded ie liquid options On theother hand tight bid ask spreads are associated with these options In dataset 3 we collect the 56most active options and run our strategy on this subset The results are shown in Table 55

556 Alternative one-day trading strategy

As prediction quality theoretically decreases for longer prediction horizons we test a one-daytrading strategy The trading rule is as follows

1 If uα(sm τ) gt d(τ) + b buy one contract today ie at t = 0 Sell the contract after onetime unit ie at t = τ and apply the strategy on that day again

2 If uα(sm τ) le d(τ)+ b do not buy anything and apply the strategy after time τ has passed

We use d(t) = ua(t) In Table 56 the results of this strategy can be seen for all samples from alloptions (dataset 1) for options with significant bid-ask spread (dataset 2) and for most activelytraded options

Measure Dataset 1 Dataset 2bc quad dcregression

emptyb(nt minus 2) 8973$ 9396$empty decisions made 121 113empty profitable trades 19 9empty losing trades 52 32

bc regression dcquademptyb(nt minus 2) 9118$ 9285$empty decisions made 122 112empty profitable trades 14 7empty losing trades 54 35

bc regression dcregressionemptyb(nt minus 2) 9847$ 9899$empty decisions made 88 115empty profitable trades 2 1empty losing trades 4 1

Table 53 Impact of different combinations of boundary conditions and decision criteria as presen-ted in Section 343

Measure Dataset 1 Dataset 2bc quad dc quad

bc quad dc regemptyb(nt minus 2) 9322$ 9396$empty decisions made 128 122empty profitable trades 20 9empty losing trades 55 36

Table 54 Experiment Linear regression function as volatility

Measure Dataset 3bc quad dc quad

emptyb(nt minus 2) 9740$empty decisions made 100empty profitable trades 7empty losing trades 12

Table 55 Experiment Most actively traded options

Measure Dataset 1 Dataset 2 Dataset 3bc quad dc quad

emptyb(nt minus 2) 9775$ 9938$ 9913$empty decisions made 98 115 100empty profitable trades 5 3 3empty losing trades 11 6 7

Table 56 Experiment One-day trading strategy

557 Forecasting with locally reconstructed implied volatility Futurework

Note that we are leaving the inverse Black-Scholes framework presented in Section 3 due to thefact that the volatility coefficient now depends both on time t and stock price s Since the retrievalof volatility in Isakov [11] is not based on the same model as in Klibanov [29] in this case the testspresented here are of experimental nature only However the authors of [29] believe that based onthe theory of globally convergent algorithms which was successfully applied in Klibanov [28 26]it should be possible to construct functionals such that convergence can be proved The key ideais to modify the Tikhonov-like functional to become globally strictly convexSince the results for the volatility reconstruction in Chapter 4 are not satisfactory we refrain fromproviding further commentary here

Chapter 6

Conclusions

What has been done

The study was set out to investigate the possibility of forecasting options prices via the solution ofthe time inverse Black-Scholes equation To this end fundamental terminology and basic conceptsregarding the derivative form of options were introduced Since the problem we are dealing withis ill-posed in the sense of Hadamard some basic theory on the treatment and analysis of suchproblems was presented in Section 33 The regularization technique applied here is Tikhonovregularization We cited a convergence proof of the minimizer of the Tikhonov functional to theexact solution of the time inverse problem and analyzed different parameter selection criteria fortheir applicability to the inverse time option pricing problem The analogy to the heat equationserved as an accessible example to point out some of the particularities one encounters in the con-text of (ill-posed) inverse problems A numerical example confirmed the theoretic result of solvingthe backward time heat equation for up to T = 05s which is a remarkable result considering theinstantaneous failure of standard Euler scheme like solvers for this problemOne aspect of this thesis was to test the conjecture of profitability of the reconstruction algorithmwhen applied with a simple trading strategy to real market data To do so data was collected fromYahoo Finance and accumulated using Pythonrsquos pandas library The queried data was importedinto a MongoDB database filtered using the statistical computing software R and formatted foruse with Matlab Finally backtesting was done using different boundary conditions and decisioncriteria and a two day strategy was compared to a one day strategy The number of optionsconsidered (samples taken from a database of roughly 300000 different options) was significantlyhigher than the 20 options tested in the original paper by Klibanov [29]

What the results mean

The reconstruction algorithm works when a reasonable regularization parameter is chosen How-ever for the computational domain we are interested in it is difficult to find automated parameterselection procedures At the moment good heuristic approaches seem to be the best way of choos-ing such parameters in this case The results obtained in Sections 551-554 should not be seen interms of absolute return but as insight into which changes the boundary conditions trading rulesand decision criteria have on the overall performance of the algorithm Since the test sets onlycomprised one monthrsquos worth of data prolonged observations are imperative for a final assessmentof the strategies performance The results support our conjecture of the prediction to be ratherpointless for options whose underlyingrsquos bid-ask spreads are too tight In that case the predictiongenerated by the solution of the time inverse problem essentially corresponds with the interpola-tion function F (s t) One basically tries to predict price by extrapolation of a polynomial in thatcase and the solution of the inverse problem is superfluous

What can be done in future work

As already mentioned before the algorithm should be tested for longer periods of time than onemonth preferably more than one year to observe long term behavior of the algorithm Additionallythe data filtering can probably improved based on heuristics and experience with option tradingFurthermore in some cases better results were achieved when weighting the regularization termsin (367) differently In this context it might be interesting to have a look at multi parameterTikhonov regularization techniques Note however that the convergence results from Section 34are not valid in this case

CHAPTER 6 CONCLUSIONS

Next the performance of the algorithm for shorter time intervals for example hourly trading couldbe investigated In theory results should be betterAdditionally improved volatility coefficients could further improve the reliability of the predictionUnfortunately numerical results for the linearization approach presented in Chapter 4 were notsatisfactory and the volatility therefore not incorporated into the modelAlso we observed that the largest jumps in option price usually occur on the day of a companyrsquosearning report This might seem obvious but one could try to map estimated earning call datesto stock symbols and pause the algorithm on these days This is also in line with the fact thatthe Black-Scholes framework is per assumption not able to predict or account for sudden marketeventsAnother possibility might be to come up with more advanced trading strategies that make use ofthe information obtained by the solution of the inverse problem in a better way than our ratherprimitive strategy In this context one could introduce confidence levels for certain symbols oroptions based on experience ie past decisions Finally machine learning techniques might beapplied therefor as well as for the selection of regularization parameters

Bibliography

[1] BATS Press Release April 2015 hhttpcdnbatstradingcomresourcespress_

releasesBATSOptions_April2015_Volume_FINALpdf Accessed 2015-08-11 39

[2] CBOE Market Statistics httpwwwcboecomdatamarketstats-2013pdf Accessed2015-08-11 39

[3] Pablo Amster Corina Averbuj Maria Mariani and Diego Rial A BlackndashScholes optionpricing model with transaction costs Journal of Mathematical Analysis and Applications(2)688ndash695 2005 8

[4] John Au Chapter 3 Inverse Problems An Ab Initio Approach to the Inverse Problem-BasedDesign of Photonic Bandgap Devices pages 20ndash38 2007 1

[5] Marco Avellaneda Craig Friedman Richard Holmes and Dominick Samperi Calibrat-ing volatility surfaces via relative-entropy minimization Applied Mathematical Finance4(1)37ndash64 1997 32

[6] Christopher Baker Leslie Fox David Mayers and K Wright Numerical solution of Fredholmintegral equations of first kind The Computer Journal 7(2)141 ndash148 1964 36

[7] Anatolii Bakushinskii Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion USSR Computational Mathematics and Mathematical Physics24(4)181ndash182 1984 13 26

[8] Fischer Black and Myron Scholes The Pricing of Options and Corporate Liabilities TheJournal of Political Economy 81(3)637ndash654 1973 5

[9] Ilia Bouchouev Derivatives valuation for general diffusion processes In The InternationalAssociation of Financial Engineers (IAFE) Conferences volume 292 1998 32

[10] Ilia Bouchouev and Victor Isakov Uniqueness stability and numerical methods for the inverseproblem that arises in financial markets Inverse Problems 15(3)R95ndashR116 1999 34

[11] Ilia Bouchouev Victor Isakov and Nicolas Valdivia Recovery of volatility coefficient bylinearization Quantitative Finance 2257ndash263 2002 iii 32 44

[12] Julianne Chung Malena I Espanol and Tuan Nguyen Optimal Regularization Parametersfor General-Form Tikhonov Regularization pages 1ndash21 2014 27 28

[13] Timothy A Davis Algorithm 915 SuiteSparseQR ACM Transactions on MathematicalSoftware 38(1)1ndash22 2011 22

[14] Bruno Dupire et al Pricing with a smile Risk 7(1)18ndash20 1994 32 33

[15] Franklin R Edwards Hedge funds and the collapse of long-term capital management Journalof Economic Perspectives 13189ndash210 1999 8

BIBLIOGRAPHY

[16] William N Goetzmann and Geert K Rouwenhorst The origins of value The financial innov-ations that created modern capital markets Oxford University Press 2005 2 3

[17] Gene Golub and William Kahan Calculating the singular values and pseudo-inverse of amatrix Journal of the Society for Industrial amp Applied Mathematics Series B NumericalAnalysis 2(2)205ndash224 1965 21

[18] Jacques Hadamard Sur les problemes aux derivees partielles et leur signification physiquePrinceton university bulletin 13(49-52)28 1902 10

[19] Per C Hansen Rank-deficient and discrete ill-posed problems numerical aspects of linearinversion volume 4 Siam 1998 27

[20] Espen G Haug and Nassim N Taleb Option traders use (very) sophisticated heuristicsnever the Black-Scholes-Merton formula Journal of Economic Behavior and Organization77(2)97ndash106 2011 3 5 8

[21] Victor Isakov Recovery of volatility coefficient by linearization 2013 iii 32 34

[22] Kazufumi Ito Bangti Jin and Tomoya Takeuchi Multi-parameter Tikhonov regularizationarXiv preprint arXiv11021173 2011 28

[23] Steven L Jones and Jeffry M Netter Efficient capital markets the concise encyclopedia ofeconomics 2008 library of economics and liberty 2008 [Online accessed 31-July-2015] 1

[24] Joseph P Kairys and Nicholas Valerio The market for equity options in the 1870s TheJournal of Finance 52(4)1707ndash1723 1997 3

[25] Ioannis Karatzas Brownian motion and stochastic calculus (graduate texts in mathematics)page 150 2001 6

[26] Michael V Klibanov Recovering of dielectric constants of explosives via a globally strictlyconvex cost functional pages 1ndash22 2014 32 44

[27] Michael V Klibanov Carleman estimates for the regularization of ill-posed Cauchy problemsApplied Numerical Mathematics 9446ndash74 2015 9 13 14

[28] Michael V Klibanov and Vladimir G Kamburg Globally strictly convex cost functional foran inverse parabolic problem pages 1ndash10 2010 32 44

[29] Michael V Klibanov and Andrey V Kuzhuget Profitable forecast of prices of stock optionson real market data via the solution of an ill-posed problem for the Black-Scholes equationPreprint pages 1ndash13 2015 iii 1 9 11 12 28 44 45

[30] Michael V Klibanov and Alexander V Tikhonravov Estimates of initial conditions of parabolicequations and inequalities in infinite domains via lateral Cauchy data Journal of DifferentialEquations 237(1)198ndash224 2007 14

[31] Michael V Klibanov and Aleksa Timonov Carleman Estimates for Coefficient Inverse Prob-lems and Numerical Applications 2004 14 32

[32] Ronald Lagnado and Stanley Osher A technique for calibrating derivative security pri-cing models numerical solution of an inverse problem Journal of computational finance1(1)13ndash25 1997 32

[33] Minqiang Li You Don t Have to Bother Newton for Implied Volatility Available at SSRN952727 2006 32

[34] Beth Marie Campbell Hetrick Rhonda Hughes and Emily McNabb Regularization ofthe backward heat equation via heatlets Electronic Journal of Differential Equations2008(130)1ndash8 2008 11

BIBLIOGRAPHY

[35] Scott Mixon The foreign exchange option market 1917-1921 Available at SSRN 13334422009 3

[36] Lyndon Moore and Steve Juh Derivative pricing 60 years before blackndashscholes Evidencefrom the Johannesburg stock exchange The Journal of Finance 61(6)3069ndash3098 2006 3

[37] Christopher C Paige and Michael A Saunders LSQR An Algorithm for Sparse Linear Equa-tions and Sparse Least Squares ACM Transactions on Mathematical Software 8(1)43ndash711982 20 21

[38] Lothar Reichel Fiorella Sgallari and Qiang Ye Tikhonov regularization based on generalizedkrylov subspace methods Applied Numerical Mathematics 62(9)1215ndash1228 2012 28

[39] Christoph Reisinger Calibration of volatility surfaces (June) 2003 32

[40] Mark Rubinstein Implied binomial trees The Journal of Finance 49(3)771ndash818 1994 31

[41] Serap Sarikaya Gerhard-Wilhelm Weber and Yesim S Dogrusoz Combination of conven-tional regularization methods and genetic algorithm for solving the inverse problem of elec-trocardiography 2010 1

[42] Trevor J Saunders Aristotle Politics books I and II Clarendon Press 1995 2

[43] Thomas I Seidman Optimal Filtering for the Backward Heat Equation SIAM Journal onNumerical Analysis 33(1)162ndash170 1996 11

[44] Sebastien Soulan Maxime Besacier Patrick Schiavone et al Resist trimming etch processcontrol using dynamic scatterometry Microelectronic Engineering 86(4)1040ndash1042 2009 1

[45] Andrey N Tikhonov Solution of incorrectly formulated problems and the regularizationmethod In Soviet Math Dokl volume 5 pages 1035ndash1038 1963 13

[46] Andrey N Tikhonov and Vasiliy Y Arsenin Solutions of ill-posed problems Vh Winston1977 13

[47] Joseph de la Vega Confusion de confusiones Dialogos curiosos entre un filosofo agudo unmercader discreto y un accionista erudito (1688) 2a ed 1958 Madrid Publicaciones BancoUrquijo 1957 1 2

[48] Anatoly G Yagola Aleksandr S Leonov and Vadim N Titarenko Data Errors and an ErrorEstimation for Ill-Posed Problems Inverse Problems in Engineering 10(2)117ndash129 200211 12 13

[49] Jorge P Zubelli Inverse Problems in Finances A Short Survey of Calibration Techniquespages 1ndash12 2006 32

Contents

List of Figures

List of Tables

Introduction

Preliminaries

Black-Scholes model (BSM)

On implied volatility

Model shortcomings and criticism

Time Inverse Parabolic Problem Option Price Forecasting

In analogy to the heat equation

Option price forecasting Time inverse BSM

Solution approaches

Error estimation

Tikhonov regularization

Convergence results Carleman estimates

Carleman estimate

Stability estimates and Regularization

Discretization

LSQR

Minimization via Conjugate gradient method

Multifrontal Sparse QR factorization

Numerical Experiments

The backward-time problem Crank-Nicolson

Verification test time inverse heat solution

Assessment How to include boundary conditions

Bid-Ask spread requirements

Choice of regularization parameter

Numerical instability of gradient method and lsqr algorithm

Alternative Boundary conditions

Parameter Identification Implied Volatility as an Inverse Problem

Ill-posedness of the inverse problem and literature overview

Linearization approach by Isakov

Continuous time formula Dupire

Linearization at constant volatility

Uniqueness and Stability

Numerical solution of the forward problem

Numerical tests reconstruction of given volatility surfaces

Towards a real world scenario Smooth interpolation of market prices and remarks

Putting everything into action A large-scale real market data study

Option exchanges amp BATS

Data gathering

Data filtering

Trading strategy

Backtesting

Restriction on options with significant bid-ask spread A Comparison

Modification Negativity of extrapolated ask price

Impact of boundary conditions and decision criterion

Influence of volatility coefficient

Test on most actively traded options

Alternative one-day trading strategy

Forecasting with locally reconstructed implied volatility Future work

Conclusions

Bibliography

Inverse Problems inBlack-Scholes OptionPrice Forecasting withLocally Reconstructed

Volatility

Masterrsquos Thesis

Jan-Philipp Wolf

Department of Mathematics and Computer ScienceCentre for Analysis Scientific computing and Applications

SupervisorsProf dr Sorin Pop TU Eindhoven (NL) University of Bergen (NO)

Dr Michiel Hochstenbach TU Eindhoven (NL)Drir Jan ten Thije Boonkkamp TU Eindhoven (NL)Prof dr Christian Rohde Universitat Stuttgart (DE)

SimTech No 6

Eindhoven 30th November 2015

Abstract

Preface

The Bhagavad Gita

Contents

Contents vii

List of Figures ix

List of Tables xi

1 Introduction 1

CONTENTS

6 Conclusions 45

Bibliography 47

List of Figures

List of Tables

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Abstract

Preface

The Bhagavad Gita

Contents

Contents vii

List of Figures ix

List of Tables xi

1 Introduction 1

CONTENTS

6 Conclusions 45

Bibliography 47

List of Figures

List of Tables

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Preface

The Bhagavad Gita

Contents

Contents vii

List of Figures ix

List of Tables xi

1 Introduction 1

CONTENTS

6 Conclusions 45

Bibliography 47

List of Figures

List of Tables

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Contents

Contents vii

List of Figures ix

List of Tables xi

1 Introduction 1

CONTENTS

6 Conclusions 45

Bibliography 47

List of Figures

List of Tables

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

CONTENTS

6 Conclusions 45

Bibliography 47

List of Figures

List of Tables

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

List of Figures

List of Tables

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

List of Tables

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 1

Introduction

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

0 05 1 15 2

Pcpayoff

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

0 05 1 15 2

minus1

minus05

Ppayoff

0 05 1 15 2

minus1

minus05

payoff

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 2

Preliminaries

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

E[dW ] = 0 E[dW 2] = dt (22)

toradic

V ar[ds]

sradicdt

(partf

partt+ microt

partx+σ2t

part2f

partx2

partxdBt (26)

du =partf

parttdt+

partsds+

part2f

(partu

partt+partu

part2u

parts2(s2

))+O(dt3 dW 3)

(partu

partt+ smicro

parts+

2σ2s2

part2u

parts2

)dt+ σs

partsdW (29)

partsδs (210)

∆D =(minus partu

parttminus σ2

2s2part2u

parts2

)∆t (211)

∆t= rD (212)

partt+σ2

2s2part2u

parts2+ rs

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

d1 =ln sK + (r + σ2

2 )(T minus t)

σradicT minus t

N(x) =

minusinfin

1radic2π

expminusz2

2dz (220)

r =V1 minus V0V0

1 +R = (1 + r)t (222)

Rl = lnV1V0 (223)

rl =Rl

t (224)

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

ksumi=0

ri (226)

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 3

u(x t) =

infinsumk=1

int π

f(x) sin(kx)dx (35)

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

minus1000

u(x t) =

infinsumk=1

u(x t)2L2=

infinsumk=1

f2k exp2k2t (310)

f(x) =

100 |x| lt 1

10 else(311)

dx2 =10minus4

( 160 )

1 A solution exits

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Lu =partu

partt+σ(t)2

2s2part2u

and with

sb minus sas+

isin H2(QT ) (318)

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Jλ(u) =

int sa

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

u(s 0) = g(s) (324)

Lparu =

nsumij=1

aij(s t)uij +

nsumj=1

bj(s t)uj + b0(s t)u (326)

L0paru =

nsumij=1

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

micro1|η|2 lensum

bjC(QT ) leM j = 0 n (329)

partt(minusu2

2ϕ2λ) (331)

(ut + L0paru)2ϕ2

λ+ (333)

λ minus ϕ2λ

nsumij=1

aijuiuj

)+ divU2

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

(ut + L0paru)2ϕ2

λ ge 2

λ minus Cϕ2λ

nsumij=1

aijuiuj

(1 + uδH2(QT )) (340)

2) (341)

le C10(1 + uδH2(QT ))δβ (342)

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

∥∥L2(QTminusε)

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

2||Avλ||2L2(QT )+

343 Discretization

Jα(u) = dtds

NMminusMsumj=M+1

2dt+σ(t(

lfloorjminus1M

rfloor+ 1))2

αdsdt

NMminusMsumj=M+1

(uj minus Fj)2 +

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Dtlowast =1

minus1 1 0 0

minus 12 0 1

minus 1

2 0 12

0 0 minus1 1

isin RNtimesN (369)

AotimesB =

a11B a1nB

am1B amnB

Dslowast =1

minus1 1 0 0

minus 12 0 1

2 0 12

0 0 minus1 1

isin RMtimesM (373)

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Dttlowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RNtimesN (376)

+ dt2part2u1partt2

+O(dt3) (377)

+ 4dt2part2u1partt2

+O(dt3) (378)

part2u1partt2

=u1 minus 2u2 + u3

2dt2+O(dt) (379)

Dsslowast =1

12 minus1 1

1 minus2 1 0

0 1 minus2 10 0 1

2 minus1 12

isin RMtimesM (383)

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

]uminus

[bradicαDF

]∥∥∥∥2

Alternative method

uminus

0radicαDFγubd

∥∥∥∥∥∥2

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

344 LSQR

Lanczos process

αi = vTi wi

Tkyk = β1e1 (394)

xk = Vkyk

Tk for example

]∥∥∥∥2

](396)

BTK minusλ2I

] [tk+1

[β1e10

[rkλI

[Uk+1 00 Vk

] [tk+1

] (397)

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

∣∣∣∣∣∣∣∣ [Bk

]yk minus

[β1e10

] ∣∣∣∣∣∣∣∣2

Using the relation

partujpartuj

= ujδjj and

MNsumj=1

ujδjj = uj (399)

partuj

s Ds[uminus F ]

ssDss[uminus F ]

s Ds +DTttDtt +DT

ssDss]uminusAT b

s Ds +DTttDtt +DT

ssDss]F (3102)

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

R11 R12

](3103)

partt+ αd

part2T

T (x 0) = ψ(x) x isin [0 π] (3105)

T (x t) = 0 x = 0 π t isin [0 Te] (3106)

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

minus10 0 10 20 30 40 50 60 70 80 90 100 110

S price underlying

2 from

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

0 1 2 3

Ttemperature

Te = 25

0 1 2 3

Ttemperature

Te = 15

0 1 2 3

x position

Ttemperature

Te = 110

0 1 2 3

x position

Ttemperature

Te = 120

01 middot 10minus2

time t

spot price s

option

priceu(st)

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

||Auminus f ||2

||uminusF|| H

L-curve

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

0 02 04 06 08 1

G(λ)GCV

function

0 2 4 6 8

middot10minus5

middot10minus17

λG(λ)GCV

function

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Linear Regression

Constant Average

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

20minus20

option

priceu(st)

02 middot 10minus2

4 middot 10minus218

option

priceu(st)

middot10minus3

t time

option

priceu(t)

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 4

partt+

2s2σ(s t)2

part2u

parts2+ smicro

Time to expiry T

Strike K

latilit

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

2σ(s t)2 =

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

u(KT ) =

int infin

results in

φT (K) =part2u

partK2(KT ) (47)

b2(KT )

part2u

partK2=partu

partT(48)

the Dupire equation

partTminus 1

2K2σ(KT )2

part2u

partK2+ microK

y = ln(K

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

u(y τj) = uj(y) y isin ω j = 1 2 (414)

2a2(y τ) =

2σ20 + f0(y) + τf1(y) (415)

lowastey) (416)

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

partτminus 1

part2V

party2+ (

2σ20 + micro)

V (y 0) = 0 y isin R (418)

eminus y2

2τσ20+cy+dτ

d = minus 1

partτminus 1

part2W

W (y 0) = 0 y isin R (423)

α(y τ) =slowast

radic2πτe

minus y2

2τσ20

τ21 + τ22 +radicτ1τ2(τ1 + τ2)3b

lt 1 (427)

(radic

τ1τ2

+radic

τ2τ1

+ 2) b2

σ20+ 2

radic2π(

radicτ1 +

radicτ2)

2(τ2 minus τ1)lt 1 (428)

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

is then shown where

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ (431)

radicτradic2σ0

2τσ20 + (

x2 minus 2xy

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

dτ) (432)

radicπ

partτminus 1

2a(y τ)2

part2u

party2+ (

2a(y τ)2 + micro)

un+1j minus unjδt

2a(y τ)2[θ

un+1j+1 minus 2un+1

j + un+1jminus1

δx2+ (1minus θ)

δx2] (434)

2a(y τ)2[θ

un+1j+1 minus un+1

jminus1

2δx+ (1minus θ)

un+1j + 2σθun+1

jminus1 = (435)

j = 2 N minus 1

N = 0 which yields

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Aun+1 = Bun (440)

int infin

|xminusy|+|y|σ0

radic2τ

eminusτ2

radicπ

2Slowast erfc

radic2τ

) (441)

Nsumk=1

with entries

such that

[K10 K11

K20 K21

] (446)

Kf = W (447)

f = Kminus1W (448)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

05minus05 0 05 1

minus1

minus05

volatilitya(yτ)

Numerical example

20 + 1

2y2 Let s = 20 σ0 = 1

nsumi=1

int xn

(d2micro(x)dx2

dx (449)

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

20 25 30 35 40 45 50 55 600

strike price K

option

priceu(KT

lowast )

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 5

52 Data gathering

53 Data filtering

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

54 Trading strategy

55 Backtesting

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

b(i) = b0 +

nosuma=1

ntminus2sumb=1

Xi1d1(a b) +Xi

2d3(a b) i = 1 2 nt minus 2 (51)

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Chapter 6

Conclusions

What has been done

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

BIBLIOGRAPHY

Contents

List of Figures

List of Tables

Introduction

Preliminaries







Solution approaches

Error estimation



Carleman estimate


Discretization

LSQR






















Data gathering

Data filtering

Trading strategy

Backtesting








Conclusions

Bibliography

Documents

Eindhoven University of Technology MASTER Inverse problems in Black-Scholes … · Abstract The Black-Scholes equation for option pricing is a famous model in nancial mathematics