Jorge P. Zubelli - UFSCmtm.ufsc.br/~aleitao/public/slides-cmrs03/slides-zubelli.pdf · Jorge P. Zubelli IMPA, Rio de Janeiro, Brazil Thanks to A. Leitao and W. Muniz˜ May 1, 2014

Inverse Problems and Stochastic Volatility Models

Jorge P. Zubelli

IMPA, Rio de Janeiro, BrazilThanks to A. Leitao and W. Muniz

May 1, 2014

Stochastic Volatility Models c©J.P.Zubelli (IMPA) May 1, 2014 1 / 69

Outline

1 Intro and Background

2 Problem Statement and Results on Local Vol Models

3 Main Technical Results

4 Numerical Examples w/ Synthetic and w/ Real Data

5 Connections with Exponential Families and Risk Measures

6 Conclusions


Historical RemarksOptions and Derivatives

600BC Greek Philosopher and Geometer Thales from Miletus: Optionsand Futures of Olives (for Olive Oil) Call options on olive presses ninemonths ahead of the next harvest

XVI century Dutch fisherman made forward contracts before departing

XVII Netherlands - Options on prices of tulips

XVII century future and option contracts on rice were negociated inAmsterdan and Osaka on rice.

XIX century USA Contracts of to arrive type on flour started to be traded(CBOT) in 1849-1850, standard future contracts on grains wereintroduced in 1865.


Historical Remarks

1972, CME started to negociate the first future contracts on currencies (6currencies)

Since the mid 80’s the growth of derivative markets overcame the growthof the underlyings

Theoretical Developments: 70’s F. Black - M. Scholes - R. Merton


Derivative Contracts

European Call Option: a forward contract that gives the holder the right, butnot the obligation, to buy one unit of an underlying asset for an agreed strikeprice K on the maturity date T . Its payoff is given by

h(XT ) =

XT −K if XT > K ,

0 if XT ≤ K .


Derivative Contracts

European Put Option: a forward contract that gives the holder the right to sella unit of the asset for a strike price K at the maturity date T . Its payoff is

h(XT ) =

K −XT if XT < K ,

0 if XT ≥ K .

At other times, the contract has a value known as the derivative price. Theoption price at time t with stock price Xt = x is denoted by P(t,x).


Main Contributions

Historical Remarks

Figure: Thales

Thales (Miletus)

L. Bachelier (Paris)

P. Samuelson (Nobel 1970)

F. Black

M. Scholes (Nobel 1997)

R. Merton (Nobel 1997)


Main Contributions

Figure: L. Bachelier

Thales (Miletus)


P. Samuelson

F. Black

M. Scholes

R. Merton


Figure: P. Samuelson

Thales (Miletus)


P. Samuelson

F. Black

M. Scholes

R. Merton


Some Background, Terminology, and NotationClassical Black-Scholes-Merton

Under highly simplifying assumptions, the call option price C on an underlyingX is given by

CBS(X , t;K ,T , r ,σ) = XN(d+)−Ke−r(T−t)N(d−) (1)

where N is the cumulative normal distribution.

d± =log(Xer(T−t)/K )

σ√

T − t± σ√

T − t2

. (2)

Some of the Assumptions:

Non dividend paying (just for simplicity)

Complete and Frictionless Markets

Exponential Brownian motion dynamics

Constant Volatility


Plot of the Black-Scholes Price of a Call


However...

Volatility is not deterministic! It is even a multi-scale phenomena!

It is not true that the underlying undergoes an Exponential BrownianMotion

Even more so in high frequency contexts...

Implied Volatility: The value of the volatility that should be used in theBlack-Scholes formula to give the quote market price of a derivative.


The Concept of Implied Volatility

RecallCBS(X , t;K ,T , r ,σ0) = XN(d+)−Ke−r(T−t)N(d−) (3)

where N is the cumulative normal distribution function and

d± =log(Xer(T−t)/K )

σ0√

T − t± σ0

√T − t2

. (4)

Notion of Implied Volatility: Fix everything else and consider

σ 7−→ CBS(X , t;K ,T , r ,σ)

The implied volatilty is the inverse to this map.IMPLIED VOL: ”wrong number that when plugged into the wrong equationgives the right price”


Figure: Implied Volatility Surface- (From Bruno Dupire - IMPA talk)


Remarks

1 Implied Vol is only a way of renaming the price (change of variables).2 Crucial to find the connection with the underlying dynamics.3 Crucial to introduce the concept of LOCAL VOLATILITY

Concept of Stochastic Volatility:

dXt

Xt= µtdt + σtdWt

Fundamental Issue: How to model the process σt?One Possibility: Following Dupire... (Local Volatility Model) assume thatσt = σ(t,Xt)Another Possibility: Assume that the process σt = f (Yt) undergoes its owndynamics according to a SDE.


Figure: Example of Data from IBOVESPA


Limitations of Classical Black-Scholes

log-normality of asset prices is not verified by statistical tests

option prices are subjet to the smile effects

volatility tends to fluctuate with time


Local Volatility Models

Idea: Assume that the volatility is given by

σ = σ(t,x)

i.e.: it depends on time and the asset price.Easy to check that the Black-Scholes eq. holds.

∂P∂t

+12

σ(t,x)2x2 ∂2P∂x2 + r

(x

∂P∂x−P

)= 0 (5)

P(T ,x) = h(x) (6)

or in the case you have dividends:

∂P∂t

+12

σ(t,x)2x2 ∂2P∂x2 + (r −d)x

∂P∂x− rP = 0

P(T ,x) = h(x)


Motivation and Goals

FocusDupire [Dup94] local volatility models

Goal:

Present a unified framework for the calibration of local volatility models

Use recent tools of convex regularization of ill-posed Inverse Problems.

Present convergence results that include convergence rates w.r.t. noiselevel in fairly general contexts

Go beyond the classical quadratic regularization.

Applicationsrisk management

hedging

evaluation of exotic derivatives.


The Direct and the Inverse Problem

The Direct ProblemGiven σ = σ(t,x) and the payoff information, determine P = P(t,x ,T ,K ;σ)

The Inverse ProblemGiven a set of observed prices

P = P(t,x ,T ,K ;σ)(T ,K )∈S

find the volatility σ = σ(t,x).

The set S is taken typically as [T1,T2]× [K1,K2].In Practice: Very limited and scarce dataNote: To price in a consistent way the so-called exotic derivatives one has toknow σ and not only the transition probabilities


The Smile Curve and Dupire’s Equation

Assuming that there exists a local volatility function σ = σ(x , t) for which (5)holds Dupire(1994) showed that the call price satisfies

∂T C− 12 σ2(K ,T )K 2∂2

K C + rS∂K C = 0 , S > 0 , t < TC(K ,T = 0) = (X −K )+ ,

(7)

Theoretical: way of evaluating the local volatility

σ(K ,T ) =

√2

(∂T C + rK ∂K C

K 2∂2K C

)(8)

In practice To estimate σ from (7), limited amount of discrete data and thusinterpolate. Numerical instabilities! Even to keep the argument positive is hard.


LiteratureVery vast!!!

Avellaneda et al.[ABF+00, Ave98c, Ave98b,Ave98a, AFHS97]

Bouchev & Isakov [BI97]

Crepey [Cre03]

Derman et al. [DKZ96]

Egger & Engl [EE05]

Hofmann et al. [HKPS07, HK05]

Jermakyan [BJ99]

Achdou & Pironneau (2004)

Roger Lee (2001,2005)

Abken et al. (1996)

Ait Sahalia, Y & Lo, A (1998)

Berestycki et al. (2000)

Buchen & Kelly (1996)

Coleman et al. (1999)

Cont, Cont & Da Fonseca (2001)

Jackson et al. (1999)

Jackwerth & Rubinstein (1998)

Jourdain & Nguyen (2001)

Lagnado & Osher (1997)

Samperi (2001)

Stutzer (1997)


Problem Statement

Starting Point: Dupire forward equation [Dup94]

−∂T U +12

σ2(T ,K )K 2

∂2K U− (r −q)K ∂K U−qU = 0 , T > 0 , (9)

K = X0ey , τ = T − t , b = q− r , u(τ,y) = eqτU t,X (T ,K ) (10)

and

a(τ,y) =12

σ2(T − τ;X0ey ) , (11)

Set q = r = 0 for simplicity to get:

uτ = a(τ,y)(∂2y u−∂y u) (12)

and initial conditionu(0,y) = X0(1−ey )+ (13)


Problem Statement

The Vol Calibration ProblemGiven an observed set

u = u(t,S,T ,K ;σ)(T ,K )∈S

find σ = σ(t,S) that best fits such market data

Noisy data: u = uδ

Admissible convex class of calibration parameters:

D(F) := a ∈ a0 + H1+ε(Ω) : a≤ a≤ a. (14)

where, for 0≤ ε fixed, U := H1+ε(Ω) and a > a > 0.

Parameter-to-solution operator

F : D(F)⊂ H1+ε(Ω)→ L2(Ω)

F(a) = u(a) (15)


Setting of the problem

Theorem (H. Egger-H. Engl[EE05] Crepey[Cre03])The parameter to solution map

F : H1+ε(Ω)→ L2(Ω)

is

weak sequentialy continuous

compact and weakly closed

Consequences:

The inverse problem is ill-posed.

We can prove that the inverse problem satisfies the conditions to apply theregularization theory.


Well-Posed and Ill-Posed Problems

Hadamard’s definition of well-posedness:

Existence

Uniqueness

Stability

The problem under consideration: Ill-posed.Equation:

F(a) = u

Need Regularization:


Approach

Convex Tikhonov RegularizationFor given convex f minimize the Tikhonov functional

Fβ,uδ(a) := ||F(a)−uδ||2L2(Ω) + βf (a) (16)

over D(F), where, β > 0 is the regularization parameter.

Remark that f incorporates the a priori info on a.

||u−uδ||L2(Ω) ≤ δ , (17)

where u is the data associated to the actual value a ∈D(F).

Assumption (very general!)

Let ε≥ 0 be fixed. f : D(f )⊂ H1+ε(Ω)−→ [0,∞] is a convex, proper, coerciveand sequentially weakly lower semi-continuous functional with domain D(f )containing D(F).


Questions

Theoretical Questions:Does there exist a minimizer of the regularized problem?

Suppose that the noise level goes to zero... How fast does the regularizedgo to the true solution?

Results obtained in joint work with D. Cezaro and O. Scherzer.Published in J. Nonlinear Analysis, 2012 [DCSZ12]


Questions

Practical Questions:Can we devise an iterative algorithm to compute the solution?

Does this algorithm converge?

Can we regularize by stopping the iteration judiciously?

We proved:1 A tangential cone condition that ensures convergence of the

Landwebber iteration. Joint work with D. Cezaro. (IMA J. of AppliedMath.)

2 Obtained a Morozov-type criterium to stop the iteration.


Main Theoretical ResultF(a) = u(a) (15) F

β,uδ(a) := ||F(a)−uδ||2L2(Ω) +βf (a) (16)

Theorem (Existence, Stability, Convergence)

For the regularized inverse problem

F(a) = u (18)

we have:

∃ minimizer of Fβ,uδ .

If (uk )→ u in L2(Ω), then ∃ a seq. (ak ) s.t.

ak ∈ argmin

Fβ,uk (a) : a ∈D

has a subsequence which converges weakly to a

a ∈ argmin

Fβ,u(a) : a ∈D


Main Theoretical Result (cont)F(a) = u(a) (15) F

β,uδ(a) := ||F(a)−uδ||2L2(Ω) +βf (a) (16)

Theorem (cont.) NOISY CASE

Take β = β(δ) > 0 and assume

β(δ) satisfies

β(δ)→ 0 andδ2

β(δ)→ 0 , as δ→ 0 . (19)

The seq. (δk ) converges to 0, and that uk := uδk satisfies ‖u−uk‖ ≤ δk .

Then,1 Every seq. (ak ) ∈ argminFβk ,uk , has weak-convergent subseq. (ak ′).2 The limit a† := w− limak ′ is an f -minimizing solution of (15), and

f (ak )→ f (a†).3 If the f -minimizing solution a† is unique, then ak → a† weakly.


Bregman distance

Let f be a convex function. For a ∈D(f ) and ∂f (a) the subdifferential of thefunctional f at a.We denote by D(∂f ) = a : ∂f (a) 6= /0 the domain of the subdifferential.The Bregman distance w.r.t ζ ∈ ∂f (a1) is defined on D(f )×D(∂f ) by

Dζ(a2,a1) = f (a2)− f (a1)−〈ζ,a2−a1〉 .

Assumption (1)

We assume that1 ∃ an f -minimizing sol. a† of (15), a† ∈DB(f ).2 ∃β1 ∈ [0,1), β2 ≥ 0, and ζ† ∈ ∂f (a†) s.t.

〈ζ†,a†−a〉 ≤ β1Dζ†(a,a†) + β2∥∥F(a)−F(a†)

∥∥L2(Ω)

for a ∈Mβmax(ρ) ,(20)

where ρ > βmaxf (a†) > 0.


Convergence rates [SGG+08]

Theorem (Convergence rates [SGG+08])

Let F , f , D , H1+ε(Ω), and L2(Ω) satisfy Assumption 1. Moreover, letβ : (0,∞)→ (0,∞) satisfy β(δ)∼ δ. Then

Dζ†(aδ

β,a†) = O(δ) ,

∥∥∥F(aδ

β)−uδ

∥∥∥L2(Ω)

= O(δ) ,

and there exists c > 0, such that f (aδ

β)≤ f (a†) + δ/c for every δ with

β(δ)≤ βmax.

Example: The regularization functional f as the Boltzmann-Shannon entropy

f (a) =∫

Ωa log(a)dx , a ∈D(F) ,


Example of the Convergence Rate


How about the assumptions?

Although Assumption 1 may seem too restrictive, the next result reveals that itcan be obtained from rather classical ones:

Proposition

Assume that ∃ an f -minimizing solution a† of F(a) = u and that F is Gateauxdifferentiable at a†.Moreover, assume that ∃γ≥ 0 and ω† ∈ L2(Ω)

∗ with γ∥∥ω†

∥∥ < 1, s.t.

ζ† := F ′(a†)∗ω† ∈ ∂f (a†) (21)

and ∃βmax > 0 satisfying ρ > βmaxf (a†) such that∥∥F(a)−F(a†)−F ′(a†)(a−a†)∥∥ ≤ γDζ†(a,a†) , for a ∈Mβmax(ρ) . (22)

Then, Assumption 1 holds.


How about algorithms?NOTE: We have proved

We have also proved a tangential cone condition for this problem, which impliesthat the Landwever iteration converges in a suitable neighborhood. LandweberIteration [EHN96]:

aδk+1 = aδ

k + cF ′(aδk )∗(uδ−F(aδ

k )) . (23)

Discrepancy Principle:∥∥∥uδ−F(aδ

k∗(δ,yδ))∥∥∥ ≤ rδ <

∥∥∥uδ−F(aδk )∥∥∥ , (24)

where

r > 21 + η

1−2η, (25)

is a relaxation term.If the iteration is stopped at index k∗(δ,yδ) such that for the first time, theresidual becomes small compared to the quantity rδ.


Numerical Examples with Simulated DataDescription of the Examples

Using a Landweber iteration technique we implemented the calibration.

Produced for different test variances a the option prices and addeddifferent levels of multiplicative noise.

The examples consisted of perturbing a = 1 during a period ofT = 0, · · · ,0.2 and log-moneyness y varying between −5 and 5.

Initial guess: Constant volatility.


Numerical Examples - Exact Solution


Numerical Examples - Exact Solution


Numerical Examples 1 - noiseless - 4000 steps


Numerical Examples 1 - error - 100 steps













Numerical Examples 2 - 5% noise level - 100 steps








Numerical Examples 2 - 5% noise level - Stopping criteria



Numerical Examples 2 - 5% noise level - 2000 iterationsToo many!!!


Local Vol Surface from Heston Model

Figure: Local Vol Surface associated to Heston Model Calibrated on PBR data


Local Vol Surface from Heston Model

Figure: Local Vol Surface associated to Heston Model Calibrated on SPX data


Local Vol Surface for WTI Crude Oiltotally nonparametric

Figure: Local Vol Surface associated to Heston Model Calibrated on SPX data


Numerical Examples: with Real DataReconstruction of a = σ2/2 with PBR Stock Data (implemented by Vinicius L. Albani/IMPA)

Figure: Minimal Entropy functional / Landweber Method / a priori Implied Vol /maturities: 2010-11


Numerical Examples: with Real DataReconstruction of a with PBR Stock Data (implemented by Vinicius L. Albani/IMPA)

Figure: Minimal Entropy functional / Minimization (Levenberg-Marquadt) Method /


Comparison Kullback-Leibler × Quadratic RegularizationWTI Data

Figure: Left Kullback-Leibler Regularization and Right Quadratic Regularization


Comparison Kullback-Leibler × Quadratic RegularizationWTI Data

Figure: Left Kullback-Leibler Regularization and Right Quadratic Regularization


Local Vol Surface for Henry Hub Natural Gas

In the next plots we show an online approach (joint work w/ V. Albani). Weperformed the following:

We consider the evolution of prices of futures and options for several daysbut kept the maturity dates and all the other features of the options.

Calibrated using the extra information.

This is part of an extension of the above results that leads to incorporatingthe flow of information.


Figure: Local Vol Surface associated to Henry Hub Gas Prices












Connection with Statistics and Exponential Families

Regular Exponential Families:family of probability distribution functions pψ,θ : R→ R+ defined by

pψ,θ(s) := exp(s ·θ−ψ(θ))p0(s)

where ψ : R→ R∪+∞ is convex and p0 : R→ R+ is continuous.

Example:Gaussians parametrized by the mean.

The Darmois-Koopman-Pitman Thm: Under certain regularity conditions onthe probability density, a necessary and sufficient condition for the existence ofa sufficient statistic1 of fixed dimension is that the probability density belongs tothe exponential family [And70].

1no other statistic which b can be calculated from the same sample provides anyadditional information as to the value of the parameter


Connection with Statistics and Exponential Families(cont.)

Recall the Fenchel Conjugate

Given a function f : X → R∪+∞, the Fenchel dual f ∗X ∗→ R∪+∞ isdefined by

f ∗(x∗) := sup〈x∗,x〉− f (x) | x ∈ X

Theorem (Banerjee et al. [BMDG05])

Let ψ∗ denote the Fenchel transform of ψ, which we assume to bedifferentiable. Using the Bregman distance w.r.t. ψ∗

Dψ∗(a, a) = ψ∗(a)−ψ

∗(a)−ψ∗′(a)(a− a) ,

if we assume that a(θ) ∈ int(dom(ψ∗)), then

pψ,θ(a) = exp(−Dψ∗ (a,a(θ))

)exp(ψ∗(a)

)p0(a) . (26)



Example (Exponential Families and their Fenchel conjugates)

For a Gaussian distribution ψ(θ) = ϖ2

2 θ2, then ψ∗(a) = a2

2ϖ2 . For Poissondistribution ψ(θ) = exp(θ) we have ψ∗(a) = a log(a)−a.

ExampleAccording to Example 1, if we use the exponential family associated to Poissondistributions, we obtain Kullback-Leibler regularization, consisting inminimization of

a 7−→ Fβ,uδ(a) :=

∥∥∥F(a)−uδ

∥∥∥2

L2(Ω)+ βKL(a,a) , (27)

whereKL(a,a) =

∫Ω

a log(a/a)− (a−a)dx .

We note that the Kullback-Leibler distance is the Bregman distance associatedto the Boltzmann-Shannon entropy

G(a) :=∫

Ωa log(a)dx . (28)



Lemma (3)

Let Ω be a bounded subset of R2 with Lipschitz boundary. Moreover, assumethat F is continuous w.r.t. the weak topologies on L1(Ω) and L2(Ω), respec.

1 Let a,b ∈D(G). Then

‖a−b‖2L1(Ω) ≤

(23‖a‖L1(Ω) +

43‖b‖L1(Ω)

)KL(a,b) . (29)

(Convention: 0 · (+∞) = 0)

2 Let 0 6= a ∈DB(G), then the sets

Mβ,uδ(M) := a ∈DB(G) : F

β,uδ(a)≤M

are τU sequentially compact.



An important consequence of (29) and Theorem 2 is that∥∥∥aδ

β−a†

∥∥∥L1(Ω)

= O(√

δ) . (30)

Now, let δk be a sequence converging to zero and ak = aδkβk

the respective

minimizers of the Tikhonov functional (16). Take bk = a† for all k ∈ N. Then,from Lemma 2 ∥∥ak −a†∥∥

L1(Ω)→ 0 , as δk → 0 .


Conclusions

Volatility surface calibration is a classical and fundamental problem inQuantitative FinanceWe developed a unifying framework for the regularization that makes useof tools from Inverse Problem theory and Convex Analysis andestablished:

1 Convergence and convergence-rate results.2 Convergence of the regularized sol. w.r.t the noise level in L1(Ω)

(when Ω is compact)3 The connection with exponential families.4 The source condition connection with convex risk measures5 Implemented a Landweber type calibration algorithm.

Developed an Online Calibration MethodologyFuture Possibilities:

1 Use a priori asymptotic behavior (e.g.: following Roger Lee’s results for theimplied volatility)

2 Enhance the model to incorporate jumps and perhaps more sofisticatedmodels such as cogarch (for example).

.Stochastic Volatility Models c©J.P.Zubelli (IMPA) May 1, 2014 68 / 69

THANK YOU FOR YOUR ATTENTION!!!

Collaborators:V. Albani (IMPA), A. de Cezaro (FURG), O. Scherzer (Vienna)


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Jorge P. Zubelli - UFSCmtm.ufsc.br/~aleitao/public/slides-cmrs03/slides-zubelli.pdf · Jorge P. Zubelli IMPA, Rio de Janeiro, Brazil Thanks to A. Leitao and W. Muniz˜ May 1, 2014