Luiss

Quantitative Finance

Quantitative FinanceLuiss Business SchoolMASTER EMERGES

Mario Dell’Era

External Professor at Pisa [email protected]

October 29, 2013

Mario Dell’Era Quantitative Finance


Postulate of yield of money over the time

Every realistic model of Financial market can not regardless of thecost and the return of money, that is usually quantified by theinterest rates.

It is easy to see that the cost of money directly affects the strategy ofan agent, that must decide which is the more profitable investmenton market.




In economic and financial practice the cost of money has a centralrole. Daily experience teaches that those who deposit a euro at abank expects its euro grows over time at a rate determined by thecurrent interest rate, or in other words:

those who give up today to a financial availability, shifting overtime, requests that he be paid an appropriate fee called interest;who today requires the availability of a sum of which can have ata given future, it must match an appropriate reward calleddiscount;




to availability of a euro today is to start a process ofcapitalization, such as through a bank deposit, and thusgenerate a built-in one year as capital employed increased by aninterest. Therefore, the possibility of capitalization makes theeuro value of the euro possessed today’s most realized one year;the value of money is a function of the time in which is available;the postulate of time preference (or impatience or postulatepostulate of return on money) according to which any rationalindividual would prefer to have an amount immediately ratherthan in a subsequent age.




Taking as a basis the postulate of impatience, you can groped toanswer questions such as the following:

What is the value of a euro of today in the future?What it is value today of a euro which one takes in the future?



Capitalization and Discounting operations

From the beginning, financial mathematics (and actuarial) hasfocused on operations that allow you to move the money over timeaccording to two main types of operations:

Capitalization: the value of money is forward shifted over thetime;Discounting : the value of money is backward shifted over thetime.



Capitalization Operations

T = {t : 0 = t0 < · · · < tn = T}

1 Simple Capitalization

Mtn = M0(1 + nr)

2 Composed Capitalization

Mtn = M0(1 + r)n

3 Continuous Capitalization

MT = M0erT



Discounting Operations

T = {t : 0 = t0 < · · · < tn = T}

1 Simple Discounting

Zt0 = Ztn (1 + nr)−1

2 Composed Discounting

Zt0 = Ztn (1 + r)−n

3 Continuous Discounting

Z0 = ZT e−rT



Common Economic Sense, and Financial Laws

Intuitively, a market is efficient if it has no possibility of achievingcertain receipts without any risk, this circumstance and also definedopportunities for arbitrage. We define below the set of financial lawsunderlying the mathematical modeling of a market, able to guaranteethe formation of rational prices (arbitrage pricing), which excludepossibility of arbitrage.



Common Economic Sense, and Financial LawsEconomists identify, driven by common economic sense, the followinglaws as necessary and sufficient for the purpose:

Postulate of yield of money over the time;Law of one priceLaw of the linearity of the amountsLaw of the monotonicity of the amounts



Law of one Price

To illustrate the financial laws consider assigned a securities marketwhichM on the time horizon [0,T ] with a schedule T = {0,T},having assumed for the hypothesis postulated yield money.For a first definition of efficient market (or rational or coherent)economists assume the validity of a very important financial law,known as law of one price, according to which:

if two contracts have the same payoff to the same expirationdate in the future, then the cost of acquisition of the contracts isthe same.



Law of one PriceLet A and B two contracts with the pay table given by:

t = 0 t = T

Long A −p (X ) X

Long B −p (Y ) Y

where the payoffs X and Y are, in general, random numbergenerator.



Law of one PriceSuppose that the law of one price is violated: p(X) > p(Y) and,whatever happens, X = Y. Then we can create a contract Cgiven by the intersection of sale of A and B of the purchase withcash flow given by:

t = 0 t = T

Short A p (X ) −X

Long B −p (Y ) Y

C p (X )− p (Y ) > 0 −X + Y = 0



Law of one PriceIn summary, a marketM is efficient if it is truth that:

all contracts which have certainly the same payoff at maturityt = T , have also the same price at the time t = 0:

X = Y ⇔ p (X ) = p (Y )



Law of the linearity of the amounts

Another important Financial law is the following:

If the price of payoff X is p(X ) then for each c ∈ R the price ofthe payoff cX is cp(X ). With more generality:

∀a,b ∈ R p (aX + bY ) = a p (X ) + b p (Y )



Law of the linearity of the amounts

Consider the cash flow of three contracts:

t = 0 t = T

A −ap (X ) aX

B −bp (Y ) bY

C p (aX + bY ) −aX − bY

D = A + B + C p (aX + bY )− ap (X )− bp (Y ) 0



Law of the monotonicity of the amounts

The last important Financial law is the following:

If the price of payoff X is p(X ) and the price of payoff Y is p(Y )and, with certainly, at maturity T will be X > Y , then the pricep(X ) can not be lower than the p(Y ). In other words:

X > Y ⇔ p (X ) > p (Y )

We prove that if it is not valid the law of monotony of the amounts,then there exists an opportunity for arbitrage.



Law of the monotonicity of the amounts

Consider the pay table:

t = 0 t = T

A −p (X ) X

B p (Y ) −Y

C = A + B p (Y )− p (X ) X − Y

Since X > Y , the payoff of the operation C is certainly positive atmaturity date t = T , then it is an opportunity for arbitrage.



Financial Risks

The risk is the central concept of modern finance. As we shallsee, every financial transaction may be viewed as trading risk.Every economic activity is, as always, influenced by risk factors.In order to eliminate or at least reduce the Financial Risk havebeen invented several types of financial contracts, and they arecontinually invented new ones.



Financial RisksThe sources of risk are many, we list a few:

Market Risk: which depends on the risk factors that affect theoverall progress in market prices.Credit Risk: risk incurred by a party for the eventual inability(partial or total) of the counterparty to fulfill the commitmentsassumed in a contract.Liquidity Risk: risk due to the mismatch between supply anddemand on the market and that makes it impossible, or delayed,purchase and sale transactions.



Market Risk and Forward contracts

ForwardA type of contract that, since ancient times, has been used toeliminate the risk of market is the Forward contract in which twoparties are organizing a private market for the exchange deferred toan activity. More precisely:

A Forward contract is a financial contract consisting of thestipulation between two parties, for the purchase/sale of anasset/commodity at a predetermined price at the time t0 that itwill be exchanged at a specified future date T > t0.



ForwardIn a Forward contract:

Who decided to purchase the asset is one that takes a longposition or being long side of the contract.Who decided to sell the asset is one who takes a short positionor which is the short side of the contract. It is said that the sale isshort sale if, at the conclusion of the contract, the seller does nothave the right to deliver.The price agreed by the parties is called delivery price orForward price or Strike price.



Financial Markets

Over the time have been born organized markets, with officesdedicated to trading regulation, and its respect. In other words wepassed from Private accords, as well as is (OTC) to a regulated andorganized market in which every days are traded several million ofcontracts.

Roles of Traders on MarketsAssume that the main figures that give rise to financial contracts are:

Hedgers: It’s who covering a market position, assuming a longor short position, by opening another.Speculators: It’s who creates new position on market.Arbitrageurs: It’s who by crossing several positions on market,attempts to take a gain without any risk.



Financial Engineering

DerivativesOne of the most fertile of financial areas is the creation of financialinstruments, known as derivatives, whose value is derived from anasset or index. Derivatives are abstract financial instruments, whosePayoffs are functions of the underlyings. We can show hereafter themain kind of derivatives contracts as:

BondsForwardFuturesOptions



Monetary Market Model

In the following we will refer to a market multi-period time horizon[0,T ] with calendar:

T = {t0, . . . , tn}

, dove 0 = t0 < · · · < tn = T .In an efficient market model, we assumethat there are default free bonds on the market for each maturity ofT. We’ll ignore the credit risk and liquidity. Assume perfect divisibilityof securities.We will assume:

the possibility of short sales: each agent can take on debtpositions of each title being transferred.

In particular, each agent can borrow by issuing bonds.



Bonds

We will call the bond a contract in which two parties at the time t0agree to exchange an amount fixed future dates from a schedule:

T = {t0, . . . , tn}

, dove 0 = t0 < · · · < tn = T .For simplicity, we will consider onlyfixed-income securities in which the issuer of the bond receives attime t0 amount P and agrees to pay periodically to the creditor, thedates of the calendar residual {t1, · · · , tn}, interest on and therepayment at maturity T = tn of invested capital at the date of signingof the contract.



Bonds

CBBA typical title of the fixed-interest or coupon bearing bonds CBBwith investment horizon [0,T ] generates, for the buyer, adeterministic flow amounts given by:

X = −P I{t0} +n∑

i=1

cF I{ti} + F I{tn} (1)

or cash flow

O =t0 t1 · · · tn−1 tn

−P cF · · · cF (c + 1) F(2)



Bonds

ZCBOne ZCB can be considered one CBB with zero coupon rate andtherefore cash flow for the buyer is given by:

X = −Z n0 I{t0} + I{tn} (3)

or cash flow

O =t0 t1 · · · tn−1 tn

−Z (t0, tn) 0 · · · 0 1(4)



Term Structure of ZCB.

Given a market in which are negotiable ZCBs, if at the time t0 theprices recorded for maturities {t1, t2, · · · , tn}, are respectively:

Z (t0, t1),Z (t0, t2), · · · ,Z (t0, tn)

this sequence is the term structure of prices at time t0, thatsatisfies the property of monotonicity:

1 = Z (t0, t0) > Z (t0, t1) > · · · > Z (t0, tn) > 0



Efficient Market model

Hypothesis for an Abstract or Ideal market model

The theoretical goal that we are aiming, is the definition of a marketmodel which, starting from a limited set of simple hypotheses:

Market Frictionless,Competitive Market,Markov Processes,Absence of Arbitrage opportunities,

that allow to modeling a mathematical structure in which to evaluatethe price of any financial contract.



Complete Market Model

Replication Strategies

An efficient market model is also defined complete, if the valueof every traded contracts can be replaced by strategies onmarket. Following this methodology, we are able to obtain theno-arbitrage price for every Derivatives contract by a suitablestrategies, where the fair price is the actualized expected futurecash flow.



Asset Pricing Theorems

1st Asset Pricing Theorem

Given a Markov’s market model, the following conditions:absence of arbitrage and the existence at least of one Risk-Neutralprobability measure are equivalent to each other.

2nd Asset Pricing Theorem

Given a complete Markov’s market model, the following conditions:absence of arbitrage and the existence of a unique Risk-Neutralprobability measure are equivalent to each other.



Pricing for Arbitrage: Forward price

Considering a contract Forward a game with zero-sum, we have:

Replication Strategy

t T

Short Forward 0 F Tt − ST

Buy S −St ST

Sell F Tt zcb Z T

t F Tt −F T

t

Net Z Tt F T

t − St 0



no-arbitrage Forward price

It is intuitively clear that a transaction with no final amountcertainly must have a rational cost of acquisition null andtherefore the equilibrium price of the Forward contract is givenby:

F Tt =

St

Z Tt



Pricing for Arbitrage: Futures

FuturesFutures are standard contracts, whose Payoff is given by thedifference between the spot price St of the underlyingasset/commodity, and the futures price f T

t at time t and maturity T as:

(St − f Tt ).

Futures can be traded only on regulated markets, in which there isthe cash compensation that adjusts the margins at the end oftrading days.



Cox-Ross-Rubinstein

no-arbitrage Futures price

To determine the relationship between spot prices and Futures priceswe can follow the following trading strategy suggested byCox-Ross-Rubinstein:

t0 t1 t2 t3

−f30 M10 f30 + M1

0

“f31 − f30

”0 0

0 −M10 f31 M1

0 M21 f31 + M1

0 M21

“f32 − f31

”0

0 0 −M10 M2

1 f32 M10 M2

1 M32 f32 + M1

0 M21 M3

2

“f33 − f32

”

−f30 0 0“

M10 M2

1 M32

”∗ X

thusf 30 =

(M1

0 M21 M3

2)∗ X



Example

Forward and FuturesAs result of climate changes, a major Company that produces Fuel bywheat, worried about the price of wheat in a year, in order to hedgeoneself against the risk of appreciation of the latter, it decides to buythe grain necessary for the production, to meet the needs of an entireyear, using the Forward and Futures contracts.Be given a schedule It = {0,T1,T2,T3,T4}, where T1 = 3−month,T2 = 6-month, T3 = 9-months and T4 = 12-months. Let S0 = 5.00$ thespot price of wheat in kg. Supposing to enter into a contract Forwardand in a contract Futures, for delivering of 100.000 kg of wheat inone year and in six months respectively. The interest rates for bondswith spot stipulation in T = 0 and maturity T1, T2, T3 and T4 arerespectively: r1 = 3%, r2 = 4%, r3 = 5% and r4 = 6%.



Compute:(1) the Forward price for delivering of 100.000 kg of wheat in one

year;(2) the Futures price for delivering of 100.000 kg of wheat in six

months;

Suppose lending rate and borrowing rate equal to each other.



(1) The Forward price is given by the following equation based onarbitrage principle: F T4

0 =(S0er4(T4−0) × 100.000

)$.

(2) For the Futures price we have to consider the forward rate rfamong T2 and T1; which can be obtained by: rf = r2T2−r1T1

T2−T1. Thus

using the Cox-Ross-Rubinstein law, one has the Futures price asrequired: f T2

0 =(er1(T1−0)+rf (T2−T1)S0 × 100.000

)$.



Derivatives

Options can be of different style:European style;American style;Bermuda style.

Options of European Style

An Option is a contract that gives to the holder the right and do notthe obligation, to buy or to sell at maturity T , the underlying asset, atthe strike price established at the signing time.This right has a price, that is the price of the option which has to beevaluated with great careful from the issuer.



Vanilla Options

Call Option

A Call option is a financial contract between two parties, the buyerand the seller of this type of option. The buyer of the call option hasthe right, but not the obligation to buy an agreed quantity of aparticular commodity or financial instrument (the underlying) from theseller of the option at a certain time (the expiration date) for a certainprice (the strike price). The seller (or ”writer”) is obligated to sell thecommodity or financial instrument should the buyer so decide. Thebuyer pays a fee (called a premium) for this right. Indicate its payoffas follows:

ψCall = (ST − K )+ = max (0,ST − K )



Vanilla Options

Call Option

Figure:



Vanilla Options

Put Option

A Put option is a contract between two parties to exchange an asset(the underlying), at a specified price (the strike), by a predetermineddate (the expiry or maturity). One party, the buyer of the put, has theright, but not an obligation, to re-sell the asset at the strike price bythe future date, while the other party, the seller of the Put, has theobligation to repurchase the asset at the strike price if the buyerexercises the option. Indicate its payoff as follows:

ψPut = (K − ST )+ = max (0,K − ST )



Vanilla Options

Put Option

Figure:



Complete Market Model in discrete time

Binomial ModelConsider a single time step δt . We know the asset price S0 at thebeginning of the time step; the price S1 at the end of the period isa random variable. We start with price S0, at the next instant weassume that the price may take either value S0u or S0d , whered < u, with probabilities pu and pd respectively.

S0u↗

S0↘

S0d

t = 0 δt



Binomial Modelits future value, depending on the realized state, will be either

Π1 = ∆S0u + βeδt or Π1 = ∆S0d + βeδt

Let us try to find a portfolio which will exactly replicate the optionpayoff:

∆S0u + βeδt = fu∆S0d + βeδt = fd

Solving this system of two linear equations in two unknownvariables ∆, β.



Binomial Modelwe get

∆ =fu − fd

S0(u − d)

β = e−rδ(

ufu − dfdu − d

)

In order to avoid the arbitrage, the initial value of this portfoliomust be exactly to f0:

f0 = ∆S0 + β

=

(fu − fdu − d

)+ e−rδ

(ufu − dfd

u − d

)= e−rδt

{erδt − du − d

fu +u − erδt

u − dfd

}Mario Dell’Era Quantitative Finance


Risk Neutral Measure

Binomial ModelIt is important to note that this relationship is independent on theobjective probabilities pu and pd ; we can nevertheless interpret theabove equation as an expected value if we set

qu =erδt − du − d

, qd =u − erδt

u − d

where d < u. We may notice thatqu + qd = 1;qu and qd are positive if d < erδt < u, which must be the case ifthere is no arbitrage strategy involving the riskless and the riskyasset; hence we may interpret qu and qd as probabilities.



Binomial Modelthe option price can be interpreted as the discounted expectedvalue of payoff under those probabilities:

f0 = e−rδtEQ[f1|F0 = S0] = e−rδt (qufu + qd fd )

It’s worth noting that the expected value of S1 under probabilitiesqu and qd is

EQ[S1|S0] = quS0u + qdS0d = S0erδt

The last observation explains why the “artificial probabilities” quand qd are called risk-neutral.



Multi-periodic Binomial Model

Cox-Ross-RubinsteinConsider to repeat the above scheme on a trading calendar

T = {t0, t1, t2, t3}.

Suppose S is a risk asset, that follows the Markov process

Sn (ξ1, . . . , ξn) = Sn−1 (ξ1, . . . , ξn−1)

(u

1 + ξn

2+ d

1− ξn

2

)=

= Sn−1 (ξ1, . . . , ξn−1) u(1+ξn)/2d (1−ξn)/2

(5)and Bn = ertn , a riskless bond, where n=0,1,2,3.




where 0 < d < ertn < u

Q {ξn = +1} = qu =ertn − du − d

Q {ξn = −1} = qd =u − ertn

u − d

(6)

for which we have our process is a martingale

u qu + d qd = 1

EQ (Sn|Fn−1) = Sn−1 (uqu + dqd ) = Sn−1




S0 u u u↗

S0 u u↗ ↘

S0 u S0 u u d↗ ↘ ↗

S0 S0 u d↘ ↗ ↘

S0 d S0 u d d↘ ↗

S0 d d↘

S0 d d d

t0 t1 t2 t3



Option Pricing


Consider a Binomial Market model, thus the price of an option,whose payoff is a function f (Sn) of the underlying asset S, isgiven by:

f0 = f (S0) = e−rtnEQ[f (Sn)|F0]

= e−rtnn∑

k=0

(nk

)qk

u qn−kd f

(S0uk dn−k

)



Option Pricing

Vanilla Options

C0 = e−rTn∑

k=0

(nk

)max

(0,S0uk dn−k − K

)qk

u qn−kd

P0 = e−rTn∑

k=0

(nk

)max

(0,K − S0uk dn−k

)qk

u qn−kd



Complete Market Model in continuous time

Black-Scholes market modelSuppose that on markets there exist only risk securities as St ,riskless or fixed income bonds as Bt and derivatives, whosevalue is a function f (ST ,T ) also said Payoff .Considering thedynamic of St follows a Geometric Brownian motion, whoseanalytical form is given by the following SDE (stochasticdifferential equations):

dSt = µStdt + σStdWt

and the dynamic of the fixed income as:

dBt = rBtdt



Black-Scholes market modelDefine Black-Scholes market model, the set of equations:

dSt = µStdt + σStdWt →asset

dBt = rBtdt →fixed income

f = f (ST ,T )→Payoff

in which µSt is the drift term, σSt is the diffusion term and r is theinterest rate.



Risk Neutral MeasureBy Asset Pricing theorems we know that in any market models ofMarkov, in which there is no arbitrage opportunities, a risky assetcan not gain in mean more than a fixed income security.Thus weneed to change the probability measure of Geometrical Brownianmotion following Girsanov’s theorem:

dWt = γtdt + dWt

such that for γt = r−µσ , one has:

dSt = rStdt + σStdWt



Risk Neutral Black-Scholes market modelDefine risk neutral Black-Scholes market model, the set ofequations:

dSt = rStdt + σStdWt →asset

dBt = rBtdt →fixed income

f = f (ST ,T )→Payoff

Since we have supposed that the underlying asset follows aGeometrical Brownian motion, and the value over the time of anyderivative is built as function of the underlying asset f (t ,St ), then todescribe its dynamic we need to use the Ito’s lemma as follows:



Ito’s Lemma

Let be given a function f = f (t ,S) ∈ C1,2, i.e. so that, f is afunction continuos one time with respect to the variable t and twotimes with respect to the variable S. Computing the Taylorexpansion at first order for t and at second order for S of f (t ,S)one has:

df =∂f∂t

dt +∂f∂S

dS +12∂2f∂S2 dS2

Therefore substituting in the latter dS = rSdt + σSdWt , we have:

df (t ,S) =∂f∂t

dt +∂f∂S

“rSdt + σSdWt

”+

12∂2f∂S2

“rSdt + σSdWt

”2

=∂f∂t

dt +∂f∂S

“rSdt + σSdWt

”+

12∂2f∂S2

“(rSdt)2 + 2rSdt × σSdWt + (σSdWt)

2”



Ito’s Lemma

=∂f∂t

dt +∂f∂S

“rSdt + σSdWt

”+

12∂2f∂S2

“(rSdt)2 + 2rSdt × σSdWt + (σSdWt)

2”

=

„∂f∂t

+∂f∂S

rS«

dt +∂f∂S

σSdWt +12∂2f∂S2 (σSdWt)

2

+12∂2f∂S2

“(rSdt)2 + 2rSdt × σSdWt

”=

„∂f∂t

+∂f∂S

rS +12∂2f∂S2 σ

2S2«

dt

+12∂2f∂S2

“(rSdt)2 + 2rσS2dt

32

”+∂f∂S

σSdWt



Ito’s Lemma

=

»„∂f∂t

+∂f∂S

rS +12∂2f∂S2 σ

2S2«

dt

+∂2f∂S2 (rσS2)dt

32 +

12∂2f∂S2 (rS)2dt2 +

∂f∂S

σSdWt .

in which we have used the famous relation by which is possibleto build the Wiener process Wt beginning to the Random walkprocess εt :

dWt = εt√

dt



Ito’s LemmaIn other words, Ito considers worthless the terms with grade in dtgreater than one of the Taylor expansion of the function f (t ,S); thusone has the following relation whose name in literature is known likeIto’s lemma:

df (t ,S) =

„∂f (t ,S)

∂t+∂f (t ,S)

∂SrS +

12∂2f (t ,S)

∂S2 σ2S2«

dt

+∂f (t ,S)

∂SσSdWt



Ito’s Lemma

We can conclude that given a function f (t ,S) ∈ C1,2 such that Sfollows a Geometric Brownian motion, thus also f (t ,S) follows aGeometric Brownian motion with drift coefficient:(

∂f (t ,S)

∂t+∂f (t ,S)

∂SrS +

12∂2f (t ,S)

∂S2 σ2S2).

and diffusion coefficient:

∂f (t ,S)

∂SσS.



Black-Scholes PDETherefore as we have seen before, by Asset Pricing theorems,the expected value of the stochastic process df (t ,St ) has to beequal to the fixed income return:

EQ[df (t ,St )] = rf (t ,St )

namely

EQ

[∂f (t ,S)

∂t+ rS

∂f (t ,S)

∂S+σ2S2

2∂2f (t ,S)

∂S2 + σS∂f (t ,S)

∂SdWt

]= rf (t ,St )

Black− ScholesPDE :

∂f (t ,S)

∂t+ rS

∂f (t ,S)

∂S+σ2S2

2∂2f (t ,S)

∂S2 = rf (t ,St )



Vanilla Options

Black-Scholes Approach: Vanilla Options Pricing

Let be given the Black-Scholes PDE, for which we suppose thata generic underlying asset S follows a geometric Brownianmotion, and the price of the money creases with an interest rateequal to r , that we suppose to be a constant. In order to price aEuropean Call (Put) option we have to solve the followingCauchy problem, for which the initial condition is our payofff (T ,S) = (S − K )+ = max(S − K ,0):

∂f (t ,S)

∂t+σ2S2

2∂2f (t ,S)

∂S2 + rS∂f (t ,S)

∂S= rf (t ,S)

f (T ,S) = (S − K )+

S ∈ [0,+∞) t ∈ [0,T ]

where K is the strike price.



Vanilla Options


Changing the variable as follows:

x = ln S +

(r − 1

2σ2)

(T − t) x ∈ (−∞,+∞)

τ =σ2

2(T − t) τ ∈

[0,σ2

2T]

f (t ,S) = e−r(T−t) f (τ, x)



Vanilla Options


we have the one-dimension heat equation:

∂ f (τ, x)

∂τ=∂2 f (τ, x)

∂x2

f (0, x) = (ex − K )+

x ∈ (−∞,+∞) τ ∈[0,σ2

2T]

its solution is known in literature (see Handbook of Linear PartialDifferential Equation by A.Polyanin) and is given by:

f (τ, x) =1√4πτ

∫ +∞

−∞dx ′ f (0, x ′)e−

(x′−x)2

4τ

where x ′ is the integration variable at the time T :Mario Dell’Era Quantitative Finance


Vanilla Options


f (τ, x) =1√2πτ

∫ +∞

−∞dx ′

(ex′ − K

)+

e−(x′−x)2

4τ

=1√2πτ

∫ +∞

ln Kdx ′

(ex′ − K

)e−

(x′−x)2

4τ

=1√2πτ

∫ +∞

ln Kdx ′ex′e−

(x′−x)2

4τ − 1√2πτ

K∫ ∞

ln Kdx ′e−

(x′−x)2

4τ

=1√4πτ

∫ +∞

ln Kdx ′e−

(x′−x)2−4τ(x′−x)+4τ2

4τ ex+τ

− 1√4πτ

K∫ ∞

ln Kdx ′e−

(x′−x)2

4τ



Vanilla Options


=1√4πτ

ex+τ

∫ +∞

ln Kdx ′e−

[(x′−x)−2τ ]2

4τ − 1√4πτ

K∫ ∞

ln Kdx ′e−

(x′−x)2

4τ

Let us impose: ξ = [(x′−x)−2τ ]√2τ

and η = (x′−x)√2τ

, thus we have:

dξ =dx ′√

2τ, ξ =

[(ln K − x)− 2τ ]√2τ

=[(ln K − ln S −

(r − 1

2σ2)

(T − t)− σ2(T − t)]√σ2(T − t)

,



Vanilla Options


dη =dx ′√

2τ, η =

[ln K − x ]√2τ

=[ln K − ln S −

(r − 1

2σ2)

(T − t)]√σ2(T − t)

.

Therefore substituting the new variable in the previous integral,we have:

f (τ, x) =1√4πτ

ex+τ

∫ +∞

ξ

√2τdξe−

ξ22

− 1√4πτ

K∫ ∞η

√2τdηe−

η22



Vanilla Options


=1√2π

ex+τ

∫ +∞

ξ

dξe−ξ22 − 1√

2πK∫ ∞η

dηe−η22

Remembering that f (t , s) = e−r(T−t) f (τ, x), and

x = ln S +

(r − 1

2σ2)

(T − t), τ =12σ2(T − t),



Vanilla Options


we have:

f (t , s)

= e−r(T−t){

1√2π

ex+τ

∫ +∞

ξ

dξe−ξ22 − 1√

2πK∫ ∞η

dηe−η22

}

= e−r(T−t)eln S+(r− 12σ

2)(T−t)+ 12σ

2(T−t) 1√2π

∫ +∞

ξ

dξe−ξ22

− e−r(T−t) 1√2π

K∫ ∞η

dηe−η22

= S1√2π

∫ +∞

ξ

dξe−ξ22 − K

1√2π

∫ ∞η

dηe−η22



Vanilla Options


where

N(−ξ) =1√2π

∫ ∞ξ

dξe−ξ2

2 , N(ξ) =1√2π

∫ ξ

−∞dξe

−ξ22 ,

and N(ξ) + N(−ξ) = 1.



Vanilla Options


Thus we can write the price of European Call option as follows:

f (t ,S) = S ∗ N(d1)− e−r(T−t)K ∗ N(d2)

where d1 = −ξ =[(ln S/K+(r+ 1

2σ2)(T−t)]√

σ2(T−t),

d2 = −η =[ln S/K+(r− 1

2σ2)(T−t)]√

σ2(T−t)and d2 = d1 −

√σ2(T − t).



Vanilla Options


For a European Put option we are able to repeat the samecalculus, for which our payoff is f (T ,S) = (K − S)+, and one hasthe following result:

f (t ,S) = e−r(T−t)K ∗ N(−d2)− S ∗ N(−d1) .


Education

Luiss