Financial Engineering and Stochastic Calculas

7/31/2019 Financial Engineering and Stochastic Calculas

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Financial Engineering with Stochastic Calculus I

Johannes Wissel

Cornell UniversityFall 2010
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0. Introduction
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Motivation: Some examples of financial derivatives

an airline seeks protection against rising oil prices ( forwardcontract)

a company wants to hedge the risk in a payment obligation inforeign currency at a future time ( call option)

a fund manager wants to protect a stock position againstlosses ( put option)

Main objectives of financial engineering (FE) development of quantitative models for financial markets

design, pricing, and hedging of financial derivatives

development of quantitative methods of risk management

Stochastic calculus

provides math. framework for continuous time models in FE

asset values are modeled by stochastic processes

trading strategy values are modeled by stochastic integrals
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Course syllabus

I: Introduction: financial engineering, binomial model

II: Background in probability: information and -algebras,

independence, general conditional expectations, martingales,fundamental theorem of asset pricing

III: Brownian motion (BM): scaled random walks, definition ofBM, distribution of BM, filtration for BM, martingale property

of BM, quadratic variationIV: Stochastic calculus: stochastic integral, Ito processes,

Ito-Doeblin formula, Black-Scholes-Merton equation,multivariable stochastic calculus

V: Risk-neutral pricing: Girsanovs theorem, risk-neutral measure,martingale representation, fundamental theorems of assetpricing

VI: Miscellaneous topics (if time permits): dividends, forwardsand futures, PDE pricing techniques.


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1. The binomial model

Main goals

Introduce some fundamental ideas and concepts in FE

Introduce a math. model which later serves as a main building

block for Brownian motion and continuous time models
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Motivating problem: Pricing of options

Consider a financial asset S, e.g. a stock, which is traded on an

exchange. Denote the market price of the asset at time t by St. Call option on the asset S: A derivative which gives the

holder the right, but not the obligation, to buy the asset Sat a future time T for a pre-arranged price K from the issuer(to exercise the option).

Terminology: S underlying, K strike price, T maturityof the option.

Value of the option at maturity? If ST K, option is worthless. If ST > K, exercise the option (buy S for K) and sell S on the

exchange for ST, making a profit ST K.Thus, option value at maturity is

ST K

+(option payoff).

Question: What is the value of the option prior to T?


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Motivating problem: Pricing of options

Consider a financial asset S, e.g. a stock, which is traded on an

exchange. Denote the market price of the asset at time t by St. Put option on the asset S: A derivative which gives the

holder the right, but not the obligation, to sell the asset Sat a future time T for a pre-arranged price K to the issuer(to exercise the option).

Terminology: S underlying, K strike price, T maturityof the option.

Value of the option at maturity? If ST K, option is worthless. If ST < K, exercise the option (sell S for K) and buy S on the

exchange for ST, making a profit K ST.Thus, option value at maturity is

K ST

+(option payoff).

Question: What is the value of the option prior to T?


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The market

We consider a market with two assets:

Risky asset (stock, commodity, foreign exchange rate, ...)

Riskless asset (money market, bank account or bond withfixed interest rate)

To develop a concept of a fair value of financial assets andderivatives, throughout this course we make the standardassumption that agents may

borrow and invest at the same interest rate in the moneymarket

take long and short positions in all traded assets trade without transaction costs and feedback effects on prices

(frictionless market)


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Simplest example: One-period binomial model

We assume that prices for the bond Bt and stock St are quoted attimes t = 0 (today) and t = 1 (option maturity).Suppose B0 = B1 = 1, S0 = 100, and S1 is a random variable

taking values S1 = 115 or S1 = 90 each with probability p = 12 .Consider a call with strike K = 100. Value at time 0?

First guess: Expectation of the payoff

S1 100)+? GivesES1 100)+ = 12(115 100)+ + 12(90 100)+ = 7.5

Turns out to be too expensive: A suitable investment strategycan generate the option payoff out of less capital!

Indeed, suppose at time 0 we invest into shares of bond and shares of stock, and require that at time 1

+ 115 = (115 100)+, (1.1) + 90 = (90 100)+. (1.2)

We find = 0.6, =

54. This requires +

100 = 6

initial capital and does the same job as the call.


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The initial capital C0 = 6 is the fair value at time 0 of the optionin the above example.

Any price C0 = 6 for the option would introduce an arbitrageopportunity into the market:

If C0 > 6, the seller could make a certain profit of C0 6 byselling the option and simultaneously employing the aboveinvestment strategy.

If C0 < 6, the buyer could make a certain profit of 6 C0 bybuying the option and simultaneously employing the reversestrategy ( = 0.6, = 54).

Such arbitrage opportunities (possibility of a riskless profit without

net investment of capital) are unrealistic in most markets.

Absence of arbitrage is a fundamental concept in financialmarket models and in the theory of derivative pricing.
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Let C1 =

S1 K)+ denote the option payoff. We can rewrite(1.1), (1.2) in one equation as

+ S1 = C1,

which must hold for both possible values of S1.The fair option value at time 0 is the initial value of the aboveinvestment strategy

+ S0 = C0.

In the above example we have E[S1] > S0 and therefore E[C1] > C0.

Valuation via expectations. Now suppose we change theprobabilities of the values of S1 to p and 1 p such that

E[S1] = p 115 + (1 p) 90 = S0.We have to take p = 0.4. Then we obtain

C0 = +S0 = +E[S1] = E[+S1] = E[C1] = ES1K+

.
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Conclusion:

Our first guess expectation of the payoff works, but onlyunder a new probability measure!

The crucial property of the new measure is

E[S1] = S0.

The new measure is known as a risk-neutral or pricing measure.

The concept of pricing assets by computing expectations underrisk-neutral measures is another fundamental concept in financialengineering. We will later generalize this method to dynamic(multi-period) models using stochastic processes known asmartingales.

There is a deep relationship between absence of arbitrage andasset pricing via risk-neutral measures, known as the fundamentaltheorem of asset pricing. We will formulate and discuss severalversions of this result in later chapters of the course.


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The multi-period binomial model

Discrete-time model: time t = 0, 1,..., N

Asset prices Bt (bond) and St (stock) at time t

Bond price process:

B0 = 1

Bt+1 = Bt(1 + r), t = 0,..., N 1

where r 0 is a constant interest rate.Bt = (1 + r)

t for all t (no randomness riskless asset). Stock price process: initial value S0

St+1 = St u, probability pSt d, probability 1 p

for t = 0,..., N 1where 0 < p < 1 and 0 < d < r + 1 < u.St is a random variable for t

1

risky asset
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Pricing derivatives in the binomial model

A European-type derivative on an underlying asset S ischaracterized by a maturity date N a payoff function h(S)

The instrument pays to the holder h(SN) at time N Examples:

call option with strike K: h(S) = (S K)+ put option with strike K: h(S) = (K S)+

The key to derivative pricing is the idea of payoff replication viaself-financing trading strategies (sfts).
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Definition 1.1Consider a trading strategy which invests into t units of bondand t units of stock at time t = 0, 1, ..., N 1, so that theportfolio value at each time t is

Xt = tBt + tSt. (1.3)

The strategy is self-financing if for any t = 0, 1, ..., N 1

Xt+1 = t+1Bt+1 + t+1St+1 = tBt+1 + tSt+1. (1.4)

(1.4) says that the portfolio is rearranged at time t+ 1 from(t, t) to (t+1, t+1) without inserting or withdrawing capital.

The value process Xt of a sfts is computed as follows:

Xt+1Bt+1

XtBt

=tBt+1 + tSt+1

Bt+1 tBt + tSt

Bt= t

St+1Bt+1

StBt

Xn

Bn=

XkBk

+n1

t=ktSt+1Bt+1

StBt (1.5)

N
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Note:

The value process Xn at any time n > k is controlled by theportfolio value Xk at initial time k and the stock positionsk,..., n

1. (Bond positions are determined via (1.3)).

Xn is a random variable which depends on the (random)values of the stochastic process St, t = 0, ...., n.

Replication via a sfts

goal: find a sfts with XN = h(SN)

recursive procedure: Suppose we have a function ht+1(St+1).Can we find ht(St) and t(St) such that the one-period sftswith initial value ht(St) and stock position t(St) from t to

t+ 1 yields the value ht+1(St+1) at t+ 1? By (1.5),

ht+1(St+1)

Bt+1=

ht(St)

Bt+ t(St)

St+1Bt+1

StBt

(1.6)

must hold for any values of the random variables St, S

t+1.
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Given St there are two possibilities for St+1:

ht+1(Stu)

Bt+1=

ht(St)

Bt+ t(St)

Stu

Bt(1 + r) St

Bt

,

ht+1(Std)Bt+1

= ht(St)Bt

+ t(St) Std

Bt(1 + r) St

Bt

Solving for t(St), ht(St) gives

t(St) = ht+1(Stu) ht+1(Std)

St (u d) ,ht(St)

Bt=

1

Bt+1

pht+1(Stu) + (1 p)ht+1(Std)

where p is defined by

pStu

Bt(1 + r)+(1 p) Std

Bt(1 + r)=

StBt

p = 1 + r du d (1.7)

In summary we obtain
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Theorem 1.2For a derivative with payoff h(SN) at time N, define recursivelyfunctions hN,..., h0 via

hN(SN) = h(SN), (1.8)

ht(St)

Bt=

1

Bt+1

pht+1(Stu) + (1 p)ht+1(Std)

(1.9)

for t = N 1,..., 0 and p in (1.7). Then there exists a sfts withvalue process Xt = ht(St), t = 0, ..., N, and stock positions

t(St) =ht+1(Stu) ht+1(Std)

St (u d) (1.10)

for t = 0, ..., N 1.

We say we can replicate the payoff h(SN) from initial capitalhk(Sk) at time k within the binomial model, using the sfts (1.10).

We call Xk the fair price of the derivative at time k.
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2. Background inprobability theory

Main goals

to introduce concepts from probability theory on infinitespaces which are needed for continuous time models(probability measures, -algebras, independence, generalconditional expectations)

to further analyze the binomial model using elementaryprobability theory
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Interpretation

(, F, P) is a model for a random experiment. is the set of possible outcomes of the random experiment.

Fmodels the information obtained from observing the resultof the experiment. It contains the sets of outcomes which we

can distinguish by the observation.The observation may not tell us the exact outcome of theexperiment, but for any set A Fwe can tell whether or not A.

For any A

F, the number P[A] is the probability with

which we expect to be in A.

Example 2 2


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Example 2.2

Finite independent coin-toss space: Let

N = = (1...N) i {H, T} i = 1,..., N,F= {A | A N}, and P[A] = |A|2N for all A F.Example 2.3 (Another probability measure)

Finite independent coin-toss space (unfair coin), binomial model:

Let p (0, 1) andN =

= (1...N)

i {H, T} i = 1,..., N,F

=

{A

|A

N

}, and P[

{

}] = p#H(1...N)(1

p)#T(1...N)

for all N. This defines a probability on Fby settingP[A] =

A

P[{}].

It is easy to check the axioms for Fand P in Definition 2.1.


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The power set

F=

{A

|A

}is a -algebra for every space .

However this is not always the appropriate choice: For infinite spaces , it is not always possible to put a

reasonable probability measure on the power set, see Example2.4 below.

In some situations smaller -algebras model the relevantinformation, see Examples 2.31 and 2.33.

The trivial -algebra F0 = {, } also satisfies the axioms forevery space . This corresponds to the situation where no

information is available on the outcome of the random experiment.
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Example 2.4

Infinite independent coin-toss space: Let

= = (12...) i {H, T} i 1.F= {A | A } is a -algebra, but it is impossible to constructa meaningful probability measure on F.

For x n let Ax denote the set of all sequences beginningwith x. We want P[Ax] =

12n for our measure P. But this implies

P[{12...}] = P[A1 A12 ...] = limnP[A1...n ] = limn

1

2n= 0

(see [Shr04], Theorem A.1.1.), i.e. individual have probability zero.

We cannot build P from the bottom up, starting withelements .

Idea: Top down approach.


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1) For each n 1, let An consist of the sets Ax with x n.Examples:

A1 =

{AH, AT

},

A2 = {AHH, AHT, ATH, ATT}, etc.2) Collect all sets in A1, A2,..., and add all sets required to make

the collection a -algebra. We call the result F. Note that

F F.3) It can be shown1 that specifying the probability on the sets in

A1, A2,... (via P[Ax] = 12n for any x n) uniquelydetermines a probability measure P on F.

Note: The same construction works for p (0, 1) andP[Ax] = p

#H(x)(1 p)#T(x).

1

Caratheodorys extension theorem, see e.g. [Dur95], Theorem A.1.1.


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The last example shows that for infinite probability spaces,P[A] = 1 does not necessarily imply A = .

Definition 2.5Let (,

F, P) be a probability space. If A

Fsatisfies P[A] = 1,

we say that A occurs almost surely (a.s.).

Random variables and distributions


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Random variables and distributions

Definition 2.6Let (, F, P) a probability space. A function X : R is calledF-measurable if

{X b} := { | X() b} F

for all b R. We also say X is a random variable on (, F).IfF is the power set, then every function on is F-measurable.If X and Y are F-measurable, then f(X, Y) is also F-measurablefor every reasonable2 function f.

The distribution function FX of X is

FX(x) = P[X x].

2The function f must be Borel-measurable. Every function f : R2 R we

shall ever encounter is Borel-measurable.

Take the intervals [a, b] and all subsets ofR required to make the
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collection a -algebra (called the Borel--algebra B(R) on R).Let X be a random variable on (, F, P). Then {X B} Fforevery B

B(R). The distribution X of X under P is defined by

X(B) := P[X B], B B(R)X defines a probability measure on

R, B(R).

The distribution determines the distribution function, and vice versa.

Example 2.7 (Stock price in binomial model)

Consider the binomial model (N, F, P) (Example 2.2) and let

St() = S0u#H(1...t)d#T(1...t)

for t = 0, ..., N. Choose for instance S0 = 1 and u= 2, d =12 .

Then the distribution S2 of S2 under P is determined by

S2{4} = S2{14

} =14 , S2{1} =

12 .

M di d i bl X h d i i
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Many non-discrete random variables X have a density, i.e., afunction fX 0 on R such that

X(B) = B fX(x)dx for all B B(R).In particular, the distribution function FX satisfies

FX

(b) = X(, b] =

b

fX

(x)dx

and sofX(x) = F

X(x).

Example 2.8Let a < b and suppose X(B) = P[X B] =

B

1ba I[a,b](x)dx.

Then X has uniform distribution on [a, b]. Its density isf(x) = 1

ba I[a,b](x).
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Example 2.9

We say that X has exponential distribution with parameter if

FX(x) = P[X x] = 1 ex

for x 0, and FX(x) = 0 for x < 0. Then X has density

fX(x) = ddx1 ex = ex

for x 0 and fX(x) = 0 for x < 0.Example 2.10

Suppose P[X x] = (x) := x (z)dz with (x) = 12 e x22 .We say X has standard normal distribution. X has density .

Expectations


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Let X be a random variable on a probability space (, F, P). Theelementary definition of the expectation

E[X] =

X()P[]

cannot be used if is uncountably infinite (as in Ex. 2.4).

If X can only assume finitely many different values, we canwrite X =

ki=1 xiIAi with Ai F, using the indicator

function

IA() =

1 ( A)0 ( /

A)

Such a random variable is called a simple function. We thendefine

E[X] =k

i=1xiE

IAi

=

k

i=1xiP[Ai].


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If X 0, let Xn (n 1) be any sequence3 of simple functionssuch that Xn X (n ) a.s. Then define

E[X] = limnE[Xn].

One can show that this limit always exists in [0, ], and doesnot depend on the choice of the approximating sequence.

Finally for a general random variable X, note thatX+ = max{X, 0} and X = max{X, 0} are nonnegativeand satisfy X = X+ X. So we define

E[X] = E[X+] E[X],

provided that E[X+], E[X] < In this case we say that Xis integrable.

3There always exists such a sequence, e.g. Xn =n2n

j=0j

2nI{ j

2n X 0, and suppose we know that theoutcome of the random experiment will be in A. What isour estimate for X in this case?

The conditional expectation of X given A is

E[X|A] =E[X IA]

P[A] .

Example: Take the binomial model (cf. Example 2.2), where

S = S0

nYi and E [S ] = S0

nE [Yi ] = S0

u+dn


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Sn = S0i=1

Yi and E[Sn] = S0i=1

E [Yi] = S0

2

.

Now suppose we know the results of the first two coin tosses or

the values of Y1, Y2 or the information in F2. We find

E[Sn|AHH] =E

S0n

i=1 Yi

I{Y1=Y2=u}

P[AHH]= S0u

2u+d2

n2,

E[Sn|AHT] = S0udu+d2 n2 ,E[Sn|ATH] = S0du

u+d2

n2,

E[Sn|ATT] = S0d2u+d2

n2.

Note: On each of the individual sets, the conditional expectationsare equal to the value of the random variable S2

u+d2

n2. Thus

E

Sn

A

= E

S2u+d2

n2

A

for any set A {

AHH

, AHT

, ATH

, ATT}

, and hence for all A F2

.

In summary, the random variable S2 u+d2 n2 is F2-measurable


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2

for any A F2 satisfies

ESn A = ES2 u+d2 n2 AIt is an estimate of Sn based on the information in F2.Definition 2.36

Let X be an integrable random variable on (, F, P) and G be asub--algebra ofF. A conditional expectation of X given G isany random variable, denoted E[X|G], that satisfies

(i) (Measurability) E[X|G] is G-measurable,

(ii) (Partial averaging)

E

X IA

= E

E[X|G] IA

for any A G.

In the example above, E[Sn | F2] = S2 u+d2 n2.

A conditional expectation always exists ([Shr04], Theorem B.1).It is unique in the following sense. Let Y and Z be conditional

t ti f X i G Th A {Y Z 0} G b (i)
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expectations of X given G. Then A = {Y Z > 0} G by (i),and so by (ii)

E[(Y Z)IA] = E[YIA] E[ZIA] = E[XIA] E[XIA] = 0,which implies P[Y Z > 0] = 0. Reversing the roles of Y and Z,we obtain P[Z Y > 0] = 0. Hence Y = Z a.s.As for the expectation we have

Theorem 2.37

a) Linearity: For , R and r.v.s X, Y

E[X + Y | G] = E[X | G] + E[Y | G]

b) Monotonicity: If X Y a.s., then E[X | G] E[Y | G].c) Jensens inequality: If is a convex function then

E(X) G E[X | G].
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P f [Sh 04] Th 2 3 2 d [R 99] i (10 17)


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Proofs: [Shr04] Theorem 2.3.2, and [Res99] equation (10.17).
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Definition 2 40


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Definition 2.40a) A filtration on a space is a family of -algebras(

Ft)t=0,1,2,...,N (discrete time) or (

Ft)t

[0,T] (continuous time)

such thatFs Ft for all s t.b) A stochastic process (Xt)t[0,T] is a family of random variablesXt indexed with time t. We say that (Xt)t[0,T] is adapted to thefiltration (Ft)t[0,T] if Xt isFt-measurable for each t.

In discrete time, we replace [0, T] with {0, 1,..., N}.Intuition: Ft models the information available at time t.A filtration models the flow of information.

Adapted processes are obtained as follows. For a given stochastic

process (Xt), let Ft = (Xs | s t). This is called the filtrationgenerated by the process (Xt). Clearly (Xt) is adapted to (Ft).We can then construct further adapted processes from (Xt).

Example: Let the infinite coin toss space and


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p pFt = (Y1,..., Yt) for t = 0, 1, 2,... as in Example 2.31. (Y

t)t=0,1,2,...

is adapted.

(St)t=0,1,2,... with St = S0Y1 Yt is adapted. For any sequence of deterministic functions ft(),

t = 0, 1, 2,..., the process

ft(St)

t=0,1,2,...

is adapted.

Remark: In Theorem 1.2 we found that the stock positions for thereplication strategy of a European derivative are of the form

t = ft(St),

i.e., t is an adapted process. Intuitively, this means that t canbe determined with information available at time t. In particular,we do not look into the future to make the investment decision.

Definition 2 41


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Definition 2.41Let (Ft)t[0,T] be a filtration on and (Xt)t[0,T] a stochasticprocess on (,

F, P). Suppose(Xt)t

[0,T] is adapted and all Xt are

integrable.

(ii) The process (Xt)t[0,T] is a martingale ifE[Xt | Fs] = Xs for all 0 s t T.

(iii) The process (Xt)t[0,T] is a submartingale ifE[Xt | Fs] Xs for all 0 s t T.

(iv) The process (Xt)t[0,T] is a supermartingale ifE[Xt

| Fs]

Xs for all 0

s

t

T.

In the same way we define (sub-/super-) martingales in discretetime for a discrete time filtration (Ft)t=0,1,2,...,N.


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So StBt

t=0,1,2,...,Nis a martingale on (N, F, P) if


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pu+ (1 p)d = 1 + r p = 1+rdud (2.1)

(a submartingale if p > 1+rdud and a supermartingale if p < 1+rdud ).

Note:

p in (2.1) is equal to the parameter p in (1.7).

So p in (1.7) can be interpreted as that up-move probability

for which

StBt

t=0,1,2,...,N

becomes a martingale.

Suppose StBtt=0,1,2,...,N is a martingale under P. We musthave p [0, 1], which is satisfied if d < 1 + r < u.

This condition is equivalent to absence of arbitrage, see below.

Example 2.43Given initial wealth X0, let (Xt)t=0,1,2,...,N be the value process
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XtBt

= X0B0

+t1

n=0n Sn+1Bn+1

SnBn

of a sfts (see (1.5)), where n is Fn-measurable for each time n(investment decisions based on available information).

IfStBtt=0,1,2,...,N is a martingale , then so is

XtBtt=0,1,2,...,N:

it is adapted

XtBt

= XsBs

+t1n=s

n

Sn+1Bn+1

SnBn

, and for n s

En Sn+1Bn+1 SnBn Fs = EEn Sn+1Bn+1 SnBn Fn Fs= E

nE

Sn+1Bn+1

SnBn

Fn Fs= 0.


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Fundamental theorem of asset pricing

Consider a market with one riskless asset Bt and m risky assets Sit

for i = 1 m (stocks derivatives ) Asset prices are modeled


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for i = 1,..., m (stocks, derivatives,...). Asset prices are modeledby nonnegative adapted stochastic processes on a space (, F, P)with a filtration (Ft)t=0,1,2,.... We assume Bt > 0 for all t.Definition 2.45a) Consider a trading strategy which invests into t units of Band it units of S

i at time t = 0, 1, ..., N 1, so that the portfoliovalue at each time t is

Xt = tBt +m

i=1

itSit. (2.3)

We demand that t and it are bounded and

Ft-measurable.

b) The strategy is self-financing if for any t = 0, 1, ..., N 1

Xt+1 = t+1Bt+1 +m

i=1

it+1Sit+1 = tBt+1 +

mi=1

itSit+1. (2.4)
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Theorem 2.47 (FTAP in discrete time)

There is no arbitrage opportunity in the market B, S1, ..., Sd if,and only if there exits a measure P equivalent to P on ( F)


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and only if, there exits a measure

P equivalent to P on (, F)

such that all discounted price processesSitBtt=0,1,...,N fori = 1,..., m are martingales under P.

Proof of if part: Suppose there exists

P as above. Then (2.6)

implies that the value process XtBt

of any sfts is a

P-martingale.

Now if we had an arbitrage opportunity, its value process wouldsatisfyEXNBN

= X0

B0= 0 (2.7)

and P[XN < 0] = 0, thus also P[XN < 0] = 0. So (2.7) impliesP[XN > 0] = 0, and thus also P[XN > 0] = 0, a contradiction.

The proof of the only if part requires tools from functionalanalysis beyond the scope of this course, see e.g. Follmer andSchied, Stochastic Finance, Theorem 5.17. A measure

P as the

FTAP is called a martingale measure


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Motivation

Goal: constructing a continuous-time limit of the binomial model

L t ( F P) i E l 2 4 (i d d t i t


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Let (, F, P) as in Example 2.4 (independent coin tosseswith head and tail probability 12 . Define

Xj() =

1 (j = H)

1 (j = T)and a symmetric random walk by M0 = 0,

Mk =k

j=1

Xj, k = 1, 2,...

Note

#H(1...k) =kj=1 1+Xj()2 = 12k + Mk()#T(1...k) =

kj=1

1Xj()2 =

12

k Mk()

so

Sk = S0u#Hd#T = S0u

12(k+Mk)d

12(kMk). (3.1)


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We obtain

S (n)(t) = S0e12

n(log unlog dn)W(n)(t)+ 12 (log un+log dn)nt (3 4)


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S (t) = S0e2 2 (3.4)

We want to construct S(t) via taking n . This works ifwe have limits

W(n)(t) W(t)12

n(log un

log dn)

12(log un + log dn)n c

for n . In this case we obtain

S(t) = S0

eW(t)+ct

Now suppose we know that W(n)(t) W(t) for n .(We can hope that we get convergence since Var[W(n)(t)] = tfor all n.)
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Brownian motion

B i i i b i d h li i f W (n)( ) f


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Brownian motion is obtained as the limit of W(n)(t) for n .

Definition 3.3A stochastic process W(t), t 0 on a space (, F, P) is aBrownian motion (BM) if it satisfies

(i) W(0) = 0 a.s.

(ii) t W(t) is continuous a.s.(iii) For all times 0 = t0 < t1 < ... < tm, the increments

W(t1) = W(t1)W(t0), W(t2)W(t1), ..., W(tm)W(tm1)

are independent, and W(tj) W(tj1) N(0, tj tj1)for all j = 1, ..., m.

Distribution of Brownian motion

For 0 = t0 < t1 < ... < tm


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W(t1) W(t0),..., W(tm) W(tm1)=t1 t0Z1, ..., tm tm1Zmwith independent standard normal Z1, ..., Zm by (iii).

So W(t1),..., W(tm) has multivariate normal distribution(as linear combination of a multivar normal).For all times 0 s < t

E

W(s)

= E

W(t)

= 0,

CovW(s), W(t) = EW(s)W(t)= E

W(s)

W(t) W(s)+ W(s)2= s = min(s, t).


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I l d l F i D fi i i 3 4 i


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In general models, Ft in Definition 3.4 can contain moreinformation than F

W

t .However we only work with the filtration Ft = FWt , t 0 in ourmodels, so we drop the superscript and only write Ft as before.


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Geometric Brownian motion

S(t) = S0eW(t)+( 122)t


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(i) S(t) is continuous in t, a.s.(ii) For 0 s < t,

S(t) = S0eW(s)+( 122)se

W(t)W(s)

+( 122)(ts)

= S(s)eW(t)W(s)+( 122)(ts). (3.7)So the logarithmic return log S(t)

S(s) is independent ofFs (byproperty (iii) of BM).

(iii) For 0 s < t, the logarithmic return logS(t)S(s) has a normal

distribution. S(t) has a lognormal distribution.

Geometric BM is used as a model for the stock price in theBlack-Scholes model (see Chapter 4).
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Basic properties. 1) If f(t) is continuous and g(t) has finitefirst-order variation (e.g. if g(t) has an integrable derivative) then

[f, g](T) = 0 for all T.


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2) Thus if f(t) is continuous and has finite first-order variation,

[f, f](T) = 0 for all T.

3) Covariation is bilinear and symmetric. In particular,

[f + g, f + g](T) = [f, f](T) + [g, g](T) + 2[f, g](T).

4) If f1, f2 are continuous and g1, g2 are continuous and have finitefirst-order variation, then

[f1 + g1, f2 + g2] = [f1, f2].

5) The function [f, f](t) is increasing in t, and thus has finitefirst-order variation.

Quadratic variation of Brownian motion

Theorem 3.11Let W be a BM. Then [W, W](T) = T for all T 0 a.s.
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It follows that BM paths W(t), t 0, do not have finite first-ordervariation. W(t), t 0, is not differentiable (a.s.).

Intuition:BM accumulates quadratic variation at rate one per unit time.

Note: By definition,

[W, W](T)() = lim

||

0

n1

j=0 W(tj+1)() W(tj)()2

.

Theorem 3.11 says that the random variablesn1j=0

|W(tj+1) W(tj)|2

converge to T when

|

|0.


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So if S(t) is a geometric BM, then for small ||hist[0, T] (3.11)

for every T.


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Let S(t) be a stochastic process adapted to a filtration (Ft)t0,and > 0 a small time increment. The (annualized) volatility at t

is the standard deviation of 1

log S(t+)

S(t) conditional on Ft.

Assume again S(t) is a geometric BM. Then

1

log S(t+)S(t) =

W(t+)W(t)

+( 1

22)

N( 122), 2,

so the volatility is equal to .

we can identify from price observations (approximately)via (3.11).

Remark. In the second semester of the course we will analyzemodels which have non-constant (stochastic) volatility.

Excursion: Existence of BMDoes there exist a stochastic process as in Definition 3.3?

Existence proof via Donskers theorem: Let S be the space of allcontinuous functions on [0, T ] with (0) = 0. Define a norm on
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continuous functions on [0, T] with (0) 0. Define a norm on

S by = supt[0,T] |(t)|, and let Sbe the -algebra generatedby the open subsets of S with respect to the norm . For eacht [0, T] define a random variable on S by Wt() = (t), S.We define probability measures n on (S, S) by

n(B) = P[W(n)

() B], B S,where W(n)(t) is the scaled symmetric random walk in (3.3).

Theorem 3.12 (Donskers theorem)

There exists a unique probability measure on (S,

S) such that

n weakly (n ),

and the stochastic process Wt, t [0, T], is a BM on (S, S, ).Proof: See [Dur95] Section 7.6.

A consequence of Donskers theorem is the following:

C ll 3 13
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Corollary 3.13

Let W(n) be the scaled symmetric random walk and W be a BM.Then for any times 0 < t1 < ... < tm and any bounded continuousfunction h : Rm R we have

limnEhW(n)(t1),..., W(n)(tm) = EhW(t1), ..., W(tm).This result implies that when the number n of time steps per yeargoes to infinity, derivative prices in the binomial model converge toderivative prices in the Black-Scholes model (see Chapter 5).
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4. Stochastic calculus

Main topics

Stochastic integrals as value processes of self-financingtrading strategies

Ito-Doeblin formula

Application: Black-Scholes-Merton equation
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Continuous trading

We now consider strategies with continuously changing positions.A continuous trading strategy (t) is approximated by asequence of simple strategies (t) (trading at discrete times) for


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sequence of simple strategies n(t) (trading at discrete times) for

which the re-balancing frequency becomes small when n .That is,

(t) = limnn(t), t [0, T]. (4.3)

Remark: One can show that every continuous adapted process

(t) can be approximated by simple processes as in (4.3).

We will see that this approximation is a stable procedure in thesense that the value processes Xn(t) corresponding to the simpleself-financing trading strategies n(t) satisfy

Xn(t) X(t) (n )

for some limit value X(t).

Definition 4.2a) Let (t) = limn n(t) be a continuous adapted processapproximated by simple processes n(t). The stochastic integral

of (t) with respect to M(t) := S(t) is
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of (t) with respect to M(t) := ( )B(t) ist

0(u)dM(u) := lim

n

t0

n(u)dM(u), t [0, T]. (4.4)

b) The value process X(t) of the continuous self-financing

trading strategy (t) with initial value X(0) is given by

X(t)

B(t)= X(0) +

t0

(u)d

S(u)

B(u)

. (4.5)

We now specify, in three steps, a class of processes M(t) = S(t)B(t)

for which the limit in (4.4) exists.

Step 1: SI with respect to differentiable processes

Let M(t) be a stochastic process which is differentiable in t.

Theorem 4.3If is adapted and (t)M(t) integrable as a function of t, then


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the limit in (4.4) exists a.s., andt0

(u)dM(u) =

t0

(u)M(u)du.

Proof: For a simple process (t) =m1j=0 tjI[tj,tj+1)(t),t0

(u)dM(u) =m1j=0

tj

M(tj+1 t) M(tj t)

=m

1

j=0

tj tj+1ttjt

M(u)du =m

1

j=0

tj+1ttjt

(u)M(u)du

=

t0

(u)M(u)du.


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Excursion: existence and uniqueness of SI. Let L2 be the space

of all Ft-measurable random variables X with X = E[X2] 12 < ,and L2W be the space of all adapted processes with

W = Et

(u)2du

12

< .


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W 0 ( )

Let Sbe the space of all simple processes. Then the map

t0

(u)dW(u)

defines a linear isometry S L2 by Theorem 4.7. Since Sis densein L2W, and L

2 is a Banach space, there is a unique way to extendthis map7 to a linear isometry L2W L2 by setting

t0 (u)dW(u) := limnt

0 n(u)dW(u) in L

2

for a sequence n Swith n in L2W.7See e.g. Kreyszig, Introductory Functional Analysis with Applications, p. 100.

The last result provides a positive answer to the existence and uniqueness

question in (4.4) in the case of M(t) = W(t) BM

specifies a space L2W of admissible integrands for BM:


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all adapted processes with Et0 (u)2du < Note: If L2W adapted, then by definition

t0 (u)dW(u) := limnt

0 n(u)dW(u) in L2

when n are simple processes with

n

in L2W (n

).

The last condition is fulfilled for instance if n a.s. (n )and |n| || a.s. for all n. A sequence n of this type can befound for every adapted L2W.

Basic properties of SI. Let W be a BM.

Theorem 4.8The stochastic integral I(t) =

t0 (u)dW(u) satisfies

(i) I(t) is linear in .
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(ii) I(t) is Ft-measurable.(iii) I(t)t[0,T] is a martingale.(iv) I(t) is continuous in t.

Proof: (i) - (iv) are immediate for simple processes . For an

arbitrary adapted process L2W we choose simple n such thatn in L2W. Then (i) and (ii) generalize from n to . For(iii) we note that for A Ft, the L2-convergence gives

ET0 dW IA = limn ET0 ndW IA= limn E

t0 ndW IA

= Et

0 dW IA

.

The proof of (iv) requires results from martingale theory, see[Dur96] Chapter 2 Theorems 4.3a and 6.3.
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Stochastic integrals. The limit in the definition (4.4) of the SIexists in L2 if the integrator M is an Ito process:

Theorem 4.11 (Associativity)

Let adapted and X(t) an Ito process


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X(t) = X(0) +t0

(u)du+t0

(u)dW(u).

Then under the appropriate integrability conditions

t0

(u)dX(u) = t0

(u)(u)du+ t0

(u)(u)dW(u).

Implications:

1) If X(t) is an Ito process, then I(t) = t0 (u)dX(u) is againan Ito process.

2) SIs w.r.t. Ito processes can be reduced to SIs w.r.t. Brownianmotion plus ordinary (Lebesgue) integrals.
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Important Example: The SI X dXLet X an Ito process as in (4.6). We compute8

t0 X(u)dX(u).

Let 1, 2,... partitions of [0, t] with |n| 0 and definen(t) = X(tj)I[tj,tj+1)(t).


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tjnThen n(t) X(t) and so

2

t0

X(u)dX(u) = limn

tjn2X(tj)

X(tj+1) X(tj)

= limn

tjn

X(tj+1)2 X(tj)2 limn

tjn

X(tj+1) X(tj)2= X(t)2 X(0)2 [X, X](t). (4.7)

For X = W BM we have [W, W](t) = t by Theorem 3.11. Thust0

W(u)dW(u) =1

2W(t)2 1

2t.

8The SI exists ifEt

0(u)4du


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0shows that the chain rule from ordinary calculus does not hold forthe stochastic integral with respect BM: Let g(t) be differentiable.

chain rule:d

dug(u)2 = 2g(u)g(u) integral form:

g(t)2

g(0)2 = 2

t

0

g(u)g(u)du = 2t

0

g(u)dg(u)

The additional term t in (4.8) appears because BM has non-zeroquadratic variation, in contrast to differentiable functions.

Let f : R R be a C2 function. We want a chain rule forf

W(t)

. (Previously we had f(x) = x2). By Taylors theorem

f W(tj+1) f W(tj) = f W(tj) W(tj+1) W(tj)1 2
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+ 2 fW(tj )W(tj+1) W(tj)2with some tj (tj, tj+1). For a partition = {t0,..., tm} of [0, t]take

m1j=0 and let || 0, then

m1j=0 f

W(tj)W(tj+1) W(tj) t0 fW(u)dW(u)m1j=0

12 fW(tj )W(tj+1) W(tj)2 12 t0 fW(u)du

We obtain the Ito-Doeblin formula for BM

f

W(t) fW(0) = t

0f

W(u)

dW(u) +1

2

t0

f

W(u)

du.

Main results on Ito processes

We ultimately want to extend the Ito-Doeblin formula to multi-variate functions f

X1(t),..., Xd(t)

of Ito processes X1,..., Xd.

To this end we need the quadratic variation of an Ito process.
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Theorem 4.12Let X(t) = X(0) +

t0 (u)du+

t0 (u)dW(u) an Ito process.

Then its quadratic variation is

[X, X](t) = t

0

(u)2du.

Let Y(t) = Y(0) +t0 (u)du+

t0 (u)dW(u) another Ito process.

The covariation of X and Y is then (symmetric bilinear form)

[X, Y](t) = 12 [X + Y, X + Y](t) 12 [X, X](t) 12 [Y, Y](t)= 12t0

(u) + (u)

2du 12

t0 (u)

2du 12t0 (u)

2du

=t

0(u)(u)du.


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Proof of Theorem 4.12. We have to show [X, X](t) = t0 (u)2du.We give a proof under the assumption that is bounded.By remarks 1) and 3) after Definition 3.10, [X, X](t) = [Y, Y](t)for Y(t) =

t

0 (u)dW(u), so it suffices to show


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[Y, Y](t) = t0

(u)2du.

To this end, we use (4.7) to compute

[Y, Y](t) t0

(u)2du = Y(t)2 2t0

Y(u)dY(u) t0

(u)2du

= Y(t)2 t0

(u)2du 2t0

Y(u)(u)dW(u).

RHS is a continuous martingale by Theorem 4.14 and the martingaleproperty of Brownian integrals. LHS has finite first order variation.So Theorem 4.13 yields [Y, Y](t) t0 (u)2du = 0 for all t.

Unique representation of Ito processes. Let X(t) be an Ito

process and suppose that we have two representations

X(t) = X(0) +

t0

1(u)du+

t0

1(u)dW(u)

( )

t

( )

t

( ) ( )
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= X(0) + 0 2(u)du+ 0 2(u)dW(u)with adapted processes i, i for i = 1, 2 satisfying the requiredintegrability conditions. Then

Z(t) := t01(u) 2(u)du = t

02(u) 1(u)dW(u)

for all t. So Z(t) is a continuous martingale (RHS) with finite firstorder variation (LHS), hence Z(t) = 0 for all t by Theorem 4.13.We obtain Z(t) = 1(t)

2(t) = 0. Also

0 = E t

0

1(u)2(u)

dW(u)

2= E t

0

1(u)2(u)

2du

,

which implies 1(t) 2(t) = 0.


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Special cases

1) For d = 2 with X1(t) = t, X2(t) = W(t) we obtain theIto-Doeblin formula for BM (general case)

f W ( ) f 0 W (0)t

f W ( ) d


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ft, W(t) = f0, W(0)+ 0

ftu, W(u)du+

t0

fx

u, W(u)

dW(u) +1

2

t0

fxx

u, W(u)

du.

(4.10)

2) Let X, Y be Ito processes and f(x, y) = xy. Then we obtainthe product rule

X(t)Y(t) = X(0)Y(0)+t

0 X(u)dY(u)+t

0 Y(u)dX(u)+[X, Y](t).(4.11)

Proof of Theorem 4.15: See [Dur96] Section 2.10.

Differential notation

We often write the Ito formula

f X ( ) f X (0)

d t fX ( ) dXi ( )
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fX(t) = fX(0)+i=10 xiX(u) dXi(u)

+1

2

d

i,j=1t0

2f

xixj

X(u)

d[Xi, Xj](u).

in a differential form:

df

X(t)

=

d

i=1f

xiX(t)

dXi(t) +

1

2

d

i,j=12f

xixjX(t)

d[Xi, Xj](t).

For an Ito process

X(t) = X(0) +

t0

(u)du+

t0

(u)dW(u)

l t Y (t)t

0 ( )dX ( ) B Th 4 11 h
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let Y(t) = 0 (u)dX(u). By Theorem 4.11 we haveY(t) = Y(0) +

t0

(u)(u)du+

t0

(u)(u)dW(u).

We write this in differential form as

dX(t) = (t)dt+ (t)dW(t),

dY(t) = (t)dX(t)

implies

dY(t) = (t)(t)dt+ (t)(t)dW(t).

Examples

1) Generalized geometric BM. Let W(t) be a BM for afiltration (Ft)t0, let (t), (t) adapted processes, and

X(t) =t

0

(u)1

2

(u)2 du+t

0

(u)dW(u).
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0 2 0By the Ito formula S(t) = S0e

X(t) is an Ito process. f(x) = S0ex

yields f

X(t)

= f

X(t)

= f

X(t)

= S(t) and so

dS(t) = S(t)dX(t) +

1

2 S(t)d[X, X](t).

By Theorem 4.12, [X, X](t) =t0 (u)

2du, so Theorem 4.11 yields

dS(t) = S(t)(t) 1

2

(t)2dt+ S(t)(t)dW(t) +1

2

S(t)(t)2dt

= S(t)(t)dt+ S(t)(t)dW(t).

For constant and we obtain X(t) =

122

t+ W(t), so

S(t) = S0eW(t)+( 122)t is geometric BM.
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The Black-Scholes-Merton PDE. We have X(t) = ct, S(t).Compute differential on both sides. By (4.5)X(t)B(t) = X(0) +

t0 (u)d

S(u)B(u)

.

Also S(t)B(t) = S0e

W(t)+((r) 122)t is a geometric BM, so


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d

S(t)B(t)

=

S(t)B(t)

( r)dt+

S(t)B(t)

dW(t). (4.14)

Thus

dX(t)B(t) = (t)d S(t)B(t)= (t)

S(t)B(t)

( r)dt+ (t)

S(t)B(t)

dW(t),

(4.15)

dX(t) = dX(t)B(t) B(t) = X(t)B(t)dB(t) + B(t)dX(t)B(t)= X(t)rdt+ (t)S(t)( r)dt+ (t)S(t)dW(t)= X(t) (t)S(t) rdt+ (t)dS(t). (4.16)

SodX(t) =

X(t) (t)S(t)rdt + (t)dS(t).

By Itos formula,

dc t S(t) =
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dct, S(t) == ct

t, S(t)

dt+ cx

t, S(t)

dS(t) +1

2cxx

t, S(t)

d[S, S](t)

= ctt, S(t)+1

2

cxxt, S(t)S(t)22dt+ cxt, S(t)dS(t)Equating dX(t) and dc

t, S(t)

, we obtain

(t) = cx

t, S(t)

, (4.17)

rcxt, S(t)S(t) = rct, S(t)+ ctt, S(t)+ 12 cxxt, S(t)S(t)22.

So c(t, x) is the solution to the Black-Scholes-Merton PDE

ct(t, x) + rxcx(t, x) +1

22x2cxx(t, x) = rc(t, x) (4.18)

for all t [0, T), x > 0, with terminal condition

c(T x) = (x K )+ for all x 0 (4 19)
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c(T, x) = (x K)+ for all x 0. (4.19)The Black-Scholes formula is the solution to this PDE. It is

c(t, x) = xd+(t, x) Ker(Tt)d(t, x) (4.20)

where

d(t, x) = 1Tt

log xK

+

r 22

(T t)

.

This can be checked by direct verification10. The fair price at time t

of a European call with strike K and maturity T in the Black-Scholesmodel is X(t) = c

t, S(t)

. The replication strategy is given by

X(0) = c(0, S0) and (t) = cx

t, S(t)

.

10See [Shr04], Ex. 4.9. We shall see another derivation of (4.20) in Chapt. 5.

Remarks. 1) The Black-Scholes formula c(t, x) is a deterministic

function. The price X(t) = ct, S(t) and the stock position(t) = cx

t, S(t)

, t [0, T], are adapted stochastic processes.

2) The Black-Scholes formula does not depend on the drift ofthe stock. c t, S(t) depends on r, (model parameters), K, T

(option characteristics) and t S(t) (state variables)
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(option characteristics), and t, S(t) (state variables).3) The above analysis applies to every European type derivativewith a payoff h(ST) at time T for some function h(x), x 0. Thefair price is given by the solution c(h)(t, x) to the PDE (4.18) withterminal condition c(h)(T, x) = h(x) for all x

0.

Example. Let C(t), P(t) and F(t) be prices of a European call, a

European put, and a forward contract with payoffs

S(T) K+,

K S(T)

+

, and S(T) K, respectively. In any model we haveC(t) = F(t) + P(t) (put-call parity) under absence of arbitrage.

Also F(t) = S(t) er(Tt)K in the Black-Scholes model (checkthe PDE). Indeed, in the Black-Scholes model

P(t) = c t, S(t) S(t) + er(Tt)K.

Delta hedging and the greeks

The partial derivatives of ct, S(t) w.r.t. t, S(t), , r are calledthe greeks. In particular,

(t) = cx t, S(t) = d+(t, S(t)) > 0

(t) = cxx t S(t) = d+(t,S(t))S( )

T > 0
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(t) cxxt, S(t) ( , ( ))S(t)Tt > 0(t) = ct

t, S(t)

= S(t)

d+(t,S(t))

2

Tt rKer(Tt)

d(t, S(t)) < 0

Example: Hedging a short position in a call option. At time t0:(1) shares of option, (t0) shares of bond, (t0) shares of stock.Total portfolio value at t t0:

V

t, S(t)

= (1)c

t, S(t)

+ (t0)e

rt + (t0)S(t)

delta-neutral: V(t,S)S = 0

short gamma:2V(t,S)

S2< 0

long theta: V(t,S)t > 0


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Ito processes for multi-dimensional BM

Definition 4.19Let W(t) =

W1(t),..., Wd(t)

a d-dimensional BM with filtration

(Ft), t 0. An Ito process is a stochastic process of the form

X (t) = X (0) +t

(u)du +

d t

i (u)dWi (u) (4.21)


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X(t) X(0) + 0

(u)du+i=1

0

i(u)dWi(u) (4.21)

with a constant X(0) and adapted processes and i satisfyingthe appropriate integrability conditions.

Most results on Ito processes for 1-dimensional BM carry over.

Theorem 4.20 (Associativity)

Let adapted and X(t) an Ito process as in (4.21) Then under theappropriate integrability conditionst

0(u)dX(u) =

t0

(u)(u)du+d

i=1

t0

(u)i(u)dWi(u).

We again need quadratic variation and covariation of Ito processes.

Theorem 4.21Let

X(t) = X(0) +

t

0(u)du+

d

i=1 t

0i(u)dWi(u),

t d t
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Y(t) = Y(0) +

t0

(u)du+d

i=1

t0

i(u)dWi(u).

Then

[X, Y](t) =d

i=1

t0

i(u)i(u)du.

Proof: The idea is to first verify the claim for simple processes

h1, h2. For adapted processes h1, h2, one approximates theintegrands h1, h2 with simple processes.Details: See [Dur96] Chap. 2, Theorems (4.2c) (simple processes)and (5.4), (6.5), (8.7) (general integrands).

The Ito formula (4.9) again holds.

Theorem 4.22Let f : Rn R be a C2 function and X1(t), ..., Xn(t) Ito processes.Set X(t) = X1(t),..., Xn(t). Then


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( ) ( ), , ( )f

X(t)

= f

X(0)

+n

i=1

t0

f

xi

X(u)

dXi(u)

+ 12

ni,j=1

t0

2

fxixj

X(u) d[Xi, Xj](u).Proof: See [Dur96] Section 2.10.

The next result provides an important characterization of BM.

Theorem 4.23 (Levy)

Let M1(t), ..., Md(t) be continuous martingales for a filtration(Ft)t0 with Mi(0) = 0, [Mi, Mi](t) = t, and [Mi, Mj](t) = 0 fori = j. Then M(t) = M1(t),..., Md(t), t 0, is a d-dim BM.Proof (for d = 1, sketch): We show M(t)

M(s)

N(0, t

s)

and independent ofFs for s t. Take u R and define
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p X(t) = eiuM(t)+

12u2t.

By Itos formula for continuous martingales (see [Dur96] Chap. 2)

dX(t) = X(t)diuM(t) + 12u2t+ 12X(t)d[iuM, iuM](t)= X(t)iudM(t) + X(t)12u

2dt+ 12X(t)(1)u2d[M, M](t)= X(t)iudM(t).

It follows that X(t) is also a martingale. Hence

EeiuM(t)+ 12u2t Fs = eiuM(s)+ 12u2s,E

eiu(M(t)M(s))Fs = e 12u2(ts).

This implies M(t) M(s) N(0, t s) and independent ofFs.
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We have

Corr

log S1(t+)

S1(t), log S2(t+

)S2(t)

=Corr

1

W1(t+ ) W1(t)

+ (1 1221),

2W3(t+ ) W3(t)+ (2 1222)C [ ( ) ( ) ( ) ( )]


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( ) ( ) ( ) =Corr [W1(t+ ) W1(t), W3(t+ ) W3(t)]=Corr

W1(t+ ) W1(t),

W1(t+ ) W1(t)+1 2W2(t+ ) W2(t)=Corr[W1(t+ ) W1(t), W1(t+ ) W1(t)]=.

That is, logarithmic returns of S1

and S2

have correlation .

In summary, starting from a 2-dim BM, we constructed twogeometric BM asset price processes S1 and S2 with correlatedlogarithmic returns.

5. Risk-Neutral Pricing
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Main topics

Change of measure: Girsanovs theorem Risk-neutral measure

Martingale representation theorem

Fundamental theorems of asset pricing

Motivation

Discrete time: If the discounted stock process is a martingale, sois the discounted value process of any sfts (Example 2.43).

Continuous time: Let B(t) = ert, S(t) = S0eW(t)+( 122)t

(Black-Scholes model) and X(t) the value process of a sfts in thisS( )


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market. By Theorem 3.6, S(t)B(t) is a martingale if and only if = r.

In this case X(t)B(t) is also a martingale by (4.15) and Theorem 4.8.

If in addition X(T) = S(T) K+ (replication strategy), thenX(t)

B(t)= E

X(T)

B(T)

Ft = E

S(T) K+B(T)

Ft

. (5.1)

Problem: In general = r.Idea: Find a new measure P on (, F) such that S(t)

B(t) , t 0,becomes a martingale under P. Then replace E with E in (5.1).

Change of measure

Let Z 0 a random variable on (, F, P) with E[Z] = 1. Then

P[A] = E[ZIA] for all A F (5.2)

defines a new probability measure P on (, F) (check!). Note thatfor each A F, if P[A] = 0 then also P[A] = 0. Conversely, we have


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Theorem 5.1 (Radon-Nikodym)

Suppose P and

P are probability measures on (, F) such that for

each A F

, if P[A] = 0 then also P[A] = 0. Then there exists arandom variable Z 0 on (, F) such that E[Z] = 1 andP[A] = E[ZIA] for all A F.

Proof: See [Dur95] Appendix A.8.

Z is called a Radon-Nikodym derivative of P w.r.t. P anddenoted Z = d

PdP

. If Z is another random variable with theproperties in Theorem 5.1, then P[Z = Z] = 1 (check!).

Examples

1) Finite space. Let finite, F= power set of , and P, Pprobability measures such that P[A] = 0 implies P[A] = 0. Then

Z() = P[]P[] I{P[]>0},


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is a Radon-Nikodym derivative of P w.r.t P (check!).For with

P[] > 0 we have

P[] = Z()P[],so P is obtained by weighting the probabilities under P with Z.2) Binomial model. P and

P on (N, F) as in Examples 2.2

and 2.42. Then by 1)

Z() =P[]P[]

= 2Np#H()(1 p)#T(), N.

3) Normal distributions. Let X a random variable on (,

F, P)

with P[X x] = (x) and Y = X + for some R. HenceY N(, 1) under P.Find a probability P on (, F) such that Y N(0, 1) under P.Solution: Define Z = eX

122 . Then Z > 0 and

E [Z ] = 1

e12x2ex

122dx =

1e

12(x+)2dx = 1
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E[Z]

2

e e dx

2

e dx 1.

So

P[A] = E[ZIA] for A Fdefines a probability measure on (, F),

and for bR

P[Y b] = EZI{Yb} = EeX 122 I{Xb}=

b

12

e12x2ex

122dx =

b

12

e12(x+)2dx

=b

12

e12y2dy = (b),

so Y N(0, 1) under P.

Expectations under change of measure. Let P and P as before,Z = d

PdP

, and (Ft)t[0,T] be a filtration on (, F).The Radon-Nikodym derivative process of P w.r.t. P is

Z(t) := EZFt, t [0, T]. (5.3)B it t d diti i it i ti ti l
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By iterated conditioning it is a nonnegative martingale.

Lemma 5.2Let Y be a P-integrable random variable. ThenE[Y] = E[YZ]. (5.4)If moreover Y isFt-measurable, then also

E[Y] = E[YZ(t)], (5.5)

Z(s)E

YFs = EYZ(t) Fs for all s t. (5.6)

Proof.

For (5.4): If Y =

ki=1 yiIAi with Ai

Fthen

E[Y] =k

i=1

yiP[Ai] = ki=1

yiE[ZIAi] = E

Zk

i=1

yiIAi

.

If Y is nonnegative, take simple Yn with Yn Y, then bymonotone convergence

E [Y ] = lim E [Y ] = lim E [Y Z ] = E [YZ ]
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E[Y] = limnE[Yn] = limnE[YnZ] = E[YZ].

If Y is integrable, use Y = Y+ Y.

For (5.5):

E[Y](5.4)

= E[YZ] = E

E[YZ | Ft]

= E

YE[Z | Ft]

= E[YZ(t)].

For (5.6): Z(s)E[Y | Fs] is Fs-measurable, and for A Fs

EZ(s) E[Y | Fs]IA = EE[YIA | Fs]Z(s) (5.5)= EE[YIA | Fs]= E[YIA]

(5.5)= E[YIAZ(t)],

proving the partial averaging property for Z(s)E[Y | Fs] and YZ(t).

Girsanovs Theorem

In view of the role of martingales in risk-neutral pricing we ask:

How do martingales behave under change of measure?

How does BM behave under change of measure?

Definition 5.3Probability measures P and P on ( F) are equivalent if they
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Probability measures P and P on (, F) are equivalent if theyagree which sets in Fhave probability zero, that is if

P[A] = 0

P[A] = 0 for each A F.If P, P equivalent then

Z = dP

dP> 0 P-a.s.

Z(t) = E[Z | Ft] > 0 P-a.s.since P[Z = 0] = EI{Z=0}Z = 0 and P[Z(t) = 0] = EI{Z(t)=0} =(5.5)

= E I{Z(t)=0}Z(t) = 0, hence P[Z = 0] = P[Z(t) = 0] = 0.

Theorem 5.4Let P, P be equivalent and Y(t), t [0, T] an adapted process.Y(t) is a

P-martingale if and only if Y(t)Z(t) is a P-martingale.

Proof: First note E[|Y(t)|] = E[|Y(t)|Z(t)] = E[|Y(t)Z(t)|], soY (t) i P i t bl if d l if Y (t)Z (t) i P i t bl
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Y(t) is P-integrable if and only if Y(t)Z(t) is P-integrable.

If Y(t) is a

P-martingale then for s t

E[Y(t)Z(t) | Fs](5.6)

= Z(s)E[Y(t) | Fs] = Y(s)Z(s). If Y(t)Z(t) is a P-martingale then for s t

E[Y(t)

| Fs]

(5.6)=

1

Z(s)

E[Y(t)Z(t)

| Fs] =

1

Z(s)

Y(s)Z(s) = Y(s).

Let W(t), t [0, T] be a Brownian motion on a space (, F, P)with a filtration (Ft)t[0,T]. For an adapted (t) define

Z(t) := exp

t0

(u)dW(u) 12

t0

(u)2du

, t [0, T].

(5.7)

This is a generalized geometric BM with
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Z(t) = Z(0) t0

Z(u)(u)dW(u),

see Chap. 4. Assuming some integrability11

, Z(t) is a martingale. SoE[Z(T)] = Z(0) = 1 and we can define a probability measure P by

P[A] = E[Z(T)IA] for all A F. (5.8)

Theorem 5.5 (Girsanov)The processW(t) := W(t) + t0 (u)du, t [0, T], is a BM under P.

11A sufficient condition is E

exp12

T0(u)2du


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= W(t)Z(t)(t)dW(t) + Z(t)dW(t) + Z(t)(t)dt Z(t)(t) 1dt= Z(t)

1 W(t)(t)

dW(t),

so W(t)Z(t) is a P-martingale. Hence W(t) is a P-martingale byTheorem 5.4. Also W(t) is continuous, we have W(0) = 0, and[

W,

W](t) = [W, W](t) = t. Therefore

W(t) is a BM under

P.

Risk-neutral measure

Take a BM W(t) on a space (, F, P) with a filtration (Ft)t[0,T].The market model for bank account B(t) and stock S(t):

dB(t) = B(t)R(t)dt, B(0) = 1, (5.9)

dS(t) = S(t)(t)dt+ S(t)(t)dW(t), S(0) = S0, (5.10)

( ) ( ) ( ) ( )
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R(t) (instantaneous interest rate), (t) (mean rate of return) and(t) (volatility) are adapted processes. We assume (t) > 0a.s. for all t. Then

B(t) = expt

0R(u)du

,

S(t) = S0 exp

t

0(u)dW(u) +

t

0 (u) 1

2(u)2

du

,

S(t)B(t)

= S0 expt

0(u)dW(u) + t

0

(u) R(u) 12

(u)2du .Thus S(t)

B(t) is a generalized geometric BM with
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Let V(T) be an FT-measurable random variable which representsthe payoff of a derivative at time T. We allow path-dependence,so this includes but is not limited to payoffs V(T) = hS(T).We say that V(T) is attainable if it can be replicated, i.e.,if there exists a sfts with terminal value

X(T) = V(T) a.s.The fair price V(t) of the derivative at t is equal the value X(t)


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p ( ) q ( )of the replication portfolio at t.

Corollary 5.7 (Risk-neutral pricing)

Suppose V(T) is attainable. Then

V(t) = B(t)EV(T)B(T)

Ft = Ee Tt R(u)duV(T) Ft . (5.16)Proof: By (4.5), which holds for a general asset price B(t),

X(t)B(t) = X(0) +

t0 (u)d

S(u)B(u)

.

So X(t)B(t) is a P-martingale by (5.12), and

V(t)B(t) =

X(t)B(t) .

Risk-neutral pricing in the Black-Scholes model

Let bond and stock satisfy

dB(t) = B(t)rdt, B(0) = 1,

dS(t) = S(t)(t)dt+ S(t)dW(t), S(0) = S0,

where r and constant and (t) adapted. Let (t) =(t)r

and
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Z(t) and P as in (5.7), (5.8). From (5.14) we haveS(t) = S0 expW(t) + r 122t (5.17)

where W(t) = W(t) + t0 (u)du is a P-BM. Results in Chap. 4say that the European call payoff V(T) =

S(T) K+ can be

replicated (the analysis is the same when the is replaced by an

adapted process). By Corollary 5.7

c

t, S(t)

= V(t) = E

er(Tt)

S(T) K+ Ft. (5.18)

We write

S(T) = S(t)eW(T)W(t)+(r1

22

)(Tt) = S(t)eY+(r12

2

)

with = T t, Y = W(T)W(t)Tt N(0, 1) and independent of Ft

under

P. So by (5.18)

ct, S(t) = V(t) = EerS(t)eY+(r 122) erK+ FtE

Y 12 rK+
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= E

xeY

2 erK

x=S(t)

.

Notexe

Y 1

22

erK

0

log x Y 122 r + log K Y 1

log x

K+ (r 122)

= d(t, x) .

Hence

c(t, x) = ExeY 122I{Yd(t,x)} KerI{Yd(t,x)}= x

d(t, x) +

Kerd(t, x)

= x d+(t, x) Ker d(t, x) .

Martingale representation theorem and hedging

In the Black-Scholes model, a European call can be replicated viathe sfts X(0) = c

0, S(0)

, (t) = cx

t, S(t)

(delta hedging, see

Chap. 4). What about other models and derivatives?

Take a BM W(t) with filtration (Ft)t0 generated by this BM,and consider the model (5.9), (5.10)
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dB(t) = B(t)R(t)dt, B(0) = 1,

dS(t) = S(t)(t)dt+ S(t)(t)dW(t), S(0) = S0,

with adapted R(t), (t), and (t) > 0. Define (t) =(t)R(t)

(t)

and

P via the R-N derivative process Z(t) as in (5.7), (5.8). Also

let W(t) = W(t) + t

0 (u)du. This is a P-BM.The next result guarantees the existence of replication strategies.For a proof see e.g. Revuz and Yor, Continuous Martingales andBrownian motion, Theorem V.3.4.

Theorem 5.8 (Martingale representation theorem)Let W(t), t [0, T] be a BM and (Ft)t[0,T] the filtrationgenerated by this BM. For every P-martingale M(t) w.r.t.(Ft)t[0,T] there exists an adapted process (t) such that

M(t) = M(0) + t0

(u)dW(u), t [0, T].
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Corollary 5.9

For every P-martingale M(t), t [0, T] w.r.t. (Ft)t[0,T] thereexists an adapted process(t) such that

M(t) =

M(0) +

t

0 (u)d

W(u), t [0, T].

Remark: This is not trivial from Theorem 5.8 since the filtration isgenerated by W (and not by W). But it can easily be derived fromthe martingale representation theorem by using Theorem 5.4.

The hedging problem. Recall that a sfts with initial value X(0)

and stock position (t) has discounted value processX(t)B(t) = X(0) +

t0 (u)d

S(u)B(u)

, t [0, T]. (4.5)

Given an FT-measurable derivative payoff V(T), the task is to findX(0) and (t) such that the value process satisfies X(T) = V(T).

We have d S(t)B(t) = S(t)

B(t)(t)dW(t), see (5.12). Define
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V(t) = B(t)E

V(T)B(T)

Ft

.

V(t)

B(t) is a P-martingale, so by Corollary 5.9 there exists (t) withV(t)B(t) =

V(0)B(0) +

t0(u)dW(u)

= V(0) +

t

0

(u)B(u)S(u)(u) S(u)B(u)(u)d

W(u)

= V(0) + t0 (u)B(u)S(u)(u)d S(u)B(u) , t [0, T].Therefore taking X(0) = E

V(T)B(T)

and (t) =

(t)B(t)S(t)(t) in (4.5)

yields the desired sfts with X(T) = V(T).

Remarks.

1) We have shown that in a model of the form (5.9), (5.10), everyFT-measurable derivative payoff is attainable, i.e. can be

replicated by self-financing trading in bank account and stock.Such a model is called complete.

2) Th k i h h fil i i d b
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2) The key assumptions are that the filtration is generated by aone-dimensional BM W(t), and that (t) is positive.

3) The martingale representation theorem justifies the use of therisk-neutral pricing formula (5.16) by showing that every derivativeis attainable. But it does not provide a method of finding (t) in(t) =

(t)B(t)S(t)(t) . Under further assumptions on the processes R(t)

and (t), the strategy (t) can be found using PDE methods

(Feynman-Kac theorem, see [Shr04] Chap. 6).

The fundamental theorems of asset pricing

We consider a market with a bank account process B(t),

dB(t) = B(t)R(t)dt, (5.19)

and m risky assets Si(t), i = 1,..., m,

dSi(t) = Si(t)i(t)dt+ Si(t)d

j=1

ij(t)dWj(t). (5.20)
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j=1

Here W(t) =

W1(t), ..., Wd(t)

is a d-dim BM on a space

(,

FT, P) with a filtration (

Ft)t

[0,T], and R(t), i(t), and ij(t)

are adapted processes.

Assume i(t) =d

j=1 ij(t)2 > 0 for i = 1,..., m and let

Bi(t) =

d

j=1 t

0

ij(u)

i(u)dWj(u).

ThendSi(t) = Si(t)i(t)dt+ Si(t)i(t)dBi(t).

The processes Bi(t) =d

j=1

t0

ij(u)i(u)

dWj(u) are martingales with

d[Bi, Bk](t) =

dj1=1

dj2=1

ij1(t)kj2(t)

i(t)k(t) d[Wj1 , Wj2](t) =

dj=1

ij(t)kj(t)

i(t)k(t) dt
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= ik(t)dt.

So d[Bi, Bi](t) = dt, and Bi(t) is a 1-dim BM by Levys theorem.

Also ik(t) [1, 1] is the instantaneous correlation of Bi(t) andBk(t) for i = k.This means that the logarithmic returns of Si and Sk at time thave conditional correlation ik(t), cf. [Shr04] Exercise 4.17.

Arbitrage and the first FTAP

Let X(t) be the value process of a sfts which holds i(t) shares ofasset Si at time t, where i(t) are adapted processes. Then

dX(t) = X(t) m

i=1 i(t)Si(t)R(t)dt+m

i=1 i(t)dSi(t).


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The product rule and d

1

B(t)

= 1

B(t)R(t)dt imply

X(t)B(t)

= X(0) +m

i=1

t0

i(u)dSi(u)B(u)

. (5.21)Definition 5.10

a) A sfts is admissible if there exists a constant b 0 such thatits value process satisfies X(t) b for all t, a.s.b) A sfts is an arbitrage if it is admissible, and its value processsatisfies X(0) = 0 a.s., X(T) 0 a.s., and P[X(T) > 0] > 0.

Theorem 5.11 (First fundamental theorem of asset pricing)

If there exists a measure P equivalent to P such that S1(t)B(t) ,..., Sm(t)B(t)are P-martingales, then the market model B, S1,..., Sm does notadmit arbitrage.

A measure P as in Theorem 5.11 is called a risk-neutral orequivalent martingale measure for the market B, S1,..., Sm. One can show that a market model without a risk-neutral


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One can show that a market model without a risk neutralmeasure violates the no free lunch with vanishing riskproperty, which is close to admitting arbitrage. See Delbaen

and Schachermayer, The mathematics of arbitrage (2006).Proof (sketch): Let X(t) be the value process of an admissible

sfts. Then X(t)B(t) is a

P-martingale12 as a sum of stochastic integrals

w.r.t. the P-martingalesS1(t)B(t) ,...,

Sm(t)B(t) . So if X(0) = 0 and

X(T) 0, then EX(T)B(T) = 0 and hence P[X(T) > 0] = 0. By

equivalence P[X(T) > 0] = 0, so the sfts cannot be an arbitrage.

12This requires an integrability condition on (t).

Existence of a risk-neutral measure

Let W(t) =

W1(t),..., Wd(t)

a d-dim BM as above. Take ad-dim adapted (t) =

1(t),...,d(t)

and define for t [0, T]

Z(t) := expt0

(u) dW(u) 1

2 t0 (u)2du . (5.22)


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If E

exp12

T0 (u)2du

< , then Z(t) is a martingale and

we can define a probability measure P byP[A] = E[Z(T)IA] for all A F.The same proof as in the 1-dimensional case yields

Theorem 5.12 (Girsanov)

The processW(t) := W(t) + t0 (u)du, t [0, T], is a d-dim BMunder P.

Writing the model (5.19), (5.20) in discounted prices we have

dSi(t)B(t) = Si(t)

B(t)i(t) R(t)dt+ Si(t)B(t) d

j=1

ij(t)dWj(t)

for i = 1,..., m. Now S1(t)B(t) , ...,

Sm(t)B(t) are

P-martingales if

dSi(t)B(t) = Si(t)B(t) d

j=1

ij(t)j(t)dt+ dWj(t)


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= Si(t)

B(t)

d

j=1ij(t)d

Wj(t)

with Wj(t) = Wj(t) + t0 j(u)du, and this is the case if and only ifi(t) R(t) =

d

j=1ij(t)j(t), i = 1, ..., m. (5.23)

This is a linear system of m equations for d unknowns 1(t),...,d(t).

Conclusion: If (5.23) has a solution (t) =

1(t),...,d(t)

, thenthe model (5.19), (5.20) does not admit arbitrage.

If (5.23) has no solution, there is an arbitrage in the model.

Instead of a proof we give anExample. Take m = 2, d = 1, and constant coefficients R, i, i.Then (5.23) becomes

1 R = 1,

2 R = 2.This has a solution if and only if 1R1 =

2R2

. Suppose this


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1 2

does not hold, say 1R1 >2R2

. We can realize an arbitrage via

the sfts X(0) = 0, 1(t) =1

S1(t)1, 2(t) =

1

S2(t)2. By (5.21),

its discounted value process is

X(t)B(t) =

t0 1(u)d

S1(u)B(u)

+t0 2(u)d

S2(u)B(u)

=

t

01

S1(u)1

S1(u)B(u) (1 R)du+ 1dW(u) t0 1S2(u)2 S2(u)B(u) (2 R)du+ 2dW(u)

=t0

1B(u)

1R1

2R2

du > 0.

Risk-neutral pricing and the second FTAP

So far we defined prices of derivatives via replication arguments.We now introduce the concept of pricing by arbitrage arguments.

Suppose the market B, S1, ..., Sm given by (5.19), (5.20) admits arisk-neutral measure P. For a derivative with FT-measurable payoff V(T) at T define

V (T )


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V(t) = B(t)EV(T)B(T)

Ft

. (5.24)

ThenV(t)B(t) , t [0, T] is a P-martingale. So if the derivative is

traded for V(t) at t, the market B, S1,..., Sm, V does notadmit arbitrage by the first FTAP. We call V(t) a fair orarbitrage-free price.

If (5.23) has more than one solution (typical situation ford > m), then there is more than one risk-neutral measure.Different measures usually yield different prices V(t) in (5.24).

When do we have a unique fair price?


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Proof (sketch). (i) (ii): Assume completeness and let

P1,

P2

risk-neutral measures. For A F= FT define V(T) = IAB(T).By completeness V(T) can be replicated via a sfts with some initialvalue V(0). Its discounted value process is a martingale under

P1 and

P2, so

Pi[A] = Ei[IA] = Ei

V(T)B(T)

= V(0) for i = 1, 2.

(ii)

(iii): Assume uniqueness of P and let (t) and (t)solutions of (5.23). Then Z(T) and Z(T) in (5.22) with (t) and

(t) both define a R-N derivative for a risk-neutral measure. ThusZ (T ) Z (T ) d ( ) ( )


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Z(T) = Z(T) a.s. and so (t) = (t).

(iii) (i): Given an integrable derivative payoff V(T) we have toshow that there are i(t) with

V(T)B(T) = V(0) +

mi=1

T0 i(t)d

Si(t)B(t)

= V(0) +

mi=1

T

0 i(t)Si(t)B(t)

dj=1 ij(t)d

Wj(t).

Take a risk-neutral measure and define V(t) via (5.24). We obtainV(T)B(T) = V(0) +

dj=1

T0j(t)dWj(t)

for suitable j(t) by multi-dim martingale representation, s. below.

So we have show that there is a solution 1(t),..., m(t) tom

i=1

i(t) Si(t)

B(t)ij(t) = j(t), j = 1,..., d, (5.25)

Since (5.23) has a unique solution, the matrix ij(t) has rank d(and thus m d). This implies that (5.25) has a solution.Theorem 5.15 (Martingale representation theorem)

Let

W1(t) Wd(t)

t [0 T ] be a BM and (Ft)t[0 T ] the


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Let

W1(t),..., Wd(t)

, t [0, T] be a BM and (Ft)t[0,T] thefiltration generated by this BM. For every P-martingale M(t) w.r.t.(

Ft)t

[0,T] there exists an adapted1(t),..., d(t) such that

M(t) = M(0) +d

j=1

t0

j(u)dWj(u), t [0, T].

For every P-martingale M(t), t [0, T] w.r.t. (Ft)t[0,T] thereexists an adapted 1(t),..., d(t) such thatM(t) = M(0) + d

j=1

t0

j(u)dWj(u), t [0, T].

Discussion of the multi-dimensional asset price model.

1) d = m = 1: The model (5.19), (5.20) is arbitrage-free andcomplete if 1(t) = 0 for all t.

2) d = m > 1: The model (5.19), (5.20) is arbitrage-free andcomplete if the matrix

ij(t)

i,j=1,...,d

is nonsingular for all t.

3) d Th d l (5 19) (5 20) i bit f if th t i


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3) d > m: The model (5.19), (5.20) is arbitrage-free if the matrix

ij(t)

i=1,...,m,j=1,...,dhas rank m for all t.

It is incomplete in this case.4) d < m: The model (5.19), (5.20) is arbitrage-free if the vector

1(t) R(t),...,m(t) R(t)

is in the image of the linear mapdefined by the matrix

ij(t)

i=1,...,m,j=1,...,d

(drift conditions).

It is complete if moreover the matrix ij(t)i=1,...,m,j=1,...,d hasrank d.

Dividend-Paying Stocks

Let B(t) = et0 R(u)du a bank account process, and S(t) the priceprocess of a stock with cumulative dividend process D(t) (= sumof all dividends paid between times 0 and t). D(t) is an increasingprocess. If we hold the stock from 0 to t and invest all dividends in

the bank account, our wealth at time t isY(t) = S(t) +

t0

B(t)B(u)dD(u).
http://find/


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We conclude

dY(t)B(t) = d S(t)B(t) + t0 1B(u)dD(u)= 1

B(t)dS(t) S(t) 1B(t)R(t)dt+ 1B(t)dD(t)= 1

B(t)

d

S(t) + D(t)

S(t)R(t)dt

. (5.26)

The value process X(t) of a sfts in the stock and bank accountwith stock position (t) at time t satisfies

dX(t) =

X(t) (t)S(t)R(t)dt+ (t)dS(t) + D(t).
http://find/


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In summary we obtain

d

X(t)B(t)

= (t)d

Y(t)B(t)

, (5.28)

d

Y(t)B(t)

= S(t)

B(t)(t)d

W(t) (5.29)

for the value process X(t) of a sfts. Threrefore, as in the case ofzero dividends, a derivative payoff V(T) at time T has fair price


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V(t) = B(t)E

V(T)B(T)

Ft

(5.30)

and can be replicated via a sfts (martingale representation theorem).

The difference between dividend and no-dividend case is in thestock model, now given by (5.27)

dS(t) + D(t) = S(t)(t)dt+ S(t)(t)dW(t).We solve the equation for S(t) in two important cases.

Examples.

1) Continuously paying dividend. Here we assume

D(t) =

t0

A(u)S(u)du

for an adapted rate process A(t)

0. Then (5.27) becomes

dS(t) = S(t)(t) A(t)dt+ S(t)(t)dW(t)and we obtain
http://find/


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and we obtain

S(t) = S0 expt

0

(u)dW(u) + t

0 (u) A(u) 1

2(u)2du ,

= S0 exp

t0

(u)dW(u) + t0

R(u) A(u) 1

2(u)2

du

.

(5.31)

For constant coefficients R(t) = r, (t) = , and A(t) = a,

S(t) = S0 exp

W(t) + r a 1

22

t

.

Using the risk-neutral pricing formula (5.30) we can computederivative prices as in the Black-Scholes model without dividends.

With S(t) = S0 exp

W(t) +

r a 122

t

, t [0, T], we

obtain

V(t) = E

er(Tt)

S(T) K+ Ft
http://find/


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= S(t)

d+

t, S(t)

er(Tt)K

d

t, S(t)

with S(t) = S(t)ea(Tt) via a similar computation as for theno-dividend case in (5.18).

When a = 0 we recover the original Black-Scholes price formulaV(t) = ct, S(t).

2) Lump payments of dividends. Here we assume

D(t) =tjt

ajS(tj)

with Ftj-measurable random variables aj [0, 1] for j = 1, ..., n,and a finite number of payment dates 0 < t1 < ... < tn < T.Then D(t) is constant in t between consecutive payment dates,and jumps up by ajS(tj) at time tj.
http://find/


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By (5.27) the sum S(t) + D(t) is continuous in t. Therefore S(t)jumps down by ajS(tj

) at time tj. Also

dS(t) = S(t)(t)dt+S(t)(t)dW(t) = S(t)R(t)dt+S(t)(t)dW(t)for t (tj, tj+1) between consecutive payment dates. Thus

S(t) = S(tj)ettj (u)dW(u)+ttj R(u) 12(u)2du if t (tj, tj+1),

S(tj+1) = (1 aj+1)S(tj)etj+1tj

(u)dW(u)+tj+1tjR(u) 1

2(u)2du

.

By recursion it follows for all t [0, T] thatS(t) = S0

tjt

(1 aj) et0(u)dW(u)+t0

R(u) 1

2(u)2du (5.32)

For constant coefficients R(t) = r, (t) = , and aj,

S(t) = S0

(1 aj) eW(t)+r 1

22t.
http://find/


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( )tjt

( j)

The risk-neutral pricing formula (5.30) then yields for the price V(t)of the European call with strike K and maturity T

V(t) = S(t)

d+

t, S(t)

er(Tt)K

d

t, S(t)

where S(t) = S(t) t


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) ( )

tjt j ( j ) ( )consecutive pa

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