Introduction to Mathematical Finance II

Lecture of Dr. Joseph Najnudel

August 27, 2010

Notes of Michael Hartmann

Universitat Zurich

Contents

1 Introduction: Topics of the Lecture 3

1.1 Example of Financial Assets in the Market . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Some Basic Example of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 How Realistic Can a Model for a Stock Price Be? . . . . . . . . . . . . . . . . . . . 4

1.4 Binomial Model and Its Continuous-Time Limit . . . . . . . . . . . . . . . . . . . . 4

2 Brownian Motion and Stochastic Calculus 6

2.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Interpretation of the Relation with Binomial Model . . . . . . . . . . . . . 8

2.2 Filtrations and Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Natural Filtration of a Process . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Stopping Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Financial Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 Properties of Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.5 Brownian Motion with Respect to a Filtration . . . . . . . . . . . . . . . . 10

2.3 Continuous-Time Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Stochastic Integral and Ito Calculus . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Ito Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 The Black-Scholes Model 19

3.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Explicit Description of the Prices . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.2 Self-Financing Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.3 Precise Condition of Boundedness . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Change of Probability, Representation of Martingales . . . . . . . . . . . . . . . . . 22

3.2.1 Representation of Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Pricing and Hedging Options in the Black-Scholes Model . . . . . . . . . . . . . . . 24

3.3.1 Particular Case where h is a Function of ST . . . . . . . . . . . . . . . . . . 27

3.3.2 Hedging call and puts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 Perpetual Put, Critical Price . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.2 Description of the Best Strategy for an American Put . . . . . . . . . . . . 34

3.5 Implied Volatility and Local Volatility Models . . . . . . . . . . . . . . . . . . . . . 34

3.6 The Black-Scholes Model with Dividends, and Call/Put Symmetry . . . . . . . . . 35

3.6.1 Self-Financing Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1

4 Option Pricing and PDE 37

4.1 European Option Pricing and Diffusions . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Infinitesimal Generator of a Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Conditional Expectations and PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Application to Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 PDE on Bounded Open Sets and Computations of Expectations . . . . . . . . . . 43

4.6 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6.1 Example of Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . 47

4.7 Solving the PDE Numerically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2

1 Introduction: Topics of the Lecture

Goal of the lecture: the most simple model which are not “totally realistic”. Binomial model is

not realistic (at least for simple num. values)

144

120

kkkkkk

SSSSSS

100

kkkkkk

SSSSSS 96

80

kkkkkkk

SSSSSSS

64

1.1 Example of Financial Assets in the Market

• Bonds: (evolve like “money put into a bank account”). The interest rate is fixed (or changes

slowly). It is an unrisky asset (at least less risky than other assets).

• Stocks: A stock is a part for a company.

• Commodities: Based on a physical product (oil, gold, etc.).

• Currencies: Exchange ¤, $, CHF, etc.

Remark. Stock, commodities and currencies are risky assets (i.e. can evolve fastly). There are

more complicated financial products (in particular: derivatives: products based on other financial

products).

Example. A product which depends on the evolution of a stock price. [derivatives ∼= options]

1.2 Some Basic Example of Options

• Call/Put options: (the most non-trivial derivative). It is the right but not the obligation to

buy/sell a financial product (typically a stock) at a given price, at a given date in the future.

Example. A call (put) option on UBS at strike price K=10 CHF and maturity T=1.4.2010

is the right to buy (sell) the stock UBS for 10 CHF at the date 1.4.2010. Call and put options

are exchanged in the market: the price is known.

• Other options: The question is: “Can we determine a price for these options, which are not

directly traded in the moment?” We need to model the stock prices to answer this question.

Example. 1. We suppose that the price of UBS is 5 CHF and never changes. No interest

rate. If I have the option, I can buy the stock a the strike price 10 CHF, and immediately

sell it in the market for 15 CHF. I earn 5 CHF at time T. No interest rate⇒ Fair price:

5 CHF

3

2. Price of UBS is 15 CHF now, 8 CHF at T (a.s.). Call option is useless. Fair price is 0

CHF.

3. Let us suppose that the price is 15 CHF now and can be 5 CHF or 30 CHF at time T.

No interest rate. It is a binomial model. In the first case, the option is useless. In the

second case, you buy the stock for 10 CHF and sell it for 30 CHF. You earn 20 CHF. I

borrow 4 CHF and buy 0.8 stocks (in real markets the number of stocks is an integer).

– Initial value of the portfolio: 0.8·1512 − 4 = 8 CHF

– Terminal value: In the first case: 5 · 0.8 = 4 (by giving the borrowed 4 CHF back,

the terminal value is 0 CHF). In the second case: 30 · 0.8 − 4 = 20. The terminal

value is equal to the payoff of the option. The fair price of the option should be

equal to the initial value of the portfolio: 8 CHF.

Conclusion. The fair price of the option depends on the model.

1.3 How Realistic Can a Model for a Stock Price Be?

Model 2) is unrealistic. Imagine you have a stock at the origin. You can sell the stock at price 15

CHF and buy it for 8 CHF at time T . You have earned 7 CHF without taking any risk. In this

case, a lot of trader do the same operation. A lot of people want to sell the stock at the origin.

(Nobody wants to buy). So the price decreases before the sellers can sell until the possibility of

earning money with no risk disappears (this possibility is called arbitrage opportunity). At this

moment the price is 8 CHF. The model 2) is impossible.

One generally considers the viable markets: markets in which one cannot earn money without

taking any risk (i.e. no arbitrage opportunity). Expression: “No free lunch”. (In real markets, the

assumption is not perfectly true).

1.4 Binomial Model and Its Continuous-Time Limit

Let us consider a financial market with one bound and one stock. The price of the bond is constant

(no interest rate). The price of the stock is described as follows:

• The initial value is 1 (time 0).

• For t ∈ N the value is constant in the interval of time [t, t+ 1), at time t+ 1, it is multiplied

by a > 1 with probability 1/2, and by 1/a < 1 with probability 1/2, independently of the

past. One can prove that there is no arbitrage opportunity in this model.

4

The stock price can be described by the following picture:

α2

α

1/2 jjjjjjjj1/2

UUUUUUUU

1

1/2 kkkkkkk1/2

PPPPPP 1 · · ·

1/α

1/2 kkkkkkk1/2

QQQQQQ

1/α2

It is a particular case of the binomial model. The stock price at t is given by: St = α

[t]∑k=1

ζk, where

[t] = supn ∈ N|n ≤ t is the integer part of t and (ζk)k∈N are i.i.d. random variables with

probability P[k ≡ −1] = P[3k = −1] = 12 . This is a discrete model. The price does not change in

(n, n+ 1)

Question. This model is not totally unrealistic if the frequency of steps is high. Does there exist

a “continuous time” model which is the “limit of binomial model when the frequency of steps goes

to infinity”?

Answer. Yes!

Suppose that the number of steps by unit of time is N. (The changes of price one traded at times

kN , k ∈ N∗). Before time t, the number of steps is [Nt]. The stock price is given by S

[N ]t = α

[Nt]∑k=1

ζk

Can we take the limit, when N goes to infinity?

Problem. If α is constant huge fluctuations occur.

At infinity nothing realistic happens. In order to make something realistic, make the fluctuations

smaller when N goes to infinity. Replace α by αN , where αN −→ 1 (N →∞). We obtain:

S[N ]t = α

[Nt]∑k=1

ζk

N = exp

βN [Nt]∑k=1

ζk

, βN := log(αN ), βN −→ 0 (N →∞)

Question. How can we compute a good value of βN?

Answer. We need to have βN[Nt]∑k=1

ζk convergent “in some sense”.

5

E

βN [Nt]∑k=1

ζk

= βN

[Nt]∑k=1

E [ζk] = 0

V ar

βN [Nt]∑k=1

ζk

= E

βN [Nt]∑

k=1

ζk

2

= E

[Nt]∑k1=1

[Nt]∑k2=1

ζk1 · ζk2

β2N

= β2N

[Nt]∑k1=1

[Nt]∑k2=1

E [ζk1 · ζk2 ]

!= β2

N

[Nt]∑k1=1

[Nt]∑k2=k1

1

= [Nt] · β2N

≈ Ntβ2N (t→∞)

!: For k1 = k2: E [ζk1 · ζk2 ] = E[ ζ2k1︸︷︷︸

=1 a.s.

] = 1

For k1 6= k2: E [ζk1 · ζk2 ] = E [ζk1 ] · E [ζk2 ] = 0

For t fixed, one needs: Nβ2N convergent for N →∞. One chooses naturally: βN = 1√

N

S[N ]t = exp

1√N·

[Nt]∑k=1

ζk

One can prove that the process (S

[N ]t )t≥0 has the limit (in a sense which can made more precise):

(St)t≥0, St = exp(βt). (βt)t≥0 is called Brownian motion.

Black-Scholes-model: Stock price is based on the exponential of a Brownian motion.

2 Brownian Motion and Stochastic Calculus

Brownian motion is a stochastic process, i.e. a family of random variables indexed by the time.

Definition 1. A continuous stochastic process with values in Rd is a family of random variables

with values in Rd, indexed by an interval of time I (generally I = [0, T ] or I = R+).

∀t ∈ I : Xt : (Ω,F ,P) −→ Rd, such that ∀ω ∈ Ω the function t 7→ Xt(ω) is continuous.

Example. Ω = 1, 2, 3, F = P(Ω), P probability measure such that P(1) = 12 , P(2) = 1

3 ,

P(3) = 16 . We define for all t ∈ I := [0, 20], Xt such that: Xt(1) = 0, Xt(2) = sin(t − 1),

Xt(3) = (3t− 4)(t− 3)(2t2 − 2t).

What is X10, X0,X1? (the law)

• X10: X10(0) = 0, X10(2) = sin(9), X10(3) = 360360. X10 ∈ 0, sin(9), 360360 with proba-

bility 1.

6

P(X10 = 0) = P(1) = 12

P(X10 = sin(9)) = P(2) = 13

P(X10 = 360360) = P(3) = 16

• X0: X0(1) = 0, X0(2) = − sin(1), X0(3) = 0. X0 ∈ 0,− sin(1) almost surely.

P(X0 = 0) = P(1) + P(3) = 12 + 1

6 = 23

P(X0 = − sin(1)) = P(2) = 13

• X1: X1 ∈ 0 almost surely.

P(X1 = 0) = 1

The process is continuous because the three functions are continuous:

• t 7→ 0

• t 7→ sin(t− 1)

• t 7→ (3t− 4)(t− 3)(2t3 − 2t)

are continuous. This example has not much interest because the probability space is too small.

2.1 Brownian Motion

Definition 2. A Brownian motion of dimension d ≥ 1 is a continuous stochastic process:

(Ω,F ,P) −→ Rd defined on an interval of time I and satisfying the following assumptions:

• For all t ∈ I, the coordinates X(1)t , . . . , X

(d)t of the random variables Xt are independent

Gaussian variables of variance t and expectation 0. In particular X0 = 0 almost surely.

• For 0 < t1 < · · · < tk, the increments of (Xt)t∈I , i.e. Xt1 , Xt2 − Xt1 , Xt3 − Xt2 , . . . are

independent.

• (condition implied by the two conditions above). For all s, t ∈ I, s < t, Xt−Xs has the same

law as Xt−s.

Xt

t

d = 1

(One can prove that the set (X(1)t , X

(2)t )t∈R+ is dense in the plane, but not all the plane).

Proposition 1. Brownian motion exist (admitted)

Remark. We can also define left-continuous, right-continuous, etc process in the same way.

7

2.1.1 Interpretation of the Relation with Binomial Model

Let (ζk)k≥1 i.i.d. random variables such that P[ζk = 1] = P[ζk = −1] = 12 . The stock price

obtained before is an exponentially increasing function of the following quantity:

W[N ]t =

1√N

[Nt]∑k=1

ζk

Let us consider the family of random variables (W[N ]t )t≥0 for N ≥ 1 we know that

[Nt]∑k=1

ζk · (√

[Nt])−1

converges in law to a standard Gaussian variable, by central limit theorem. [(ζk)k≥1 are i.i.d

random variables, expectation 0, variance 1, in particular square-integrable ⇒ 1√n

n∑k=1

ζk −→ N .

(0, 1)]. Now

W[N ]t =

√[Nt]√N︸ ︷︷ ︸

→√t

1√[Nt]

[Nt]∑k=1

ζk︸ ︷︷ ︸→N(0,1)???

−→√tN (0, 1) = N (0, t) (N →∞)

(Compare with the first property of the BM). Let 0 < t1 < · · · < tk:

W[N ]t1 =

1√N

[Nt1]∑k=1

ζk

depends only on ζ1, . . . , ζ[Nt1].

W[N ]t2 −W

[N ]t1 =

1√N

[Nt2]∑k=1

ζk −[Nt1]∑k=1

ζk

=

1√N

[Nt2]∑k=[Nt1]+1

ζk

depends only on ζ[Nt1]+1, . . . , ζ[Nt2].

Since the variables (ζk)k≥1 are independent, a family of variables which depend on disjoint sets of

(ζk)k≥1 are necessarily independent. Hence W[N ]t1 , Wt2

[N ] −Wt1[N ], ... are independent.

The second assumption satisfied by the Brownian motion is satisfied by (W[N ]t )t≥0. The jumps of

W[N ]t are of the size 1√

N, in some sense (W

[N ]t )t≥0 is “almost continuous”.

The properties of the Brownian motion are “almost satisfied”. It is natural to expect that (W[N ]t )t≥0

“tends in law” to a Brownian motion. It is possible to give a rigorous definition of these heuristics.

That is why Brownian motion is used in finance and in other areas of application.

2.2 Filtrations and Stopping Times

Definition 3. Let I = R+ or I = [0, T ], and let (Ω,A,P) be a probability space. A filtration

on this space is a family of sub-σ-algebras of A, indexed by I (generally denoted by (Ft)t∈I) and

increasing, i.e. Fs ⊂ Ft ∀s ≤ t.

8

• A filtration (Ft)t∈I is right-continuous iff for all t ∈ I, such that t < T if I = [0, T ],

Ft =⋂

s∈I>tFs.

• A filtration (Ft)t∈I is complete iff for all sets A ∈ A, B ⊂ A, such that P[A] = 0 ⇒ B ∈ Ft∀t ∈ I (i.e. B ∈ F0).

Remark. The sets B are called negligible.

• One say that a filtration (Ft)t∈I satisfies the usual conditions iff it is right-continuous and

complete.

Definition 4 (Natural filtration). Let (Xt)t∈I be a continuous process defined on a probability

space (Ω,A,P). Let us define Ft := σ(Xs|s ≤ t). Ft is increasing ⇒ it is a filtration. It is not

complete or right-continuous in general. Let us define Ft+ =⋂

s∈I>tFs (and if I = [0, T ] : FT+ =

FT ), then N := negligible subsets of Ω and define: Ft := σ(Ft+ ∪N ). Then (Ft)t∈I is again a

filtration, Ft ⊂ Ft and it satisfies the usual assumptions. (Ft)t∈I is called natural filtration of the

process (Xt)t∈I .

Remark. For all t ∈ I, Ft is, by construction, measurable with respect to Ft and hence, with

respect to Ft. One says that (Xt)t∈I is adapted with respect to the filtration (Ft)t∈I .

Intuitive meaning: The σ-algebra Ft represents the information available at time t. (typically,

the evolution of the stock price before time t). In general, the stock price at time u > t is not in

Ft.

2.2.1 Natural Filtration of a Process

Canonical filtration σ(Xs|s ≤ t) at time t. Add the negligible sets → right continuous version →

we have defined the natural filtration of X (it satisfies the natural conditions).

When we consider a stochastic process and we don’t precise the filtration, it is a priori the natural

filtration of the process.

2.2.2 Stopping Time

Let (Ft)t∈I be a filtration.

1. If I is a finite interval, a random variable τ in (Ω,A,P) is an (Ft)t∈I -stopping time iff τ ∈ I,

and for all t ∈ I the event τ ≤ t is in Ft.

2. If I = R+, then a random variable τ from (Ω,A,P) to R+ ∪ +∞ such that for all t ∈ R :

τ ≤ t ∈ (Ft)t∈I is called (Ft)t∈I -stopping time. In this case, necessarily, τ is measurable

to F∞ = σ(Ft|t ∈ R+), which is a priori smaller than A (in any case, F∞ ⊂ A).

9

2.2.3 Financial Interpretation

An instance of trading (for example the time when the trader decides to buy a stock) should be

known only by using the information available at this time. If τ is such an instant, the event

τ ≤ t needs to be in Ft ⇒ τ has to be a stopping time.

Definition 5. If τ is a stopping time, then the σ-algebra Fτ is defined as follows:

Fτ := A ∈ A|∀t ∈ I : A ∩ τ ≤ t ∈ Ft

Interpretation. Fτ represents the information available at time τ . If τ is deterministic, i.e.

τ = t0 ∈ I, then τ is a stopping time and Fτ = Ft0 .

2.2.4 Properties of Stopping Times

• If S is a stopping time, finite almost surely, if X is a continuous process defined on a filtered

probability space satisfying the usual conditions, and ifX is adapted (i.e. Xt is Ft-measurable

for all t ∈ I), then Xs is Fs-measurable.

• If S and T are two stopping times and if S ≤ T , Fs ⊂ FT .

• If S and T are two stopping times, then inf(S, T ) and sup(S, T ) are also stopping times.

In particular, if T is a stopping time and t is deterministic, then inf(t, T ) and sup(t, T ) are

stopping times.

2.2.5 Brownian Motion with Respect to a Filtration

Definition 6. Let (Ω,A,P) be a probability space, let (Ft)t∈I be a filtration. Then a process with

values in Rd, defined on (Ω,A,P) is an (Ft)t∈I -Brownian motion iff:

• The process is continuous.

• Xt is Ft-measurable for all t ∈ I. (Xt is (Ft)t∈I -adapted).

• Xt −Xs is independent of Fs if s < t; for all A ∈ Fs, B ∈ B(Rd), P(A ∩ Xt −Xs ∈ B) =

P(A)P(Xt −Xs ∈ B).

• For s ≤ t , the d coordinates of Xt −Xs are independent, centered variables, with variance

t− s.

Properties.

• By the second point Ft contains necessarily the σ-algebra generated by Xs for s ≤ t.

• A Brownian motion with respect to a filtration is a Brownian motion.

• A Brownian motion is a Brownian motion with respect to its neutral filtration.

10

2.3 Continuous-Time Martingales

Definition 7. Let (Ω,A,P) be a probability space and (Ft)t∈I a filtration. Let (Mt)t∈I be an

adapted process (Mt is Ft-measurable for all t ∈ I) such that E[|Mt|] <∞ for all t ∈ I (i.e. Mt is

integrable). (Mt)t∈I is:

• A submartingale iff ∀s, t ∈ I, s ≤ t : E[Mt|Fs] ≥Ms a.s.

• A supermartingale iff ∀s, t ∈ I, s ≤ t : E[Mt|Fs] ≤Ms a.s.

• A martingale iff ∀s, t ∈ I, s ≤ t : E[Mt|Fs] = Ms a.s.

In particular E[Mt] is:

• non-decreasing in t, if (Mt)t∈I is a submartingale.

• non-increasing in t, if (Mt)t∈I is a supermartingale.

• constant with respect to t, if (Mt)t∈I is a martingale.

Proposition 2. If (Xt)t∈I is a (Ft)t∈I -Brownian motion of dimension 1.

• (Xt)t∈I is a martingale.

• (X2t − t)t∈I is a martingale.

• (exp(λXt − tλ2/2))t∈I is a martingale for all λ ∈ C.

One of the reasons of dealing with stopping time is the following: Let (Mt)t∈I be a [super,-,sub]

martingale, let τ1 and τ2 be two stopping times with respect to the filtration on which (Mt)t∈I is

defined, such that there exists K ∈ R+ with τ1 ≤ τ2 ≤ K. Then Mτ2 is integrable and E[Mτ2|Fτ1 ]

[≤,=,≥] Mτ1 a.s.

In particular E[Mτ2] [≤,=,≥] E[M0] if τ2 is a bounded stopping time.

This result is an extension of the property which define [super,-,sub] martingale to random times.

Application.

Let (Xt)t∈I be a Brownian motion and let Ta := infs|Xs = a. Then Ta is a.s. finite (i.e. almost

surely there exists s ∈ R+ : Xs = a) and the law of the probability of Ta is given by the Laplace

transform:

E[e−λTa ] = e−|a|√

2λ, ∀λ ≥ 0

Remark. the result above is called “optional stopping theorem”, or “optional stopping time”, or

“stopping theorem”.

Lemma 3 (Dook’s inequality). Let (Mt)t∈I be a continuous martingale and p > 1. Then:

• E[ sup0≤s≤t

|Ms|p] ≤(

pp−1

)pE[|Mt|p], ∀t ∈ I

11

• E[sups∈I|Ms|p] ≤

(pp−1

)psups∈I

E[|Ms|p]

This result (admitted) gives some control on the supremum of |M |, only from the control on

each variable (Mt).

2.3.1 Stochastic Integral and Ito Calculus

Discrete-time market: We consider a self-financing portfolio with bonds (constant price = 1) and

stocks (price St at time t ∈ N). We suppose that, in the portfolio, there are Hγ stocks on the

interval of time (γ − 1, t], γ ∈ N∗.

If Vt is the value of the portfolio at time t, then

Vt = V0 +

t∑γ=1

Hγ(Sγ − Sγ−1)

In continuous-time, we expect to have: Vt = V0 +t∫

0

HsdSs (infinitesimal increment of s).

The problem is that the integral is not well-defined in the usual sense (the increments of S should

be “too irregular”). That’s why we define the so called stochastic integral.

Definition 8 (Simple processes). Let I = [0, T ] be a finite interval, (Ω,A,P) a probability space

and (Ft)t∈I a filtration on this space. A process (Ht)t∈I is simple with respect to the filtered

probability space (Ω,A, (Ft)t∈I ,P) iff there exists 0 = t0 < t1 < · · · < tp = T and φ1, . . . , φp such

that φi is Fti−1-measurable and bounded for all i = 1, . . . , p and such that: Ht =p∑i=1

φi ·1(ti−1,ti)(t).

φ1

φ2

φ3

t0 = 0 t1 t2 t3 · · ·

Remark. In particular, a simple process is adapted.

Definition 9 (Stochastic integral of simple processes (with respect to a Brownian motion)). Let

(Wt)t∈I be an (Ft)t∈[0,T ]-Brownian motion, and H a simple process. The stochastic integral of H

with respect to W is the process (I(H)t)t∈[0,T ] such that:

• (I(H))0 = 0

• For t ∈ (tk, tk+1) for some k ∈ 0, . . . , p−1: (I(H))t =k∑i=1

φi(Wti−Wti−1)+φk+1(Wt−Wtk).

Another possibility to define (I(H))t: (I(H))t =∑

1≤i≤pφi(Wti∧t −Wti−1∧t).

It is the “intuitive” version oft∫

0

Hs dWs.

12

The following proposition is the main tool of the extension of stochastic integrals to more general

processes H.

Proposition 4. If (Ht)t∈I is a simple process,

• ((I (H))t)t∈[0,T ]is an (Ft)-martingale.

• E[[(I (H))t]

2]

= E[t∫

0

H2s ds

].

• E[supt≤T

[(I (H))t]2

]≤ 4E

[T∫0

H2s ds

].

Proof. omitted

(Wt)t∈[0,T ] a Brownian motion, (Hs)s∈[0,T ] simple function. Random process ((I (H))t)t∈[0,T ],

(I (H))t =t∫

0

Hs dWs

•(t∫

0

Hs dWs

)t≤T

continuous (Ft)t≤T -martingale.

• E

[(t∫

0

Hs dWs

)2]

= E[t∫

0

H2s ds

].

• E

[supt∈I

∣∣∣∣ t∫0

Hs dWs

∣∣∣∣2]≤ 4E

[T∫0

H2s ds

].

We first extend the stochastic integral to the space

H :=

(Ht)t∈I |(F)t∈I -adapted process, progressively measurable, such that E

T∫0

H2s ds

<∞

Definition 10. Progressively measurable means that for all t ∈ [0, T ] = I the map (s, w) 7→

(Hs(w)) from [0, t]× Ω to R is measurable with respect to the σ-algebra B([0, t])⊗Ft

Remark.

• It will not be needed explicitly in this lecture.

• This property is for example satisfied if (Ht)t∈I is adapted, and left or right continuous (or

continuous).

H contains the simple processes. All the processes in H are adapted.

Proposition 5. Let (Wt)t≤T be an (F)t≤T -Brownian motion. Then there exist a map J from H

to the space of continuous (F)t≤I -martingales such that:

• If it is a simple process, then (J(H))t = (I(H))t for all t ≤ T , P-almost surely.

• J is linear in H. I.e. J(H + λK) = J(H) + λJ(K) P-almost surely ∀H,K ∈ H, λ ∈ R.

13

• If t ∈ I,H ∈ H then E[(J(H))2t ] = E[

t∫0

H2s ds].

J is unique in the following sense: If J1 and J2 satisfy the conditions above, then almost surely:

• (J1(H))t = (J2(H))t for all H ∈ H,∀t ≤ T .

• J(H) is consistently denoted by:t∫

0

Hs dWs

Properties. If (Ht)t≤T is in H:

• E[supt≤T|t∫

0

Hs dWs|2]≤ 4E

[T∫0

H2s ds

]• If τ is an (Ft)t≤T -stopping time, then P-almost surely:

τ∫0

Hs dWs =

T∫0

1s≤τHs︸ ︷︷ ︸∈H

dWs

(In particular for deterministic τ).

One can extend the stochastic integral to a still larger set, but one looses some properties.

H :=

(Ht)t≤T |(Ft)t≤T -adapted, progressively measurable process,

T∫0

H2s ds <∞,P− a.s.

Proposition 6. There exist a map J from H to the space of continuous processes such that:

• If (Ht)t≤T is in H, then P-almost surely J(H) = J(H).

• If (H(n))n≥0 is a sequence of processes in H such thatT∫0

(H(n)s )2 ds→ 0 (n→∞) in probabil-

ity. (i.e. P

[∣∣∣∣∣T∫0 (H(n)s )2 ds

∣∣∣∣∣ ≥ ε]→ 0 (n → ∞) ∀ε > 0) then, sup

t≤T|(J(H(n)))t| → 0 (n → ∞)

in probability.

• J is linear.

The map is unique in the sense explained above. We can again denote: (J(H))t =t∫

0

Hs dWs

(consistent with the previous notation).

2.3.2 Ito Calculus

Once the stochastic integral is constructed, it is natural to ask how we can do some computation

on them. For example

• Can we compute the integralt∫

0

Ws dWs?

• Can we write, for example, exp(−W 4t ) as a stochastic integral?

14

Let f be a continuous differentiable function from R to R, we want to write f(Wt) as a stochastic

integral. (Second question above).

If W were differentiable with respect to the time, we would expect the following computation:

f(0) + f(Wt) =

t∫0

d

ds(f(Ws)) ds

=

t∫0

W ′sf′(Ws) ds

∗=

t∫0

f ′(Ws) dWs

*: W ′s ds = dWS

In fact, W is not differentiable.

Question. Do we have f(Wt) = f(0) +t∫

0

f ′(Ws) dWs?

For f : x 7→ x2, we would have: W 2t = 2

t∫0

Ws dWs.

Since

E

t∫0

W 2s ds

=

t∫0

E[W 2s

]ds =

t∫0

sds =t2

2<∞

the previous W is in H.

Then (2t∫

0

Ws dWs)t≤T is a martingale.

Problem. (W 2t )t≤T is not a martingale. ((W 2

t − t)t≤T is a martingale). The formula above is

wrong.

Suppose f is analytic: one has Taylor formula: f(x+ h) =∞∑k=0

f (k)(x)hk

k! .

Let n ≥ 1 integer:

f(Wt)− f(W0) =

n−1∑k=0

[f(W k·t

n + tn

)− f(W k·tn

)]

=

n−1∑k=0

[ ∞∑l=0

f (l)(W k·tn

)(W k·t

n + tn−W k·t

n)l

l!

]

=

∞∑l=0

[n−1∑k=0

f (l)(W k·tn

)

l!(W k·t

n + tn−W k·t

n)l

]

• l = 1:∞∑k=0

f (1)(W k·tn

)(W k·tn + t

n−W k·t

n) −→

t∫0

f ′(Ws) dWs (n→∞)

(in a sense which has to be made precise). The sum is the stochastic integralt∫

0

f ′(W [n·s]t

tn

) dWs

• l = 2:n−1∑k=0

f(2)(W k·tn

)

2 (W k·tn + t

n−W k·t

n)2

The terms W k·tn + t

n− W k·t

nare Gaussian with expectation 0 and variance t

n , and are in-

dependent. Hence, their squares are independent, with expectation tn . By “some” kind

15

of law of large numbers, we can replace (W k·tn + t

n− W k·t

n)2 by t

n . In this case we obtain

tn

n−1∑k=0

f ′′(W k·tn

) · 12 . This (Riemann) sum converges to 1

2

t∫0

f ′′(Ws) ds

• l ≥ 3: 1l!

n−1∑k=0

f (l) (W k·tn + t

n−W k·t

n)l︸ ︷︷ ︸

≈(t/n)l2

≈ n ·(tn

) l2 −→ 0 (n→∞)

One has the “first heuristic formula” + an additional term due to the “quadratic variation”

of (Wt)t≤T .

A rigorous statement can be written:

Proposition 7. If (Wt)t≥0 is on??? (Ft)t≥0-Brownian motion, and if f is a function from R to

R, continuously twice differentiable, then almost surely for all t ≤ T :

f(Wt) = f(0) +

t∫0

f ′(Ws) dWs +1

2

t∫0

f ′′(Ws) ds

This formula, fundamental in stochastic calculus, is called Ito formula.

Examples:

1. W 2t = 2

t∫0

Ws dWs +t∫

0

ds = 2t∫

0

Ws dWs + t

As stated before (W 2t − t)t≥o is a martingale.

2. Let us take f(x) = ex = f ′(x) = f ′′(x).

eWt = 1 +t∫

0

eWs dWs + 12

t∫0

eWs ds

Here, the stochastic integral is only defined with respect to a Brownian motion.

Question. In finance, how can we define the integral with respect to dSt, where St is the stock

price?

St is defined by using Brownian motion, but it is not exactly a Brownian motion.

Answer. We need to extend stochastic integral and Ito formula.

Definition 11. Let (Ω,A,P) be a probability space, (Ft)t≤T a filtration, and (Wt)t≤T a one

dimensional (Ft)t≤T -Brownian motion. A real valued process (Xt)t≤T is an Ito process iff almost

surely Xt = X0 +t∫

0

Ks ds+t∫

0

Hs dWs where

• X0 is F0-measurable

• (Kt)t≤T and (Ht)t≤T are progressively measurable processes (for example, right or left con-

tinuous and adapted)

•T∫0

|Ks|ds <∞ P-almost surely

•T∫0

|Hs|2 ds <∞ P-almost surely

16

The processes K and H are uniquely determined by X in the following sense:

Proposition 8. If (Mt)t≤T is a continuous martingale such that Mt =t∫

0

Ks ds, with P-a.s.

t∫0

|Ks|ds <∞, then a.s. for all t ≤ T : Mt = 0.

This implies: If there are two decompositions of an Ito process

Xt = X0 +

t∫0

Ks ds+

t∫0

Hs dWs = X′

0 +

t∫0

K′

s ds+

t∫0

H′

s dWs

then

• X0 = X′

0 P-a.s.

• Hs(w) = H′

s(w) almost everywhere with respect to the measure ds ⊗ dP. This means

(s, w)|Hs(w) 6= H′

s(w) has measure zero.

• Similar for Ks and K′

s.

Moreover, if a martingale (Xt)t≤T has the form: Xt = X0 +t∫

0

Ks ds +t∫

0

Hs dWs then Kt = 0

dt⊗ dP almost everywhere.

Proposition 9. Let (Xt)t≤T be an Ito process: Xt = X0 +t∫

0

Ks ds+t∫

0

Hs dWs (formally dXt =

Ktdt+HtdWt). Let f be a function, continuously, twice differentiable. Then

f(Xt) = f(X0) +

t∫0

f ′(Xs) dXs +1

2

t∫0

f ′′(Xs) d〈X,X〉s

where

•t∫

0

f ′(Xs) dXs =t∫

0

f ′(Xs)Ks ds+t∫

0

f ′(Xs)Hs dWs

• 〈X,X〉s =t∫

0

H2s ds (it is called “the quadratic variation of X”)

⇒t∫

0

f ′′(Xs) d〈X,X〉s =t∫

0

f ′′(Xs)H2s ds

Generalization for functions depending also on the time. Let f be a function from I×R, which

is twice differentiable with respect to the second variable (space); once differentiable with respect

to the first variable (time); and all the corresponding partial derivatives are continuous on I × R.

(we also write: f ∈ C1,2).

f(t,Xt) = f(0, X0) +

t∫0

f ′I(s,Xs) ds+

t∫0

f ′R(s,Xs) dXs +1

2

t∫0

f ′′R(s,Xs) d〈X,X〉s

Another form: (integration by parts formula). Let Xt, Yt be two Ito processes:

• Xt = X0 +t∫

0

Ks ds+ 12

t∫0

Hs dWs

17

• Yt = Y0 +t∫

0

K′

s ds+ 12

t∫0

H′

s dWs

Then, XY is also an Ito process, more precisely:

XtYt = X0Y0 +

t∫0

Xs dYs +

t∫0

Ys dXs + 〈X,Y 〉t

where 〈X,Y 〉 is the quadratic variation of X and Y , it is defined by:

〈X,Y 〉t =

t∫0

HsH′

s ds

(It is application of Ito formula for X, Y and X + Y , with the function f : x 7→ x2).

These three forms of Ito formulas are particular cases of the multidimensional Ito formula stated

below:

Proposition 10. Let (Wt)t∈I be a d-dimensional Ft-Brownian motion. Xt is an Ito process with

respect to (Wt)t∈I iff

Xt = X0 +

t∫0

Ks ds+

d∑i=1

t∫0

H(i)s dW (i)

s

with

• (Kt)t∈I , (H(i)t )t∈I adapted.

•T∫0

|Ks|ds <∞.

•T∫0

(H(i)s )2 ds <∞ .

Then: If X(1)t , . . . , X

(n)t are n Ito processes

X(i)t = X

(i)0 +

t∫0

K(i)s ds+

d∑j=1

t∫0

H(i,j)s dW ij

s

If f ∈ C1,2,...,2 from I × Rn to R, then

f(t,X(1)t , . . . , X

(n)t ) = f(0, X

(1)0 , . . . , X

(n)0 ) +

t∫0

∂f

∂xi(s,X(1)

s , . . . , X(n)s ) ds

+

n∑i=1

t∫0

∂f

∂xi(s,X(1)

s , . . . , X(n)s ) dX(i)

s

+1

2

n∑i,j=1

t∫0

∂2f

∂xi∂xj(s,X(1)

s , . . . , X(n)s ) d〈X(i), X(j)〉s

where formally dX(i)s = K

(i)s ds+

d∑i=1

H(i,j)s dW

(j)s

d〈X(i), X(j)〉s =d∑

m=1H

(i,m)s H

(j,m)s ds

18

3 The Black-Scholes Model

3.1 Description of the Model

As informally seen before, the simplest nontrivial binomial model admits a “kind of limit” when

the frequency of the steps tends to infinity.

This limit is called Black (and) Scholes model (described in 1973, with Merton; some ideas go back

to Bachelier in 1900).

This model (in its simplest non-trivial form) can be rigorously stated as follows.

The corresponding model contains only two assets (as in the binomial model): a stock (risky) and

a bond (risk-less).

• The bond price satisfies the following equation:

dS0t = rS0

t dt

i.e.

S0t = S0

0 + r

t∫0

S0s ds

where S0t is the price at time t.

• The stock price satisfies the stochastic differential equation:

dSt = St(µdt+ σdBt)

where St is the price at time t.

This equation means that:

St = S0 + µ

t∫o

Ss ds+ σ

t∫0

Ss dBs

Here r,µ,σ > 0 are real parameters and (Bs)s≥0 is a one dimensional Brownian motion

(Generally µ > r > σ).

Intuitively.

• The bond is a risk-less asset, which gives a fixed interest rate r, generally positive.

• The stock is a risky asset, which gives “in average” a gain of µ per unit of time, but with

fluctuations, driven by the Brownian motion B. (Generally µ > r, because one expect a

better gain in average if one takes some risk).

• The parameter σ represents the amplitude of the fluctuations of the stock price, it is called

volatility

19

3.1.1 Explicit Description of the Prices

• The equation satisfied by the bond price can be easily solved. One obtains S0t = S0

0 exp(rt).

• By using Ito formula, one can check that for S0 > 0 given: St = S0 exp((µ− π2

2 )t+σBt) is a

solution of the stochastic differential equation stated before. In fact, this solution is unique

(exercise).

Now, since we have a model for stock prices, we have also a model for trading in the financial

model described above. We then use this model to replicate options, and hence, to evaluate option

prices.

3.1.2 Self-Financing Strategies

A strategy will be defined as a process φ = (φt)0≤t≤T = (Ht0, Ht)0≤t≤T with values in R2, adapted

to the natural filtration (Ft)0≤t≤T of the Brownian motion (Bt)0≤t≤T .

H0t and Ht represent the quantities of bonds and stocks hold in the portfolio at time t. The time

T ∈ R∗t is the time when one stops trading (typically the maturity of the option one needs to

replicate).

Remarks.

• H0t and Ht can change at each instant, i.e. it is possible to trade “continuously” on the

market.

• H0t and Ht can be negative. For H0

t it corresponds to “borrowing money”, and for Ht, it is

“short-selling” of the stock.

One can now define the value of the portfolio (H0t , Ht)0≤t≤T as the process

(Vt(φ) = H0t S

0t +HtSt)0≤t≤T

Question. How can we define a self-financing portfolio (or strategy)?

Intuitive answer. The value of the portfolio does not change at a time of trading. More con-

cretely, let us consider instants t and t+ dt (dt > 0 “small”).

• Between t and t + dt let us suppose that there is no trading. One has H0t bonds and Ht

stocks, which gives a value at t+ dt.

Vt+dt(φ) = H0t S

0t+dt +HtSt+dt

• Let us trade just after t + dt. The quantity of bonds is H0t+dt and the number of stocks is

Ht+dt. Value after trading:

Vt+dt(φ) = H0t+dtS

0t+dt +Ht+dtSt+dt

20

• By self-financing property, the value does not change during trading, so

Vt+dt(φ) = Vt+dt(φ)

= Vt+dt(φ)

= H0t S

0t+dt +HtSt+dt

= H0t (S0

t + dS0t ) +Ht(St + dSt)

= H0t S

0t +HtSt +H0

t dS0t +HtdSt

= Vt(φ) +H0t dS0

t +HtdSt

So

dVt(φ) = H0t dS0

t +HtdSt

In other words the portfolio φ is self-financing (by definition) iff

Vt(φ) = V0(φ) +

t∫0

H0s dSs +

t∫0

Hs dSs

where the stochastic integrals have supposed to be well-defined (H0s , Hs adapted and H0

s ,

Hs “sufficiently bounded“).

3.1.3 Precise Condition of Boundedness

dH0s = rH0

sds, dHs = µHsds+ σHsdBs

T∫0

|H0s |ds <∞ and

T∫0

|Hs|ds <∞ and

T∫0

(Hs)2 ds <∞ almost surely

Let us denote by St = e−rtSt the discounted price of the risky asset. Then the following result

holds:

Proposition 11. Let φ = (H0t , Ht)0≤t≤T be an adapted process with values in R2, satisfying:

T∫0

|H0t |dt+

T∫0

(Ht)2 dt <∞ almost surely

Let Vt(φ) = H0t S

0t +HtSt and Vt(φ) = e−rtVt(φ) (= H0

t S00 +HtSt). Then, φ defines a self-financing

strategy iff for all t ∈ [0, T ]:

Vt(φ) = V0(φ) =

t∫0

Hs dSs almost surely

21

Proof. Suppose that φ is a self-financing strategy. From the equality:

dVt(φ) = −rVt(φ) + e−rtdVt(φ)

(No extra term in the Ito formula here, because e−rt has bounded variation). We deduce:

dVt(φ) = −re−rt(H0t ert +HtSt)dt+ e−rtH0

t d(ert) + e−rtHtdSt

= Ht(−re−rtStdt+ e−rtdSt) = HtdSt

In the other direction, the proof is similar.

Intuitively. The variation of the discounted value of the portfolio is the product of the quantity

of stocks hold by the ”discounted gain” per stock.

3.2 Change of Probability, Representation of Martingales

Recall (Equivalent Probabilities). Let (Ω,A,P) be a probability space. Recall that a probability

Q on (Ω,A) is absolutely continuous with respect to P iff for all A ∈ A : P(A) = 0⇒ Q(A) = 0.

Theorem 12 (Radon-Nikodym). A probability measure Q is absolutely contains with respect to

P iff there exists a non-negative random variable Z on (Ω,A) such that for all A ∈ A:

Q(A) =

∫A

Z(ω) dP(ω)

Z is called density of Q with respect to P and is denoted by dQdP .

Informally. x ∈ Ω : Q(“around x“) ∼= Z(x)P(“around x“)

Definition 12. Two probabilities P and Q are equivalent if they are defined on the same measur-

able space and if they are absolutely continuous with respect to each other.

Criterion. If Q is absolutely continuous with respect to P, with density Z, then P is equivalent

to Q iff P is absolutely continuous with respect to Q, or equivalently iff P(Z > 0) = 1.

If P is equivalent to Q, a property holds P-almost surely iff it holds Q-almost surely.

Let us describe a particular case of probability change, which is related to Brownian motion.

Theorem 13 (Girsanov). Let (Ω,F , (Ft)t≤T ,P) be a filtered probability space, and (Bt)t≤T a

one-dimensional (Ft)-Brownian motion. We suppose that (θt)t≤T is an adapted process, satisfyingT∫0

θ2s ds < ∞ almost surely, and such that the process (ιt)t≤T defined by ιt := exp(−

t∫o

θs dBs −

12

t∫0

θ2s ds) is a martingale. (It is the case if (θs)s≤T ) satisfies ”good integrability conditions“).

Then under the probability P(ι) with density ιT with respect to P (recall that EP[ιT ] = EP[ι0] by

assumption) the process (Wt)t≤T defined by:

Wt := Bt +

t∫0

θs ds

is an (Ft)t≤T -Brownian motion.

22

Remarks.

• If P is replaced by P(ι), one needs to correct B by a ”drift term“ (θsds) in order to obtain

again a Brownian motion under the new probability measure.

• By Ito-Formula:

ιt = ι0 + [a stochastic integral with respect to dBs]

Hence (ιt)t≤T is a martingale with respect to P if it satisfies some integrability conditions.

It is, for example, sufficient to have: E

[exp

(12

T∫0

θ2t dt

)]<∞ (called: Novakov’s criterion)

Proof. Admitted (exercise for θ constant, a simple but important case)

3.2.1 Representation of Martingales

Let (Bt)t≤T be a one-dimensional Brownian motion built on a probability space (Ω,F ,P) and

let (Ft)t≤T be its natural filtration. Recall that if (Ht)t≤T is an adapted process such that:

E

[T∫0

H2t dt

]< ∞ then the process

(t∫

0

HS dBS

)t≤T

is a square-integrable martingale, equal to

zero. The result stated below shows that any ”Brownian martingale“ can be represented in term

of stochastic integral.

Theorem 14. Let (Mt)t≤T be a square-integrable martingale, with respect to the filtration

(Ft)t≤T (recall that (Ft)t≤T is the natural filtration of the Brownian motion B). There exists

an adapted process (Ht)t≤T such that

E

T∫0

H2t dt

<∞and for all t ∈ [0, T ] :

Mt = M0 +

t∫0

HS dBs almost surely

Remarks.

• This theorem does not apply for a ”general filtration“.

• From this theorem, if U is an (Ft)t≤T -measurable, square-integrable random variable, it can

be written as:

U = E [U ] +

T∫0

HS dBS almost surely

23

where (Ht)t≤T satisfies the conditions above. (consider: Mt = E [U |Ft]). One can also

represent martingales which are not square-integrable. In this case,

T∫0

H2t dt <∞

but not necessarily the expectation.

3.3 Pricing and Hedging Options in the Black-Scholes Model

One begins this section by proving that there exists a probability, equivalent to P, under which

the discounted stock price (St := e−rt)t≤T is a martingale.

Indeed, one obtains, from the stochastic differential equation satisfied by (St)t≤T :

dSt = −re−rtSt dt+ e−r dSt = St((µ− r) dt+ σ dBt)

(The ”drift term“ is shifted by −r in the s.d.e., with respect to St). Let us set:

Wt := Bt +(µ− r)t

σ(σ > 0)

We check immediately that: dSt = Stσ dWt.

By applying Girsanov’s theorem, with (θt) constant, equal to (µ− r)/σ, one obtains a probability

P∗ equivalent to P, under which (Wt)t≤T is a standard Brownian motion. Moreover, one can

prove that the definition of the stochastic integral is invariant by a change of probability which is

equivalent. Then, under P∗, one has the same s.d.e.:

dSt = StdWt (here W is a Brownian motion)

which has a unique solution (up to a multiplicative constant):

St = S0 exp

(σWt −

tσ2

2

)In particular, (St)t≤T is a martingale under P∗. This ”equivalent martingale measure“ is very

important when we want to compute option prices.

More precisely, let us focus on European contingent claims (more simply said: European option).

They are defined as non-negative, Ft-measurable random variables, which will generally be denoted

by h in this lecture. Quite often , h can be written as a function f of ST : for example, for a call

option with strike price K and maturity T , f(x) = (x−K)+, then

h = f(ST ) = (ST −K)+

The value of such an option will be defined by a replication argument. For technical reasons, we

will limit our study to the following admissible strategies:

24

Definition 13. A strategy φ = (H0t , Ht)t≤T is admissible iff it is self-financing and if the discounted

value:

Vt(φ) = Vt(φ)e−rt = H0t +HtSt

of the corresponding portfolio is, for all t, non-negative, and such that: supt∈[0,T ]

Vt(φ) is square-

integrable under P∗.

An option is said to be replicable iff its payoff is equal to the terminal value of an admissible

strategy. It is clear that, for the option defined by h, it can be replicable only if it is non-negative,

and square-integrable under P∗.

Example. A call option satisfies the condition above:

EP∗[(St −K)2

+

]<∞

For a put, h is even bounded.

Theorem 15. In Black-Scholes-model, any option defined by a non-negative, FT -measurable

random variable, which is square-integrable under the probability P∗, is replicable and the value

at time t of any replicating portfolio is given by:

Vt := EP∗[e−r(T−t)h|Ft

]Thus, by basic ”non-arbitrage argument“, the option value at time t can be naturally defined by

the expression giving Vt.

Proof. First, assume that there is an admissible strategy (H0, H) replicating the option. The value

at time t of the portfolio (H0t , Ht) is given by:

Vt = H0t S

0t +HtSt

and by assumption, we have VT = h. Let Vt = Vte−rt be the discounted value:

Vt = H0t +HtSt

Since the strategy is self-financing, we get:

Vt = Vo +

t∫0

Hu dSu = V0 +

t∫0

HuσSu dWu

Under P∗, supt∈[0,T ]

Vt is square-integrable, by definition of admissible strategies.

Moreover, (Vt)t≤T is a stochastic integral with respect to dWu. It follows (admitted) that (Vt)t≤T

is a square-integrable martingale under P∗.

Hence,

Vt = E∗[VT |Ft]

25

where E∗ denotes the expectation under P∗. Equivalently:

Vt = E∗[e−r(T−t)h|Ft]

We have proved that if a portfolio (H0, H) replicates the option defined by h, its value is given by

the expression in the theorem.

• The value Vt was found and proved last time.

• To complete the proof of the theorem, it remains to show that the option is replicable, i.e.

to find process (H0t )t≤T and (Ht)t≤T defining an admissible strategy such that:

Vt = H0t S

0t +HtSt = E∗[e−r(T−t)h|Ft]

(In this case VT = E∗[h|FT ] = h).

Under the probability P∗, the process defined by

Mt := E∗[e−rTh|Ft]

is a martingale, square-integrable because h is square-integrable with respect to P∗. The filtration

(Ft)t≤T , which is the natural filtration of (Bt)t≤T is also a natural filtration of (Wt)t≤T , and from

the martingale representation theorem, there exists an adapted process (Kt)t≤T such that:

E∗ T∫

0

K2s ds

<∞and for all t ∈ [o, T ]:

Mt = M0 +

t∫0

KS dWS

P∗-almost surely, (which is equivalent to P-almost surely, since P∗ is equivalent to P). The strategy

φ = (H0, H), with

Ht :=Kt

σStand H0

t := Mt −HtSt

is a self-financing strategy, with value at t:

Vt(φ) = H0t S

0t +HtSt

= (Mt −HtSt)S0t +Ht · St

= MtS0t −HtSte

rt +HtSt︸ ︷︷ ︸=0

= MtS0t

= E[e−r(T−t)h|Ft

]26

Check the self-financing property:

dVt = d(Mtert)

= Md(ert) + ertdMt(!)

= rMtertdt+ ertKtdWt

(!): No extra term because ert has finite variation.

On the other hand:

H0t S

0t +HtSt = H0

t rertdt+Htd(ertSt)

= H0t re

rtdt+Ht(ertdSt + rertStdt)

= (Mt −HtSt)rertdt+

Kt

σSt(ert(σStdWt) + rertStdt)

= rMtertdt−HtStre

rtdt+KtertdWt +

Kt

σrertdt

= rMtertdt−HtStre

rtdt+KtertdWt +HtStre

rtdt

= rMtertdt+ ertKtdWt

⇒ the two increments are equal.

With the expression of Vt, we also check that Vt(φ) ≥ 0 (since h ≥ 0), with supt∈[0,T )

Vt(φ) square-

integrable (by Doob inequality applied to the martingale (Mt)t≤T ). We have found an admissible

strategy replicating h.

3.3.1 Particular Case where h is a Function of ST

We have h = f(ST ) (f ≥ 0 such that E∗[(f(ST ))2

]<∞). We can express the option value Vt at

time t as a function of t and St. We have indeed:

Vt = E∗[e−r(T−t)f(St)|Ft

]∗= E∗

[e−r(T−t)f(Ste

r(T−t)eσ(WT−Wt)−σ2

2 (T−t))|Ft]

*: by the explicit expression of ST : St = S0e(r−σ22 )t+Wt

The random variable St is Ft-measurable, and under P∗, (WT −Wt) is independent of Ft. One

can deduce Vt = F (t, St), where

F (t, x) = E[e−r(T−t)f(xer(T−t)eσNT−t−

σ2

2 (T−t))]

where NT−t represents WT −Wt and is a normal variable with expectation zero and variance T − t.

In other words:

F (t, x) = e−r(T−t)∞∫−∞

f(x · e(r−σ22 )(T−t)+σy

√T−t) e−y22

√2π

dy

where y√T − t represents NT−t.

The function F , which depends linearly on f , can be explicitly computed for call (f(x) = (x−K)+)

27

and put (f(x) = (x−K)−).

Case f(x) = (x−K)+:

F (t, x) = E[(xeσ√θN−σ22 −Ke−rθ

)+

]where N is the standard Gaussian variable and θ = T − t.

Let us set:

d1 :=log( xK ) + (r + σ2

2 )θ

σ√θ

d2 := d1 − σ√θ =

log( xK ) + (r − σ2

2 )θ

σ√θ

F (t, x) = E[(xeσ√θg−σ22 θ −Ke−rθ

)1g+d2≥0

](∗)

=

∞∫−d2

(xeσ√θy−σ22 θ −Ke−rθ

) e−y22

√2π

dy

=

d2∫−∞

(xe−σ

√θy−σ22 θ −Ke−rθ

) e−y22

√2π

dy

(∗): g = −d2 corresponds to the change of the expression xeσ√θg−σ22 θ −Ke−rθ

We can then split the integral into two parts:

F (t, x) =

d2∫−∞

x · e−σ√θy−σ22 θ

e−σ2

2

√2π

dy−Ke−rθd2∫−∞

e−y22

√2π

dy

︸ ︷︷ ︸−Ke−rθN (d2)

where N (d2) =d2∫−∞

e−y22√2π

dy = P [N ≤ d2] (repartition function N (d2) of the standard Gaussian

variable).

By doing the change of variable z = y + σ√θ in the first integral we obtain:

F (t, x) = xN (d1)−Ke−rθN (d2)

Case f(x) = (x−K)−:

we similarly obtain:

F (t, x) = Ke−rθN (−d2)− xN (−d1)

These explicit expressions which give the price of a call and a put option for Black-Scholes-models

(the simplest case, studied here) are called Black-Scholes formulae

3.3.2 Hedging call and puts

We have used the martingale representation theorem in order to prove the existence of replicating

portfolio of a call and a put. It is not satisfactory to have only a theorem of existence. We need

28

to find explicitly a replicating strategy, to hedge the call and put option. We shall build such a

strategy. The strategy can be extended to general functions f . The value of a replicating portfolio

should be equal to F (t, St) where F is described above:

F (t, x) = E∗[e−r(T−t)f(xer(T−t)eσNT−t−

σ2

2 (T−t))]

The discounted value is:

Vt = e−rtF (t, St)

For a ”large class“ of functions f (containing the puts and calls), F is C∞ on [0, T )× R.

If we set, F (t, x) = e−rtF (t, xert) (F is also C∞), we have:

Vt = F (t, St)

and for t < T , by the Ito formula:

F (t, St) = F (0, S0) +

t∫0

∂F

∂x(u, Su) dSu +

t∫0

∂F

∂t(u, Su) du+

1

2

t∫0

∂2F

∂x2(u, Su) d〈S, S〉u

where ∂x is the derivative with respect to the second variable.

From dSt = σStdWt, we deduce d〈S, S〉u = σ2S2udu so that F (t, St) can be written:

F (t, St) = F (0, S0) +

t∫0

σ∂F

∂x(u, Su)Su dWu +

t∫0

Ku du

for some process K (coming from the last two terms).

The discounted value of the self-financing portfolio is a martingale under P∗, the termt∫

0

Ku du

vanishes.

⇒ F (t, St) = F (0, S0) +

t∫0

σ∂F

∂x(u, Su)Su dWu

= F (0, S0) +

t∫0

σ∂F

∂x(u, Su) dSu

It is natural to suppose that the number of stocks for the replicating portfolio is:

Ht :=∂F

∂x(t, St)⇔ Ht :=

∂F

∂x(t, St)

Let us define H0t := F (t, St) − HtSt, the portfolio (H0

t , Ht) has discounted value Vt = F (t, St).

One can check that it is self-financing.

Remarks.

• In fact it is not necessary the martingale representation theorem to deal with the particular

case h = f(St)−

29

• In the case of the calls and the puts, by using the notation of the Black-Scholes formulae, we

obtain:

· ∂F∂x (t, x) = N (d1) for the call, and

· ∂F∂x (t, x) = −N (−d1) for the put.

This derivation is called the delta of the option.

More generally, when the value at time t of an option is Ψ(t, St),(∂Ψ∂x

)(x, St) measures the sen-

sitivity of the option to the variation of the stock price, is called the delta of the option (also

available for portfolio).

The second order derivative ∂2Ψ∂x2 (t, St) is called gamma.

The derivative ∂Ψ∂t (t, St) is the theta.

The derivative with respect to the volatility (σ) is called vega.

Remarks.

• One can easily see that the parameter µ does not appear giving option prices.

– dSt = StdWt

– dSt = StσdWt + rdt (no µ) under P∗

This can be considered as a counterintuitive fact. (One could expect, for example, the call

price is higher if the stock price tends to increase than if the stock price tends to decrease).

However, it is a consequence of the fact that we always work with the probability P∗, under

which the parameter µ disappears (analogous to the binomial model).

• For call option, it is possible to check that the option price is decreasing in t (theta is

negative), increasing in r, increasing in St (delta is positive) and increasing in σ (important:

vega is positive).

3.4 American Options

An American option is an option which can be exercised at any time until maturity T . The deci-

sion to exercise or not at time t is made by the owner of the option, using only the information

available at time t (i.e., the time of exercise has to be a stopping time, chosen by the owner; it

has also to be smaller than T ). In our model, an American option can be defined by an adapted

non-negative process (ht)t≤T . ht represents the payoff at time t if the option is exercised at this

time. For the sake of simplicity, we will only consider payoff processes of the form ht = Ψ(St),

where Ψt is a continuous function from R+ to R+ satisfying Ψ(x) ≤ Ax+B for some A,B ≥ 0.

An American call option, corresponds to Ψ(x) = (x − K)+ and an American put option, corre-

sponds to Ψ(x) = (K − x)+.

30

• A call is the right, but not the obligation to buy the stock at price K at any time t ≤ T .

• A put is the right, but not the obligation to sell the stock at price K at any time t ≤ T .

In order to give an information on the ”fair price“ of an American option, we need to hedge it by

trading in the market. Let us state the following

Definition 14. A trading strategy with consumption is an adapted process φ = (H0t , Ht)t≤T with

values in R2, satisfying the following properties:

1.T∫0

|H0t |dt+

T∫0

(Ht)2 dt <∞ almost surely

2. H0t S

0t + HtSt =: Vt(φ) = V0(φ) +

t∫0

H0u dS0

u +t∫

0

Hu dSu − ct almost surely for all t ∈ [0, T ],

where (ct)t≤T is an adapted, continuous non-decreasing process, null at time t = 0. ct

represents the cumulative consumption up to time t.

With this definition, let us say that a trading strategy with consumption hedges the American

option defined by ht = Ψ(St) if, for all t ∈ [0, T ], Vt(φ) ≥ Ψ(St) almost surely. We will denote

by ΦΨ the set of all trading strategies with consumption hedging the American option defined by

ht = Ψ(St). If the writer (i.e. the seller) of the option follows a strategy φ ∈ ΦΨ, he possesses at

any time t a wealth at least equal to Ψ(St), which is precisely the payoff if the option is exercised

at time t. Hence, he can pay the owner in all possible cases with the wealth corresponding to his

portfolio.

The following theorem introduces the minimal value of an hedging strategy for an American option.

with finite time

Theorem 16. Let u be the map from [0, T ]× R+ to R

u(t, x) := supτ∈Ct,T

E∗[e−r(τ−t)Ψ(xe(r−σ22 )(τ−t)+σ(wτ−wt))

]where Ct,T is the set of all the stopping times with values in [t, T ]. There exists a strategy φ ∈ ΦΨ,

such that Vt(φ) = u(t, St) for all t ∈ [0, T ]. Moreover, this strategy is minimal in the following

sense:

Vt(φ) ≥ u(t, St) for all t ∈ [0, T ].

Sketch of the proof. One can show that the process (e−rtu(t, St))t≤T is the smallest right continu-

ous P∗-supermartingale which dominates (e−rtΨ(St))t≤T . Now, it is provable (and quite intuitive)

that the discounted value of a trading strategy consumption is a P∗-supermartingale (because of

the decreasing term −ct). Hence, the discounted stock price of a hedging strategy φ (for the pay-

off Ψ), which dominates (e−rtΨ(St))t≤T by assumption, has to be greater than (e−rtu(t, St))t≤T

which implies Vt(φ) ≥ u(t, St) for any strategy φ ∈ ΦΨ. [Due to the supermartingale property of

the discounted value of φ, applied to all the stopping times τ ∈ Ct,T ].

The existence of φ such that Vt(φ) = u(t, St) can be proved by using the two following ingredients:

31

• The decomposition of a supermartingale into a martingale and a decreasing process.

• The martingale representation theorem.

It is (quite) natural to consider u(t, St) as a price for the American option at time t, since it

is the minimal value of a strategy hedging the option. The price u(t, St) can also be considered in

the owners point of view, as follows:

For simplicity, let us assume that t = 0. The owner of the option has to optimize the time when

the option is exercised by choosing the best possible stopping time. Now, for any stopping time τ

(in Ct,T ), the discounted value of the payoff Ψ(Sτ ) obtained by exercising the option at time τ is

e−rτΨ(Sτ ), and hence, the value at time zero of an admissible strategy with value Ψ(Sτ ) at time

τ is

E∗[e−rτΨ(Sτ )

].

Therefore, the price of the American option would be

E∗[e−rτΨ(Sτ )

]if the owner were obliged to exercise it at time τ .

By optimizing in τ , one obtains the ”fair price“ u(0, S0) of the American option.

The following proposition proves that in fact, there is no difference between European and American

call option, if the interest rate r is positive.

Proposition 17. The fair price of an American option corresponding to the function Ψ : x 7→

(x−K)+ (i.e. the American call option with strike price K and maturity T ) is equal to the price

of a European call of strike price K and maturity T , if the interest rate is positive.

Proof. It is sufficient to show that ”the stopping time T is optimal“ (i.e. it is never useful to

exercise the American call strictly before maturity). It is equivalent to prove (let us assume t = 0)

that for all stopping times τ :

E∗[e−rτ (Sτ −K)+

]≤ E∗

[e−rT (ST −K)+

]= E∗

[(S∗T −Ke−rT )+

]On the other hand, we have:

E∗[(ST −Ke−rT )+|Fτ

]≥ E∗

[(ST −Ke−rT )|Fτ

]Using the martingale property:

E∗[(ST −Ke−rT )|Fτ

]= Sτ − e−rTK

32

since (St)t≤T is a P∗-martingale and τ is a (bounded) stopping time.

Since r ≥ 0, one deduces:

E∗[(ST −Ke−rT )+|Fτ

]≥ Sτ − e−rτK

Since the left hand side is non-negative:

E∗[(ST −Ke−rT )+|Fτ

]≥ (Sτ − e−rτK)+

By taking the expectations, we obtain:

E∗[(ST −Ke−rT )+

]≥ E∗

[(Sτ −Ke−rτ )+

]

3.4.1 Perpetual Put, Critical Price

In the case of the put, the American option price is not equal to the European one and there does

not exist a closed form for the function u giving the price:

Recall.

u(t, x) = supτ∈Ct,T

E∗[(Ke−r(τ−t) − xeσWt−τ σ

2

2

)+

]By replacing T by T − t, it is always possible to come down to the case t = 0 which gives:

u(0, x) = supτ∈C0,T

E∗[(Ke−rτ − xeσWτ−τ σ

2

2

)+

]Let us now consider a probability space (Ω,F ,P), and let (Bt)t≥0 be a standard Brownian

motion (with infinite time horizon) defined on this space. The function u is now equal to the

following:

u(0, x) = supτ∈C0,T

E[(Ke−rτ − xeσBτ−τ σ

2

2

)+

]≤ supτ∈C0,∞

E[(Ke−rτ − xeσBτ−τ σ

2

2

)+1τ<∞

]where C0,∞ denotes the set of all stopping times of the natural filtrations of (Bt)t≥0 and C0,Tconsists of all the stopping times with values in [0, T ]. The right hand side of the inequality above

can be interpreted naturally as the value of a ”perpetual put“ (i.e., which can be exercised at any

time). (Here, we have no rigorous justification of this interpretation). The following proposition

gives an explicit formula for the price at a ”perpetual put“.

Proposition 18. The function u∞(x) := supτ∈C0,∞

E[(Ke−rτ − xeσBτ−τ σ

2

2

)+1τ<∞

]is given by

the formula:

u∞(x) =

K − x for x ≤ x∗

(K − x∗)( xx∗ )γ for x > x∗

with γ = 2rσ2 and x∗ = k·γ

1+γ . In particular 0 < x∗ < K.

33

Proof. Admitted.

Example. t = 1 ⇒ σ2 = 2r ⇒ x∗ = K2

u∞(x)put price with maturity zeroK

2

K

K2

We see, that the put price is K − x for x ≤ x∗ for all T .

Remark. T = 0 gives (K − x)+

3.4.2 Description of the Best Strategy for an American Put

(T <∞).

One can prove that for any t ∈ [0, T ), there exists a real number s(t) ∈ [0,K] satisfying:

∀x ≤ s(t) : u(t, x) = K − x and ∀x > s(t) : u(t, x) > (K − x)+

(T =∞, s(0) = x∗).

The number s(t) is interpreted as the ”critical price“ at time t. If the price of the underlying asset

at time t is less then or equal to s(t), the buyer of the option should exercise it immediately, in

the opposite case, he should wait.

3.5 Implied Volatility and Local Volatility Models

One of the main features of the Black-Scholes model (and one of the reasons for it success) is the

fact that the pricing formula, as well as the hedging formula, depend on only one ”non-observable“

parameter: the volatility σ (recall that the drift parameter µ does not appear!).

Question. How can we evaluate σ?

• Historical Method: for all t, log(St) has variance σ2t and the variables log(StS0

), log

(S2t

St

),

log(

SNtS(N−1)t

)are independent, identically distributed. Therefore σ can be estimated by

statistical means using past observation of the asset prices.

This method is essentially not used by practitioners.

Problem. In order to make a good statistics, you need to quite far in the past (ex. t = 1

day, N = 1000, you need to consider ∼= 3 years of data) of taking very short intervals of time

(Nt = 1 day, t < 1 min).

– case 1: Market changes a lot.

34

– case 2: The price is not continuous at very short scale in practice.

• ”Implied Volatility” Method: Some options (calls and puts for some particular maturities and

strike prices) are directly traded in the market. The “fair price” is observable. In Black-

Scholes model this price is an increasing function of the volatility σ. (Fair price is a certain

function of the volatility which is known by the market). (fair price︸ ︷︷ ︸known

= function︸ ︷︷ ︸known

( σ︸︷︷︸unknown

))

By inverting Black-Scholes formulae, we can remove σ. This method is used by practitioners.

Problem. The volatility observed in this way depends on the option which is considered.

The model for the stock price depends on the option (the parameter σ changes). This is due

to the fact that Black-Scholes model does not fit in the real market dates. (for example: the

tails of distribution of log(St) are larger than Gaussian tails in practice).

Despite of this problem, the Black-Scholes model is used as a reference by practitioners. The

implied volatility of an option being “more meaningful than its price“.

One has constructed some models which are more consistent with market data. For example, one

can replace the constant volatility σ by a stochastic process (σt)t≥0, and then obtain:

dSt = St(µdt+ σtdBt).

If (σt)t≥0 is adapted to the natural filtration of B, and if σt, r/St are bounded, then the approach

of this chapter can be extended. In particular one can construct a martingale measure P∗, replicate

European options, and give their prices as an expectation of the discounted payoff with respect to

P∗. However, we have no explicit formulae (as Black-Scholes formulae).

A particular case of this model is the local volatility model where σt = σ(t, St), σ being a de-

terministic function. Dupire, Derman and Kani have observed that, given market prices of call

options, one can construct a local volatility model which provides the same prices as the market.

More precisely, if C(T,K) is the market price of a call option with strike price K and maturity T ,

observed at time 0, the local volatility is given by the following equation (Dupire’s formula):

∂C

∂T(T,K) =

σ2(T,K) ·K2

2· ∂

2C

∂K2(T,K)− rK ∂C

∂K(T,K)

In particular, this formula involves the derivatives of the option prices⇒ it is difficult to implement.

(Not so many options are traded).

Remark. The class of stochastic volatility models generally refers to models where (σt) is driven by

another Brownian motion. These markets are not complete, i.e. one cannot, as in the Black-Scholes

model, replicate all the European options.

3.6 The Black-Scholes Model with Dividends, and Call/Put Symmetry

In the models described in this chapter, there is no distributed dividends. In fact, Black-Scholes

methodology can be extended to options on dividend paying stock, when dividends are paid in

35

continuous time, at a constant rate δ. This means that, in the infinitesimal interval of time [t, t+dt],

the holder of one share (i.e. stock) receives δStdt. The interest of this rather unrealistic assumption

is that it leads to closed formulae. (Foreign currencies: The model is more realistic, and δ can be

interpreted as an interest rate).

In the context of a dividend paying asset, the self-financing condition takes the following form:

dVt = H0t dS0

t +Ht(dSt + δStdt)

and there is the discounted version:

dVt = Ht(dSt + δStdt) = HtσStdW(δ)t

where

W(δ)t = Bt +

(µ+ δ − r)σ

t

3.6.1 Self-Financing Property

dVt = H0t dS0

t +Ht(dSt + δStdt)

dVt = HtσStdWδt

where

W δt = Bt +

µ+ δ − rσ

t

The pricing measure is now the probability Pδ under which (W δt )t≤T is a Brownian motion. Under

Pδ, one can check that (e(δ−r)tSt)t≤T is a martingale. One can write the option prices as expecta-

tions under this probability measure Pδ. One has a symmetry between call and put prices.

Let us denote by Ce(t, x,K, r, δ) (resp. Pe(t, x,K, r, δ)) the prices at time t of a European call

(resp. put) option with maturity T , exercise (strike) price K, for a current stock price x (St = x)

when the interest rate is r and the dividend yield is δ. By similar computation as for the option

without dividends, one has:

Ce(t, x,K, r, δ) = Eδ[e−r(T−t)

(xe(r−δ−σ22 )(T−t)+σ(W δ

T−Wδt ) −K

)+

]Let us also define Ca(t, x,K, r, δ) as the price of the American option with the same parameters

(also define Pa(t, x,K, r, δ)).

Proposition 19. • Ce(t, x,K, r, δ) = Pe(t,K, x, δ, r)

• Ca(t, x,K, r, δ) = Pa(t,K, x, δ, r)

36

Proof. We prove the result for American options (the case of European option is similar). We

assume that t = 0 for simplicity. We have (similarly to the case of options without dividends):

Ca(0, x,K, r, δ) = supτ∈C0,T

Eδ[e−rτ

(xe(r−δ−σ22 )τ+σW δ

τ −K)

+

]For τ ∈ C0,T , we have, with the notation W δ

t := W δt − σt, and let us denote by Pδ the probability

measure with density given by:

dPδ

dPδ= exp(σW δ

T − Tσ2

2)

one has:

Eδ[e−rτ

(xe(r−δ−σ22 )τ+σW δ

τ −K)

+

]=Eδ

[e−δτeσW

δτ−σ

2

2 τ(x−Ke(δ−r+σ2

2 )τ−σW δτ

)+

]=Eδ

[e−δτeσW

δτ−σ

2

2 τ(x−Ke(δ−r−σ22 )τ−σW δ

τ

)+

]=Eδ

[e−δτeσW

δT−σ

2

2 T(x−Ke(δ−r−σ22 )τ−σW δ

τ

)+

]The last equality comes from the fact that

(eσW

δt −tσ

2

2

)t≤T

is a martingale under Pδ. Therefore

Eδ[e−rτ

(xe(r−δ−σ22 )τ+σW δ

τ −K)

+

]=Eδ

[e−δτ

(x−K(δ−r−σ22 )τ−σW δ

τ

)+

]under Pδ.

Now, under the probability Pδ, the process (W δt )t≤T is a standard Brownian motion (because of

the Girsanov theorem), as well as (−W δt )t≤T :

We can deduce, by looking to the last formula that:

Ca(t, x,K, r, δ) = Pa(t,K, x, δ, r)

4 Option Pricing and PDE

In the previous chapter we have obtained a formula for European option prices in the case of Black-

Scholes model. However, there are not anymore explicit expressions for more general options (e.g.

as American option) or for more complex models. In this chapter we study option prices in a more

general framework, and we prove that they satisfy some PDE. Despite that, these equations do

not give explicit formulas, they can be used in order to do numerical computations.

37

4.1 European Option Pricing and Diffusions

Recall that in Black-Scholes model the price of an European call or put (vanilla) options is given

by:

Vt = E[e−r(T−t)f(ST )|Ft

]with

f(x) = (x−K)+ for calls, and

f(x) = (K − x)+ for puts, and

ST = S0 exp

((r − σ2

2

)T + σWT

)(where r is the interest rate, σ the volatility, T the maturity, K the strike price and W is a standard

Brownian motion).

The problem of computing Vt can be generalized with the following way: Let (Xt)t≤T be a stochas-

tic process which satisfies the stochastic differential equation:

dXt = b(t,Xt)dt+ σ(t,Xt)dWt

where b and σ are functions from R× R to R, satisfying the following technical conditions:

(a) |b(t, x)− b(t, y)|+ |σ(t, x)− σ(t, y)| ≤ K|x− y| for some K > 0 independent of t.

(b) |b(t, x)|+ |σ(t, x)| ≤ K(1 + |x|) for some K > 0 independent of t.

The rigorous meaning of this stochastic differential equation is the following: Consider the filtered

probability space (Ω,F , (Ft)t≥0,P), and (Wt)t≥0 an (Ft)t≥0-Brownian motion. A solution of the

equation above is an (Ft)t≥0-adapted continuous stochastic process (Xt)t≥0 that satisfies:

1. ∀t ≥ 0, the integralst∫

0

b(s,Xs) ds,t∫

0

σ(s,Xs) dWs exist, i.e.

•t∫

0

|b(s,Xs)|ds <∞

•t∫

0

|σ(s,Xs)|2 dWs <∞

2. There exists an F0-measurable random variable Z such that

Xt = Z +

t∫0

b(s,Xs) ds+

t∫0

σ(s,Xs) dWs

If b and σ satisfy the conditions (a) and (b) above, and if Z is a square integrable F0-measurable

random variable, then there exists a unique solution (to the stochastic differential equation above)

(Xt)t≤T on the interval [0, T ] such that X0 = Z almost surely.

(Uniqueness: Two solutions are indistinguible, i.e. a.s. there are equal at each time).

38

(For Black-Scholes model under P∗: b = rXt and σ(t,Xt) = Xt · volatility).

If (Xt)t≥0 satisfies the stochastic differential equation above, let us try to compute:

Vt = E

e− T∫0

r(s,Xs) dsf(Xt)|Ft

where r : (t, x) 7→ r(t, x) is a bounded, continuous function. In this model, r is the (non-constant)

interest rate and σ is the (non-constant) volatility, multiplied by the current stock price.

⇒ Vt represents the value at time t of an European option with payoff f(Xt).

In the same way as in the Black-Scholes model, Vt can be written as a function of t and Xt:

Vt = F (t,Xt) where

F (t,Xt) = E

e− T∫t

r(s,X(t,x)s ) ds

f(X(t,x)T )

and (X

(t,s)s )s∈[t,T ] is the solution of the stochastic differential equation which starts at the value of

x and at time t. Intuitively,

F (t, x) = E

e− T∫t

r(s,Xs) dsf(Xt)|Xt = x

(non rigorous a priori).

As a consequence, the option price is known, if we are able to compute the function F . Now, we

will find that F satisfies a certain partial differential equation, related with an operator, called

infinitesimal generator of the process (Xt)t≥0. (One says that X is a diffusion, which justifies the

title of this section).

4.2 Infinitesimal Generator of a Diffusion

Let us now assume that b and σ do not depend on the time. Hence (Xt)t≥0 is the solution of the

equation:

dXt = b(Xt)dt+ σ(Xt)dWt.

We deduce the following result:

Proposition 20. Lef f be continuously twice differentiable function (f ∈ C2) with bounded

derivatives, and let A be the differential operator that maps a C2-function f to Af such that:

(Af)(x) =[σ(x)]2

2f ′′(x) + b(x)f ′(x)

Then, the process (Mt)t≥0, Mt = f(Xt)−t∫

0

(Af)(Xs) ds is an (Ft)t≤T -martingale.

39

Proof. Ito formula yields: (f ∈ C2)

f(Xt) = f(X0) +

t∫0

f ′(Xs) ds+1

2

t∫0

f ′′(Xs)σ2(Xs) ds

σ2(Xs)ds = d〈X,X〉s because of the stochastic differential equation satisfied at (Xt)t≥0.

⇒ f(Xt) = f(X0) +

t∫0

f ′(Xs)σ(Xs) dWs +

t∫0

(1

2σ2(Xs)f

′′(Xs) + b(Xs)f′(Xs)

)︸ ︷︷ ︸

(Af)(Xs)

ds

⇒ f(Xt)−t∫

0

(Af)(Xs) ds =

t∫0

f ′(Xs)σ(Xs) dWs

To check that the right-hand side is a martingale, it is sufficient to check that the expectation:

E

t∫0

|f ′(Xs)|2 · |σ(Xs)|2 ds

<∞This is a consequence of the technical assumptions made on f ′, σ and of some properties of the

solutions of stochastic differential equation (unfortunately, we cannot detail the proof here).

Remark. If we denote by X(x)t the solution of the stochastic differential equation above, such that

X(x)0 = x, we obtain, from the proposition above:

E[f(X

(x)t )

]= f(x) + E

t∫0

(Af)(X(x)s ) ds

Moreover, since the derivatives of f are supposed to be bounded by a constant Kf , and since

|b(x)|+ |σ(x)| ≤ K(1 + |x|) we have:

E[

sups≤T|Af(X(x)

s )|]≤ Kf

(1 + E

[sups≤T|X(x)

s |2])

where the right-hand side can be proved to be finite (properties of solutions of stochastic differential

equation). Therefore, since x 7→ (Af)(x) and s 7→ X(x)s are continuous, the dominated convergence

theorem is applicable and yields:

d

dtE[f(X

(x)t )

]∣∣∣∣t=0

=d

dt

E

t∫0

(Af)(X(x)s ) ds

∣∣∣∣∣∣t=0

= limt→0

E

1

t

t∫0

(Af)(X(x)s ) ds

= limt→0

E

1∫0

(Af)(X(x)tu ) du

= (Af)(x)

40

f(Xt)x

t

The differential operator A is called infinitesimal generator of the diffusion X.

The proposition above can be generalized in the time-dependent case, with major difficulties.

Proposition 21. If u(t, x) is a C1,2-function (i.e. once differentiable with respect to t and twice

with respect to x and with continuous partial derivatives) with bounded derivatives with respect

to x, and if (Xt)t≥0 is the solution of the stochastic differential equation above:

dXt = b(t,Xt)dt+ σ(t,Xt)dWt,

(with the technical assumption above)

Mt = u(t,Xt)−t∫

0

(∂u

∂t+Asu

)(s,Xs) ds

is a martingale where ∂u∂x denotes the first derivative of u and the operator As is defined by:

(Asu)(x) =σ2(s, x)

2

∂2u

∂x2+ b(s, x)

∂u

∂x

Proof. The proof is similar to the time-independent case. (We apply Ito formula for a function of

the time and an Ito process).

In order to deal with discounted values (i.e. multiplied by a factor exp

(−t2∫t1

r(s,Xs) ds

)), we

need to generalize again a little bit.

Proposition 22. Under the assumption of the proposition above and for r : (t, x) 7→ r(t, x) a

bounded, continuous function from R+ × R to R, the process

Mt = e−

t∫0

r(s,Xs) dsu(t,Xt)−

t∫0

e−

s∫0

r(v,Xv) dv(∂u

∂t+Asu− ru

)(s,Xs) ds

is a martingale.

Sketch of the proof. We use integration by parts formula to differentiate the product e−

t∫0

r(s,Xs) ds

u(t,Xt) and then apply Ito formula to the process u(t,Xt).

4.3 Conditional Expectations and PDE

In this section, we emphasize the link between pricing European option and solving a partial

differential equation.

Recall. We want to compute:

Vt = E

e− T∫t

r(s,Xs) dsf(XT )|Ft

41

which is equal to F (t,Xt), where:

F (t, x) = E

e− T∫t

r(s,X(t,x)s ) ds

f(X(t,x)T )

The following result gives a characterization of F as a solution of a partial differential equation.

Proposition 23. Let u be a C1,2-function, with a bounded derivation with respect to the second

variable defined on [0, T )×R. If u satisfies: ∀x ∈ R : u(T, x) = f(x) and(∂u∂t +Atu− ru

)(t, x) = 0

for all (t, x) ∈ [0, T ) × R, where u(T, x) = limt→T

u(t, x) (by definition). Then: ∀(t, x) ∈ [0, T ) × R :

F (t, x) = u(t, x).

Proof. Let us prove the equality F (t, x) = u(t, x) for t = 0 (for t > 0, use time shift with a similar

proof). From the previous proposition, we know that the process:

Mt = e−

t∫0

r(s,X(0,x)s ) ds

u(t,X(0,x)t )

is a martingale (the non-written term here is equal to zero by the partial differential equation

satisfied by u). Therefore E [M0] = E [MT ], which yields:

u(0, x) = E

e− T∫0

r(s,X(0,x)s ) ds

u(T,X(0,x)T )

= E

e− T∫0

r(s,X(0,x)s ) ds

f(X(0,x)T )

since u(T, x) = f(x).

Remark. This result suggests the following method to price the options: In order to compute:

F (t, x) = E

e− T∫0

r(s,X(t,x)s ) ds

f(X(t,x)T )

for a given f , we first need to find u such that:

∂u∂t +Atu− ru = 0 in [0, T )× R

u(T, x) = f(x) for x ∈ R

This is a partial differential equation with final condition (at terminal time T ). If we find a

solution u of this equation and if it satisfies the regularity assumptions of the last proposition, of

the previous section, we conclude that F = u.

4.4 Application to Black-Scholes Model

We are working on the probability measure P∗, under which the discounted stock price is a mar-

tingale (P∗ is called risk neutral probability measure).

42

Under P∗, the stock price St satisfies dSt = St(rdt+σdWt), where (Wt)t≤T is a Brownian motion.

The operator At is time-independent and is equal to:

At = A(bs) =σ2

2x2 ∂

2

∂x2+ rx

∂

∂x

(bs for Black-Scholes: A(bs)f(x) = σ2

2 x2f ′′(x) + rxf ′(x))

It is straight forward (but a little bit painful) to check that the call price given by

F (t, x) = xN(d1)−Ke−r(T−t)N(d1 − σ√T − t)

with

d1 =log(x/K) + (r + σ2/2)(T − t)

σ√T − t

, N(d1) =1√2π

d1∫−∞

e−x2/2 dx

is a solution of the partial differential equation∂u∂t +A(bs)u− ru = 0 in [0, T )× R∗+ (only R∗+ because the price are > 0)

u(T, x) = (x−K)+ in R+ (as limit of u(t, x) to t→ T )

The same result holds for the put price.

4.5 PDE on Bounded Open Sets and Computations of Expectations

In this section we assume that there is only one asset and that b(x), σ(x), r(x) are all time-

independent. The quantity r(x) can be interpreted as the instantaneous interest rate.

Consider the differential operator A, given by:

(Af)(x, t) =1

2(σ(x))2 ∂

2

∂x2f(x, t) + b(x)

∂

∂xf(x, t)

Let us denote by A the ”discount operator“, defined by:

(Af)(x, t) = Af(x, t)− r(x)f(x, t).

The equation giving the option price becomes∂u∂t (t, x) + (Au)(t, x) = 0 on [0, T )× R

u(T, x) = f(x) for x ∈ R (or x ∈ R∗+)

Let us first try to solve the problem on the bounded ”space“ interval O = (a, b), a < b.

We then use some boundary conditions at a and b in order to have an equation with a unique solu-

tion. Let us assume that u vanishes at the boundaries of O (called Dirichlet boundary conditions).

We now deal with the following problem: solve:∂u∂t (t, x) + (Au)(t, x) = 0 on [0, T )×O

u(T, x) = f(x) ∀x ∈ O

u(t, a) = u(t, b) = 0 ∀t ∈ (0, T )

43

It is now more convenient to deal with bonded interval O = (a, b). And we can hope that the

solution is ”close“ to the solution of the initial problem. In fact the solution of this modified

problem can be expressed in terms of the diffusion X(t,x).

Proposition 24. Let u be a continuous function of [0, T ] × [a, b]. Assume that u is C1,2 on

(0, T ) × O and that ∂u∂x is bounded on (0, T ) × O. Then, if u satisfies the conditions given in the

”modified“ problem, then for all (t, x) ∈ [0, T ]×O:

u(t, x) = E

1∀s∈[t,T ],X(t,x)s ∈O · e

−T∫t

r(X(t,x)s ) ds

f(X(t,x)T )

Proof. We prove the result for t = 0 since the argument is similar at the other times. In order to

avoid technicalities, we assume that there exists an extension of the function u from [0, T ]×O to

[0, T ]× R, that is still of class C1,2. We also denote by u such extension. We know that:

Mt = exp

− t∫0

r(X(0,x)s ) ds

u(t,X

(0,x)t

)−

t∫0

e−

s∫0

r(X(0,x)v ) dv

(∂u

∂t+Au− ru

)(s,X(0,x)

s

)ds

is a martingale. Now let

τ (x) := inf0 ≤ s ≤ T |X(0,x)s 6∈ O ∧ T (with the convention inf ∅ =∞).

Note that τ (x) is a (bounded) stopping time, because τ (x) = T(x)a ∩ T (x)

b ∩ T where:

T(x)b = inf0 ≤ s ≤ T |X(t,x)

s = b,

and indeed T(x)b is a stopping time according to general results on cont. stochastic processes. By

applying the optimal stopping theorem with the stopping time τ (x), we get E [M0] = E [Mτ(x) ].

Thus, by noticing that for s ∈ [0, τ (x)], (Af)(X(0,x)s ) = 0, it follows that:

u(0, x) = E

e− τ(x)∫0

r(s,X(0,x)s ) ds

u(τ (x), X(0,x)

τ(x) )

= E

1∀s∈[t,T ],X(t,x)s ∈Oe

−T∫0

r(s,X(0,x)s ) ds

u(T,X(0,x)T )

+ E

1∃s∈[t,T ],X(t,x)s 6∈Oe

−τ(x)∫0

r(s,X(0,x)s ) ds

u(τ (x), X(0,x)

τ(x) )

Furthermore, f(x) = u(T, x) and u(τ (x), X

(0,x)τ(x) ) = 0 on the event ∃s ∈ [t, T ]|X(t,x)

s 6∈ O (because

of the boundary conditions). The second expectation vanishes

u(0, x) = E

1∀s∈[t,T ],X(t,x)s ∈Oe

−T∫0

r(s,X(0,x)s ) ds

u(T,X(0,x)T )︸ ︷︷ ︸

f(X(t,x)T )

which completes the proof for t = 0.

44

u(t, x) = E [ solution of SDE ]→ solution of a partial differential equation.

Remark. An option defined by the Ft-measurable random variables:

1∀s∈[t,T ],X(t,x)s ∈O exp

− T∫t

r(X(t,x)s ) dsf

(X

(t,x)t

)is called extinguishable. Indeed, as soon as the asset price exists the open set O, the option becomes

worthless (payoff 0).

In the Black-Scholes model, if O is of the form (0, l) or (l,∞) we are ale to compute explicit

formulae for the option price.

4.6 American Options

The analysis of American options in continuous time is not straight-forward. In the Black-Scholes

model, we obtained the following formula for the price of American call (f(x) = (x −K)+) or of

American put (f(x) = (K − x)+):

Vt = Φt(t, St)

where

Φ(t, x) = supτ∈Ct,T

E∗[e−r(τ−t)f(x · e(r−σ22 )(τ−t)+σ(wτ−wt))

]and under P∗, (Wt)t≥0 is a standard Brownian motion, Ct,T is the set of all stopping times with

values in [t, T ]. We showed that American call price (on a stock offering no dividends) is equal to

the European call price. For the American put price, there is no explicit formula and numerical

methods are needed. The problem to be solved is a particular case of the following general problem,

given a ”good“ function f and a diffusion (Xt)t≥0 in R, the solution of the s.d.e.:

dXt = b(t,Xt)dt+ σ(t,Xt)dWt,

compute the function:

Φ(t, x) = supτ∈Ct,T

E

e− T∫t

r(s,X(t,x)s ) ds

f(X(t,x)τ )

By considering the stopping time τ = t, we get Φ(t, x) ≥ f(x). Also note that, for t = T , we

clearly have Φ(T, x) = f(x).

Remark. It can be proved that the process e−

t∫0

r(s,Xs) dsΦ(t,Xt) is the smallest supermartingale

which dominates the process f(Xt) at all times.

For European options, we have constructed some corresponding PDEs. For American options, we

will obtain some partial differential inequalities. The following properties tries to explain that.

45

Proposition 25. Assume that u is a ”regular“ solution of the following system of partial differ-

ential inequalities (we don’t give here the meaning of ”regular“):∂u∂t +Atu− ru ≤ 0, u ≥ f in [0, T )× R

∂u∂t +Atu− ru = 0 or u = f in [0, T )× R

u(T, x) = f(x)

Then, u(t, x) = Φ(t, x) = supτ∈Ct,T

E

e− T∫t

r(s,X(t,x)s ) ds

f(X(t,x)τ )

Sketch of the proof. We suppose t = 0, and we denote Xx

t the solution of the s.d.e. satisfied by X,

which starts at x. The process

Mt = exp

−t∫

0

r(s,X(x)s ) ds

u(t,X

(x)t

)−

t∫0

exp

−s∫

0

r(v,X(x)v ) dv

(∂u

∂t+Asu− ru

)(s,X(x)

s

)ds

is a martingale. By applying the optimal stopping theorem to this martingale, we get E [M0] =

E [Mτ ], and since ∂u∂t +Asu− ru ≤ 0:

u(0, x) ≥ E

[e−τ∫0

r(s,X(x)s ) ds

u(τ,Xxτ )

]

Recall that u(t, x) ≥ f(x), thus:

u(0, x) ≥ E

[e−τ∫0

r(s,X(x)s ) ds

f(X(x)τ )

]

This proves that

u(0, x) ≥ supτ∈C0,T

E

[e−τ∫0

r(s,X(x)s ) ds

f(X(x)τ )

]= Φ(0, x)

(one side of the ”double“ inequality).

We need to prove the reverse inequality.

Let τopt = inf0 ≤ s ≤ T |u(s,X(x)s ) = f(X

(x)s ).

It can be proved that τopt is a stopping time. Also, for s between 0 and τopt, we have:(∂u

∂t+Atu− ru

)(s,X(x)

s

)= 0

The optimal stopping theorem yields:

u(0, x) = E

e− τopt∫0

r(s,X(x)s ) ds

u(τopt, X(x)τopt)

but at time τopt: u(τopt, X

(x)τopt) = f(X

(x)τopt), so that

u(0, x) = E

e− τopt∫0

r(s,X(x)s ) ds

f(X(x)τopt)

46

This proves

u(0, x) ≤ supτ∈C0,T

E

[e−τ∫0

r(s,X(x)s ) ds

f(X(x)τ )

]= Φ(0, x)

⇒ u(0, x) = Φ(0, x). We have also proved that τopt is an optimal stopping time for the owner of

the option.

Remark. The precise meaning of differential inequalities given in the proposition is awkward be-

cause even for a ”regular” function f , the solution is generally not in C2. (And hence the second

derivative is not well-defined a priori). This is a problem to define the infinitesimal generator of

X on a and then one cannot apply directly Ito formula. (The sketch of the proof above is not

rigorous). However, it is possible to give a meaning to everything here, and to make rigorous

statements and proofs.

4.6.1 Example of Black-Scholes Model

We work under the measure P∗ ⇒ dSt = St(rdt + σdWt). Let us introduce Xt := log(St) =

log(S0) + (r − σ2/2)t+ σWt. Its infinitesimal generator A is actually time-independent, and

ABS-log :=σ2

2

∂2

∂x2+ (r − σ2

2)∂

∂x

Let us define:

ABS-log := ABS-log − r

(multiplication of functions by r).

If we consider Φ(x) := (K − ex)+, the partial differential inequality corresponding to the price of

the stock price of the American put is:

∂v∂t (t, x) + ABS-logv(t, y) ≤ 0 a.e. in [0, T )× R

v(t, x) ≥ Φ(x)

v(t, x) = Φ(x) or ∂v∂t (t, x) + ABS-logv(t, x) = 0 a.e. in [0, T )× R

v(T, x) = Φ(x)

The following proposition states existence and uniqueness results for a solution of this system and

establishes the connection with American put price.

Proposition 26. The system of inequalities above has a unique continuous solution v(t, x) such

that its partial derivatives in the distribution sense (extension of the notion of derivative to more

general framework; not precised here):

∂v

∂x,∂2v

∂x2

47

are locally bounded. Moreover, this solution satisfies:

v(t, log x) = Φ(t, x) = supτ∈Ct,T

E∗[e−r(τ−t)f(xe(r−σ22 )(τ−t)+σ(wτ−wt))

](formula for the price of American option: put with strike price K)

4.7 Solving the PDE Numerically

(case of European options).

We saw under which conditions the option price coincided with the solutions of the PDE:∂u∂t (t, x) + Au(t, x) = 0 on [0, T )× R

u(T, x) = f(x) ∀x ∈ R

(for European options)

We have seen that it is more convenient to deal with a bounded interval instead of R. It is possible

by using the expression of the solution as an expectation of a function f(X(x)s )s≤t (see above) to

give some bound for the difference between the solution with R and the solution with (−l, l). We

have to solve the following system:∂u∂t (t, x) + Au(t, x) = 0 on [0, T )× (−l, l)

u(t, l) = u(t,−l) = 0 if t ∈ [0, T )

u(T, x) = f(x) if x ∈ (−l, l)

The finite difference method: basically, this is a discretization of time and space of this system.

We shall start by discretizing the differential operator A on (−l, l). In order to do this, we need to

associate to a function (f(x))x∈(−l,l), which is an element of an infinite-dimensional vector space,

a vector (fi)1≤i≤N in RN . We proceed as follows: for i = 0, 1, . . . , N , let xi = −l + 2il(N+1) . The

number fi is supposed to be an approximation of f(xi). The boundary conditions can be stated

as f0 = 0, fN+1 = 0. Let h := 2l(N+1) be the mesh of the subdivision of (−l, l) comming from the

xi’s. The discretization of the operator A is the operator Ah on RN , defined as follows:

Think of a vector un = (u(i)n )1≤i≤N in RN as the discrete approximation of a function u (i.e.

u(i)n = u(xi)) and replace the first derivative ∂u

∂x (xi) with

∂nuin =

u(i+1)n − u(i−1)

n

2h

”discrete derivative“.

Similarly, replace the second derivative ∂2u∂x2 (xi) by

∂2nu

in =

ui+1n −uinh − uin−u

i−1n

h

h=ui+1n − 2uin + ui−1

n

h2

The discretized operator Anun is defined by:

(Anun)i =σ2(xi)

2∂2nu

in + b(xi)∂nu

in − ruin, i = 1 : N

48

Black-Scholes case: (after logarithmic change of variable).

ABS-logu(x) =σ2

2

∂2u

∂x2(x) +

(r − σ2

2

)∂u

∂x(x)− ru(x)

(ABS-logn un

)i

=σ2

2h2

(ui+1n − 2uin + ui−1

n

)+

(r − σ2

2

)1

2h

(ui+1n − ui−1

n

)− ruin

ABS-logn can be represented by the following matrix:

((ABS-logn )ij)

Ni,j=1 =

β γ 0 · · · 0

α β γ 0 · · · 0

0 :...

. . .. . .

. . .. . . 0

γ

0 · · · 0 α β

where

α =σ2

2h2− 1

2h

(r − σ2

2

)β =

−σ2

h2− r

γ =σ2

2h2+

1

2h

(r − σ2

2

)By doing the discretization above, we replace the initial PDE by an ODE:

dun(t)dt + ABS-log

n un(t) = 0 if 0 ≤ t ≤ T

un(T ) = fn

where fn = (f in)1≤i≤N is the vector f in = f(xi).

The time discretization can be obtained, using the so called ”Θ-Schemes“. Consider Θ ∈ [0, 1]

and let k be a time step such that T = Mk (M is a large positive integer). We approximate the

solution un of the previous equation at the time nk by unh,k where the sequence (unh,k)Mn=0 solves

the following recursive equation:uMh,k = fn

un+1h,k −u

nh,k

k + ΘABS-logn unh,k + (1 + Θ)ABS-log

n un+1h,k = 0 0 ≤ n ≤M − 1

We can solve the system by going backwards from n = M to n = 0, since the vector uMh,k is known.

Particular values of Θ:

• Θ = 0 (”explicit scheme“), unh,k is computed directly from un+1h,k (without solving a system of

equations).

• Θ = 1 (”completely implicit scheme“), we need to solve systems of equations.

• Θ = 12 , more symmetric, we also need to compute systems. (More precise).

49