stochastic calculus and applications to mathematical finance · stochastic calculus and...

Stochastic Calculus and Applications to

Mathematical Finance

by

GREG WHITE

Mihai Stoiciu, Advisor

A thesis submitted in partial fulfillment

of the requirements for the

Degree of Bachelor of Arts with Honors

in Mathematics

WILLIAMS COLLEGE

Williamstown, Massachusetts

May 16, 2012

Abstract

In this paper, we review fundamental probability theory, the theory of stochastic

processes, and Ito calculus. We also study an application of Ito calculus in math-

ematical finance: the Black-Scholes option pricing model for the European call

option. We study the development of the model and the assumptions necessary to

arrive at the Black-Scholes no arbitrage rational price for a European call option.

We supplement the simple Black-Scholes model by relaxing the assumption that

trading can be performed continuously in time, and studying the deviation the

Black-Scholes replicating portfolio exhibits from the self-financing characteristic of

the continuous-time portfolio. We term this deviation the cumulative correction

of the portfolio and explain in detail its construction. We study the cumulative

correction of Black-Scholes portfolios by performing a numerical analysis of the

cumulative correction for outcomes of the stock price stochastic process. While

finding a closed form probability distribution representing the cumulative correction

proves difficult and we do not pursue that route in this paper, the numerical

analysis indicates that the second central moment of the distribution of cumulative

corrections decreases as the number of discrete time steps at which the portfolio is

rebalanced increases. Additionally, we analyze the cumulative correction required

to replicate the European call option for the historical stock price data series of

certain actual stocks, finding examples of a stock that would have required a positive

cumulative correction and a stock that would have required a negative cumulative

correction.

Greg White

[email protected]

Williams College

Williamstown, Massachusetts 01267

1

mailto:[email protected]

Acknowledgements

I cannot possibly do justice to the role that Professor Mihai Stoiciu’s advising has

played in shaping this thesis and my study of the topics involved. His insights,

explanations, enthusiasm and patience have made working with him an incredibly

rewarding experience, and without him, this thesis would not have been possible.

Professor Thomas Garrity’s guidance as the second reader of this thesis has also

been invaluable. I am only beginning to understand the wisdom of Professor Gar-

rity’s favorite mathematical principle, that “functions describe the world” (although

my work during this thesis did further that understanding), but his teaching style

and the courses I have taken with Professor Garrity have fundamentally shaped my

experience studying at Williams, for which I am grateful.

The Professors of Williams College have provided an incredible learning environ-

ment during my time here, and because of that, I have gained so much more than

an understanding of academic topics from my time working with them.

I would also like to thank Zhaoning Wang, Williams ’11, whose thesis served as

an excellent learning example of creating a project in LATEX.

Finally, I would like to thank my friends and family, whose generosity, support,

and friendship are vital parts of my life. To say that this thesis would not have

been possible without them is an incredible understatement of the impact they have

on me: without them I would not be who I am today.

2

CONTENTS

Contents

1 Background 6

1.1 Probability spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 The Borel σ-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Discrete random variables . . . . . . . . . . . . . . . . . . . 11

1.3.2 Continuous random variables . . . . . . . . . . . . . . . . . 12

1.3.3 Properties of expected value of random variables . . . . . . . 13

2 Convergence of Random Variables 16

2.1 Definitions of modes of convergence . . . . . . . . . . . . . . . . . . 16

3 Characteristic Functions and Applications 17

3.1 Characteristic functions . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Limit theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 The Multivariate Normal Distribution 20

4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Stochastic Processes 24

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2 Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Ito Calculus 25

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2 Ito integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.2.1 Motivation for Ito integral . . . . . . . . . . . . . . . . . . . 25

6.2.2 Characteristics of integrator Ws . . . . . . . . . . . . . . . . 26

6.2.3 Conditions for integrand ψs . . . . . . . . . . . . . . . . . . 27

6.2.4 Defining

∫ ∞0

ψsdWs for random step functions ψs . . . . . . 28

6.2.5 Sequences of predictable step functions approaching ψ . . . 32

6.2.6 Existence and uniqueness of limit Ito integral I(ψ) . . . . . 35

6.2.7 Example:

∫ t

0

WsdWs . . . . . . . . . . . . . . . . . . . . . . 36

6.3 The Ito Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3

CONTENTS

6.3.1 Key idea of the Ito Lemma: (dWt)2 m.s.−−→ dt as dt→ 0 . . . . 39

6.3.2 The Ito Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.3.3 Ito processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.3.4 Geometric Brownian motion . . . . . . . . . . . . . . . . . . 42

7 Application: Black-Scholes Option Pricing Model 42

7.1 Black-Scholes option pricing model . . . . . . . . . . . . . . . . . . 42

7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

7.1.2 Set-up of the model . . . . . . . . . . . . . . . . . . . . . . . 43

7.1.3 European call option . . . . . . . . . . . . . . . . . . . . . . 45

7.1.4 Assumptions of the model . . . . . . . . . . . . . . . . . . . 51

7.2 Approximating the Black-Scholes portfolio at discrete time steps . . 54

7.2.1 Introduction and motivation . . . . . . . . . . . . . . . . . . 54

7.2.2 Deviation of a discrete-time portfolio from the self-financing

property: cumulative correction . . . . . . . . . . . . . . . . 54

7.2.3 Cumulative correction of sample paths of St . . . . . . . . . 56

7.2.4 Numerical analysis of cumulative correction over several sam-

ple paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2.5 Cumulative correction of portfolios based on historical stock

price data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.2.6 Summary of study of cumulative correction . . . . . . . . . . 68

8 References 69

A Appendix: Mathematica Models 70

A.1 European call option for GBM sample path . . . . . . . . . . . . . 70

A.2 Numerical analysis of cumulative correction . . . . . . . . . . . . . 72

A.3 European call option for historical data . . . . . . . . . . . . . . . . 74

4

LIST OF TABLES

List of Tables

1 Important discrete random variables . . . . . . . . . . . . . . . . . 15

2 Important continuous random variables . . . . . . . . . . . . . . . . 15

5

1 BACKGROUND

1 Background

This section is intended to provide the foundation for later work in the paper. We

define probability spaces and random variables and outline important properties

that we will use to investigate topics later in the paper.

1.1 Probability spaces

Definition 1.1.1. The sample space Ω of an experiment is the set of all possible

outcomes.

Definition 1.1.2. A σ-field F on a set S is a collection of subsets of S that satisfies

the following properties:

1. ∅ ∈ F ,

2. A1, A2, . . . ∈ F ⇒⋃∞i=1Ai ∈ F ,

3. A ∈ F ⇒ Ac ∈ F .

For the remainder of this section, F denotes a σ-field on Ω.

Definition 1.1.3. An event A is a subset of Ω with A ∈ F . We denote the

complement of a subset A of Ω as Ac.

Definition 1.1.4. A probability measure P on (Ω,F) is a function

P : F → [0, 1],

such that:

1. P(∅) = 0 and P(Ω) = 1,

2. If A1, A2, . . . satisfy Ai ∩ Aj = ∅ ∀ i 6= j, then

P

(∞⋃i=1

Ai

)=∞∑i=1

P(Ai).

Definition 1.1.5. A probability space is a triple (Ω,F ,P), where Ω is a sample

space, F is a σ-field on Ω, and P is a probability measure on (Ω,F).

6

1 BACKGROUND

Definition 1.1.6. An event A is a null event if P(A) = 0.

Definition 1.1.7. A probability space (Ω,F ,P) is complete if all subsets of null

events are elements of F .

Definition 1.1.8. A family of events Ai : i ∈ I is independent if

P

(⋂i∈J

Ai

)=∏i∈J

P(Ai),

for all finite subsets J of I.

Definition 1.1.9. If P(B) > 0 then the conditional probability that A occurs

given that B occurs is:

P(A|B) =P(A ∩B)

P(B).

Definition 1.1.10. A family B1, B2, . . . , Bn of events is a partition of Ω if:

Bi ∩Bj = ∅ when i 6= j andn⋃i=1

Bi = Ω.

Lemma 1.1.11. If B1, B2, . . . , Bn is a partition of Ω such that P(Bi) > 0 ∀ i, then

P(A) =n∑i=1

P(A|Bi)P(Bi).

Proof. A =⋃ni=1(A ∩Bi). This is a disjoint countable union, so

P(A) =n∑i=1

P(A ∩Bi)

=n∑i=1

P(A|Bi)P(Bi) by Definition 1.1.9.

1.2 The Borel σ-field

Lemma 1.2.1. If Fi : i ∈ I is a family of σ-fields of subsets of Ω, then G =⋂i∈I Fi is a σ-field as well.

7

1 BACKGROUND

Proof. We see that G satisfies the following:

1. Since ∅ ∈ Fi ∀ i ∈ I, ∅ ∈ G,

2. Suppose A1, A2, . . . ∈ G. Then ∀ i ∈ I,⋃∞i=1Ai ∈ Fi. Therefore

⋃∞i=1Ai ∈ G.

3. Suppose A ∈ G. Then A,Ac ∈ Fi ∀ i ∈ I. Therefore Ac ∈ G.

Definition 1.2.2. Let J be the collection of all open intervals of R. Let Bi : i ∈ Ibe the collection of σ-fields that contain J . Then the Borel σ-field B is defined

as B =⋂∞i=1 Bi. It is the unique smallest σ-field that contains J . Elements of B

are called Borel sets on R.

Note that B is a σ-field by Lemma 1.2.1. When we study functions of

random variables, it will be important to understand which functions of random

variables are themselves random variables. We will see later that functions that are

B-measurable will satisfy this criteria.

Definition 1.2.3. A function g : R→ R is Borel-measurable (B-measurable)

if we have, for all B ∈ B, g−1(B) ∈ B.

1.3 Random variables

Definition 1.3.1. A random variable X is a function X : Ω→ R such that for

every x ∈ R, ω ∈ Ω : X(ω) ≤ x ∈ F (the function is F-measurable).

Theorem 1.3.2. A function X : Ω→ R has the property that for all x ∈ R, ω ∈Ω : X(ω) ≤ x ∈ F (X is F-measurable) if and only if for all B ∈ B, ω ∈ Ω :

X(ω) ∈ B ∈ F .

Proof. (⇒) Suppose X is F -measurable. Let C = C ⊆ R : X−1(C) ∈ F . We will

show that C is a σ-field of R. We know that ∅ ∈ C because X−1(∅) = ∅ ∈ F .

Suppose C1, C2, . . . ∈ C. Then

X−1

(∞⋃n=1

Cn

)=∞⋃n=1

X−1(Cn) ∈ F .

8

1 BACKGROUND

Therefore,⋃∞n=1 Cn ∈ C. Now let A ∈ C. We see that X−1(Ac) = [X−1(A)]

c, so

Ac ∈ C. Thus C is a σ-field on R. Note that for all x ∈ R, we have (−∞, x] ∈ C.

We know that B is the smallest σ-field on R that contains all open intervals of R by

construction. We now need to show that this is equivalent to the smallest σ-field

that contains (−∞, x] ∀ x ∈ R. We need to show that a σ-field on R contains all

open intervals on R if and only if it contains (−∞, x] ∀ x ∈ R. Let D be a σ-field

on R.

(⇒) Suppose D contains all open intervals I ⊆ R.

Then for any x ∈ R, let Di = (−∞, x+ 1i). Since D is a σ-field we have

(−∞, x] =∞⋂i=1

Di =

(∞⋃i=1

Dci

)c

∈ D.

(⇐) Suppose D contains (−∞, x] ∀ x ∈ R.

We need to show that for all I ⊆ R, I ∈ D. We have the following 3 cases:

Case 1: I = (−∞, b) for some b ∈ R.

Let Ai = (−∞, b− 1i]. Then each Ai ∈ D, so

(−∞, b) =∞⋃i=1

Ai ∈ D.

Case 2: I = (a,∞) for some a ∈ R.

We know (−∞, a] ∈ D, so

(a,∞) = (−∞, a]c ∈ D.

Case 3: I = (a, b) for some a, b ∈ R with a < b.

We know (−∞, a] ∈ D and [b,∞) = (−∞, b)c ∈ D by Case 1. Then

(a, b) =(

(−∞, a]⋃

[b,−∞))c∈ D.

Then B is the smallest σ-field that contains (−∞, x] ∀ x ∈ R. Therefore

B ⊆ C, so for all B ∈ B, we have that X−1(B) ∈ F . Then

ω ∈ Ω : X(ω) ∈ B ∈ F .

9

1 BACKGROUND

(⇐) Suppose we have that ∀ B ∈ B, ω ∈ Ω : X(ω) ∈ B ∈ F . We know that ∀ x ∈R, (x,∞) ∈ B, so (−∞, x] = (x,∞)c ∈ B. Then ∀ x ∈ R, ω ∈ Ω : X(ω) ≤ x ∈F .

Definition 1.3.3. Every random variable X has a distribution function F :

R→ [0, 1] defined by F (x) = P(X ≤ x).

Given a random variable X, we can associate events A(x) ⊆ Ω with our

random variable by letting A(x) = ω ∈ Ω : X(ω) ≤ x. We will henceforth denote

A(x) as X ≤ x.

Definition 1.3.4. Random variables X and Y are independent if X ≤ x and

Y ≤ y are independent events.

We will often concern ourselves with functions of random variables. To do

so, we will define the conditions for which a function of a random variable is itself

a random variable.

Theorem 1.3.5. If X is a random variable and we have a function g : R → R,

then g(X) : Ω → R, defined by g(X)(ω) = g(X(ω)) is also a random variable if g

is B-measurable.

Proof. g(X) clearly maps Ω → R, so we only need to show that g(X) is F -

measurable. Suppose g is B-measurable. Then ∀ B ∈ B, g−1(B) ∈ B. By Theo-

rem 1.3.2, sinceX is F -measurable, we also have that ∀ B ∈ B, ω ∈ Ω : X(ω) ∈ B ∈F . Then we have ∀ B ∈ B:

ω ∈ Ω : g(X)(ω) ∈ B = ω ∈ Ω : X(ω) ∈ g−1(B) ∈ F .

Therefore, by Theorem 1.3.2, g(X) : Ω→ R is F -measurable.

Theorem 1.3.6. If random variables X and Y are independent, and g, h : R →R are B-measurable functions, then g(X) and h(Y ) are also independent random

variables.

10

1 BACKGROUND

Proof. We know that g(X) and h(Y ) are random variables by Theorem 1.3.5. Then:

P(g(X) ≤ x, h(Y ) ≤ y) = P(g(X) ∈ (−∞, x], h(Y ) ∈ (−∞, y])

= P(X ∈ g−1((−∞, x]), Y ∈ h−1((−∞, y]))

= P(X ∈ g−1((−∞, x]))P(Y ∈ h−1((−∞, y]))

= P(g(X) ∈ (−∞, x])P(h(Y ) ∈ (−∞, y])

= P(g(X) ≤ x)P(h(Y ) ≤ y).

We will also need to be able to understand multivariate random variables,

so we need to define the joint distribution function of a vector of random variables.

Definition 1.3.7. Suppose we have a vector of random variables X = (X1, X2, . . . , Xn)

on (Ω,F ,P). Then the joint distribution function of X is the function FX :

Rn → [0, 1] given by:

F (x) = P (X1 ≤ x1, X2 ≤ x2, . . . , Xn ≤ xn) .

1.3.1 Discrete random variables

Definition 1.3.8. A discrete random variable is a random variable that only

takes values in some countable subset x1, x2, . . . of R.

Definition 1.3.9. The (probability) mass function of a discrete random vari-

able f : R→ [0, 1] is defined by f(x) = P(X = x).

Definition 1.3.10. The expected value of a discrete random variable X with

mass function f is defined as:

E(X) =∑

x:f(x)>0

xf(x),

as long as this sum converges absolutely.

We now introduce indicator functions, a specific type of Bernoulli random

variable. Indicator functions will be useful in many ways, including allowing us to

understand properties of the expectation operator that will make it easier to work

with.

11

1 BACKGROUND

Definition 1.3.11. Let A be an event. Let the indicator function of A, IA :

Ω→ R be defined as follows:

IA(ω) =

1 ω ∈ A0 ω ∈ Ac

.

Then IA is a Bernoulli random variable with probability mass function:

f(x) =

P(A) x = 1

1− P(A) x = 0.


on (Ω,F ,P). X is jointly discrete if it only takes values in some countable subset

of Rn. X has the joint mass function f : Rn → [0, 1] defined by:

f(x1, x2, . . . , xn) = P(X1 = x1, X2 = x2, . . . , Xn = xn).

1.3.2 Continuous random variables

Definition 1.3.13. A random variable X is a continuous random variable if

its distribution function F (x) can be expressed as:

F (x) =

∫ x

−∞f(u)du,

for some integrable f : R → [0,∞). The function f is called the (probability)

density function of X.

Definition 1.3.14. The expected value of a continuous random variable X with

probability density function f is:

E(X) =

∫ ∞−∞

xf(x)dx,

if this integral exists.


on (Ω,F ,P). X is jointly continuous if its joint distribution function can be

12

1 BACKGROUND

expressed as

FX(x) =

∫ xn

un=−∞· · ·∫ x2

u2=−∞

∫ x1

u1=−∞f(u1, u2, . . . , un)du1du2 . . . dun,

for some integrable function f : Rn → [0,∞). This function f is called the joint

density function of X.

1.3.3 Properties of expected value of random variables

In this section, we state without proof some characteristics of the expectation

operator E that make it easier to work with.

Theorem 1.3.16. For random variables X and Y , if a, b ∈ R, then:

E(aX + bY ) = aE(X) + bE(Y ).

Theorem 1.3.17. If random variables X and Y are independent, then

E(XY ) = E(X)E(Y ).

When we need to calculate the expectation of a function of a random variable

g(X), rather than attempting to find the probability mass or probability density

function fg(X) of g(X), if we know the probability mass or probability density

function fX of X, we can calculate the expectation of g(X) using the following

lemma.

Lemma 1.3.18. If a discrete random variable X has probability mass function f

and we have a B-measurable function g : R→ R, then

E(g(X)) =∑x

g(x)f(x),

whenever this sum absolutely converges.

If a continuous random variable X has probability density function f and we have

a B-measurable function g : R→ R, then

E(g(X)) =

∫ ∞−∞

g(x)f(x),

13

1 BACKGROUND

whenever this integral exists.

We often describe characteristics of probability distributions in terms of the

moments and central moments of the distribution.

Definition 1.3.19. For k ∈ N, the kth momentmk ofX is defined bymk = E(Xk)

and the kth central moment σk is defined by σk = E((X −m1)k).

The most commonly referenced moments are the mean m1 and the variance

σ2 of a random variable.

14

1B

AC

KG

RO

UN

D

Table 1: Important discrete random variablesName Parameter(s) Distribution function Mass function m1 σ2

Bernoulli 0 ≤ p ≤ 1 F (x) =

0 x < 01− p 0 ≤ x < 11 x ≥ 1

f(x) =

1− p x = 0p x = 10 x /∈ 0, 1

p p(1− p)

Binomialn ∈ N

0 ≤ p ≤ 1F (x) =

∑k∈0,1,...,n,k≤x

f(k) f(x) =

(n

x

)px(1− p)n−x x ∈ 0, 1, . . . , n

0 x /∈ 0, 1, . . . , nnp np(1− p)

Poisson λ > 0 F (x) =∑

k∈0,1,2,...,k≤x

f(k) f(x) =

λx

x!e−λ x ∈ 0, 1, 2, . . .

0 x /∈ 0, 1, 2, . . .λ λ

Geometric 0 < p < 1 F (x) = 1− (1− p)bmaxx,0c f(x) =

p(1− p)x−1 x ∈ N0 x /∈ N

1

p

1− pp2

Table 2: Important continuous random variablesName Parameter(s) Distribution function Density function m1 σ2

Uniform [a, b] ∈ R F (x) =

0 x ≤ ax− ab− a

a < x ≤ b

1 x > b

f(x) =

1

b− ax ∈ [a, b]

0 x /∈ [a, b]

b+ a

2

(b− a)2

12

Exponential λ > 0 F (x) =

0 x < 01− e−λx x ≥ 0

f(x) =

0 x < 0λe−λx x ≥ 0

1

λ

1

λ2

Normalµ ∈ Rσ2 > 0

F (x) = 12

(1 + erf

(x−µ√

2σ2

))f(x) =

1√2πσ2

e−(x−µ)2

2σ2 µ σ2

Gamma λ, t > 0 F (x) =γ(t, λx)

Γ(t)f(x) =

λt

Γ(t)xt−1e−λx x ≥ 0

t

λ

t

λ2

15

2 CONVERGENCE OF RANDOM VARIABLES

2 Convergence of Random Variables

In order to understand stochastic calculus and its applications, we will need to

be able to make statements regarding the convergence of a sequence of random

variables to a limit random variable.

2.1 Definitions of modes of convergence

There are multiple types of convergence a sequence of random variables can exhibit,

and we define them here.

Definition 2.1.1. Let X,X1, X2, . . . be random variables on (Ω,F ,P). Then:

1. Xn → X almost surely, denoted Xna.s.−−→ X, if:

A = ω ∈ Ω : Xn(ω)→ X(ω) as n→∞ ∈ F with P(A) = 1.

2. Xn → X in rth mean, denoted Xnr−→ X, if:

E (|Xrn|) <∞ ∀ n and E (|Xn −X|r)→ 0 as n→∞.

For r = 2, we say that Xn → X in mean square, denoted Xnm.s.−−→ X.

3. Xn → X in probability, denoted XnP−→ X, if:

P (|Xn −X| > ε)→ 0 as n→∞ for all ε > 0.

4. Xn → X in distribution, denoted XnD−→ X, if:

P(Xn ≤ x)→ P(X ≤ x) as n→∞

for all x at which FX(x) = P(X ≤ x) is continuous.

16

3 CHARACTERISTIC FUNCTIONS AND APPLICATIONS

3 Characteristic Functions and Applications

3.1 Characteristic functions

Characteristic functions provide a method of studying functions of random variables

and moments of random variables, and will also provide a method of proving the

law of large numbers and the central limit theorem.

Definition 3.1.1. The characteristic function φX of a random variable X is the

function φX : R→ C defined by:

φX(t) = E(eitX).

Characteristic functions allow independent random variables and linear trans-

formations of random variables to be handled easily, as the next two theorems

demonstrate.

Theorem 3.1.2. If X and Y are independent random variables, then

φX+Y (t) = φX(t)φY (t).

Proof. We have:

φX+Y (t) = E(eit(X+Y ))

= E(eitXeitY )

= E (cos(tX) cos(tY )− sin(tX) sin(tY ) + [cos(tX) sin(tY ) + sin(tX) cos(tY )]i)

= E(cos(tX))E(cos(tY ))− E(sin(tX))E(sin(tY )) +

iE(cos(tX))E(sin(tY )) + iE(sin(tX))E(cos(tY ))

= [E(cos(tX)) + iE(sin(tX))][E(cos(tY )) + iE(sin(tY ))]

= E(cos(tX) + i sin(tX))E(cos(tY ) + i sin(tY ))

= φX(t)φY (t).

Theorem 3.1.3. If a, b ∈ R, and Y = aX + b is a random variable, then

φY (t) = eitbφX(at).

17


Proof. We have:

φY (t) = E(eit(aX+b))

= E(eitbei(at)X)

= eitbφX(at).

The characteristic function of a random variable reveals a great deal of

information about the moments of the random variable. The following theorem is

quite similar to Taylor’s theorem for a function of a complex variable, and the proof

is omitted here.

Theorem 3.1.4. 1. If φ(k)(0) exists, then

E|Xk| <∞ if k is even,

E|Xk−1| <∞ if k is odd.

2. If E|Xk| <∞, then

φ(t) =k∑j=0

E(Xj)

j!(it)j + o(tk).

We will also state a theorem relating the convergence of distribution func-

tions and characteristic functions of random variables without proof here.

Theorem 3.1.5. Suppose that F1, F2, . . . is a sequence of distribution functions

with corresponding characteristic functions φ1, φ2, . . .. Then:

1. If Fn → F for some distribution function F with characteristic function φ,

then φn(t)→ φ(t) ∀ t.

2. If φ(t) = limn→∞

φn(t) exists and is continuous at t = 0, then φ is the charac-

teristic function of some distribution function F , and Fn → F .

3.2 Limit theorems

With the theory of characteristic functions and convergence of random variables,

we can state and prove two limit theorems regarding sequences of random variables,

the law of large numbers, and the central limit theorem.

18


Theorem 3.2.1. Let X1, X2, . . . be a sequence of independent, identically dis-

tributed random variables with finite means µ. The partial sums Sn =∑n

k=1Xk

satisfy:Snn

D−→ µ as n→∞.

Proof. Let φ be the characteristic function of each of the Xk, and let φn be the

characteristic function of Snn

.

By Theorem 3.1.2, φ∑Xk = φ(t)n. Then by Theorem 3.1.3, φn =φ( t

n)n

.

By Theorem 3.1.4, φ(t) = 1 + iµt+ o(t). Therefore,

φn(t) =

1 +

itµ

n+ o

(t

n

)n→ eitµ as n→∞.

Note that eitµ is the characteristic function of the constant µ, so by Theorem 3.1.5,Snn

D−→ µ as n→∞.

The law of large numbers asserts that Sn is approximately equal to nµ

for large n. The central limit theorem provides information about the difference

between Sn and nµ.

Theorem 3.2.2. Let X1, X2, . . . be a sequence of independent identically distributed

random variables with finite mean µ and finite non-zero variance σ2, and let Sn =∑nk=1. Then

Sn − nµ√nσ2

D−→ N(0, 1) as n→∞.

Proof. Let Yk = Xk−µσ

, and let φY be the characteristic function of the Yk.

By Theorem 3.1.4, we have:

φY (t) =2∑j=0

E(Y j)

j!(it)j + o(t2)

= 1 + itE(Y )− t2E(Y 2)

2+ o(t2)

= 1 + it(µσ− µ

σ

)− t2

2E(X2 − 2µX + µ2

σ2

)+ o(t2)

= 1− t2

2σ2

(E(X2)− E(X)2

)+ o(t2)

= 1− t2

2+ o(t2).

19

4 THE MULTIVARIATE NORMAL DISTRIBUTION

By Theorem 3.1.2 and Theorem 3.1.3, the characteristic function ψn(t) of

Sn − nµ√nσ2

=1√n

n∑k=1

Yk

satisfies:

ψn(t) =

φY

(t√n

)n=

1− t2

2n+ o

(t2

n

)n→ e−

t2

2 as n→∞.

Because e−t2

2 is the characteristic function of the N(0, 1) distribution, by Theo-

rem 3.1.5, we have that

Sn − nµ√nσ2

D−→ N(0, 1) as n→∞.

4 The Multivariate Normal Distribution

A multivariate distribution that will be of particular interest for the remainder of

this paper is the multivariate normal distribution. This section includes multiple

definitions of the multivariate normal distribution, and some results about the

distribution that make it easier to calculate associated probabilities. Much of this

section is based on [5].

4.1 Definitions

There are multiple definitions of the multivariate normal distribution that will

be useful to us. Before we can define the distribution, however, we must outline

notational preliminaries. We begin by defining covariance and positive definite

symmetric matrices.

Definition 4.1.1. If X and Y are random variables, the covariance of X and Y

20


is

cov (X, Y ) = E [(X − E(X)) (Y − E(Y ))] .

Definition 4.1.2. Let A be a symmetric real nonsingular matrix. There exists an

orthogonal matrix B such that A = BΛB′, where Λ is the diagonal matrix with

the eigenvalues λ1, λ2, . . . , λn of A on the diagonal. Let

Q(x) = yΛy′ =∑i

λiy2i , where y = xB.

A is a positive definite matrix if Q(x) > 0 for all x having some non-zero

coordinate. We denote this A > 0.

Let

X =

X1

X2

...

Xn

with n ≥ 2

be an n-dimensional random variable. Let µi and σ2i denote the expected value and

variance, respectively, of Xi, and let σij = cov (Xi, Xj). The mean vector µ and

covariance matrix Σ of X are then:

µ =

µ1

µ2

...

µn

,Σ =

σ2

1 σ12 · · · σ1n

σ12 σ22 · · · σ2n

......

. . ....

σ1n σ2n · · · σ2n

.

The first definition of the multivariate normal distribution is based on the joint

probability density function of X.

Definition 4.1.3. An n-dimensional random variable X with mean vector µ and

covariance matrix Σ has the multivariate normal distribution, denotedNn (µ,Σ),

if Σ is a positive definite matrix and the density function of X is given by:

f(x) =1√

(2π)n|Σ|e−

(x−µ)′Σ−1(x−µ)2 , where x ∈ Rn.

It will also be helpful to define the multivariate normal distribution in terms

of a transformation from the standard multivariate normal distribution Nr (0, Ir).

21


Definition 4.1.4. An n-dimensional random variable X with mean vector µ and

covariance matrix Σ has the multivariate normal distribution, denotedNn (µ,Σ),

if there exists an n× r matrix C with rank r ≤ n such that:

1. CC’ = Σ, and

2. X and CZr + µ, where Zr ∼ Nr (0, Ir), are identically distributed.

4.2 Change of variables

Because it is often easier to calculate probabilities from multivariate normal distri-

butions when the variables are independent, we would like to be able to apply

transformations to vectors of random variables having the multivariate normal

distribution and to understand the resulting distribution. The theory behind these

transformations actually provides a more general framework than simply applying

a transformation that yields independent normally distributed variables, which we

now describe.

We begin by considering the characteristic functions of n-dimensional ran-

dom variables.

Theorem 4.2.1. Let X be an n-dimensional random variable with characteristic

function ψX(t). Let C be any given m × n real matrix, and let b be any m× 1 real

vector. Then the characteristic function of Y = CX + b is given by:

ψY(t) = eit′bψX(C′t).

Proof.

ψY(t) = E(eit′Y)

= E(eit′(CX+b)

)= eit

′bE(ei(C

′t)X).

We would like to find define the characteristic function of the multivariate

normal variable. We know that the characteristic function of the univariate normal

22


variable is e−t2

2 . Then the characteristic function of Zr is given by:

ψZr (t) = e−t′t2 for t ∈ Rr.

Then by Theorem 4.2.1, Definition 4.1.4, and the uniqueness theorem for

characteristic functions, we have the characteristic function of the multivariate

normal variable.

Theorem 4.2.2. X ∼ Nn (µ,Σ) if and only if its characteristic function is given

by:

ψX (t) = eit′µ−t′Σt

2 for t ∈ Rn.

Theorem 4.2.3. Let C be any given m × n real matrix, and let b be any m × 1

real vector. If X ∼ Nn (µ,Σ) and Y = CX + b, then Y ∼ Nm(Cµ+ b,CΣC′).

Proof. 1. Case 1: m = n.

Follows from Theorem 4.2.2 and Theorem 4.2.1.

2. Case 2: m < n.

Consider the following transformation:

Y∗ =

(Y1

Y2

)=

(C

B

)X +

(b

0n−m

),

where B is any (n−m)× n matrix. Then:(Y1

Y2

)∼ Nn

((Cµ+ b

Bµ

),

(CΣC′ CΣB′

BΣC′ BΣB′

)).

Therefore Y = Y1 = CX+b ∼ Nm (Cµ,CΣC′) .

3. Case 3: m > n.

If r ≤ n is the rank of Σ, by Definition 4.1.4, we know there exists an n× rmatrix D such that X and DZr + µ are identically distributed. Therefore

by Definition 4.1.4 CX + b and CDZr + (Cµ+ b) both have the singular

N (Cµ+ b,CΣC′) distribution.

23

5 STOCHASTIC PROCESSES

5 Stochastic Processes

5.1 Introduction

In order to study stochastic calculus, we must first understand stochastic processes.

In this section, we define stochastic processes in general and give definitions of some

of the most important processes.

Definition 5.1.1. A stochastic process X is a family Xt : t ∈ T of random

variables which map the sample space Ω into the state space S ⊆ R.

We can observe stochastic processes in two manners: by studying fixed

realizations of the process, or by studying the distributional properties of the

process.

Definition 5.1.2. The realization (or sample path) of X at ω for a fixed

ω ∈ Ω is the collection Xt(ω) : t ∈ T of members of S.

Definition 5.1.3. Let t = (t1, t2, . . . , tn) be a vector with each ti ∈ T . Then

the vector (Xt1 , Xt2 , . . . , Xtn) has the joint distribution function Ft : Rn → [0, 1]

defined by Ft(x) = P(Xt1 ≤ x1, Xt2 ≤ x2, . . . , Xtn ≤ xn). Letting t range over all

finite-length vectors of members of T , the collection Ft is called the collection of

finite-dimensional distributions (fdds) of X.

A specific class of stochastic processes, Gaussian processes, will be of par-

ticular interest to us.

Definition 5.1.4. A real-valued, meaning that S = R, continuous-time, meaning

that T = [0,∞), stochastic process X is called a Gaussian process if each

finite-dimensional vector (Xt1 , Xt2 , . . . , Xtn) has the multivariate normal distribu-

tion N (µ(t),Σ(t)) for mean vector µ(t) and covariance matrix Σ(t) which may

depend on t.

5.2 Wiener process

The Wiener process is a stochastic process that will be important for our study of

stochastic calculus and the applications of stochastic calculus.

Definition 5.2.1. A Wiener process W = Wt : t ≥ 0 is a real-valued Gaussian

process such that:

24

6 ITO CALCULUS

1. For any n, Xj = Wtj − Wsj where 1 ≤ j ≤ n are independent variables

whenever the intervals (sj, tj] are disjoint (W has independent increments).

2. Ws+t −Ws ∼ N (0, σ2t) ∀ s, t ≥ 0, where σ2 is a positive constant.

3. The sample paths of W are continuous.

A Wiener process W is called standard if W0 = 0 and σ2 = 1.

6 Ito Calculus

6.1 Introduction

In this section, we will define the integral∫∞

0ψsdWs, where ψs is a stochastic

process. We also explore one of the fundamental ideas of Ito calculus: the Ito

Lemma. We conclude the section with a brief look at solutions to stochastic

differential equations.

6.2 Ito integral

This approach closely follows Section 13.8 of [2] and Chapter 2 of [4]. As we

explore in the next section, the Wiener process cannot be integrated path-wise

with respect to itself, so we define the stochastic integral∫∞

0ψsdWs as the mean

square limit of Riemann-Stieltjes sums. We first define the integral for a simpler

set of stochastic processes, random step functions, as a Riemann-Stieltjes sum. We

then show that, for stochastic processes ψs satisfying certain conditions, the process

can be expressed as the limit of a sequence of random step functions. We define the

Ito integral to be the mean square limit of the sequence of integrals of the random

step functions.

6.2.1 Motivation for Ito integral

The standard tool in mathematics for integrating one function with respect to

another function is the Riemann-Stieltjes integral.

Definition 6.2.1. Let f, g : [0, 1]→ R. Consider a partition τn of [0, 1]:

τn : 0 = t0 < t1 < . . . < tn−1 < tn = 1,

25

6 ITO CALCULUS

and an intermediate partition σn of τn:

σn : ti−1 ≤ yi ≤ ti for i = 1, . . . , n.

The Riemann-Stieltjes sum corresponding to τn and σn is defined by:

Sn = Sn(τn, σn) =n∑i=1

f (yi) [g(ti)− g(ti−1)] .

The Riemann-Stieltjes integral of f with respect to g is defined by the limit

S = limn→∞

Sn,

if this limit exists as mesh(τn) → 0 and if the limit is independent of τn and σn.

The Riemann-Stieltjes integral is denoted:∫ 1

0

f(t)dg(t).

Because the Wiener process Ws is a function of s ∈ [0,∞) and ω ∈ Ω, we

could use the Riemann-Stieltjes integral to integrate sample paths of stochastic

processes ψs with respect to Ws so long as the sample paths of ψs satisfy certain

criteria. However, the set of processes ψs for which the Riemann-Stieltjes integral

exists is small enough (and does not include Ws) that we will need to define the

desired integral as the mean square limit of Riemann-Stieltjes sums.

6.2.2 Characteristics of integrator Ws

Let Ws be a standard Wiener process on the complete probability space (Ω,F ,P).

Note that Ws has the following characteristics:

1. E (|Ws|2) = s <∞ for all s.

2. E (|Ws+h −Ws|2) = h→ 0 as h ↓ 0 for all s.

3. Whenever u ≤ v ≤ s ≤ t,

E ([Wv −Wu][Wt −Ws]) = E (Wv −Wu)E (Wt −Ws) = 0,

26

6 ITO CALCULUS

by Theorem 1.3.17 because W has independent increments.

The first characteristic is important because we will be working with mean-square

convergence. The second is helpful for technical reasons. The third is helpful when

establishing the existence of the limits necessary to define the integral.

6.2.3 Conditions for integrand ψs

We now outline the conditions that must be satisfied by the integrand in order to

define the Ito integral. We begin by stating the definition of a measurable function.

Definition 6.2.2. Let (Ω,F ,P) be a probability space. A function X : Ω → R is

F-measurable if ω ∈ Ω : X(ω) ≤ x ∈ F for all x ∈ R.

Note that random variables are F -measurable functions mapping Ω to R.

Define Ft to be the smallest sub-σ-field of F which contains the null events and with

respect to which Ws, for all 0 ≤ s ≤ t, are measurable. The collection Ft : t ≥ 0is called the filtration, and denoted F . Let B denote the Borel σ-field of subsets of

[0,∞).

Definition 6.2.3. A stochastic process ψ is measurable if the function ψt(ω) of

t and ω ∈ Ω is measurable with respect to the product σ-field B ⊗ F .

Definition 6.2.4. A measurable stochastic process ψ is adapted to the the

filtration F if ψt is Ft-measurable for all t.

Before we can continue, we need to be able to understand the integral∫ baµtdt,

where µt is a real-valued stochastic process. µt is a function of t ∈ [0,∞) and ω ∈ Ω.

The integral∫ baµtdt is a function from Ω → R, and we can interpret it as follows.

For any fixed ω ∈ Ω, µt becomes a deterministic function mapping [0,∞) to R.

The integral∫ baµtdt evaluated at ω, then, is the value of the integral of the sample

path at ω. This integral∫ baµtdt is itself a random variable.

Let

A =

ψt : ψ is adapted to the filtration F and E

(∫ ∞0

|ψt|2dt)<∞

.

We are able to define the Ito integral for stochastic processes ψ ∈ A. For

ψ ∈ A, define

||ψ|| =

√E(∫ ∞

0

|ψt|2dt).

27

6 ITO CALCULUS

The reason we need to impose these conditions on the integrand ψ is that we need

the integrand and the integrator Ws to match in terms of measurability.

6.2.4 Defining

∫ ∞0

ψsdWs for random step functions ψs

Having established the conditions on the process ψs, we can proceed to define the

Ito integral for a predictable step function.

Definition 6.2.5. Let 0 = a0 < a1 < . . . < an = t and let C0, C1, . . . , Cn−1 be

random variables such that:

1. For each i, E (C2i ) <∞.

2. Each Ci is Faj -measurable.

Then we define a predictable step function φ by setting:

φt =n−1∑i=0

CiI(ai,ai+1](t) =

0 if t ≤ 0 or t > an

Ci if ai < t ≤ ai+1

,

where

I(ai,ai+1](t) =

0 if t ≤ ai or t > ai+1

1 if ai < t ≤ ai+1

.

We can now define the stochastic integral of a predictable step function φ

with respect to W .

Definition 6.2.6. Let φ be a predictable step function. Then the stochastic

integral I(φ) of φ with respect to W is defined by:

I(φ) =n−1∑i=0

Ci(Wai+1−Wai).

Before considering sequences of predictable step functions, we will outline

properties of the stochastic integral of a predictable step function.

Theorem 6.2.7. Let α, β ∈ R and let φ1, φ2 be predictable step functions. Then

I(αφ1 + βφ2) = αI(φ1) + βI(φ2).

28

6 ITO CALCULUS

Proof. We have that

(αφ1 + βφ2)t =

0 if t ≤ 0 or t > an

αC1i + βC2

i if ai < t ≤ ai+1

.

Then

I(αφ1 + βφ2) =n−1∑i=0

(αC1i + βC2

i )(Wai+1−Wai)

= αn−1∑i=0

C1i (Wai+1

−Wai) + βn−1∑i=0

C2i (Wai+1

−Wai)

= αI(φ1) + βI(φ2).

To understand another important property of the stochastic integral of a

predictable step function, we must first define the L2 norm.

Definition 6.2.8. Let U be a random variable. Then the L2 norm of U , denoted

||U ||2 is defined by:

||U ||2 =√

E (U2).

We state without proof here that the L2 norm satisfies the triangle inequality,

that is, for random variables U and V :

||U + V ||2 ≤ ||U ||2 + ||V ||2.

We must also understand a more general theory of conditional expectation of

random variables. Rather than simply conditioning random variables on families

of random variables, we can condition random variables on sub-σ-fields of F .

Definition 6.2.9. Let (Ω,F ,P) be a probability space, let Y be a random vari-

able with finite second moment, and let G be a sub-σ-field of F . Let H be the

collection of G-measurable random variables with finite second moment. Then the

conditional expectation E (Y |G) is itself a G-measurable random variable such

that

E ([Y − E(Y |G)]Z) = 0 for all Z ∈ H.

29

6 ITO CALCULUS

Now, ∅,Ω ∈ G, so IΩ ∈ H, yielding:

E([Y − E(Y |G)]IΩ) = 0

E(Y )− E(E(Y |G)) = 0 because IΩ = 1 ∀ ω ∈ Ω

E(Y ) = E(E(Y |G)).

Theorem 6.2.10. If φ is a predictable step function, ||I(φ)||2 = ||φ||.

Proof. We have:

E(|I(φ)|2

)= E

(n−1∑j=0

Cj(Waj+1−Waj)

n−1∑k=0

Ck(Wak+1−Wak)

)

= E

(n−1∑j=0

C2j (Waj+1

−Waj)2 + 2

∑0≤j<k≤n−1

CkCj(Wak+1−Wak)(Waj+1

−Waj)

).

To calculate E(C2j (Waj+1

−Waj)2), we will first show that, when considering only

events in Faj , Cj and Waj+1−Waj are independent.

It suffices to show that, for any I, J ⊂ R,

P((Waj+1−Waj) ∈ I and Cj ∈ J) = P((Waj+1

−Waj) ∈ I)P(Cj ∈ J).

Denote

(Waj+1−Waj)

−1(I) = ω ∈ Ω : (Waj+1−Waj)(ω) ∈ I and C−1

j (J) = ω ∈ Ω : Cj(ω) ∈ J.

Then we need to show that

P[(Waj+1−Waj)

−1(I)∩C−1j (J)] = P[(Waj+1

−Waj)−1(I)]P[C−1

j (J)]. We know that

Waj+1−Waj and Ws are independent for any s ≤ aj. Then we have that:

P[(Waj+1−Waj)

−1(I) ∩W−1s (J)] = P[(Waj+1

−Waj)−1(I)]P[W−1

s (J)]

P[(Waj+1−Waj)

−1(I) ∩K] = P[(Waj+1−Waj)

−1(I)]P[K] for any K = W−1s (J).

Because Faj is defined to be the smallest sub-σ-field of F with respect to which Ws

is measurable for all s ≤ aj, K and (Waj+1−Waj)

−1(I) are independent events for

any K ∈ Faj . We also know that Cj is Faj -measurable, so C−1j (J) ∈ Faj , which

gives us the desired independence. We condition on Faj because, if we allow for

30

6 ITO CALCULUS

all of the events in F , we would not be guaranteed the independence of Cj and

Waj+1−Waj .

Then, because Waj+1−Waj ∼ N (0, aj+1 − aj),

E(C2j (Waj+1

−Waj)2) = E

(E(C2

j (Waj+1−Waj)

2|Faj))

= E(C2j )(aj+1 − aj).

Now, to calculate E[CjCk(Waj+1−Waj)(Wak+1

−Wak)] when j < k, we first

show that, when considering only events in Fak , CjCk(Waj+1−Waj) and Wak+1

−Wak

are independent.

Again, it suffices to show that, for any I, J ⊂ R,

P([CjCk(Waj+1

−Waj)] ∈ I and (Wak+1−Wak) ∈ J

)= P(CjCk(Waj+1

−Waj) ∈ I)P((Wak+1−Wak) ∈ J).

Denote (CjCk(Waj+1−Waj))

−1(I) = ω ∈ Ω : (CjCk(Waj+1−Waj))(ω) ∈ I and

(Wak+1−Wak)

−1(J) = ω ∈ Ω : (Wak+1−Wak)(ω) ∈ J. Then we need to show that

P[(CjCk(Waj+1−Waj))

−1(I)∩(Wak+1−Wak)

−1(J)] = P[(CjCk(Waj+1−Waj))

−1(I)]P[(Wak+1−

Wak)−1(J)]. We know that Wak+1

−Wak and Ws are independent for any s ≤ ak.

Then we have that:

P[(Wak+1−Wak)

−1(J) ∩W−1s (I)] = P[(Wak+1

−Wak)−1(J)]P[W−1

s (I)]

P[(Wak+1−Wak)

−1(J) ∩K] = P[(Wak+1−Wak)

−1(J)]P[K] for any K = W−1s (I)

Because Fak is defined to be the smallest sub-σ-field of F with respect to which Ws

is measurable for all s ≤ ak, K and (Wak+1−Wak)

−1(J) are independent events for

any K ∈ Fak . We also know that Cj, Ck, and (Waj+1−Waj) are all Fak-measurable,

so [CjCk(Waj+1−Waj)]

−1(I) ∈ Fak , which gives us the desired independence.

Then, because E(Wak+1−Wak) = 0, we have:

E(|I(φ)|2) =∑j

E(C2j )(aj+1 − aj) = E

(∫ ∞0

|φ(t)|2dt)

= ||φ||2.

31

6 ITO CALCULUS

6.2.5 Sequences of predictable step functions approaching ψ

Having defined the stochastic integral for a predictable step function φ, we now

consider the limit of a sequence of predictable step functions. Specifically, we will

show that we can find a sequence of predictable step functions converging to any

stochastic process ψ satisfying the requirements outlined above.

Theorem 6.2.11. Let ψ ∈ A have continuous sample paths. Then there exists

a sequence φ = φ(n) of predictable step functions such that ||φ(n) − ψ|| → 0 as

n→ 0.

Proof. Define the predictable step function φ(n) by:

φ(n)(t) =

n

∫ jn

j−1n

ψsds for jn< t ≤ j+1

n, 1 ≤ j < n2

0 otherwise

.

We have, for j ≥ 1:

∫ j+1n

jn

|φ(n)t |2dt =

1

n|φ(n)t |2

= n

∣∣∣∣∣∫ j

n

j−1n

ψsds

∣∣∣∣∣2

.

We state without proof here one of the variations of the Cauchy-Schwarz inequality:∣∣∣∣∫ f(x)g(x)dx

∣∣∣∣2 ≤ ∫ |f(x)|2dx∫|g(x)|2dx.

Then

∫ j+1n

jn

|φ(n)t |2dt = n

∣∣∣∣∣∫ j

n

j−1n

1ψsds

∣∣∣∣∣2

≤ n

∫ jn

j−1n

ds

∫ jn

j−1n

|ψs|2ds by Cauchy-Schwarz inequality

=

∫ jn

j−1n

|ψs|2 ds.

32

6 ITO CALCULUS

Therefore, we have, for T ≥ 0:∫ ∞T

|φ(n)s |2ds ≤

∫ ∞T− 2

n

|ψs|2ds.

We also know that:∫ ∞0

|φ(n)s − ψs|2ds =

∫ T

0

|φ(n)s − ψs|2ds+

∫ ∞T

|φ(n)s − ψs|2ds.

To complete the proof, we will to show that

limn→∞

E(∫ ∞

0

∣∣φ(n)s − ψs

∣∣2 ds) = 0.

We will first show that

∫ ∞0


∣∣2 ds a.s.−−→ 0.We will then show that

∫ ∞0


∣∣2 dsis uniformly dominated by an integrable function on [0,∞), and use an application

of the dominated convergence theorem to prove the desired result.

Note that∫ ∞0


∣∣2 ds =

∫ T

0


∣∣2 ds+

∫ ∞T


∣∣2 ds.On the interval ( j

n, j+1

n], the mean value of ψs is given by

n

∫ j+1n

jn

ψsds = φ(n)t for t ∈

(j + 1

n,j + 2

n

].

By the mean value theorem, φ(n)t = ψr for all t ∈

(jn, j+1

n

]and some r ∈(

jn, j+1

n

]. Since the sample paths of ψ are continuous on the compact interval [0, T ],

we know that the sample paths of ψ are uniformly continuous on [0, T ]. Then for

all ε > 0, there exists δ > 0 such that:

|x− y| < δ ⇒ |ψx − ψy| < ε.

Then given any ε > 0, let δ be such that whenever |x − y| < δ, |ψx − ψy| <√ε

T.

33

6 ITO CALCULUS

Let n0 satisfy n0 >T

δ. Then for any n ≥ n0, on the jth interval,

∫ j+1n

jn


∣∣2 ds < ε

nT.

On [0, T ], there are nT intervals of length 1n, so,

∫ T

0


∣∣2 ds < ε⇒∫ T

0


∣∣2 ds a.s.−−→ 0

. Now, since for x, y ∈ R, |x+ y|2 ≤ 2 (|x|2 + |y|2), we have:∫ ∞T


∣∣2 ds ≤ 2

(∫ ∞T

∣∣φ(n)s

∣∣2 ds+

∫ ∞T

|ψs|2 ds)

≤ 2

(∫ ∞T− 2

n

|ψs|2 ds+

∫ ∞T

|ψs|2 ds

)

= 4

∫ ∞T− 2

n

|ψs|2 ds−∫ T

T− 2n

|ψs|2 ds

≤ 4

∫ ∞T− 2

n

|ψs|2 ds.

Now let n→∞ and then T →∞ in:∫ ∞0


∣∣2 ds =

∫ T

0


∣∣2 ds+

∫ ∞T


∣∣2 ds.Because ψ ∈ A, ∫ ∞

0

|ψs|2 ds <∞ almost surely.

Therefore, ∫ ∞T


∣∣2 ds ≤ 4

∫ ∞T− 2

n

|ψs|2 dsa.s.−−→ 0 as T →∞.

We have shown that

∫ ∞0


∣∣2 ds a.s.−−→ 0, and we must now show that∫ ∞0


∣∣2 ds is dominated on [0,∞).

34

6 ITO CALCULUS

We know that

0 ≤∫ ∞

0


∣∣2 ds ≤ 4

∫ ∞0

|ψs|2 ds.

Then we can apply the dominated convergence theorem as follows:

limn→∞

E(∫ ∞

0


∣∣2 ds) = E[

limn→∞

∫ ∞0


∣∣2 ds] = 0.

Therefore,

||φ(n) − ψ|| → 0 as n→∞.

6.2.6 Existence and uniqueness of limit Ito integral I(ψ)

We are now able to define the Ito integral∫∞

0ψsdWs. We will first show that, if

φ is a sequence of predictable step functions converging in A to ψ, that I(φ(n))

converges in L2 to a limit random variable. We then show that this limit random

variable is almost surely unique. This limit random variable is defined as the Ito

integral of the process ψ.

Theorem 6.2.12. Let ψ ∈ A and φ = φ(n) be a sequence of predictable step

functions converging in A to ψ (in the sense of Theorem 6.2.11). Then the sequence

I(φ(n)) converges mean square to a limit random variable, which we denote I(φ).

Proof. We first show that I(φ(n) is Cauchy in L2. φ(n) − φ(m) is a predictable step

function, so we have the following:

||I(φ(n))− I(φ(m))||2 = ||I(φ(n) − φ(m))||2= ||φ(n) − φ(m)|| by Theorem 6.2.10

≤ ||φ(n) − ψ||+ ||φ(m) − ψ|| by the triangle inequality

→ 0 as m,n→∞.

Because L2 is complete, and the L2 norm corresponds to mean-square convergence,

the sequence I(φ(n)) converges in mean square to a limit random variable.

The following theorem asserts the almost sure uniqueness of the limit random

variable I(φ).

35

6 ITO CALCULUS

Theorem 6.2.13. Let φ and ρ converge in A to ψ. Then P[I(φ) = I(ρ)] = 1.

We denote this almost surely unique random variable I(ψ).

Proof. By the triangle inequality,

||I(φ)− I(ρ)||2 ≤ ||I(φ)− I(φ(n))||2 + ||I(φ(n))− I(ρ(n))||2 + ||I(ρ(n))− (ρ)||2.

As n→∞, ||I(φ)− I(φ(n))||2, ||I(ρ(n))− I(ρ)||2 →∞ by Theorem 6.2.12. We also

have that:

||I(φ(n))− I(ρ(n))||2 = ||I(φ(n) − ρ(n))||2 by Theorem 6.2.7

= ||φ(n) − ρ(n)|| by Theorem 6.2.10

≤ ||φ(n) − ψ||+ ||ρ(n) − ψ|| by the triangle inequality

→ 0 as n→∞.

Then ||I(φ)− I(ρ)||2 = 0, so P(I(φ) = I(ρ)) = 1.

Definition 6.2.14. The Ito integral of the stochastic process ψ is defined to be

I(ψ). We denote the integral

∫ ∞0

ψsdWs.

∫ t

0

ψsdWs is defined by

∫ ∞0

I(0,t](s)ψsdWs,

where

I(0,t](s) =

1 if s ∈ (0, t]

0 otherwise.

6.2.7 Example:

∫ t

0

WsdWs

We wish to find the random variable

∫ t

0

WsdWs according to the definition of the

Ito integral outlined in the previous section. For a given n, let δ = tn. We consider

the sequence of predictable step functions given by:

φ(n)s =

Wjδ for jδ < s ≤ (j + 1)δ, 0 ≤ j ≤ n− 1

0 otherwise.

36

6 ITO CALCULUS

Then I(φ(n)s ) =

n−1∑j=0

Wjδ(W(j+1)δ −Wjδ). We have:

2I(φ(n)s ) = 2

n−1∑j=0

WjδW(j+1)δ −W 2jδ

=n−1∑j=0

W 2(j+1)δ −W 2

jδ −W 2jδ −W 2

(j+1)δ + 2W(j+1)δWjδ

=n−1∑j=0

(W 2

(j+1)δ −W 2jδ

)−

n−1∑j=0

(W(j+1)δ −Wjδ

)2

= W 2t −W 2

0 −n−1∑j=0

(W(j+1)δ −Wjδ

)2.

37

6 ITO CALCULUS

Note thatn−1∑j=0

(W(j+1)δ −Wjδ

)2 m.s.−−→ t as n→∞:

E

[n−1∑j=0

(W(j+1)δ −Wjδ

)2 − t

]2 = E

(n−1∑j=0

[(W(j+1)δ −Wjδ

)2 − t

n

])2

Let Aj =(W(j+1)δ −Wjδ

)2 − t

n= E

((A0 + A1 + . . . An−1)2

)= E

(n−1∑j=0

A2j + 2

∑0≤s<t≤n−1

AmAk

)

= E

(n−1∑j=0

A2j

)+ 2

∑0≤s<t≤n−1

E(AmAk)

= E

(n−1∑j=0

A2j

)+ 2

∑0≤s<t≤n−1

E(Am)E(Ak)

= E

(n−1∑j=0

A2j

)

=n−1∑j=0

E

([(W(j+1)δ −Wjδ

)2 − t

n

]2)

=n−1∑j=0

E((W(j+1)δ −Wjδ

)4)

− 2t

nE((W(j+1)δ −Wjδ

)2)

+t2

n2

= n

(3t2

n2− 2t2

n2+t2

n2

)=

2t2

n→ 0 as n→∞.

Therefore, I(φ(n))→ 12

(W 2t − t) in mean square, yielding:∫ t

0

WsdWs =1

2

(W 2t − t

).

38

6 ITO CALCULUS

6.3 The Ito Lemma

Now that we have defined the Ito integral, we can develop a tool that will allow us

to relatively easily define Ito integrals for a set of stochastic processes ψs. This tool

is the Ito Lemma, and we will use a similar argument in its construction to that

used to justify the classical chain rule of differentiation. The argument makes use of

Taylor expansions, and for this reason, the Ito Lemma is considered the stochastic

analog of the deterministic chain rule.

6.3.1 Key idea of the Ito Lemma: (dWt)2 m.s.−−→ dt as dt→ 0

Before using Taylor expansions to construct the Ito Lemma, it is important to

understand the most important concept of the Ito Lemma: that the increment of

the Wiener process squared, (dWt)2, over dt converges to dt in mean square as dt

approaches 0.

Theorem 6.3.1. Let dWt = Wt+dt −Wt. Then

(dWt)2 m.s.−−→ dt as dt→ 0.

Proof. We have, for all t ≥ 0:

E ([(dWt)2 − dt]2) = E ([(Wt+dt −Wt)

2 − dt]2)

= E ((Wt+dt −Wt)4 − 2dt(Wt+dt −Wt)

2 + dt2)

= E ((Wt+dt −Wt)4)− 2dtE (Wt+dt −Wt)

2) + dt2

= 3dt2 − 2dt2 + dt2

= 2dt2

→ 0 as dt→ 0.

This key idea will allow us to interpret the Taylor expansions of functions

of Wiener processes by determining which arguments in the expansion will have

non-negligible contributions to the Taylor expansion.

6.3.2 The Ito Lemma

Let f : R→ R be an infinitely differentiable function. If we consider a sample path

of a Wiener process Wt(ω), we have the following Taylor expansion, again writing

39

6 ITO CALCULUS

dWt = Wt+dt −Wt:

f(Wt + dWt)− f(Wt) = f ′(Wt)dWt +1

2f ′′(Wt)(dWt)

2 + . . . .

Then, because (dWt)2 m.s.−−→ dt as dt → 0, third and higher order terms have

negligible contribution to the Taylor expansion, so we have:

f(Wt)− f(Ws) =

∫ t

s

f ′(Wx)dWx +1

2

∫s

tf ′′(Wx)dx,

where∫ tsf ′(Wx)dWx is an Ito integral, and

∫ tsf ′′(Wx)dx is a Riemann integral, and

equality is in the mean square sense.

If we let f : R2 → R have infinite partial derivatives, the Taylor expansion yields:

f(t+ dt,Wt+dt)− f(t,Wt) = f1(t,Wt)dt+ f2(t,Wt)dWt

+12

[f11(t,Wt)(dt)2 + 2f12(t,Wt)dtdWt + f22(t,Wt)(dWt)

2]

. . . .

Because (dWt)2 m.s.−−→ dt as dt→ 0, we have the following:

1. The contribution of third and higher order terms to the Taylor expansion are

negligible.

2. The contribution of the dtdWt term is negligible.

Therefore, we have:

f(t,Wt)− f(s,Ws) =

∫ t

s

[f1(x,Wx) +

1

2f22(x,Wx)

]dx+

∫ t

s

f2(x,Wx)dWx,

where again∫ tsf2(x,Wx)dWx is an Ito integral, and

∫ ts

[f1(x,Wx) + 1

2f22(x,Wx)

]dx

is a Riemann integral, and equality is in the mean square sense.

6.3.3 Ito processes

There are a subset of stochastic processes, called Ito processes, which can be

represented as the solution to a stochastic differential equation (SDE).

40

6 ITO CALCULUS

Definition 6.3.2. A stochastic process Xt is an Ito process if it is a solution to

the stochastic differential equation:

Xt = X0 +

∫ t

0

µ(s,Xs)ds+

∫ t

0

σ(s,Xs)dWs,

where µ and σ are stochastic processes,∫ t

0µ(s,Xs)ds is a Riemann integral, and∫ t

0σ(s,Xs)dWs is an Ito integral. We abbreviate this SDE to:

dX = µ(t,X)dt+ σ(t,X)dW.

We state without proof the following facts:

1. So long as weak conditions on µ,σ, and X0 are satisfied, the SDE dX =

µ(t,X)dt + σ(t,X)dW has a unique solution which has continuous sample

paths.

2. If X is an Ito process, the processes µ and σ are uniquely determined:

µ1(t,X)dt+σ1(t,X)dW = dX = µ2(t,X)dt+σ2(t,X)dW ⇒ µ1 = µ2 and σ1 = σ2.

We will now extend the Ito Lemma to stochastic processes that are functions of Ito

processes. The justification that follows is similar to that in the earlier Ito Lemma

statements: a justification for the formula rather than a rigorous proof. Suppose

Xt is an Ito process with

dXt = µ(t,Xt)dt+ σ(t,Xt)dWt.

Let f : R2 → R have infinite partial derivatives. Then a Taylor expansion for

f(t+ dt,Xt+dt)− f(t,Xt) yields:

f(t+ dt,Xt+dt)− f(t,Xt) = f1(t,Xt)dt+ f2(t,Xt)dXt

+12

[f11(t,Xt)(dt)2 + 2f12(t,Xt)dtdXt + f22(t,Xt)(dXt)

2]

. . .

= f1(t,Xt)dt+ µf2(t,Xt)dt+ σf2(t,Xt)dWt

+12

[σ2f22(t,Xt)(dWt)2]

. . .

41

7 APPLICATION: BLACK-SCHOLES OPTION PRICING MODEL

Then we have the following extension of the Ito Lemma:

f(t,Xt)− f(s,Xs) =

∫ t

s

[f1(y,Xy) + µf2(y,Xy) +

σ2

2f22(y,Xy)

]dy

+

∫ t

s

σf2(y,Xy)dWy.

6.3.4 Geometric Brownian motion

Let µ, σ ∈ R with σ > 0. Consider the SDE:

dXt = Xt(µdt+ σdWt),

where Wt is a standard Wiener process. The unique solution to this SDE is given

by:

Xt = f(t,Wt) = X0e

(µ−σ

2

2

)t+σWt ,

and Xt is called a geometric Brownian motion (note that what we have called

a Wiener process is also sometimes called a Brownian motion). The geometric

Brownian motion of this form is clearly an Ito process with

µ(s,Xs) = µXs and σ(s,Xs) = σXs,

where µ(s,Xs) and σ(s,Xs) are stochastic processes determining the Ito process,

and µ and σ are constant parameters of the geometric Brownian motion.

7 Application: Black-Scholes Option Pricing Model

To this point, we have covered the mathematical background necessary to under-

stand an application of stochastic calculus in finance: the Black-Scholes option

pricing model. In this section, we explain and explore the model.

7.1 Black-Scholes option pricing model

The derivation of the Black-Scholes option pricing model and the solution for the

European call option closely follow the explanations given in Chapter 4 of [4] and

Section 13.10 of [2].

42


7.1.1 Introduction

Subject to a nontrivial set of assumptions, the Black-Scholes option pricing model

can be used to find the no-arbitrage rational price for an option, a financial instru-

ment that derives its value from the value of an underlying asset and which we will

explain in detail later. There are a couple of key motivating ideas that are crucial

to the model:

1. The price of the asset underlying the option (henceforth stock) can be modeled

using a geometric Brownian motion stochastic process.

2. There do not exist arbitrage opportunities, or opportunities to profit without

risk.

3. There exists a unique self-financing portfolio of the stock and a risk-free

bond that replicates the payout of the option. This portfolio is continuously

rebalanced between the stock and the bond in such a manner that changes to

the value of the portfolio are caused entirely by changes in the prices of the

stock and bond rather than a cash flow into or out of the portfolio. Because

it replicates the payout of the option and because of the assumption that

arbitrage opportunities do not exist, the value of the portfolio at time t = 0

must match the no-arbitrage rational price of the option at time t = 0.

The first two of these ideas are important assumptions of the Black-Scholes model,

while the third depends upon the type of option, and is the most significant step

in the derivation of the model.

7.1.2 Set-up of the model

Suppose the price per unit of the stock that underlies the option is given by a

stochastic process S = St : t ≥ 0. Further, assume that S satisfies the geometric

Brownian motion SDE, dSt = St(µdt + σdWt). This is a vital assumption for the

model, and one that we will explore in more detail in the next subsection. We know

that the unique solution to the geometric Brownian motion SDE is

St = f(t,Wt) = e

(µ−σ

2

2

)t+σWt ,

43


where we have normalized so that S0 = 1.

It is assumed that financial market participants can buy any quantity, including

a negative quantity, of the stock without transaction costs. Buying a negative

quantity of the stock is equivalent to selling the stock short: borrowing the stock

and selling it.

In addition to the stock, financial market participants can buy any quantity of

a non-risky asset, or bond, which has a constant continuously compounded interest

rate of r. If we denote the price of one unit of the bond at time t by βt, β satisfies

the integral equation

dβt = rβtdt, which yields βt = ert after normalizing so that β0 = 1.

Buying a negative quantity of a bond is equivalent to borrowing money at an

interest rate of r.

We call a pair (at, bt) of stochastic processes which are both adapted to the filtration

of σ-fields generated by Wt a portfolio, where at and bt represent the number of

units of stock and bond in the portfolio, respectively, at time t. The value of the

portfolio is given by:

Vt(a, b) = atSt + btβt.

The portfolio is called self-financing if changes in the value of the portfolio arise

only from changes in the price of the stock and the bond, and not from the input of

money from external sources into the portfolio or taking money out of the portfolio

for external use. The self-financing condition is:

dVt(a, b) = atdSt + btdβt.

In order to price an option, we make use of the no arbitrage assumption: that there

does not exist an opportunity for a risk-free profit. We need to find a self-financing

portfolio that replicates the payout of the option in question, and the portfolio will

be specific to the type of option. Then the value of the option at time t is equal to

the value, Vt(a, b) of the self-financing portfolio (a, b) that replicates the option.

44


7.1.3 European call option

In this section, we find the no-arbitrage rational price for a European call option

using the Black-Scholes model. The European call option is a contract that enables

the holder to purchase one unit of stock at a fixed strike price, K, at a given maturity

time, T . The holder of the European call option has the ability to purchase the

share only at time T , and can decide not to purchase the share. To find the value

of the option at time T , consider the following:

1. If ST > K, the holder of the option can purchase the share for K and sell it

immediately for ST , yielding a profit of ST −K.

2. If ST < K, the holder of the option will not purchase the share for K, because

he/she has can purchase the same share for ST < K. Thus the value of the

option is 0.

Then, the value of the option at time T is

maxST −K, 0 = (ST −K, 0)+.

Because the European call option can only be exercised at time T , a portfolio

(a, b) with value function Vt(a, b) is said to replicate the option if VT (a, b) = (ST −K, 0)+ almost surely. Assume that there exists a self-financing portfolio (a, b) that

replicates the European call option. We will later revisit this assumption. Then

because financial market participants can purchase either the option or the portfolio

at any time t < T , and because the portfolio replicates the option, the cost of the

portfolio, Vt(a, b) and the cost of the option must be the same at time t.

To find the value function of (a, b), assume that there exists a smooth deterministic

function u : R2 → R such that

Vt = atSt + btβt = u(T − t, St) for t ∈ [0, T ].

Because (a, b) replicates the option, we have the following terminal condition:

u(0, ST ) = VT (a, b) = (ST −K, 0)+.

We will derive the value function as follows: on the one hand, we will apply the Ito

Lemma to obtain an Ito process expression for Vt − V0, using the fact that St is an

45


Ito process, and on the other hand, we will use the fact that (a, b) is self-financing

to obtain another Ito process expression for Vt − V0. We will combine these two

expressions to yield a partial differential equation that the value function satisfies,

and to which the solution is well understood. We will conclude the derivation by

proving that (a, b) is indeed self-financing and replicates the European call option.

Let f(t, x) = u(T − t, x). Then we have:

f1(t, x) = −u1(T − t, x), f2(t, x) = u2(T − t, x), f22(t, x) = u22(T − t, x) .

Note that St satisfies the geometric Brownian motion SDE dSt = St(µdt + σdWt),

so St is an Ito process with µ(t, St) = µSt and σ(t, St) = σSt. Then an application

of the Ito Lemma yields:

Vt − V0 = f(t, St)− f(0, S0)

=

∫ t

0

(f1(s, Ss) + µSsf2(s, Ss) +

1

2σ2S2

sf22(s, Ss)

)ds

+

∫ t

0

σSsf2(s, Ss)dWs

=

∫ t

0

(−u1(T − s, Ss) + µSsu2(T − s, Ss) +

1

2σ2S2

su22(T − s, Ss))ds

+

∫ t

0

σSsu2(T − s, Ss)dWs.

Then the value process is an Ito process with

µ(t, St) = −u1(T − s, Ss) + µSsu2(T − s, Ss) +1

2σ2S2

su22(T − s, Ss),

and

σ(t, St) = σSsu2(T − s, Ss).

46


We now find another Ito process representation of Vt beginning with the self-

financing condition:

dVt(a, b) = atdSt + btdβt

Vt − V0 =

∫ t

0

asdSs +

∫ t

0

bsdβs

=

∫ t

0

asdSs +

∫ t

0

Vs − asSsβs

rβsds, because bs =Vs − asSs

βsand dβs = rβsds

=

∫ t

0

asdSs +

∫ t

0

r (Vs − asSs) ds

=

∫ t

0

µasSsds+

∫ t

0

σasSsdWs +

∫ t

0

r (Vs − asSs) ds

=

∫ t

0

((µ− r)asSs + ru(T − s, Ss)) ds+

∫ t

0

σasSsdWs.

Therefore, we have found a representation for Vt as an Ito process with

µ(t, St) = (µ− r)asSs + ru(T − s, Ss),

and

σ(t, St) = σasSs.

Because we have two representations for the Ito process Vt, the coefficient processes

of the Riemann and Ito integrals must coincide:

σatSt = σStu2(T − t, St)at = u2(T − t, St),

and

(µ− r)atSt + ru(T − t, St) = −u1(T − t, St) + µStu2(T − t, St) + 12σ2S2

t u22(T − t, St)= (µ− r)u2(T − t, St)St + ru(T − t, St).

Then we have the following partial differential equation:

ru(t, x) =1

2σ2x2u22(t, x) + rxu2(t, x)− u1(t, x).

47


When combined with the terminal condition, this partial differential equation ac-

tually has the well-studied explicit solution given by:

u(t, x) = xΦ(g(t, x))−Ke−rtΦ(h(t, x)),

where

g(t, x) =ln(xK

)+(r + 1

2σ2)t

σ√t

,

h(t, x) = g(t, x)− σ√t,

and

Φ(x) =1√2π

∫ x

−∞e−y22 dy, for x ∈ R.

Therefore, at any time t < T , the Black-Scholes rational price of the option is given

by:

Vt = u(T − t, St) = StΦ(g(T − t, St))−Ke−r(T−t)Φ(h(T − t, St)).

Now, this function is the value function of the portfolio given by

at = Φ(g(T − t, St)),

and

bt = −Ke−rTΦ(h(T − t, St)),

because

bt =Vt − atSt

βtand βt = ert.

To conclude the derivation, we must now show that this portfolio is indeed self-

financing and that it replicates the European call option.

Let α, γ : R2 → R be smooth functions. Let (α, γ) be the portfolio that holds

α(t, St) units of stock and γ(t, St) units of bond at time t. Then the value function

is given by:

Vt(α, γ) = v(t, St) = Stα(t, St) + γ(t, St)ert.

Theorem 7.1.1. Suppose v(t, x) is twice continuously differentiable. (α, γ) is self-

financing if and only if the following hold:

xαx + ertγx = 0,

48


1

2σ2x2αx + xαt + ertγt = 0.

Proof. Applying the Ito Lemma to v(t, St) = Stα(t, St) + γ(t, St)ert yields:

dv(t, St) = vx(t, St)dSt +

[vt(t, St) +

1

2σ2S2

t vxx(t, St)

]dt.

Now, (α, γ) is self-financing if and only if:

dv(t, St) = α(t, St)dSt + rγ(t, St)ertdt.

Then we have two representations of vt as an Ito process, so coefficient processes

coincide, and (α, γ) is self-financing if and only if:

α = vx

= α + xαx + ertγx

0 = xαx + ertγx,

andrγert = xαt + γte

rt + rγert + 12σ2x2(xαxx + 2αx + ertγxx)

0 = xαt + γtert + 1

2σ2x2(xαxx + 2αx + ertγxx).

.

Differentiate 0 = xαx + ertγx with respect to x:

0 = xαxx + αx + ertγxx,

so

0 = xαt + ertγt +1

2σ2x2αx.

Theorem 7.1.2. Let v(t, x) be twice continuously differentiable. v(t, St) is the

value function of a self-financing portfolio if and only if

1

2σ2x2vxx + rxvx + vt − rv = 0.

Proof. Suppose1

2σ2x2vxx + rxvx + vt − rv = 0.

49


Let

α = vx and γ = e−rt(v − xvx).

Then the portfolio (α, γ) has value function v(t, St), and

xαx + ertγx = xvxx + vx − vx − xvxx= 0,

and

12σ2x2αx + xαt + ertγt = 1

2σ2x2vxx + xvxt + (vt − xvxt)− r(v − xvx)

= 0 by assumption.

Therefore (α, γ) is self-financing, and v(t, St) is the value function of a self-financing

portfolio.

Now assume v(t, St) is the value function of a self-financing portfolio. Then, by

Theorem 7.1.1,

v(t, x) = xα(t, x) + ertγ(t, x)

for some (α, γ) such that:

xαx + ertγx = 0,

1

2σ2x2αx + xαt + ertγt = 0.

Thenvx = xαx + α + ertγx

= α,

and

γ = e−rt(v − xvx),

so0 = 1

2σ2x2αx + xαt + ertγt

= 12σ2x2vxx + xvxt + (vt − xvxt)− r(v − xvx)

= 12σ2x2vxx + rxvx + vt − rv.

Now, by Theorem 7.1.1 and Theorem 7.1.2, the portfolio given by

at = Φ(g(T − t, St))

50


and

bt = −Ke−rTΦ(h(T − t, St))

is self-financing.

We know by construction that

Vt(a, b) = StΦ(g(T − t, St))−Ke−r(T−t)Φ(h(T − t, St)).

Taking the limit as t ↑ T , we see that

g(T − t, St), h(T − t, St)→

−∞ if ST < K,

∞ if ST > K.

Therefore, if ST 6= K, we have VT (a, b) = (ST − K)+. Since P(ST = K) = 0,

VT (a, b) = (ST −K)+ almost surely, and (a, b) replicates the European call option.

7.1.4 Assumptions of the model

While we have explicitly mentioned some of the assumptions of the model in earlier

sections, we will restate the assumptions here, including some justification for the

validity of as well as potential problems with the assumptions.

Stock price is approximated by geometric Brownian motion St = e(µ−12σ2)t+σWt

This assumption is vital for multiple reasons. First, it includes the weaker as-

sumption that St is an Ito process, which is the fact that allows us to utilize the

Ito integral and Ito lemma to find solutions to the stochastic differential equations

that are derived from the model. Second, the specification of St as a geometric

Brownian motion assumes a constant mean rate of return µ and volatility σ from

time t = 0 to t = T , and determines the nature of the randomness that is modeled

into the stock price (namely that the Wiener process is a good approximation of

the randomness in a stock price over time).

St takes values in the positive reals

While this assumption is a consequence of modeling the stock price process as a

geometric Brownian motion, it is provably false in reality, because prices paid for

shares of stock cannot be irrational numbers. Therefore, in reality, stock prices

51


must be elements of the rational numbers, a countable subset of R. If we assume

that share prices are bounded above by some large finite bound, the set of values

that a share price can take is actually finite. In either case, the proof that the Black-

Scholes portfolio replicates the European call option relies upon the fact that, for a

continuous random variable X, P (X = x) = 0 ∀ x ∈ R. Of course, the strike price

K of the European call option will also be a rational number, so the assertion that

P (ST = K) = 0 is no longer trivially true.

No arbitrage

The assumption that there do not exist risk-free opportunities to make a profit is

vital to equate the value of the self-financing portfolio that replicates the European

call option, which is found using mathematical techniques described above, with the

value of the option itself. Without this assumption (or assumptions that provide

other ways to necessitate the equality of the value of the portfolio and the value

of the option), the mathematical theory would allow us to prove the existence and

calculate the value of the self-financing portfolio that replicates the option, but

would not allow us to make a conclusion about the rational price of the option

based on the portfolio.

Existence of risk-free bond with constant interest rate

This assumption, combined with the no arbitrage assumption, states that all finan-

cial market participants can borrow or lend money at the same risk-free constant

interest rate r, and is clearly not true in reality.

Ability to purchase and hold any positive or negative quantity of stock

and bond

While at ≥ 0 ∀ t ∈ [0, T ] and bt ≤ 0 ∀ t ∈ [0, T ] for the self-financing portfolio

described above, this assumption and the resulting self-financing portfolio does

imply that all financial market participants can borrow any amount of money at

the same risk-free constant interest rate r.

52


Ability to make trades in continuous time

The construction of the self-financing portfolio vitally relies upon the assumption

that financial market participants can buy and sell units of stock and bond in contin-

uous time in order to instantaneously re-balance the portfolio in response to changes

in the stock price. Clearly, this is impossible in reality. In Section 7.2, we explore

how relaxing this assumption and only allowing financial market participants to re-

balance a portfolio that replicates the European call option at discrete time steps

impacts the amount the portfolio deviates from the self-financing property that

holds for instantaneously rebalanced portfolios.

No transaction costs

While the assumption that financial market participants need not pay transaction

costs is made to simplify calculations, it is problematic because even small transac-

tion costs could have a large impact on the value of a portfolio that is rebalanced

continuously in time, as the Black-Scholes model assumes, or even on the value of a

portfolio that is rebalanced at several discrete steps of time per day in an attempt to

minimize the amount the discrete time portfolio deviates from being self-financing

(the deviation of portfolios that are re-balanced at discrete time steps from being

self-financing is studied at length in Section 7.2).

No dividends

The assumption that the stock does not pay dividends is an assumption made to

simplify calculations.

Value function Vt of portfolio is a smooth deterministic function of t and

St

This assumption is required to be able to apply the Ito lemma to the value process.

The assumption that the function is smooth is potentially problematic. Because

sample paths of the Wiener process are not differentiable, the possibility exists

that the value process of the self-financing portfolio is not differentiable, or that it

is not continuous. Other methods of deriving the Black-Scholes model, such as the

change of measure technique, rely upon stronger backgrounds of probability theory

53


and measure theory, and allow this assumption to be relaxed, however. The change

of measure derivation of the Black-Scholes model is described in Section 13.10 of [2].

7.2 Approximating the Black-Scholes portfolio at discrete

time steps

7.2.1 Introduction and motivation

One of the assumptions of the Black-Scholes model that is essential to prove

that the portfolio that replicates the European call option is also self-financing

is that financial market participants can make transactions in continuous time,

instantaneously reacting to changes in the price of the stock and bond. In reality,

however, financial transactions are conducted at discrete time steps. In this section,

we study the effect that relaxing this assumption has on the self-financing property

of the replicating portfolio. The motivation to further study the effect of rebalancing

the portfolio at discrete time steps on the self-financing property of the portfolio

stems in part from [3].

We build a model that calculates a sample path of the stock price process St at

discrete time steps, conduct a numerical analysis over several sample paths of St to

study the relationship between the frequency of the time steps and the deviation

of the portfolio from the self-financing property, and build a model to study the

deviation of portfolios based on historical stock price data from the self-financing

property. Of course, before we can properly study the deviation of a portfolio

from the self-financing property, we must define the way that we will quantify that

deviation.

7.2.2 Deviation of a discrete-time portfolio from the self-financing prop-

erty: cumulative correction

Suppose we want to use the replicating portfolio from the Black-Scholes model

to find the value of a European call option with strike price K and maturity T

over time, and that we assume that financial market transactions occur at discrete

time steps (this assumption is the only deviation from the set-up of the Black-

Scholes model described in Section 7.1). Suppose we break the interval [0, T ] into

n ∈ N intervals each of length Tn

and that financial market participants perform

54


transactions only at times t ∈ jTn for j ∈ 0, 1, . . . n. At these times t, financial

market participants solve the Black-Scholes portfolio equations:

at = Φ(g(T − t, St)), and bt = −Ke−rTΦ(h(T − t, St)),

and adjust their holdings of stock and bond accordingly. Because the stock price St

changes between tj = jTn

and tj+1 = (j+1)Tn

without the financial market participant

instantaneously re-balancing the replicating portfolio, it is possible that the value of

the portfolio that the participant must purchase at time tj+1, Stj+1atj+1

+ βtj+1btj+1

is greater than or less than the value of the actual portfolio the participant holds

before he/she performs rebalancing transactions, Stj+1atj +βtj+1

btj . We can think of

the rebalancing transaction as the financial market participant selling the portfolio

that he/she held at time tj,(atj , btj

)at the prices at time tj+1 and purchasing

the desired portfolio at time tj+1,(atj+1

, btj+1

)at the same prices. The difference

between the money gained by selling(atj , btj

)and the money necessary to purchase(

atj+1, btj+1

)is what we term the correction over the time interval (tj, tj+1], and is

the way we construct our measure of the deviation of a portfolio from the self-

financing property.

Definition 7.2.1. The correction Cj of a portfolio over the time interval (tj, tj+1]

is given by:

Cj = Stj+1atj+1

+βtj+1btj+1−(Stj+1

atj + βtj+1btj)

= Stj+1

(atj+1

− atj)+βtj+1

(btj+1

− btj).

Cj > 0 indicates that the financial market participant is required to input additional

money from an external source into the portfolio in order perform the required

transaction at time tj+1, and Cj < 0 indicates that the financial market participant

is required to remove money from the portfolio in order perform the required

transaction.

The cumulative correction Ctk of a portfolio up to time tk is given by:

k−1∑j=0

Cj.

Note that Ctn is the cumulative correction of a portfolio up to the maturity T of

55


the option. We denote Ctn as C.

The models described in the following sections study the cumulative correc-

tion of different Black-Scholes portfolios in detail. In each of the models, we use

time units such that a time interval of length 1 represents a day.

7.2.3 Cumulative correction of sample paths of St

The first model we have built to study the cumulative correction of Black-Scholes

portfolios simulates one sample path of the stock price process St, and tracks the

cumulative correction of the portfolio over time. The Mathematica code for the

model can be found in Appendix A.1. The model allows for the specification of

K,T, µ, σ, r, and n. It then calculates Stj , atj , btj , for j ∈ 0, 1, . . . , n, and Ctj for

j ∈ 1, 2, . . . , n.

The following is a sample output from the model with

S0 = 1, β0 = 1, K = 1.1, T = 250, n = 250, and annualized µ = .1, σ = .1, r = .05.

Figure 1: Stock price and value of replicating portfolio for given sample path ω0 ofSt

56


Figure 2: Composition of replicating portfolio over time for same sample path ω0

of St

Figure 3: Cumulative correction over time for same sample path ω0 of St

57


The sample path of St that generated this output is the stock price charted

in Figure 1. The composition, value, and cumulative correction of the replicating

Black-Scholes portfolio are charted in Figures 2, 1, 3, respectively. We can see that,

for this sample path of St, ST < K, aT = bT = 0, and the cumulative correction C is

positive, meaning that in order to replicate the European call option, the financial

market participant would have been required to use money from an external source

to construct the portfolio. From this output, we can make the following observation

regarding the cumulative correction and the construction of a replicating portfolio

in discrete time steps: relatively large increases in stock price over a time interval

require relatively large increases in the number of stock units held in the portfolio,

and are more likely to necessitate a positive correction over the interval. This

phenomenon is sensitive to the time remaining to maturity, the stock price relative

to the strike price, and σ.

This model allows us to study the cumulative correction for given sample paths

of St, and provides the fundamental model to perform a numerical analysis of the

cumulative correction over several sample paths, which we describe in the following

section.

7.2.4 Numerical analysis of cumulative correction over several sample

paths

Using the model described in Section 7.2.3 to calculate the cumulative correction

up to time T , C, for several sample paths, we have created a second model in order

to better understand the behavior of C for different values of nT

. The Mathematica

code for this model can be found in A.2. The ratio nT

measures the number of

times the portfolio is re-balanced per day, and intuitively, we expect that as nT

increases, C should be more likely to be close to 0. In order for our discrete

time step model to approximate the continuous time Black-Scholes model for large

n, we would like to be able to show that C is more likely to be close to 0 as

n → ∞. Our model allows for the specification K,T, µ, σ, r, the sequence of n to

be considered, and the number of iterations to be performed for each number n of

time steps. The following output was generated by our second model performing

50 iterations for each n ∈ 30, 300, 600, 900, 1200, 1500, 1800, 2100, 2400, 2700, 3000

58


with the parameters:

S0 = 1, β0 = 1, K = 1.02, T = 30, and annualized µ = .07, σ = .1, r = .05.

Figure 4: Cumulative correction over several sample paths and values of Tn

We do see that C shows a clear trend to be more likely to be close to 0 as nT

increases, however the rate at which C becomes more likely to be close to 0 is lower

for larger nT

and this simulation is far from a rigorous proof of the intuition. We

would also like to be able to study the behavior of the cumulative correction for

very large values of nT

, but computing limits necessitate a trade-off between the

number of iterations that can be performed at each nT

and values of nT

that can be

analyzed.

7.2.5 Cumulative correction of portfolios based on historical stock price

data

In order to understand the behavior of the cumulative correction when actually

attempting to create a Black-Scholes replicating portfolio in reality, we have built

a third model that creates a replicating portfolio and documents the cumulative

correction of that portfolio for a given series of historical stock price data. The

59


Mathematica code for the model can be found in A.3. We use daily close prices of

stocks, and fix n = T , allowing financial market participants to re-balance portfolios

once per day. We create the Black-Scholes portfolio over the course of one year

(T = 250, the approximate number of trading days in a year), with the following

parameters for all examples:

S0 = 1, β0 = 1, T = 250, K = 1.1, and annualized r = .05.

The creation of a Black-Scholes portfolio based on historical price data rather than

based on a geometric Brownian motion St, however, raises a problem because the

Black-Scholes portfolio depends on the parameter σ, the volatility, of the price

process. In order to create the Black-Scholes formula at time t, we need to assume

some σ of the price process for times s > t. We take two different approaches to

calculating an assumed volatility, both involving historical volatility of the stock

in question. The first approach, which we call the constant volatility approach,

calculates the historical volatility of the stock over the six-month (125 day) time

interval immediately prior to t = 0, and assumes that σ takes that constant value

over the remaining time t ∈ [0, T ]. The second approach, which we call the rolling

volatility approach, calculates the rolling historical volatility of the stock for the

six months (125 days) prior to each time t, and uses that to calculate the Black-

Scholes portfolio at t. Calculation of historical volatility (v) follows the convention

in the financial literature which is described in [1] and is performed according to

the following method:

Given daily stock prices st, st+1, . . . st+n, let ri = log(st+i)log(st+i−1)

for i = 1, 2, . . . , n. Let

r = 1n

∑ni=1 ri. Then the daily volatility vdaily is given by

vdaily =

√√√√ 1

n− 1

n∑i=1

(ri − r)2,

and we annualize the volatility by multiplying by√

250,

vannual = σ =√

250vdaily.

60


We have performed this analysis for approximately 20 stocks, and the majority

have a positive cumulative correction over the course of a year, but we would

like to highlight some interesting examples that illustrate the spectrum of possible

cumulative corrections. Historical stock price data for these stocks are data series

of daily close prices as reported by Yahoo! Finance.

The first example that we will highlight is Apple, AAPL. The output of the third

model for AAPL is as follows:

Figure 5: AAPL: annualized constant and rolling historical volatility over time

61

http://finance.yahoo.com


Figure 6: AAPL: stock price and value of replicating portfolios over time

Figure 7: AAPL: composition of replicating portfolios over time

62


For the series of historical AAPL data up through April 23, 2012, the model

charts the volatilities in Figure 5, the stock price and value of the replicating

portfolios in Figure 6, the composition of the replicating portfolios in Figure 7,

and the cumulative correction of the portfolios in Figure 8. We observe that both

the constant volatility and the rolling volatility approach to create the Black-Scholes

portfolio require a positive cumulative correction.

Figure 8: AAPL: cumulative correction of replicating portfolios over time

63


The second example that we will highlight is Dell, DELL. The output of the

third model for DELL is as follows:

Figure 9: DELL: annualized constant and rolling historical volatility over time

Figure 10: DELL: stock price and value of replicating portfolios over time

64


Figure 11: DELL: composition of replicating portfolios over time

Figure 12: DELL: cumulative correction of replicating portfolios over time

Similarly to the case with AAPL, for the series of historical DELL data

up through May 1, 2012, the model charts the volatilities in Figure 9, the stock

price and value of the replicating portfolios in Figure 10, the composition of the

replicating portfolios in Figure 11, and the cumulative correction of the portfolios

in Figure 12. We observe that the constant volatility replicating portfolio requires a

positive cumulative correction, and the rolling volatility portfolio requires a negative

cumulative correction.

65


The third and final example that we will highlight is Ford, F. The output of

the third model for F is as follows:

Figure 13: F: annualized constant and rolling historical volatility over time

Figure 14: F: stock price and value of replicating portfolios over time

66


Figure 15: F: composition of replicating portfolios over time

Figure 16: F: cumulative correction of replicating portfolios over time

Similarly to the case with AAPL and DELL, for the series of historical F

data up through May 1, 2012, the model charts the volatilities in Figure 13, the

stock price and value of the replicating portfolios in Figure 14, the composition

of the replicating portfolios in Figure 15, and the cumulative correction of the

portfolios in Figure 16. We observe that both the constant volatility and the

rolling volatility approach to create the Black-Scholes portfolio require a negative

cumulative correction. These three examples (AAPL, DELL, and F) indicate the

range of possible portfolios and cumulative correction that our third model (and

the attempt to replicate the European call option on actual stocks using the Black-

67


Scholes portfolios using historical volatilities) can output.

7.2.6 Summary of study of cumulative correction

In Section 7.2, we have proposed and defined a quantity, the cumulative correction,

to study the amount the discrete time step Black-Scholes portfolio deviates from the

self-financing condition of the continuously rebalanced portfolio. We have described

the results of models that we have built to study the behavior of the cumulative

correction:

1. Studying the cumulative correction for fixed sample paths of the stock price

stochastic process allows us to make qualitative statements regarding the

behavior of the correction over a time interval as it relates to the change in

the stock price over that interval.

2. Numerical analysis of several sample paths at different portfolio rebalancing

frequencies shows that as the frequency of rebalancing the Black-Scholes

replicating portfolio is increased, the cumulative correction is more likely to

be close to 0.

3. While the majority of historical stock return series that we have studied have

caused the discrete time step Black-Scholes replicating portfolio to have a

positive cumulative correction, we have provided examples of a stock that

requires a positive cumulative correction for both approaches used to calculate

volatility (AAPL), a stock that requires a positive cumulative correction for

one approach used to calculate volatility and a negative cumulative correc-

tion for the other approach (DELL), and a stock that requires a negative

cumulative correction for both volatility approaches (F).

68

8 References

[1] Stephen Figlewski. “Forecasting Volatility Using Historical Data”, New

York University Stern School of Business Finance Department, Working

Paper Series, 1994. http://w4.stern.nyu.edu/finance/docs/WP/1994/pdf/

wpa94032.pdf.

[2] G. Grimmett, D. Stirzaker. Probability and Random Processes, Third Edition,

Oxford University Press, 2001.

[3] Andrzej Kozlowski. “Hedging the Black-Scholes Call Option”, from the

Wolfram Demonstrations Project. http://demonstrations.wolfram.com/

HedgingTheBlackScholesCallOption/.

[4] Thomas Mikosch. Elementary Stochastic Calculus with Finance in View,

Advanced Series on Statistical Science & Applied Probability, Volume 6, World

Scientific Publishing Co. Pte. Ltd., 2004.

[5] Y.L. Tong. The Multivariate Normal Distribution, Springer Series in Statistics,

Springer-Verlag New York, 1990.

69

http://w4.stern.nyu.edu/finance/docs/WP/1994/pdf/wpa94032.pdf

http://w4.stern.nyu.edu/finance/docs/WP/1994/pdf/wpa94032.pdf

http://demonstrations.wolfram.com/HedgingTheBlackScholesCallOption/

http://demonstrations.wolfram.com/HedgingTheBlackScholesCallOption/

H*

Greg WhiteEuropean Call Replication for Sample Path of Price Process

This model replicates the European call option for a sample path of the geometricBrownian motion stock price process. The calculation of the replicatingportfolio according to the Black-Scholes equation occurs at discrete time steps,

and the correction of the portfolio is documented*L

<< PlotLegends`

thesisPath = "UsersgregwhiteDesktopThesisGreg ThesisMathematica";

H*Parameters- constants and assumptions*LnumIter = 250.0;annualR = 0.05;strike = 1.1;maturity = 250.0;annualMu = 0.1;annualSigma = 0.1;tradingDaysPerYear = 250.0;

H*Calculate sample path of Wiener process*Ldelta = maturity numIter;wiener = RandomVariate@NormalDistribution@0, Sqrt@deltaDD, numIterD;

H*Annualize parameters*Lmu = annualMu tradingDaysPerYear;sigma = annualSigma Sqrt@tradingDaysPerYearD;r = annualR tradingDaysPerYear;

H*Initiate stock price and time data series*LstockPrice = [email protected];times = [email protected];

H*Functions for the Balck-Scholes formula*Lg@t_, x_D :=

HLog@x strikeD + Hr + 0.5 * Hsigma^2.0LL * Hmaturity - tLL Hsigma * Sqrt@maturity - tDL;h@t_, x_D := g@t, xD - sigma * Sqrt@maturity - tD;

H*Initiate replicating portfolio*LstockUnits = List@CDF@NormalDistribution@D, g@times@@1DD, stockPrice@@1DDDDD;bondUnits = List@-strike * Exp@-r * maturityD *

CDF@NormalDistribution@D, h@times@@1DD, stockPrice@@1DDDDD;

H*Initiate replicating correction flow*LcorrectionIncrements = List@0D;correctionSum = List@0D;

H*Initiate the format for the the stock price to be plotted*LpricePlot = List@8times@@1DD, stockPrice@@1DD<D;strikePlot = List@8times@@1DD, strike<D;payoutPlot = List@8times@@1DD, stockPrice@@1DD - strike<D;

H*Initiate the format for the replicating portfolio to be plotted*LstockUnitPlot = List@8times@@1DD, stockUnits@@1DD<D;bondUnitPlot = List@8times@@1DD, bondUnits@@1DD<D;correctionPlot = List@8times@@1DD, correctionSum@@1DD<D;portfolioValuePlot = List@

8times@@1DD, stockUnits@@1DD * stockPrice@@1DD + bondUnits@@1DD * Exp@r * times@@1DDD<D;

H*At each discrete time step,calculate the stock price and solve the Black-Scholes equations,

*L

A APPENDIX: MATHEMATICA MODELS

A Appendix: Mathematica Models

A.1 European call option for GBM sample path

70

calculate the stock price and solve the Black-Scholes equations,creating a replicating portfolio and documenting the correction*LDo@

H*Increment the stockPrice according toa discrete approximation of a geometric Brownian motion*L

times = Append@times, times@@mDD + deltaD;stockPrice =

Append@stockPrice, Exp@HHHmu - 0.5 * Hsigma^2LL * times@@m + 1DDL + sigma * wiener@@mDDLDD;

H*Replicating portfolio construction- g and h yield infinity when t=maturity*LIf@m == numIter, If@stockPrice@@m + 1DD > strike, stockUnits = Append@stockUnits, 1D;

bondUnits = Append@bondUnits, -strike * Exp@-r * maturityDD,stockUnits = Append@stockUnits, 0D; bondUnits = Append@bondUnits, 0DD, stockUnits =

Append@stockUnits, CDF@NormalDistribution@D, g@times@@m + 1DD, stockPrice@@m + 1DDDDD;bondUnits = Append@bondUnits, -strike * Exp@-r * maturityD *

CDF@NormalDistribution@D, h@times@@m + 1DD, stockPrice@@m + 1DDDDDD;

H*Format the stock price to be plotted*LpricePlot = Append@pricePlot, 8times@@m + 1DD, stockPrice@@m + 1DD<D;strikePlot = Append@strikePlot, 8times@@m + 1DD, strike<D;payoutPlot = Append@payoutPlot, 8times@@m + 1DD, stockPrice@@m + 1DD - strike<D;

H*Format the replicating portfolio to be plotted*LstockUnitPlot = Append@stockUnitPlot, 8times@@m + 1DD, stockUnits@@m + 1DD<D;bondUnitPlot = Append@bondUnitPlot, 8times@@m + 1DD, bondUnits@@m + 1DD<D;portfolioValuePlot = Append@portfolioValuePlot, 8times@@m + 1DD,

stockUnits@@m + 1DD * stockPrice@@m + 1DD + bondUnits@@m + 1DD * Exp@r * times@@m + 1DDD<D;

H*Document correction*Lcorrection = stockPrice@@m + 1DD * HstockUnits@@m + 1DD - stockUnits@@mDDL +

Exp@r * times@@m + 1DDD * HbondUnits@@m + 1DD - bondUnits@@mDDL;correctionIncrements = Append@correctionIncrements, correctionD;correctionSum = Append@correctionSum, correctionSum@@mDD + correctionD;correctionPlot = Append@correctionPlot, 8times@@m + 1DD, correctionSum@@m + 1DD<D;

, 8m, numIter<D;

H*Export plots*LExport@FileNameJoin@8thesisPath, "GraphicsAndCode", "generalPriceValue.png"<D,

ListLinePlot@8pricePlot, strikePlot, portfolioValuePlot, payoutPlot<,AxesLabel ® 8"Time HdaysL", $<,PlotLabel ® "Stock Price and Replicating Portfolio Value over Time",PlotLegend ® 8Style@"Stock price", FontSize ® 16D, Style@"Strike price", FontSize ® 16D,

Style@"Replicating portfolio\nvalue", FontSize ® 16D,Style@"Stock price minus\nstrike price", FontSize ® 16D<,

LegendPosition ® 80.85, -0.4<, LegendShadow ® None, BaseStyle ® 8FontSize ® 18<,AspectRatio ® 7.5 10.0, PlotRange ® AllD, ImageSize ® 72 * 15D;

Export@FileNameJoin@8thesisPath, "GraphicsAndCode", "generalCorrection.png"<D,ListLinePlot@correctionPlot, AxesLabel ® 8"Time HdaysL", $<,

PlotLabel ® "Cumulative Correction of Replicating Portfolio over Time", BaseStyle ®

8FontSize ® 18<, AspectRatio ® 7.5 10.0, PlotRange ® AllD, ImageSize ® 72 * 15D;

Export@FileNameJoin@8thesisPath, "GraphicsAndCode", "generalPortfolioComposition.png"<D,ListLinePlot@8stockUnitPlot, bondUnitPlot<, AxesLabel ® 8"Time HdaysL", "Units"<,

PlotLabel ® "Replicating Portfolio Composition over Time",PlotLegend ® 8Style@"Stock units", FontSize ® 16D, Style@"Bond units", FontSize ® 16D<,LegendPosition ® 81.1, -0.4<, LegendShadow ® None, BaseStyle ® 8FontSize ® 18<,AspectRatio ® 7.5 10.0, PlotRange ® AllD, ImageSize ® 72 * 15D;

2 BlackScholesEuropeanCallReplicationvF.nb


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H*

Greg WhiteNumerical Correction Analysis

This model analyzes the cumulative correction required toreplicate the European call option. The cumulative correction isplotted versus the number of discrete time steps per day for a numberof sample paths of the geometric Brownian motion stock price process.

*L

H*Parameters- constants and assumptions*LnumIter = 50;maturityInterval = 30.0;annualR = 0.05;strike = 1.02;startingMaturity = 30.0;annualMu = 0.07;annualSigma = 0.1;tradingDaysPerYear = 250.0;

H*Generate list of maturities to be analyzed*LlistOfMaturities = List@startingMaturityD;

H*Annualize parameters*Lmu = annualMu tradingDaysPerYear;sigma = annualSigma Sqrt@tradingDaysPerYearD;r = annualR tradingDaysPerYear;

H*Functions for the Black-Scholes formula*Lg@t_, x_, maturity_D :=

HLog@x strikeD + Hr + 0.5 * Hsigma^2.0LL * Hmaturity - tLL Hsigma * Sqrt@maturity - tDL;h@t_, x_, maturity_D := g@t, xD - sigma * Sqrt@maturity - tD;

H*For each maturity to be analyzed,perform iterations of portfolio creation and plot resulting cumulative correction*LDo@

H*Generate the list of numSteps to be analyzed*LstepsPerDayInterval = 10;interval = Floor@stepsPerDayInterval * maturityD;min = 1; max = 10; step = 1;listOfNumSteps = Prepend@Table@i * interval, 8i, min, max, step<D, maturityD;listOfNumStepsPerDay = Prepend@Table@i * stepsPerDayInterval, 8i, min, max, step<D, 1D;

H*Initialize plot of cumulative corrections*LcumulativeCorrectionPlot = List@D;

H*For every value of numSteps, do numIter iterations of portfolio creation*LDo@

Do@

H*Go through one iteration of the portfolio creation*L

delta = maturity numSteps;wiener = Accumulate@RandomVariate@NormalDistribution@0, Sqrt@deltaDD, numStepsDD;

stockPrice = [email protected];times = [email protected];

H*Initiate replicating portfolio*LstockUnits =

List@CDF@NormalDistribution@D, g@times@@1DD, stockPrice@@1DD, maturityDDD;bondUnits = List@-strike * Exp@-r * maturityD *

CDF@NormalDistribution@D, h@times@@1DD, stockPrice@@1DD, maturityDDD;


A.2 Numerical analysis of cumulative correction

72

CDF@NormalDistribution@D, h@times@@1DD, stockPrice@@1DD, maturityDDD;

H*Initiate replicating correction flow*LcorrectionIncrements = List@0D;correctionSum = List@0D;

Do@H*Increment the stockPrice according

to a discrete approximation of a geometric Brownian motion*Ltimes = Append@times, times@@mDD + deltaD;stockPrice = Append@stockPrice,

Exp@HHHmu - 0.5 * Hsigma^2LL * times@@m + 1DDL + sigma * wiener@@mDDLDD;

H*Replicating portfolio construction- g and h yield infinity when t=maturity*LIf@m == numSteps, If@stockPrice@@m + 1DD > strike, stockUnits = Append@stockUnits, 1D;

bondUnits = Append@bondUnits, -strike * Exp@-r * maturityDD,stockUnits = Append@stockUnits, 0D; bondUnits = Append@bondUnits, 0DD,

stockUnits = Append@stockUnits, CDF@NormalDistribution@D,g@times@@m + 1DD, stockPrice@@m + 1DD, maturityDDD;

bondUnits = Append@bondUnits, -strike * Exp@-r * maturityD *

CDF@NormalDistribution@D, h@times@@m + 1DD, stockPrice@@m + 1DD, maturityDDDD;

H*Document correction*Lcorrection = stockPrice@@m + 1DD * HstockUnits@@m + 1DD - stockUnits@@mDDL +

Exp@r * times@@m + 1DDD * HbondUnits@@m + 1DD - bondUnits@@mDDL;correctionIncrements = Append@correctionIncrements, correctionD;correctionSum = Append@correctionSum, correctionSum@@mDD + correctionD;

, 8m, numSteps<D;

H*Add the iteration to the plot*LcumulativeCorrectionPlot =

Append@cumulativeCorrectionPlot, 8numSteps maturity, Last@correctionSumD<D;

, 8numSteps, listOfNumSteps<D;

, 8numIter<D;

H*Plot the cumulative correction versus the number of steps per day for maturity*L


Export@FileNameJoin@8thesisPath, "GraphicsAndCode", "StatisticalCumulative.png"<D,ListPlot@cumulativeCorrectionPlot,

AxesLabel ® 8"Number\nof steps\nper day", "Cumulative correction H$L"<,PlotLabel ® StringJoin@"Cumulative Correction versus Number of Steps per Day over ",

ToString@Floor@maturityDD, " Days"D, BaseStyle ® 8FontSize ® 18<,PlotRange ® 88listOfNumStepsPerDay@@1DD - stepsPerDayInterval,

Last@listOfNumStepsPerDayD + stepsPerDayInterval 2.<, All<,Ticks ® 8listOfNumStepsPerDay@@2 ;;DD, Automatic<D, ImageSize ® 72 * 15D

, 8maturity, listOfMaturities<D;

2 NumericalCorrectionAnalysisvF.nb


73

H*

Greg WhiteReplication of European Call Option using Historical Stock Prices

This model creates the European call option replicating portfolio for a seriesof historical stock price data Hwhich is imported from a csv fileL. Theportfolio composition and correction are documented.

*L

<< PlotLegends`


H*Parameters- constants and assumptions*LannualR = 0.05;strike = 1.1;tradingDaysPerYear = 250.0;r = annualR tradingDaysPerYear;

stockTicker = "DELL";

H*Import historical stock price data*LnumDays = 251;H*Import all stock price data to be used to

calculate historical volatility and stock prices*LhistoricalPrices =

Reverse@Transpose@Import@FileNameJoin@8thesisPath, "HistoricalStockData",StringJoin@stockTicker, "_Historical.csv"D<DDD@@5DD@@2 ;;DDD;

H*stockPrice is the most recent numDays' worth of price data*LstartStockPrice = Length@historicalPricesD - numDays + 1;stockPrice = historicalPrices@@startStockPrice ;;DD;stockPrice = stockPrice stockPrice@@1DD;maturity = Length@stockPriceD - 1;times = Range@0, maturityD;

H*Find historical volatility*LnumMonthsOfVolatility = 6.0;numDaysOfVolatility = Floor@HnumMonthsOfVolatility 12.0L * tradingDaysPerYearD;logarithmReturns = List@D;Do@

logarithmReturns =

Append@logarithmReturns, Log@historicalPrices@@iDD historicalPrices@@i - 1DDDD;, 8i, startStockPrice - numDaysOfVolatility, startStockPrice, 1<D;

constantHistoricalVolatility = StandardDeviation@logarithmReturnsD;

H*Functions for the Black-Scholes formula*Lg@t_, x_, sigma_D :=

HLog@x strikeD + Hr + 0.5 * Hsigma^2.0LL * Hmaturity - tLL Hsigma * Sqrt@maturity - tDL;h@t_, x_, sigma_D := g@t, x, sigmaD - sigma * Sqrt@maturity - tD;

H*Initiate replicating portfolios*LconstantVolStockUnits = List@CDF@NormalDistribution@D,

g@times@@1DD, stockPrice@@1DD, constantHistoricalVolatilityDDD;constantVolBondUnits = List@-strike * Exp@-r * maturityD * CDF@NormalDistribution@D,

h@times@@1DD, stockPrice@@1DD, constantHistoricalVolatilityDDD;

rollingVolStockUnits = List@CDF@NormalDistribution@D,g@times@@1DD, stockPrice@@1DD, constantHistoricalVolatilityDDD;

rollingVolBondUnits = List@-strike * Exp@-r * maturityD * CDF@NormalDistribution@D,h@times@@1DD, stockPrice@@1DD, constantHistoricalVolatilityDDD;

H*Initiate replicating correction flows*LconstantVolCorrectionIncrements = List@0D;constantVolCorrectionSum = List@0D;


A.3 European call option for historical data

74

constantVolCorrectionSum = List@0D;

rollingVolCorrectionIncrements = List@0D;rollingVolCorrectionSum = List@0D;

H*Initiate the format for the the stock price to be plotted*LpricePlot = List@8times@@1DD, stockPrice@@1DD<D;strikePlot = List@8times@@1DD, strike<D;payoutPlot = List@8times@@1DD, stockPrice@@1DD - strike<D;

H*Initiate the format for the replicating portfolios to be plotted*LconstantVolStockUnitPlot = List@8times@@1DD, constantVolStockUnits@@1DD<D;constantVolBondUnitPlot = List@8times@@1DD, constantVolBondUnits@@1DD<D;constantVolCorrectionPlot = List@8times@@1DD, constantVolCorrectionSum@@1DD<D;constantVolPortfolioValuePlot =

List@8times@@1DD, constantVolStockUnits@@1DD * stockPrice@@1DD +

constantVolBondUnits@@1DD * Exp@r * times@@1DDD<D;

rollingVolStockUnitPlot = List@8times@@1DD, rollingVolStockUnits@@1DD<D;rollingVolBondUnitPlot = List@8times@@1DD, rollingVolBondUnits@@1DD<D;rollingVolCorrectionPlot = List@8times@@1DD, rollingVolCorrectionSum@@1DD<D;rollingVolPortfolioValuePlot =

List@8times@@1DD, rollingVolStockUnits@@1DD * stockPrice@@1DD +

rollingVolBondUnits@@1DD * Exp@r * times@@1DDD<D;

H*Initiate the format for the volatilities to be plotted*LconstantVolPlot =

List@8times@@1DD, Sqrt@tradingDaysPerYearD * constantHistoricalVolatility<D;rollingVolPlot = List@8times@@1DD,

Sqrt@tradingDaysPerYearD * constantHistoricalVolatility<D;

Do@H*Calculate new rollingVolatility*LlogarithmReturns =

Append@logarithmReturns, Log@historicalPrices@@startStockPrice + m - 1DD historicalPrices@@startStockPrice + m - 2DDDD;

rollingVolatility = StandardDeviation@logarithmReturns@@-numDaysOfVolatility - 1 ;;DDD;

H*Add the volatilities to the plot*LconstantVolPlot = Append@constantVolPlot,

8times@@mDD, Sqrt@tradingDaysPerYearD * constantHistoricalVolatility<D;rollingVolPlot = Append@rollingVolPlot,

8times@@mDD, Sqrt@tradingDaysPerYearD * rollingVolatility<D;

H*Replicating portfolios construction- g and h yield infinity when t=maturity*LIf@m Length@timesD,

If@stockPrice@@mDD > strike, constantVolStockUnits = Append@constantVolStockUnits, 1D;rollingVolStockUnits = Append@rollingVolStockUnits, 1D;constantVolBondUnits = Append@constantVolBondUnits, -strike * Exp@-r * maturityDD;rollingVolBondUnits = Append@rollingVolBondUnits, -strike * Exp@-r * maturityDD,constantVolStockUnits = Append@constantVolStockUnits, 0D;rollingVolStockUnits = Append@rollingVolStockUnits, 0D;constantVolBondUnits = Append@constantVolBondUnits, 0D;rollingVolBondUnits = Append@rollingVolBondUnits, 0DD,

constantVolStockUnits = Append@constantVolStockUnits, CDF@NormalDistribution@D,g@times@@mDD, stockPrice@@mDD, constantHistoricalVolatilityDDD;

rollingVolStockUnits = Append@rollingVolStockUnits, CDF@NormalDistribution@D,g@times@@mDD, stockPrice@@mDD, rollingVolatilityDDD; constantVolBondUnits =

Append@constantVolBondUnits, -strike * Exp@-r * maturityD * CDF@NormalDistribution@D,h@times@@mDD, stockPrice@@mDD, constantHistoricalVolatilityDDD;

rollingVolBondUnits = Append@rollingVolBondUnits, -strike * Exp@-r * maturityD *

CDF@NormalDistribution@D, h@times@@mDD, stockPrice@@mDD, rollingVolatilityDDDD;

H*Format the stock price to be plotted*LpricePlot = Append@pricePlot, 8times@@mDD, stockPrice@@mDD<D;

;

2 BlackScholesEuropeanCallHistoricalvF.nb


75

pricePlot = Append@pricePlot, 8times@@mDD, stockPrice@@mDD<D;strikePlot = Append@strikePlot, 8times@@mDD, strike<D;payoutPlot = Append@payoutPlot, 8times@@mDD, stockPrice@@mDD - strike<D;

H*Format the replicating portfolios to be plotted*LconstantVolStockUnitPlot =

Append@constantVolStockUnitPlot, 8times@@mDD, constantVolStockUnits@@mDD<D;constantVolBondUnitPlot = Append@constantVolBondUnitPlot,

8times@@mDD, constantVolBondUnits@@mDD<D;constantVolPortfolioValuePlot = Append@constantVolPortfolioValuePlot,

8times@@mDD, constantVolStockUnits@@mDD * stockPrice@@mDD +

constantVolBondUnits@@mDD * Exp@r * times@@mDDD<D;

rollingVolStockUnitPlot =

Append@rollingVolStockUnitPlot, 8times@@mDD, rollingVolStockUnits@@mDD<D;rollingVolBondUnitPlot = Append@rollingVolBondUnitPlot,

8times@@mDD, rollingVolBondUnits@@mDD<D;rollingVolPortfolioValuePlot = Append@rollingVolPortfolioValuePlot,

8times@@mDD, rollingVolStockUnits@@mDD * stockPrice@@mDD +

rollingVolBondUnits@@mDD * Exp@r * times@@mDDD<D;

H*Document corrections*LconstantVolCorrection =

stockPrice@@mDD * HconstantVolStockUnits@@mDD - constantVolStockUnits@@m - 1DDL +

Exp@r * times@@mDDD * HconstantVolBondUnits@@mDD - constantVolBondUnits@@m - 1DDL;constantVolCorrectionIncrements = Append@constantVolCorrectionIncrements,

constantVolCorrectionD;constantVolCorrectionSum = Append@constantVolCorrectionSum,

constantVolCorrectionSum@@m - 1DD + constantVolCorrectionD;constantVolCorrectionPlot = Append@constantVolCorrectionPlot,

8times@@mDD, constantVolCorrectionSum@@mDD<D;

rollingVolCorrection =

stockPrice@@mDD * HrollingVolStockUnits@@mDD - rollingVolStockUnits@@m - 1DDL +

Exp@r * times@@mDDD * HrollingVolBondUnits@@mDD - rollingVolBondUnits@@m - 1DDL;rollingVolCorrectionIncrements = Append@rollingVolCorrectionIncrements,

rollingVolCorrectionD;rollingVolCorrectionSum = Append@rollingVolCorrectionSum,

rollingVolCorrectionSum@@m - 1DD + rollingVolCorrectionD;rollingVolCorrectionPlot = Append@rollingVolCorrectionPlot,

8times@@mDD, rollingVolCorrectionSum@@mDD<D;

, 8m, 2, Length@timesD<D;

H*Export plots*L

Export@FileNameJoin@8thesisPath, "GraphicsAndCode", StringJoin@stockTicker, "PriceValue.png"D<D,

ListLinePlot@8pricePlot, strikePlot, constantVolPortfolioValuePlot,rollingVolPortfolioValuePlot, payoutPlot<, AxesLabel ® 8"Time HdaysL", $<, PlotLabel ®

StringJoin@stockTicker, " Price and Replicating Portfolio Value over Time"D,PlotLegend ® 8Style@StringJoin@stockTicker, " price"D, FontSize ® 16D,

Style@"Strike price", FontSize ® 16D, Style@"Constant volatility\nportfolio value",FontSize ® 16D, Style@"Rolling volatility\nportfolio value", FontSize ® 16D,

Style@StringJoin@stockTicker, " price\nminus strike\nprice"D, FontSize ® 16D<,LegendPosition ® 80.85, -0.4<, LegendShadow ® None, BaseStyle ® 8FontSize ® 18<,AspectRatio ® 7.5 10.0, PlotRange ® AllD, ImageSize ® 72 * 15D;

Export@FileNameJoin@8thesisPath, "GraphicsAndCode", StringJoin@stockTicker, "Correction.png"D<D,

ListLinePlot@8constantVolCorrectionPlot, rollingVolCorrectionPlot<,AxesLabel ® 8"Time HdaysL", $<, PlotLabel ®

StringJoin@stockTicker, " Cumulative Correction of Replicating Portfolio over Time"D,

BlackScholesEuropeanCallHistoricalvF.nb 3


76

StringJoin@stockTicker, " Cumulative Correction of Replicating Portfolio over Time"D,PlotLegend ® 8Style@"Constant volatility\ncorrection", FontSize ® 16D,

Style@"Rolling volatility\ncorrection", FontSize ® 16D<,LegendPosition ® 81.1, -0.4<, LegendShadow ® None, BaseStyle ® 8FontSize ® 18<,AspectRatio ® 7.5 10.0, PlotRange ® AllD, ImageSize ® 72 * 15D;

Export@FileNameJoin@8thesisPath, "GraphicsAndCode", StringJoin@stockTicker, "Volatilities.png"D<D,

ListLinePlot@8constantVolPlot, rollingVolPlot<,AxesLabel ® 8"Time HdaysL",<, PlotLabel ®

StringJoin@stockTicker, " Annualized Constant and Rolling Volatility over Time"D,PlotLegend ® 8Style@"Constant\nvolatility", FontSize ® 16D,

Style@"Rolling\nvolatility", FontSize ® 16D<,LegendPosition ® 81.1, -0.4<, LegendShadow ® None, BaseStyle ® 8FontSize ® 18<,AspectRatio ® 7.5 10.0, PlotRange ® AllD, ImageSize ® 72 * 15D;

Export@FileNameJoin@8thesisPath, "GraphicsAndCode",StringJoin@stockTicker, "PortfolioComposition.png"D<D, ListLinePlot@

8constantVolStockUnitPlot, constantVolBondUnitPlot, rollingVolStockUnitPlot,rollingVolBondUnitPlot<, AxesLabel ® 8"Time HdaysL", "Units"<,

PlotLabel ® StringJoin@stockTicker, " Replicating Portfolio Composition over Time"D,PlotLegend ® 8Style@"Constant volatility\nstock units", FontSize ® 16D,

Style@"Constant volatility\nbond units", FontSize ® 16D,Style@"Rolling volatility\nstock units", FontSize ® 16D,Style@"Rolling volatility\nbond units", FontSize ® 16D<,

LegendPosition ® 81.1, -0.4<, LegendShadow ® None, BaseStyle ® 8FontSize ® 18<,AspectRatio ® 7.5 10.0, PlotRange ® AllD, ImageSize ® 72 * 15D;

4 BlackScholesEuropeanCallHistoricalvF.nb


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