stochastic models for asset pricing

Stochastic Models for Asset Pricing

By

Olomukoro Precious

July 2014

A RESEARCH PROJECT PRESENTED TO UNIVERSITY OF GHANA IN

PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD

OF MASTER OF SCIENCE IN MATHEMATICS

University of Ghana http://ugspace.ug.edu.ghUniversity of Ghana http://ugspace.ug.edu.gh

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DECLARATION

This work was carried out at University of Ghana in partial fulfilment of the requirements for a

Master of Science Degree.

I hereby declare that except where due acknowledgement is made, this work has never been

presented wholly or in part for the award of a degree at University of Ghana or any other

University.

Student: Olomukoro Precious

Supervisor: Prince K. Osei

i


DEDICATION

This study is dedicated to my Aunty Felicia Okoto and Prof. E. Addo.

ii


ACKNOWLEDEMENT

I thank God Almighty Creator of Heaven and Earth for his unmeasurable guidance throughout

this study period.

I offer my sincerest gratitude to my supervisor, Dr Prince. K. Osei, who has supported me

thoughout my study with his patience and knowledge whilst allowing me the room to work in

my own way. I attribute the level of my Masters degree to his encouragement and effort and

without him this study would not have been completed or written. One simply could not wish

for a better or friendlier supervisor.

Furthermore, I am grateful to Mrs Chisara Ogbogbo, for her advice and encouragement. In

addition, I thank all my classmates for everything we shared.

Many thanks go to my family for their immerse love, prayer and best wishes and in particular

Woyingimiebi Faith Caleb-Akah for her great support which was giving me more strength to

work harder every day to achieve this goal.

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Abstract

Stochastic calculus has been applied to the problems of pricing financial derivatives since 1973

when Black and Scholes published their famous paper ”The pricing of options and corporate

liabilities” in the journal of political economy. In this work, we introduce basic concepts of

probability theory which gives a better understanding in the study of stochastic processes, such

as Markov process, Martingale and Brownian motion. We then construct the Ito’s integral under

stochastic calculus and it was used to study stochastic differential equations. The lognormal

model was used to model asset prices showing its usefulness in financial mathematics. Finally,

we show how the famous Black-Scholes model for option pricing was obtained from the lognormal

asset model.

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Contents

Declaration i

Dedication ii

Acknowledgement iii

Abstract iv

Contents v

1 Introduction 1

1.1 Fundamental Concepts Of Probability Theory . . . . . . . . . . . . . . . . . . . . 2

1.2 Random Variables,Expectation And Variance . . . . . . . . . . . . . . . . . . . . 4

1.3 Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Change Of Probability Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Stochastic Processes 9

2.1 Classification Of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Stationary Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Markov Jump Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Transition Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Stochastic Calculus 29

3.1 Martingale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Discrete Time Martingale and Stopping Time . . . . . . . . . . . . . . . . . . . . 30

3.3 Continuous Time Martingales and Stopping Time . . . . . . . . . . . . . . . . . . 32

3.4 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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3.5 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.6 Ito’s Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7 Stochastic Differential Equations and Ito’s Formula . . . . . . . . . . . . . . . . . 39

4 Asset Pricing Models 44

4.1 Lognormal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Efficient Market Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Empirical Problem With The Lognormal Model . . . . . . . . . . . . . . . . . . . 46

4.4 Black-Scholes Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 Derivation Of The Black-Scholes Call Option Pricing Formula . . . . . . . . . . . 50

5 Conclusion 55

References 56

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1 Introduction

A stochastic process is a sequence of observation from a probability distribution. Rolling a dice

at regular time intervals is a stochastic process. Stochastic models are very crucial in financial

theory particularly in modern financial risk analysis.[1]

Stochastic modeling is the use of probability theory to model random real-world situations. It

plays an important role in clarifying many areas of the natural and engineering sciences. [2]

Asset models describes the prices or expected rate of returns of financial assets, since prices of

assets evolve in a random manner. This means that financial assets such as common stocks,

bonds, options and future contracts are largely unpredictable. An asset is defined as a right on

future cash flows and therefore, the value of an asset depends on these future cash flows. Asset

models prescribe financial relations between expected returns and measures of risk.[3]

The development of financial asset model theory over the years since Samuelson’s 1965 article

has enlaced the development of the theory of stochastic integration. A fundamental discovery oc-

curred in the early 1970’s when Black, Scholes and Merton proposed a method to price European

options through an explicit formula. They made use of Ito’s stochastic calculus and the Markov

property of diffusions in actualising it and their work brought order where previous pricing of

options had been done by intuition about ill defined market forces.[4]

Finance theory has produced a variety of models which affect financial decision making, in partic-

ular which affect the future prices of assets. We model asset prices for various reasons. One of the

reasons is that modelling of asset price dynamics is very crucial for the valuation of derivatives

such as equities, and index options. Secondly, it is a powerful tool for risk management. Thirdly,

simulated asset prices that keep track of historical asset return parameters such as annualised

mean and standard deviation can be employed as market data for back testing trading strategies

[5].

Other significant models include the Wilkie model as described by Wilkie [6], which is a partic-

ular example of time series model, TY model as described by Yakoubov et al [7], the binomial

models [8], the capital asset pricing model, the lognormal model and many others.

The structure of this essay is as follows. Chapter 1 gives an introduction on stochastic mod-

els for assets and introduces basic probability concepts which gives a better understanding on

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stochastic calculus. In chapter 2, we provide a brief preliminaries to stochastic processes and

some mathematical tools with the introduction of Markov processes. In chapter 3, stochastic

calculus, introducing the concept of stochastic integrals, including an important lemma :Ito’s

lemma and then the theory of stochastic differential equations. Chapter 4 focuses on stochastic

models for assets, beginning with the lognormal model, its properties and limitations and then

the derivation of the Black-Scholes call option pricing model. Finally, chapter 5 gives a conclusion

of the work.

1.1 Fundamental Concepts Of Probability Theory

A random variable is a function with random values which maybe a lifetime of an individual or

the number of car accidents in a year. In other to model such randomness mathematically, it is

convenient to assume that a random variable is a function defined somewhere on the space of all

possible states of the world W .

For the discrete case, a probability space is a unique triple (Ω,F , P ) where Ω is its sample space,

F its σ-algebra of events and P is probability measure [9].

The sample space Ω is the set of all possible samples or elementary events w : Ω = w|w ∈ Ω.

The σ-algebra F is the set of all the considered events i.e subsets of Ω : F = A|A ⊆ Ω, A ∈ F.

The probability measure P assigns a probability P (A) to every event A ∈ F : P → [0, 1]. It is

therefore necessary that P has the following properties

i.) P (Ω) = 1,P (φ) = 0

ii.) P (Ac) = 1− P (A) ∀A ∈ F

iii.) P (A ∪B) = P (A) + P (B)− P (A ∩B) ∀A,B ∈ F

Here φ is an impossible event, A and B are possible events and the superscript c denotes the

complement of an event.

Further, on a discrete probability space, one cannot construct an infinite sequence of independent

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events such as an infinite number of coin tosses, then

Ω = w : w = (a1, a2, ....), ai = 0, 1

Where ai is the result of the ith toss which represents 1 if the coin lands head up and 0 otherwise.

1.1.1 σ-algebra

Let Ω be a non-empty set and F be a collection of subsets of Ω. Then F is a σ-algebra if it

satisfies the following properties:

i.) Ω, φ ∈ F

ii.) If A1, A2, ....,∈ F , then∞⋃n=1

An ∈ F and∞⋂n=1

An ∈ F

iii.) If A ∈ F , then Ac ∈ F

The key feature here is that a σ-algebra is closed under infinite countable unions and intersections

but not closed under uncountable unions.[10]

1.1.2 Example

Let M =

A =

∞⋃i=1

[ni,mi] : ni,mi ∈ R

Then M is a σ-algebra

i.) If ni = mi, φ ∈M

ii.) ∀ A ∈M, Ac has the same form of an element in M

iii.) ∀ A,B ∈M, A ∪B has the same form of an element in M.[11]

1.1.3 Probability Measure

Let Ω be a non empty set and F be some σ-algebra of subsets of Ω. Suppose that a function

P : F → [0, 1] satisfies

1.) P (0) = 0, P (Ω) = 1

2.) P

(∞⋃n=1

An

)=∞∑n=1

P (An)

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Provided A1, A2, ....,∈ F is a collection of disjoint sets i.e Ai ∩ Aj = φ ∀i 6= j. Then P is a

probability measure and the triple (Ω,F , P ) is a probability space.

1.2 Random Variables,Expectation And Variance

1.2.1 Random Variable

Given a probability space (Ω,F , P ), a mapping

ξ : Ω→ R

is a random variable if it is measurable i.e satisfies

w : ξ(w) ∈ S ∈ F

for all Borel sets S ∈ B(R).[12]

Borel sets are the elements of a Borel σ-algebra, the minimum σ-algebra containing all open sets.

For example, the simplest random variables are indicators. If A is an event, the indicator function

of A is the random variable [13]

IA(w) =

0 ,w ∈ A

1 ,w ∈ Ac

Indicators can be used to define simple random variables represented as

ξ =N∑k=1

akIAk

The expectation of a random variable is the average of its possible values weighted according to

their probabilities i.e for some events Ak and real numbers ak, k = 1, 2, ....N

E[ξ] =N∑k=1

akE[IAk] ≡ E[ξ2]− E[ξ]2

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The variance of a random variable ξ is defined by

V ar(ξ) = E[(ξ − E[ξ])2] ≡ E[ξ2]− E[ξ]2

The square root of the variance√V ar(ξ) is called the standard deviation of ξ.

For two random variables ξ, η their covariance is defined by

Cov(ξ, η) = E[(ξ − E[ξ])(η − E[η])] ≡ E[ξη]− E[ξ]E[η]

For a sum of two random variable, we have

V ar(ξ + η) = E[((ξ − E[ξ]) + (η − E[η]))2]

= E[(ξ − E[ξ])2] + 2E[(ξ − E[ξ])(η − E[η])] + E[(η − E[]η)2]

= V (ξ) + 2Cov(ξ, η) + V (η)

Covariance is close to zero if the random variable are almost independent and equal to zero if

they are independent.

For random variables ξ and η such that V ar(ξ) and V ar(η) > 0, define the correlation of ξ and

η by

Corr(ξ, η) =Cov(ξ, η)√

V ar(ξ)√V ar(η)

The correlation equals 1 if and only if ξ = aη with some constant a 6= 0.

1.3 Probability Distribution

Given a random variable ξ, its cumulative distribution function is defined by

Fξ = P (ξ ≤ x)

Clearly, the distribution function is increasing in x and

limx→−∞

Fξ(x) = 0 , limx→+∞

Fξ(x) = 1

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A random variable ξ is called continuously distributed if its distribution function is continuous.

[13]

The expectation of a random variable can be using its distribution function by

E[ξ] =

∫ ∞−∞

xdFξ(x) =

∫ ∞−∞

xPξ(x)dx

If for some random variable ξ and function f : R → R, the expectation E[f(ξ)] exists, then

E[f(ξ)] =

∫ ∞−∞

f(x)dFξ(x) =

∫ ∞−∞

f(x)Pξ(x)dx

Where Pξ is called the probability density function of ξ

The variance of a continuous random variable is given by

V ar(ξ) =

∫ ∞−∞

(x− E[ξ])2dFξ(x) =

∫ ∞−∞

(x− E[ξ])2Pξ(x)dx

1.3.1 Independence

Two events A and B are independent if

P (A ∩B) = P (A)P (B)

For random variables, we have the following definitions

1.3.2 Independent Random Variables

Random variables ξ and η are called independent if the events

A = w ∈ Ω : a < ξ(w) < b

B = w ∈ Ω : c < η(w) < d

are independent for all real numbers a, b, c, d.[13]

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1.3.3 Mutually Independent Events

Events A1, ...., An are called mutually independent if

P (Ai1 ∩ ..... ∩ Aik) = Πks=1P (Aik)

For any k = 1, 2, ...., n and any 1 ≤ i1 < ..... < ik ≤ n

1.3.4 Mutually Independent Variables

Random variables ξ1.....ξn are mutually independent if the events

A1 = w ∈ Ω : a1 < ξ1(w) < b1

An = w ∈ Ω : a1 < ξn(w) < bn

are mutually independent for all real number ak, bknk=1

1.4 Change Of Probability Measure

If two probability measures are defined on the same set Ω and σ-algebra F , then (Ω,F , P ) and

(Ω,F , Q) are both probability spaces. In other words,change of measure implies having two

probability spaces (Ω,F , P ) and (Ω,F , Q) with the same Ω and F . Random variables are by

definition F -measurable mappings from Ω to R. So a random variable on ane probability space

can be considered as a random variable on the other probability space.[14]

1.4.1 Definition

Let (Ω,F , P ) be a probability space and Q another probability measure on F .

i.) We say that Q is absolutely continuous with respect to P (denoted Q P ) if P (A) = 0

implies Q(A) = 0 for all A ∈ F

ii.) We say that Q is equivalent to P (denoted P ∼ Q) when P (A) = 0 if and only if Q(A) = 0

for all A ∈ F . In other words,P ∼ Q if and only if Q P and P Q.

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1.4.2 Theorem (Radon Nikodym)

Let (Ω,F , P ) be a probability space and Q be another probability measure on F . If Q P ,

then there exists a random variable

dQ

dP: Ω→ [0,∞]

called the Radon-Nikodym derivative.

EQ[ξ] = EP

[ξdQ

dP

](1.4.1)

for all ξ such that EQ[|ξ|] <∞ In particular, Q(A) =

∫A

dQ

dPdp for all A ∈ F and so E[dQ

dP] = 1

Moreover, if P Q, then P(dQdP

> 0)

= 1 and dQdP

=(dQdP

)−1Vice versa, if equation (1) holds for

all bounded random variables ξ with some non-negative random variable dQdP

, then there exist a

probability measure Q P and dQdP

is its Radon-Nikodym derivative. Moreover, if dQdP

> 0, then

Q ∼ P .[15]

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2 Stochastic Processes

A stochastic process with state space S is a collection of random variables Xt, t ∈ T defined

on the same probability space (Ω,F , P ). The set T is called its parameter set. If T = N =

0, 1, 2, ...., the process is said to be a discrete parameter. If T is not countable, the process is

said to have a continuous parameter [1].

Usual examples are T = R+ = [0,∞) and T = [a, b] ⊂ R. The index t represents time and Xt

as the state or position of the process at time t.

2.1 Classification Of States

There are many classification of random process depending on the continuous or discrete nature

of the state space S and parameter set T . By definition, State space S is the set of possible

values of a random variable Xt. Thus, a random process can be classified as follows:

Discrete State Space With Discrete Time Changes:

If both T and S are discrete, the random process is called a discrete random sequence.

For example, if Xn represents the outcome of the nth toss of a fair dice, then Xn, n ≥ 1 is a

discrete random sequence, since T = 1, 2, 3, ..., and S = 1, 2, 3, 4, 5, 6

Discrete State Space With Continuous Time Changes:

If T is continuous and S is discrete, the random process is called a discrete random process.

For example, if X(t) represents the number of telephone calls received in the interval (0, t) then

X(t) is a discrete random process, since S = 0, 1, 2, 3......

Continuous State Space With Continuous Time Changes:

If both T and S are continuous, the random process is called a continuous random process.

For example, if X(t) represents the maximum temperature at a place in the interval (0, t), X(t)

is a continuous random process.

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Continuous State Space With Discrete Time Changes:

If T is discrete and S is continuous,then the random process is called a continuous random se-

quence.

For example, if Xn represents the temperature at the end of the nth hour of a day, then

Xn, 1 ≤ n ≤ 24 is a continuous random sequence, since temperature can take any value in

an interval and hence continuous.[2]

2.1.1 Filtration

Let (Ω,F , P ) be a probability space. A filtration on (Ω,F , P ) is a sequence of σ-algebra Ftt ∈ T

if Fs ⊂ Ft ⊂ F for all s, t ∈ T such that s ≤ t.

Given a filtration Ftt ∈ T on a probability space (Ω,F , P ), a random process X = Ftt ∈ T

is said to be adapted to the filtration Ftt ∈ T if the random variable Xt are Ft-measurable for

each t ∈ T

If a random process X = Ftt ∈ T is adapted to a filtration Ftt ∈ T , then for all s ≤ t ∈ T ,

the random variables Xs are Fs-measurable, which follows from the fact that Fs ⊂ Ft and so

any Fs-measurable random variable is Ft-measurable.[1]

2.1.2 Natural Filtration

If X = Ftt ∈ T is a stochastic process, the natural filtration of X is defined by

FXt = σ(Xs : s ∈ T , s ≤ t) t ∈ F

as the minimal σ-algebra such that all Xs with s ∈ T , s ≤ t are FXt -measurable.[1]

A stochastic process X is always adapted to its natural filtration FXt = σ(Xs, s ≤ t).(FXt ) is

hence the smallest filtration to which X is adapted.

The parameter t is often thought as time and the σ-algebra Ft represent the set of information

available at time t i.e events that have occurred up to time t. The filtration (F)t ≥ 0 thus

represent the evolution of the information or knowledge of the world with time. If X is an

adapted process, then Xt, its value at time t only depends on the evolution of the universe prior

to t.

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2.1.3 Predictable Process

A random process C = Ct∞t=0 is predictable with respect to a filtration Ft∞t=0 if all Ct are

Ft−1 measurable.

It follows that, if a process is predictable with respect to a filtration, then it is adapted with

respect to the same filtration.[1]

For any discrete time predictable process Ct∞t=0, we have

E[f(Ct)Xt|Ft−1] = f(Ct)E[Xt|Ft−1]

for all Xt and any continuous function f .

2.2 Markov Process

A Markov process is a stochastic model that has the Markov property. It can be used to model a

random system that changes states according to a transition rule that only depend on the current

state. It is a stochastic process in which a sequence of states is visited one after another. At

each point, the decision about which state to visit next is made at random. One of the defining

features of a Markov process is that it has no memory;the decision about which states to visit

next depends only on the identity of the current state. All the transition probabilities can be

expressed in the form of a matrix, with elements pij giving the probability of going to state j,

given that the present state is i. In particular, we distinguish between two types of stochastic

process that possess the Markov property: Markov chains and Markov jump processes. Both have

a discrete state space, but Markov chains have a discrete time set and Markov jump processes

have a continuous time set.

2.2.1 The Markov Property

A stochastic process has the Markov property if the development of the process can be predicted

from its current state; that is given the present, the future does not depend on the past. The

Markov property can be stated mathematically as:

P [Xt ∈ A|Xs1 = x1, Xs2 = x2, ...., Xsn = xn, Xs = x] = P [Xt ∈ A|Xs = x] (2.2.1)

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for all s1 < s2 < .... < sn < s < t in the time set, all states x1, x2, ..., xn, x in the state space S

and all subsets of A of S.

In the case where S is a discrete state space, the Markov property is written as:

P [Xt ∈ a|Xs1 = x1, Xs2 = x2, ...., Xsn = xn, Xs = x] = P [Xt ∈ a|Xs = x] (2.2.2)

for all s1 < s2 < .... < sn < s < t and all states a, x1, x2, ..., xn, x in S. An increment of a process

is the amount by which its value changes over a period of time. For example, Xt+u − Xt for

u > 0 is an increment. A process is said to have independent increments if for every u > 0, the

increment Xt+u −Xt is independent of all the past values of the process Xs for 0 ≤ s ≤ t

Any process with independent increments has the Markov property.[10]

2.3 Markov Chains

A Markov chain is a sequence of random variables X1, X2, X3, .... with Markov property namely

that the probability of any given state Xn only depends on its immediate previous state Xn−1.

Formally:

P (Xn = x|Xn−1 = xn−1, ...., X1 = x1) = P (Xn = x|Xn−1 = xn−1) (2.3.1)

The possible values of Xi form a countable set S called the state space of the chain. If the

state space is finite and the Markov chain time-homogeneous (i.e the transition probabilities

are constant in time), the transition probability distribution can be represented by a Matrix

P = (Pij)i,j ∈ S called the transition matrix whose elements are defined as

Pij = (n, n+ 1) = P [Xn+1 = j|Xn = i] (2.3.2)

Therefore Pij(n, n+ 1) is the probability of being in state j at time n+ 1, having been in state i

at time n.

For each fixed n = 0, 1, .... we can form a matrix of transition probabilities from time n to the

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next time step n+ 1:

P (n, n+ 1) = [Pij = (n, n+ 1)]i,j ∈ S (2.3.3)

P (n, n+ 1) is a finite matrix in the case of a finite number of states and an infinite matrix in the

case of an infinite number of states.[10]

2.3.1 Time-Inhomogeneous Markov Chains:

For a time-inhomogeneous Markov chain, the transition probability

Pij(t, t+ 1)

change with time t. The transition probability will therefore have a sequence of stochastic

matrices denoted by P (t)

P (t) = [Pij(t, t+ 1)]i,j ∈ S =

P00(t, t+ 1) P01(t, t+ 1) · · ·

P10(t, t+ 1) P11(t, t+ 1) · · ·...

......

The value of t can represent many factors such as time of year, age of policy holder or length of

time the policy has been in force.

2.3.2 Time-homogeneous Markov Chains

A Markov chain is called time-homogeneous if the transition probabilities Pij(n, n + 1) do not

depend on n and therefore we can simply write

Pij(n, n+ 1) = Pij

and call Pij the one step transition probability from state i to state j, while

P = [Pij]i,j ∈ S

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is called the one step transition matrix or simply transition matrix. Then

P (m,n) = P × P × .....× P = P n−m

for all m ≤ n and the semigroup property is now even more obvious because

P a+b = P aP b

2.3.3 The Simple (Unrestricted) Random Walk

A simple random walk is a stochastic process Xt of independent, identically distributed discrete

random variables with state space S = Z given by

Xn = Y1 + Y2 + .....+ Yn

such that

P (Yi = 1) = p and P (Yi = −1) = 1− p

The simple random walk possesses the Markov property given by:

P (Xm+n = j|X1 = i1, X2 = i2, ...., Xm = i)

= P (Xm + Ym+1 + Ym+2 + ....+ Ym+n = j|X1 = i1, X2 = i2, ...., Xm = i)

= P (Ym+1 + Ym+2 + ....+ Ym+n = j − i)

= P (Xm+n = j|Xm = i)

Thus,a simple random walk is a time homogeneous Markov chain with transition probabilities

Pij =

p if j=i+1

1− p if j=i-1

0 otherwise

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A simple random walk also possesses an additional property, namely that of spatial homogenity,

that is

P(n)ij = P (Xn = j|Xo = i) = P (Xn = j + r|Xo = i+ r) (2.3.4)

This means that the transition probability Pij should depend on i and j only through their

relative positions in space.

2.3.4 The Restricted Random Walk

The restricted random walk is a simple random with boundary conditions limited to the interval

[a, b]. The endpoints a and b are called absorbing barriers if the random walk eventually stays

there forever or reflecting barriers if the walk reaches the endpoint and bounces back.

More formally, an absorbing barrier is a value b such that

P (Xn+s = b|Xn = b) = 1 for all s > 0 (2.3.5)

That is, once state b is reached, the random walk stops and remains in this state there after.

A reflecting barrier is a value c such that

P (Xn+1 = c+ 1|Xn = c) = 1 for all s > 0 (2.3.6)

That is, once state c is reached, the random walk is pushed away

A mixed barrier is a value d such that

P (Xn+1 = d|Xn = d) = α and P (Xn+1 = d+ 1|Xn = d) = 1− α (2.3.7)

for all s > 0, α ∈ [0, 1]

That is, once state d is reached, the random walk remains in this state with probability α or

moves to the neighbouring state d + 1 with probability 1 − α, implying that it is an absorbing

barrier with probability α and reflecting barrier with probability 1− α.

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2.4 Stationary Distributions

Suppose a distribution π on S is such that, if a Markov chain starts out with initial distribution

πo = π, then we also have π1 = π. That is, if the distribution at time 0 is π, then the distribution

at time 1 is still π. Then π is called a stationary distribution for the Markov chain.[10]

2.4.1 Definition

The distribution πjj∈S is said to be a stationary distribution of a Markov chain with transition

matrix P if

1. πj =∑i∈S

πiPij for all j which can be expressed as π = πP , where π is a row vector and πP

is the usual vector-matrix product and

2. πj ≥ 0 for all j and∑j∈S

πj = 1

Given the interpretation of π is that, if the initial probability distribution i.e πi = P (Xo = i),

then at time 1, the probability distribution of X1 is still given by π. Mathematically, we have

P (X1 = j) =∑i∈S

P (X1 = j|Xo = i) P (Xo = i)

=∑i∈S

πiPij = πj

Also,

P (Xn = j) =∑i∈S

P (Xn = j|Xn−1 = i) P (Xn−1 = i)

=∑i∈S

πiPij = πj

Hence, a Markov chain started out in a stationary distribution π stays in the distribution π

forever; thats why the distribution is called stationary.

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2.4.2 Example

Consider, a Markov chain with state space S = 0, 1, 2 and transition matrix

P =

p q 0

14

0 34

p− 12

710

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(a) Calculate values for p and q

(b) Draw the transition graph for the process

(c) Calculate the transition probabilities P(3)i,j

(d) Find any stationary distribution for the process

(a) The sum of all entries in the last row must be equal to 1, as a consequence of which

p = 1− 1

5− 7

10+

1

2=

3

5

In view of the first row, we see that q = 25.

(b) We draw the transition graph for the process.

(c) We calculate the transition probabilities P(3)i,j . These are entries of the 3-step transition

matrix,P 3 which we compute using the matrix form of the Chapman-Kolmogorov equations.

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So

P 3 = P (2+1) = P (2).P = P (1+1).P = (P.P ).P

Computing these products gives us

P =

0.366 0.394 0.24

0.30625 0.195 0.49875

0.2545 0.4975 0.248

(d) Now we find any stationary distribution for the process. As mentioned, if π = (π1, π2, π3)

is a stationary distribution for the process, then it satisfies the following equations

π = πP

1 = π1 + π2 + π3

which implies that

π1 =3

5π1 +

1

4π2 +

1

10π3 (2.4.1)

π2 =2

5π1 +

7

10π3 (2.4.2)

π3 =3

4π2 +

1

5π3 (2.4.3)

1 = π1 + π2 + π3 (2.4.4)

Substituting equation (2.4.1) in equation (2.4.3) yields to the equation

4

5π3 =

3

4

(2

5π1 +

7

10π3

)

which after developing and solving for π3 gives

π3 =12

11π1 (2.4.5)

and substituting equation (2.4.5) in equation (2.4.2) and solving for π2 yields to

π2 =64

55π1 (2.4.6)

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Now we have the two unknowns π2 and π3 in terms of π1, then substituting equations

(2.4.5) and (2.4.6) in equation (2.4.4) and solving for π1 yields to

π1 =55

179

from which we obtain

π2 =64

179, π3 =

60

179

Hence, a stationary distribution of the process is given by

π =

(55

179,

64

179,

60

179

)

2.4.3 The Long-Term Behaviour Of Markov Chains

Consider a Markov chain with state space S and transition matrix P

a. A state i ∈ S is said to have period d if

gcd(n ≥ 1 : P nii > 0) = d

Where the right hand side is the greatest common divisor of the integers n ≥ 1 that satisfy

P nii > 0

b. If a state i ∈ S has period 1, then it is said to be aperiodic

c. If every state of a Markov chain is aperiodic, then the chain is said to be aperiodic

In particular, if a state i has period d, then it is impossible to go from i back to i in n steps, if

n is not divisible by d.

It turns out if a Markov chain is both irreducible and aperiodic, its long-term behaviour can be

determined.

2.4.4 Theorem

Consider an irreducible, aperiodic Markov chain with state space S = 0, 1, ..., N and transition

matrix P . Then limn→∞

P n exists. In otherwords, there exist numbers π0, π1, ..., πN such that for

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each j ∈ S

πj = limn→∞

P nij for all i ∈ S (2.4.7)

Moreover, (π0, π1, ..., πN) is the unique non negative solution to the systems of linear equations

πj =∑i∈S

πiPij for all j ∈ S

=∑j∈S

πj = 1

2.5 Markov Jump Processes

Markov jump process are continuous time Markov chains defined on continuous state space. Any

jump process has embedded a marked point process. These processes are called regular step

process.

However, the notion of a one-step transition probability does not exist in Markov jump process

due to its continuous state and therefore leads to the consideration of time intervals of arbitrarily

small length. Taking the limits of these equations lead to the reformation of the Chapman-

Kolmogorov equations in terms of differential equations.

2.5.1 Poisson Process

A stochastic process Ntt∈[0,∞) is said to be a counting process if N(t) represents the total num-

ber of events that have occurred up to time t. The Poisson process is basically a counting process.

That is, it has state space S = 0, 1, 2, ..., n, ... corresponding to the number of occurrences of

an event.

Before the definition of a poisson process, recall that a function f is said to be o(h) if

limh→0

f(h)

h= 0 (2.5.1)

That is f is o(h) if, for some small values of h, f(h) is small even in relation to f .[2]

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2.5.2 Definition

The counting process Ntt∈[0,∞), with filtration Ft is said to be a poisson process with rate λ > 0

if

1. N0 = 0

2. The process has time-homogeneous and independent increments

3. P (Nt+h −Nt = 1|Ft) = λh+ o(h)

4. P (Nt+h −Nt = 0|Ft) = 1− λh+ o(h)

5. P (Nt+h −Nt > 1|Ft) = o(h)

The process has the Markov property, that is Nt+h is independent of the number of occurrences

in (0, t].

The number of events in any interval of length t is poisson distributed with mean λt. That is for

all s, t ≥ 0

P N(t+ s)−N(s) = n = e−λt(λt)n

n!, n = 0, 1, 2, ....

Hence for any t, s > 0, Nt+s −Ns has the same probability distribution as Nt.[1]

2.5.3 Interarrival Times

Consider a Poisson process and let X1 denote the time of the first event and for n ≥ 1, let Xn

denote the time between the (n − 1)st and the nth event. It is clear that Xn for n ≥ 1 is a

continuous random variable which takes values in the range [0,∞)

The sequence Xnn≥1 is called the sequence of interarrival times.

Further, the distribution of Xn can be determined. First note that the event X1 > t takes

place if and only if, no events of the poisson process occur in the interval (0, t] and thus

P X1 > t = P N(t) = 0 = e−λt (2.5.2)

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Hence, X1 has an exponential distribution with parameter λ. To obtain the distribution of X2

conditional on X1 gives

P X2 > t|X1 = S = P 0 events in (s, s+ t]|X1 = S

= P (Nt+s −Ns = 0|X1 = S)

= P (Nt+s −Ns = 0), (by independent increments)

= Po(t) = e−λt

Therefore, from the above,X2 is also an exponential random variable with parameter λ and

furthermore, X2 is independent of X1.

Repeating the same arguement for X3, X4, .... leads to the conclusion that the interarrival times

are independent and identically distributed random variables that are exponentially distributed

with parameter λ.

2.5.4 The Time-inhomogeneous Markov Jump Process

The transition probability for a time-inhomogeneous Markov jump process is given as

Pij(s, t) = P [Xt = j|Xs = i], where Pij(s, t) ≥ 0 and s < t (2.5.3)

The transition probabilities should also satisfy the Chapman-Kolmogorov equations given as

Pij(t1, t3) =∑k∈S

Pik(t1, t2)Pkj(t2, t3), for t1 < t2 < t3 (2.5.4)

and it can be expressed in matrix form as

limt→S+

Pij(s, t) = δij =

1, i = j

0, i 6= j

This condition implies that as the time difference between the two observations approach zero,

the process will likely not change its state with probability one in the limit. It follows that this

condition is consistent with the Chapman-Kolmogorov equations and thereby taking the limits

t2 → t−3 or t2 → t+1 in the Chapman-Kolmogorov equations gives the identity.

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However, the condition does not follow from the Chapman-Kolmogorov equations.

2.5.5 Time-homogeneous Markov Jump Process

Consider the transition probabilities for a Markov process given by

Pij(s, t) = P [Xt = j|Xs = i], where Pij(s, t) ≥ 0 and s < t (2.5.5)

A Markov process in continuous time is called time-homogeneous if the transition probabilities

Pij(s, t) = Pij(0, t− s) for all i, j ∈ S s, t > 0 (2.5.6)

That is a Markov process in continuous time is called time-homogeneous if the probability

P (Xt = j|Xs = i) depends only on the time interval t− s. Therefore, we write

Pij(s, t) = P (Xt = j|Xs = i) = Pij(t− s)

Pij(t, t+ s) = P (Xt+s = j|Xt = i) = Pij(s)

Pij(0, t) = P (Xt = j|X0 = i) = Pij(t)

The transition probability Pij(s) for instance forms a stochastic matrix for every S, that is

Pij(s) ≥ 0 and∑j∈S

Pij(s) = 1

which also satisfies the continuity conditions at S = 0 given as

lims→0+

Pij(s) = Pij(0) = δij =

1, i = j

0, i 6= j

For a time-homogeneous Markov process, the transition probability Pij(s) satisfy the Chapman-

Kolmogorov equations in the form

Pij(t+ s) =∑k∈S

Pik(t)Pkj(s)

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which in matrix form becomes

P (t+ s) = P (t)P (s)

A time-homogeneous process has some properties of transition functions and transition rates

which are stated as follows

1. Transition rates qij =dPij(t)

dt|t=0 = lim

h→0

Pij(h)− δijh

exist ∀ i, j

Equivalently, as h→ 0, h > 0

Pij(h) =

hqij + o(h), i 6= j

1 + hqii + o(h), i = j

2. Transition rates are non negative and finite for i 6= j and are non-positive when i = j, that

is

qij ≥ 0 for i 6= j but qii ≤ 0 for i = j

Differentiating∑j∈S

Pij(t) = 1 with respect to t at t = 0 yields

qii = −∑j 6=i

qij

3. If the set of states S is finite, all transition rate are finite.

2.6 Transition Rates

Assume that the transition probabilities Pij(s, t) for t < s have derivatives with respect to t and

s and also that the state space S is finite. Then by the standard definiton of derivatives, we have

∂Pij(s, t)

∂t= lim

h→0

Pij(s, t+ h)− Pij(s, t)h

= limh→0

∑k Pik(s, t)Pkj(t, t+ h)− Pij(s, t)

h

= limh→0

(∑k 6=j

Pik(s, t)Pkj(t, t+ h)

h+ Pij(s, t)

Pjj(t, t+ h)− 1

h

)= lim

h→0αij

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Also, by definition

limh→0

Pij(t, t+ 1)− 1

h= qjj(t)

limh→0

Pkj(t, t+ 1)− 1

h= qkj(t) for k 6= j (2.6.1)

The quantities qjj(t) and qkj(t) are called transition rates, corresponding to the probability of

transition from state k to state j in time interval h.

From equation (2.6.1), it follows that

∂Pij(s, t)

∂t=∑k∈S

Pik(s, t)qkj(t) (2.6.2)

Where these differential equations are called the Kolmogorov forward equations which can also

be written in matrix form as

∂P (s, t)

∂t= P (s, t)Q(t) (2.6.3)

Here, Q(t) is the generator matrix with entries qij(t).

Repeating the procedure but now differentiating with respect to s, we have

∂Pij(s, t)

∂s= lim

h→0

Pij(s+ h, t)− Pij(s, t)h

= limh→0

Pij(s+ h, t)−∑

k Pik(s, s+ h)Pkj(s+ h, t)

h

= − limh→0

(∑k 6=i

Pik(s, s+ h)

hPkj(s+ h, t) +

Pii(s, s+ h)− 1

hPij(s+ h, t)

)Implying that

∂Pij(s, t)

∂s= −

∑k∈S

qik(s)Pkj(s, t) (2.6.4)

Where the differential equations (2.6.4) are called the Kolmogorov’s backward equations, written

in matrix form as

∂Pij(s, t)

∂s= −Q(s)P (s, t)

Hence, the derivative with respect to s can be expressed in terms of transition rates.

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2.7 Applications

2.7.1 Survival Model

A survival model is a two state model where the two states are alive and dead implying that

transition is just in one direction. That is, from state alive (A) to the state dead (D) with

transition rate µ(t) and has discrete state space S = A,D.

The generator matrix Q(t) is given by

Q(t) =

−µ(t) µ(t)

0 0

For the survival model, the Kolmogorov forward equation is given as

∂PAA(s, t)

∂t= −µ(t)PAA(s, t) (2.7.1)

Where the solution corresponding to the initial condition PAA(s, s) = 1 is given by

PAA(s, t) = e−∫ ts µ(x)dx

Here, PAA(s, t) is the probability that an individual alive at time s, will still be alive at time t.

Now, considering the probability for an individual aged s to survive for a further period of length

at least s+ w, denoted wPs is

wPs = PAA(s, s+ w) = e−∫ s+ws µ(x)dx = e−

∫ w0 µ(s+y)dy (2.7.2)

2.7.2 Sickness-Death Model

Considering the sickness-death model, the state of an individual is being described as healthy

(H), sick (S) or dead with the discrete state space given as S = H,S,D. It is possible that an

individual in state H can jump to either state S or state D. Also, an individual in state S can

jump to either state H or state D. Thus having the following age-dependent transition rates

H → S : σ(t)

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H → D : µ(t)

S → H : ρ(t)

S → D : ν(t)

Hence, the generator matrix is given as

Q(t) =

−(σ(t) + µ(t)) σ(t) µ(t)

ρ(t) −(ρ(t) + ν(t)) ν(t)

0 0 0

However, probabilities such as

i. The probability that an individual who is healthy at time s will still be healthy at time t

or

ii. The probability that an individual who is sick at time s will still be sick at time t can be

calculated in terms of the residual holding times as

P (Rs > t− s|Xs = H) = e−∫ ts (δ(u)+µ(u))du (2.7.3)

and

P (Rs > t− s|Xs = S) = e−∫ ts (ρ(u)+ν(u))du (2.7.4)

respectively. The sickness-death model can be extended to include the length of time an

individual has been in state S, thus leading to the ”long term care model”, where the rate

of transition out of state S will depend on the current holding time in state S.

2.7.3 Marriage Model

The state of an individual under the marriage model can be described as either;never married

(B), married (M), divorced (D), widowed (W ) or dead (∆). Hence, a markov jump process can

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be derived on the state space S as

S = B,M,D,W,∆ . (2.7.5)

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3 Stochastic Calculus

Stochastic processes with discrete or continuous time are mathematical methods used in the

study of random variables. It has applications in physical, biological and medical science, as well

as in economics and social sciences. Brownian motion is the most important stochastic process

and stochastic calculus for Brownian motion is the major part of the stochastic analysis.

3.1 Martingale

A martingale is a stochastic process for which its current value is the best estimator of its future

value. That is, a process as a function of time, has the property that its value at any time t is

the conditional expectation of the value at a future time given the information available at t.

Thus, the theory of martingale depends on the application of conditional expectation [15]

.

3.1.1 Properties (Conditional Expectation)

A number of properties of conditional expectation which are essential for the understanding and

using martingales are stated without proof as follows:

Let ξ be a random variable on the probability space (Ω,F ,P) and Ft be a filtration on F .

Then

1. The expected value of ξ and of E[ξ|F ] are equal;

E[E(ξ|F)] = E[ξ]

2. If ξ is an Ft-measurable random variable, then:

E[ξ|Ft] = ξ

3. If ξ is finite, Ft-measurable random variable, then for any random variable η:

E[ξ.η|Ft] = ξE[η|Ft]

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4. If ξ is independent of Ft, then:

E[ξ|Ft] = E[ξ]

5. Suppose F1 ⊂ F2 where F1,F2 ⊂ F . Then

E[E(ξ|F2)|F1] = E[ξ|F1]

This is known as the Tower Property of conditional expectation.

6. Let ξ be a random variable on probability space (Ω,F ,P) such that E[ξ2] < ∞. E(ξ|Ft)

is the best estimator for ξ if for any Ft-measurable random variable ζ

E[(ζ − E[ξ|Ft])2] ≤ E[(ξ − ζ)2]

3.2 Discrete Time Martingale and Stopping Time

Under the discrete time martingale, all random variables are defined on the same probability

space (Ω,F ,P).

3.2.1 Definition (Martingale)

Given a probability space (Ω,F ,P), equipped with a filtration Ft∞t=0, then a stochastic process

ξ = ξt∞t=0 with values in R is a martingale if

1. E[|ξt|] <∞ for all t = 0, 1, ...., T

2. E[ξt|Ft−1] = ξt−1 for all t = 0, 1, ..., T

The second condition is most relevant and often called the martingale property. Thus applying the

tower property of conditional expectation leads to the extrapolation of the martingale property

given as

E[ξt|Ft−k] = E[E[ξt|Ft−1]|Ft−k] = E[ξt−1|Ft−k] = ... = ξt−k (3.2.1)

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for all t ≥ k

Specifically, it is observed that

E[ξt] = E[ξt−1] = .... = E[ξ0]

inother that the mean of the martingale is constant in time.

3.2.2 Processes Relative To Martingale

Let ξ = ξtt=0 be a stochastic process adapted to the filtration F = Ft∞t=0 such that E[|ξt|] <

∞ for all t = 0, 1, ... then

a.) ξ is said to be a super-martingale if

E[ξt+1|Ft] ≤ ξt for all t = 0, 1, 2.....

b.) ξ is said to be a sub-martingale if

E[ξt+1|Ft] ≥ ξt for all t = 0, 1, 2.....

c.) ξ is said to be a martingale difference if

E[ξt+1|Ft] = 0 for all t = 0, 1, 2.....

It therfore implies that

• ξ is a sub-martingale if and only if −ξ is a super-martingale and

• ξ is a martingale differnce if and only if the processt∑

s=0

ξs is a martingale.

3.2.3 Stopping Time

Given a probability space (Ω,F ,P) anda filtration F = Ft∞t=0. A random variable τ is called

a stopping time with respect to the filtration if

1. τ takes values in 0, 1, ...∞

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2. The event τ = t ∈ Ft for all t = 0, 1, ...

3.2.4 Stopped Process

Given a random process ξ = ξt∞t=0 and a stopping time τ with respect to the natural filtration

of ξ, satisfying P (τ <∞) = 1, then the stopped process ξτt is defined as

ξτt =

ξt if t ≤ τ

ξτ if t > τ

Stopping time is essentially useful with martingales and hence leads to the following theorem

which is stated without proof.

3.2.5 Theorem (The Optional Stopping-Time)

Let ξt be a martingale and τ be a stopping time with respect to the natural filtration of ξ.

Suppose τ is bounded by a positive real k, then ξ satisfies

E[ξτ ] = E[ξ0] (3.2.2)

3.3 Continuous Time Martingales and Stopping Time

This section also introduces the concept of martingales and stopping time for a continuous time

process as earlier treated for a discrete time process in the previous section.

3.3.1 Continuous Time Martingale

Let (Ω,F ,P) be a probability space, equipped with a filtration Ftt∈τ . Then a process

ξ = ξtt∈τ is a martingale with respect to the filtration if

1. ξ is adapted to the filtration Ft

2. E[|ξt|] <∞ for all t ∈ τ

3. E[ξt|Fs] = ξs for all s < t, s, t ∈ τ

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3.3.2 Stopped Time

Given a probability space and a filtration Ftt∈[0,∞].A random variable τ is called a stopping

time if

1. τ takes values in [0,∞]

2. The event τ ≤ t ∈ Ft for all t ∈ [0,∞]

3.4 Brownian Motion

Brownian motion is an example of a continuous time stochastic process with continuous state

space. It was first introduced by a Botanist Roger Brown in 1827, when he observed the irregular

motion of pollen grain immersed in water [16]. In 1905, A.Einstein established the first mathe-

matical basis for Brownian motion, by showing that the movements of particles were randomly

caused by collisions with molecules of the solvent [17].

Brownian motion as a stochastic process, serves as a basis of most financial models for equity

prices, currencies or interest rate.

3.4.1 Standard Brownian Motion

A process Btt∈[0,∞) is said to be standard Brownian motion or Wiener process if it has the

following properties

1. Bo = 0

2. The sample paths or trajectories are continuous in t

3. Bt has stationary increments

4. The incrementsBt+δ−Bt are independent and normally distributed with mean 0 and variance

δ.

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3.4.2 Remark

A Brownian motion is a martingale with respect to its natural filtration.Thus property 4 implies

that

E[Bt+δ −Bt|FBt ] = 0 (3.4.1)

Where FBt is the σ-algebra generated by Bss∈[0,t].[18]

3.4.3 Properties of Brownian Motion

Let B = Btt∈[0,∞] be a Brownian process. Then Bt satisfies

1. Bt is a Markov process. This result from the independent property

2. Bs and Bt are jointly normally distributed random variables (for fixed s and t) and

cov(Bs.Bt) = min(s, t)

3. Even though the trajectories of Bt are continuous, Bt is not differentiable anywhere for all

t (with probability 1)

4. Once a Brownian motion has any value, it immediately hits that value again infinitely many

times

5. The process B1(t)t∈[0,∞] defined as B1(t) = 1√cBct for any c > 0, is also a standard

Brownian motion. This is the scaling property of Brownian motion

6. The process B2(t)t∈[0,∞] defined as B2(t) = tB 1t, is also a standard Brownian motion.

This is the time inverse property of Brownian motion.[18]

3.4.4 Example 1

If Bt is a Standard Brownian motion,derive the conditional expectation E(B3t |F), where s ≤ t

and Fs is the natural filtration of Bs

We have

B3t = (Bt −Bs)

3 + 3Bs(Bt −Bs)2 + 3B2

s (Bt −Bs) +B3s (3.4.2)

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By property 3 and 4 of definition (1.1.2), we have that

• E[Bt −Bs|Fs] = 0

• E[(Bt −Bs)3|Fs] = E[Bt −Bs]E[(Bt −Bs)

2|Fs] = 0

• E[(Bt −Bs)2|Fs] = V ar((Bt −Bs)

2|Fs) + (E[Bt −Bs|Fs]) = t− s

• E[B3s |Fs] = B3

s , since Bs is Fs-measurable

Hence,

E[B3s |Fs] = E[(Bt −Bs)

3 + 3Bs(Bt −Bs)2 + 3B2

s (Bt −Bs) +B3s |Fs] (3.4.3)

Using the linearity property of expectation, we have

E[B3s |Fs] = 3Bs(t− s) +B3

s (3.4.4)

2. Now, we use the result in question (1) to construct a martingale out of B3t .

Let Mt = B3t − 3tBt

Claim:Mt is a martingale. Indeed

E[Mt|Fs] = E[B3t − 3Bt.t|Fs] (3.4.5)

which from the linearity of expectation implies that

E[Mt|Fs] = E[B3t |Fs]− 3tE[Bt|Fs] (3.4.6)

Using the result in equation (1) and the fact that Bt is a martingale, we obtain

E[Mt|Fs] = 3Bs(t− s) +B3s − 3tBs (3.4.7)

which by cancelling opposite terms leads to

E[Mt|Fs] = B3s − 3sBs = Ms (3.4.8)

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3.4.5 General Brownian Motion

A process Wtt∈[0,∞] is said to be a Brownian motion if

1. Wo = 0

2. The sample paths or trajectories are continuous in t

3. Wt has stationary increments

4. The increments Wt+δ−Wt are independent and normally distributed with mean µδ and

variance σ2δ

3.4.6 Remark

In this case µ is called the drift coefficient and σ the diffusion coefficient. When µ = 0 and σ = 1,

we obtain standard brownian motion.

3.4.7 Geometric Brownian Motion

A process is called Geometric Brownian motion if it is of the form

G(t) = eµt+σBt (3.4.9)

since Bt ∼ N(µt, σ2t), it follows that lnGt = µt+ σBt ∼ N(µt, σ2t). Consequently,

Gt ∼ lnN(µt, σ2t) with mean and variance.

E[Gt] = eµt+12σ2t and V ar(Gt) = (E[Gt]

2)(eσ2t−1) (3.4.10)

3.5 Stochastic Calculus

Stochastic calculus is the area of mathematics that deals with processes containing a stochastic

component and thus allows the modelling of random systems [19]. When attempting to develop

a stochastic calculus for a continuous time stochastic process, one has to face the fact that the

sample paths are nowhere differentiable. Stochastic calculus allows a consistent theory of inte-

gration to be defined for integrals of stochastic processes and used to model systems that behave

randomly.

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Stochastic calculus grew out of the need to assign meaning to ordinary differential equations

involving continuous stochastic process and the most important of such process is the Brown-

ian motion. The stochastic integral is first introduced before introducing stochastic differential

equations.

3.5.1 Stochastic Integrals

We construct stochastic integral

It =

∫ T

0

f(t)dBt

for a non-random integrand f with respect to Brownian motion;this means that Brownian motion

will be used as a building block to achieve the objective[18].

Here, B is the Brownian motion (integrator) and f (integrand) is adapted (that is the value of

B at any point on time t is only dependent on information available up until the time).

3.5.2 Simple Function

A simple function is one of the form

Ht =n∑i=1

ai1Ii(s) (3.5.1)

where

Ii = (ti, ti+1],n⋃i=1

Ii = (0, T ], Ii ∩ Ij = φ if i 6= j

and for each i = 1, ..., n, ai : Ω→ R is an Fsi-measurable random variable with

E[a2i ] <∞.(Etheridge, 2002)

3.5.3 Simple Process

A simple process is any map H : Ω× (0,∞)→ R of the form

Ht =n∑i=1

φI(ti−1, ti](t) (3.5.2)

where 0 = t0 < t1 < ..., < tn = T, I(ti−1, ti] are the indicator function of (ti−1, ti] and the random

variables φi are FBti−1-measurable and satisfy E[φ2i ] <∞.

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The stochastic integral for a simple process is defined by

∫ T

0

HtdBt =n∑i=1

ai(Bti −Bti−1) ≡∑

Hti−1 − (Bti −Bti−1) (3.5.3)

3.5.4 Remark

• Any simple process is adapted with respect to the filtration of Brownian motion

• The process∫ t

0HsdBs

t∈[0,T ]

is adapted with respect to the filtration of Brownian motion

since the random variable ai, i = 1, 2...n are FBti−1-measurable.

3.6 Ito’s Stochastic Integral

Let f ∈ H, we define the Ito’s stochastic integral of f with respect to Btt≥0 by

I[f ] =

∫ T

0

f(t)dBt = lim∆t→0

n∑i=1

(Bti −Bti−1)f(ti) (3.6.1)

3.6.1 Properties of Ito’s Stochastic Integral

The Ito’s stochastic integral that is defined for all H ∈ H satisfies the following properties

1. Additivity:∫ T0

(aHs + bGs)dBs = a∫ T

0HsdBs + b

∫ T0GsdBs

2. Partition property:∫ T0HsdBs =

∫ s0HsdBs +

∫ TsHsdBs

3. Isometry:

E[(∫ t

0HsdBs)

2] = E[∫ t

0H2sds]

4. The process∫ t

0HsdBs

t∈[0,T ]

is a continuous martingale with zero mean i.e

E[∫ T

0Hsds] = 0

5. For a random variable f(.),the process∫ t

0HsdBs

t∈[0,T ]

is normally distributed with zero

mean and variance. That is

V ar(∫ t

0f(s)dBs) = E[(

∫ t0f(s)dBs)

2] =∫ t

0f 2(s)ds.[18]

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3.6.2 Example

State the distribution of the Ito’s integral∫ t

0(s+ 1)dBs and

∫ t0sin(s)dBs

We know that both are normal random variables with zero mean, therefore it remains to calculate

the variances:

V ar

∫ t

0

(s+ 1)dBs = E

[(∫ t

0

(s+ 1)dBs

)2]

=

∫ t

0

(s+ 1)2ds

=(t+ 1)2

3

V ar

∫ t

o

sin(s)dBs =

∫ t

0

sin2sds

=

∫ t

0

1− cos(2s)2

ds

=t

2− sin(2t)

4

Thus, ∫ t

0

(s+ 1)dBs ∼ N(0, (t+ 1)3/3)

∫ t

0

sin(2s)dBs ∼ (0, (2t− sin(2t))/4)

3.7 Stochastic Differential Equations and Ito’s Formula

3.7.1 Stochastic Differential Equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the

terms is a stochastic process. The standard example of a stochastic diferential equation is an Ito

equation for a diffusion process.

A process X = Xtt∈[0,T ] defined by

Xt = X0 +

∫ t

0

Ksds+

∫ t

0

HsdBs (3.7.1)

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is called a diffusion or an Ito process [18].

Here, Xo is a constant, Ktt∈[0,T ] and Htt∈[0,T ] are adapted processes with respect to the

filtration of Brownian motion and

∫ T

0

|Ks|ds <∞,∫ T

0

|Hs|2ds <∞

3.7.2 Stochastic Differential Equation

Let X = Xtt∈[0,T ] be a stochastic process. Then the stochastic differential equation of X is a

process defined by

dXt = µ(t, Bt)dt+ σ(t, Bt)dBt (3.7.2)

where µ and σ are referred to as the drift and diffussion process respectively.

Now, we state a crucial result for working with such stochastic differential equations. It is known

as the Ito’s formular, also known as Ito’s lemma.[18]

3.7.3 Theorem (Ito’s Formula)

Let f = f(t, x) possess continuous derivatives ∂f∂t, ∂f∂x

and ∂2f∂x2

and assume also that ∂f∂t

(s, Bs) ∈ H.

Then

f(t, Bt)− f(0, B0) =

∫ t

0

∂f

∂x(s, Bs)dBs +

∫ t

0

∂f

∂s(s, Bs)ds+

1

2

∫ t

0

∂2f

∂x2(s, Bs)ds

and therefore

df(t, Bt) =∂f

∂x(t, Bt)dBt +

(∂f

∂t(t, Bt) +

1

2

∂2f

∂x2(t, Bt)

)dt (3.7.3)

3.7.4 Derivation Of Ito’s Formula For Brownian Motion

Suppose f is a real function. We want to calculate df(Bt)

Recall the Taylor’s formular

∆f(x) = f ′(x)∆x+1

2f ′′(x)(∆x)2 +

1

3!f ′′′(x)(∆x)3 + .......

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That is

df(x) = f ′(x)dx+1

2f ′′(x)(dx)2 +

1

3!f ′′′(x)(dx)3 + .......

If B is a real variable or a real differential function x, we know that df(x) = f ′(x)dx

Because (dx)2, (dx)3e.t.c are zeros

Now,

df(Bt) = f ′(Bt)dBt +1

2f ′′(Bt)(dBt)

2 +1

3!f ′′′(Bt)(dBt)

3 + .......

Notice (dBt)2 = dt

(dBt)3 = (dBt)

2dBt = dtBt = 0, (dBt)k = 0, k = 3, 4, ....

It follows that

df(Bt) = f ′(Bt)dBt +1

2f ′′(Bt)dt

or in integral notation

f(Bt)− f(B0) =

∫ t

0

f ′(Bu)dBu +1

2

∫ t

0

f ′′(Bu)du

.

3.7.5 Stochastic Integrals With General Integrators

The integrals with respect to the integration have so far been considered and we now extend

the analysis by defining integrals for more general class of integrals.This is achieved using Ito’s

formula and therefore leads us to define Ito’s formula for general integrals, we now state without

any proof.

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3.7.6 Theorem (Ito’s Formula For General Integrator)

Let Xt be an Ito’s process given by

dXt = µ(t, Bt)dt+ σ(t, Bt)dBt

Let g(t, x) ∈ C2([0,∞)× R).Then

Yt = g(t, Bt)

is an Ito’s process and its stochastic differential equation is given by

dYt = gx(t,Xt)dXt + gt(t,Xt)dt+1

2σ2(t, Bt)gxx(t,Xt)dt (3.7.4)

where subscripts are used to denote partial derivatives.

Example (Ornstein Uhlenbeck Process) [18]

The Ornstein Uhlenbeck process appears in certain financial models for stochastic interest rates.

The process may be defined as the unique solution to the equation

Xo = xo , dXt = −αXtdt+ σdBt (3.7.5)

where α, σ ∈ R are positive constants. If Xt is the solution to the SDE, we should have

d(Xteαt) = eαtdXt + αeαtXtdt

= σeαtdBt

By definition, this means,

Xteαt −Xo = σ

∫ t

0

eαsdBs (3.7.6)

and therefore

Xt = xoe−αt + σe−αt

∫ t

0

eαsdBs (3.7.7)

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As∫ t

0eαsdBs is normally distributed, the random variable Xt is also normally distributed with

E[Xt] = xoe−αt

and

V ar[Xt] = σ2e−2αt

∫ t

0

e2αsds

= σ2 1− e−2αt

2α

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4 Asset Pricing Models

The purpose of asset pricing models is not for prediction of future price; rather, the purpose

is to provide a description of the stochastic behaviour of prices [5]. By definition, an asset is

a right on future cash flow. It therefore tries to understand the prices or values of claims to

uncertain payment, a low price implies a high rate of returns. Asset pricing models are different

ways of explaining how investors value investments. The most widely known asset price model

is the Capital Asset Price Model (CAPM). This model results in a simple view of how assets are

valued, it measures an asset risk as a covariance of the asset return with the market portfolio

return. An alternative to the CAPM is the Arbitrage Pricing Theory (APT), which is based on

different assumptions regarding how market and investors behave. The price at which an asset

trades is a ”fair market price” that reflects the actual value of the asset [7].

4.1 Lognormal Model

Stock and other prices are commonly assumed to follow a stochastic process called Geometric

Brownian Motion. Whenever quantities grow multiplicatively, the lognormal model becomes

a leading candidate for a statistical model of such quantities, which explains in good part the

persistence of the Geometric Brownian Motion model to security prices in Economics and Finance

[20]. The mainstream focuses on situations in which the lognormal price model works very well

which relys on the Efficient Market Hypothesis (EMH). EHM asserts that financial markets

are efficient, that is security prices fully reflect relevant and available information. With the

fluctuation of asset prices which can lead to loss in investments, financial analyst and engineers

have seen the use of modelling the price of an asset with the aim of forcasting its fundamental

future value in order to avoid necessary loss or negative returns [21].

If a variable is distributed in such a way that instantaneous percentage changes follow Geometric

Brownian Motion, over discrete period of time, the variable is lognormally distributed [22]. The

stock price process in the Black Scholes model is lognormal, that is, given the price at any time,

the logarithm of the price at a later time is normally distributed,but the lognormal model is more

reasonable because stocks have limited liability and cannot be negative.

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4.1.1 Definition (Lognormal Model)

The lognormal model is the financial model where the prices of a stock at any time t ≥ is given

by

S(t) = SoeσWt+(r− 1

2σ2)t (4.1.1)

where Wt is a normal random variable with mean zero and variance t and so is the price of the

stock at time t = 0

4.1.2 The Lognormal Distribution

The lognormal is a continuous distribution in which the logarithm of a variable is normally

distributed. A lognormal distribution results if the variable is the product of a large number of

independent, identically distributed variables in the same way that a normal distribution results

if the variable is the sum of a large number of independent, identically distributed variable.

4.1.3 Definition (Lognormal Distribution)

Let X be a random variable. We say that X is lognormally distributed with parameters µ and

σ2, if its logarithm lnX is normally distributed with parameters µ and σ2. It is denoted by

X lnN(µ, σ2). Furthermore, its mean and variance are given below by the following equations

E[X] = eµ+σ2

2 (4.1.2)

V ar[X] = e2µ+2σ2(eσ2−1) (4.1.3)

4.2 Efficient Market Hypothesis

An efficient market is defined as a market where there are large numbers of rational, profit

maximisers actively competing, with each trying to predict future market values of individual

securities and where important current information is almost freely available to all participant[23].

The market is made efficient by supply and demand. If it is not efficient, investors will trade to

take advantage of the inefficiencies. But, if the market is already efficient, no one will expend

resources on security analysis. Efficient Market Hypothesis (EMH) essentially says that all

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known information about investment securities, such as stocks is already factored into the prices

of these securities. The EMH is one possible reason for the random behaviour of the asset prices

and basically states that past history is fully reflected in prices and markets respond immediately

to any new inforamtion about the asset [25].

The EMH was developed indepenently by Paul A. Samuelson and Eugene F. Fama in the 1960’s.

The idea has been applied extensively to theoritical models and empirical studies of financial

security prices. For an efficient market, there are three particular form of the EMH. These

includes

• Strong form EMH: market prices include all information, both publicly available and also

that available only to insiders.

• Semi-strong form EMH: market prices includes all publicly available information.

• Weak form EMH: market prices of a particular investment incorporate all information

contained in the price history of that investment.

4.3 Empirical Problem With The Lognormal Model

The Constant Votality Assumption:

Based on Empirical studies and observations, the estimate of σ can vary significantly according

to the economic conditions. The assumption is that the parameter σ is constant, hence leads

to an incorrect assumption and models calibrated on economic data at a particular date would

quickly become inaccurate as time moves on. Therefore, due to the constant random fluctuations

of the economy conditions which constantly affect the market prices, the assumption of constant

volatility is inaccurate because it relies on past information and not on the latest information

available which affect asset prices in the market [26],[8].

The Constant Dift Assumption:

The lognormal model assumes a constant dift parameter µ and however, there are theoretical

reasons proving it is incorrect. Based on mean-reversion and momentum effects, a mean reverting

market is one in which these exist a long-term average for the return and in essence, the price

of an asset always reverts to the mean. However, the momnetum effects results from the infor-

mation that arrive in the market, where a rise one day is more likely to be followed by another

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rise the following day implying that very short-term returns are positively correlated, which is

incompatible with mean-reversion. Therefore, the assumption of constant parameter µ in the

lognormal model is incompatible [26],[8].

4.3.1 Properties Of Lognormal Model

The lognormal model has the following properties

1. The lognormal distribution is used to model continuous random quantities when the dis-

tribution is believed to be skewed, such as certain income and life time variables.

2. The lognormal process represents a product of independent random variables

3. The lognormal distribution can never be negative.

4.3.2 Example 1

Calculate the probability that the share price of PZ Plc exceeds 10 pounds after 3 years. You

are given that it has the current market price of 1.50 pounds and can be modelled using the

lognormal model with parameters µ = 10% and σ = 5% per annum.

Let St be the price process. It obeys the SDE

dSt = µStdt+ σStdWt (4.3.1)

whereWt is the standard wiener process. Equivalently, using Ito’s lemma, the log-priceXt = ln St

is the Brownian motion with the dynamics

dXt =

(µ− 1

2σ2

)dt+ σdWt (4.3.2)

and the conditional laws

Xt|Xs ∼ N

(Xs +

(µ− 1

2σ2

)(t− s), σ2(t− s)

)(4.3.3)

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Equivalently,

Xt −Xs ∼ N

((µ− 1

2σ2

)(t− s), σ2(t− s)

)(4.3.4)

Using µ = 0.1, σ = 0.05, we have

P(S3 > 10|S0 = 1.5) = P(S3|S0 > 10|1.5) = P(eX3−X0 > 6.667

)= P(X3 −X0 > ln 6.667) = P(X3 −X0 > 1.8971)

= PX3 −X0 −

(µ− 1

2

)3

σ√

3>

1.8971− 0.2963

0.0866

= P(Z > 18.4850)

4.3.3 Example 2

If the share price follows the Geometric Brownian motion process

dSt = 0.10Stdt+ 0.4StdWt (4.3.5)

where St is the price at time t and Wt is the Wiener process under the natural probability

measure. Calculate the 99% confidence interval for S10 given S5 = 5

Since St is the Geometric Brownian motion, it is natural to find the confidence interval for the

log-price Xt = ln St, with dynamics

dXt =

(µ− 1

2σ2

)dt+ σdWt

=

(0.1− 1

2(0.4)2

)dt+ 0.4dWt

= 0.02dt+ 0.4Wt

Since ln 5 = 1.6094, we have

E[X10|X5 = 1.6094] = 1.6094 + 0.02 ∗ (10− 5)

= 1.7094

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Therefore, the 99% confidence interval for X10 given X5 = 1.6094 is of the form

(1.7094−B, 1.7094 +B)

where B is determined from the condition

P(1.7094−B < X10 < 1.7094 +B|X5 = 1.6094) = 0.99

Equivalently,

P(X10 > 1.7094 +B|X5 = 1.6094)(1− 0.99)/2(0.005)

⇔ P

((X10 −X5 −

(µ− 1

2

)5)

σ√

5>

B

0.4√

5

)= 0.005

⇔ P(Z > B/0.8944) = 0.005

Since the upper 0.5% cut-off point for the standard normal is 2.58, we obtain the equation

B/0.8944 = 2.58

and find B = 2.3076. Thus, the 99% confidence interval for X10 given X5 = 1.6094 is

(1.7094− 2.3076, 1.7094 + 2.3076) = (−0.5982, 4.0170)

Exponentiating, we obtain the 99% confidence interval for S10 given

S5 = S : (e−0.5982, e4.0170) = (0.5498, 55.5343)

4.4 Black-Scholes Option Pricing

The formulation presented here was first developed in the early 1970’s by Black Scholes and

Merton and it led to the revolution in the pricing of derivative products. Its significance was

acknowledge by Merton and Scholes being awarded the Nobel Prize for Economics in 1997 [27]

An option is a contract that admits the ownes the right (not the duty) to buy (”call option”)

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or to sell (”put option”) an asset for a prespecified price. A derivative is a financial instrument

whose value depends on the value of other basic assets, such as common stock. The derivative

asset we will be most interested in is a European call option. A call option gives the owner

the right to buy the underlying asset on a certain date for a certain price. The specified date

is known as the expiration date or day until maturity while the specified price is known as the

exercise or strike price. The put option gives the owner the right to sell the underlying asset on

a certain date for a certain price [28].

4.4.1 Black-scholes Assumption

The analysis of option pricing requires one to make a number of assumptions. In particular, the

Black-scholes formulation requires the following assumptions to be made about the behaviour of

equity prices and financial markets.

• The market are efficient

• The underlying equity price follow a lognormal random walk in continuous time

• There are no arbitrage opportunities

• The risk free rate of return is constant and equal for all maturities

• The short selling of securities with full use of proceeds is permitted

• There are no transaction cost or taxes

• Equity trading is continuous

• The markets are complete

• Equities do not pay dividends during the life of the option contract.

4.5 Derivation Of The Black-Scholes Call Option Pricing Formula

The Black Scholes call option pricing model has played a central role for the growth and success

of financial mathematics. It has a huge influence on the way that traders price options [26]. We

demonstrate how the Black Scholes formula for a European call vanilla option is derived.

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Assumptions:

1. The underlying asset follow a lognormal random walk

2. Arbitrage arguments allow us to use a risk-neutral valuation approach, discounting the ex-

pected payoff of the option at expiration by the riskless rate and assuming the underlying’s

return is the risk free rate

The definition of a call option is

Co = e−rTE[CT ]

= e−rTE[max ST −X, 0] (4.5.1)

Remember that the expectation of a random variable Y can in general be computed by

EY =

∫ ∞−∞

yP(Y = y)dy (4.5.2)

That is, each possible realization is weighted by the probability of it occuring. We thus get

Co = e−rT∫ ∞

0

max y −X, 0P(ST = y)dy (4.5.3)

Note that the lower bound of integration is not −∞ but 0 since ST cannot take negative values.

Next, remember that under the risk neutral probability measure, the logarithmic terminal stock

price ST is normally distributed with

ln ST ∼ N(ln So +

(r − 1

2σ2

)T, σ2T

)(4.5.4)

An equivalent way of writing this is by defining the standard normal random variable ε ∼ N (0, 1)

and letting

ln ST = ln So +

(r − 1

2σ2

)T + σ

√Tε (4.5.5)

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or by taking the exponential

ST = So exp

(r − 1

2σ2

)T + σ

√Tε

(4.5.6)

This allows us to rewrite the expression for the call price as

Co = e−rT∫ ∞−∞

max

So exp

(r − 1

2σ2

)T + σ

√Tz

−X, 0

P ε = z dz (4.5.7)

Formally, we have made a change of variable from y = ST to

z =ln(STSo

)−(r − 1

2σ2)T

σ√T

(4.5.8)

Next, we want to eliminate the max function inside the integral. We observe that the terminal

payoff of the call option is non zero if ST > X or equivalently

z ≥ln(STSo

)−(r − 1

2σ2)T

σ√T

= −d2 (4.5.9)

Changing the lower limit of integration from −∞ to −d2,allows us to drop the max function and

we get

Co = e−rT∫ −∞−d2

(So exp

(r − 1

2σ2

)T + σ

√Tz

−X

)P ε = z dz (4.5.10)

Since integration is a linear operator,we can split up this integral into two terms that we will

evaluate separately

Co = e−rT[∫ −∞−d2

So exp

(r − 1

2σ2

)T + σ

√Tz

P ε = z dz −

∫ ∞−d2

XP ε = z dz](4.5.11)

Since ε is a standard normal random variable, its probability density is given by

P ε = y =1√2πe−

y2

2 (4.5.12)

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We now turn to the computation of the second integral and get

∫ ∞−d2

XP ε = z dz = X

∫ ∞−d2

P ε = z dz

= XP ε ≥ −d2 dz

= XP ε ≤ d2 dz

= XN (d2)

Here, we used the symmentry of the normal distribution in the second step. Using the definition

of the standard normal density, we can write the second integral as

∫ ∞−d2

So exp

(r − 1

2σ2

)T + σ

√Tz

P ε = z dz (4.5.13)

=

∫ ∞−d2

So exp

(r − 1

2σ2

)T + σ

√Tz

1√2πexp

−z

2

2

dz

= So exp

(r − 1

2σ2

)T

∫ ∞−d2

1√2πexp

−z

2 − 2σ√Tz

2

dz

= So exp

(r − 1

2σ2

)T

∫ ∞−d2

1√2πexp

−z

2 − 2σ√Tz ± σ2T

2

dz

= So erT

∫ ∞−d2

1√2πexp

−z

2 − 2σ√Tz ± σ2T

2

dz

= So erT

∫ ∞−d2

1√2πexp

−(z − σ

√T)2

2

dz (4.5.14)

We can now make a change of variable by defining x = z − σ√T and get

So erT

∫ ∞−d2−σ

√T

1√2πexp

−x

2

2

dx (4.5.15)

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= So erT

∫ ∞−d2−σ

√T

P ε = x dx

= So erTP

ε ≥ −d2 − σ

√T

= So erTP

ε ≤ d2 + σ

√T

= So erTN

(d2 + σ

√T)

(4.5.16)

Defining d1 = d2 + σ√T and combining all previous results yields

Co = So N (d1)−Xe−rTN (d2) (4.5.17)

This is the Black-Scholes formula for a European plain vanilla call option.

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5 Conclusion

Stochastic calculus is an important instrument in the pricing of derivatives and using basic

fundamental concepts of probability, we have introduced significant stochastic processes such as

Markov property, martingale property and Brownian motion. We illustrated that the process

used to describe the prices of assets is the Geometric Brownian motion which is lognormally

distributed. This led us to the concept of stochastic calculus which is based on Ito’s lemma from

Brownian motion and this was used in the illustration of stochastic differential equations.

The lognormal model was used in modeling of asset prices while expressing the influence of

the efficient market hypothesis on the determination of market prices and this was used in the

derivation of the Black-Scholes option pricing formula. However, with the usefulness of the

lognormal model in financial mathematics, it has has some limitatihatons, by assuming that the

votality coefficient and drift coefficient are constant.

The famous Black-Scholes model has been used extensively in the past, but modern markets

demand more sophisticated models that better capture the ever more complex behaviour of asset

prices and one of the most unrealistic assumption in the Black-Scholes model is that of constant

votality.

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stochastic models for asset pricing

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