**TABLE OF CONTENTS**

Preliminaries: . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 -algebra . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.2 Probability Space . . . . . . . . . . . . . . . . . . . . . 8

1.1.3 Borel -algebra . . . . . . . . . . . . . . . . . . . . . . 8

1.1.4 A random variable: . . . . . . . . . . . . . . . . . . . . 9

1.1.5 Probability distribution . . . . . . . . . . . . . . . . . 9

1.1.6 Normal distribution . . . . . . . . . . . . . . . . . . . 9

1.1.7 A d-dimensional Normal distribution . . . . . . . . . . 10

1.1.8 Log-normal Distribution . . . . . . . . . . . . . . . . . 11

1.1.9 Mathematical Expectation . . . . . . . . . . . . . . . . 11

1.1.10 Variance and covariance of random variables: . . . . . 12

1.1.11 Characteristic function . . . . . . . . . . . . . . . . . . 12

1.1.12 Stochastic process . . . . . . . . . . . . . . . . . . . . 13

1.1.13 Sample Paths . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.14 Brownian Motion . . . . . . . . . . . . . . . . . . . . . 13

1.1.15 Filtration: . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.16 Adaptedness . . . . . . . . . . . . . . . . . . . . . . . 14

1.1.17 Conditional expectation . . . . . . . . . . . . . . . . . 14

1.1.18 Martingale . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.19 Quadratic variation . . . . . . . . . . . . . . . . . . . 15

1.1.20 Stochastic dierential equations . . . . . . . . . . . . . 16

5

1.1.21 Ito formula and lemma . . . . . . . . . . . . . . . . . . 16

1.1.22 Gamma distribution . . . . . . . . . . . . . . . . . . . 17

1.1.23 Risk-neutral Probabilities . . . . . . . . . . . . . . . . 18

1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Literature Review 21

3 Financial Derivatives 24

3.0.1 Forward Contract . . . . . . . . . . . . . . . . . . . . 24

3.0.2 Future Contracts . . . . . . . . . . . . . . . . . . . . . 25

3.0.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.0.4 Hedgers . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.0.5 Speculators . . . . . . . . . . . . . . . . . . . . . . . . 28

3.0.6 Arbitrageurs . . . . . . . . . . . . . . . . . . . . . . . 29

4 Pricing of Basket option 30

4.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . 32

4.3 Methods used in pricing Basket options . . . . . . . . . . . . 34

4.3.1 Numerical Methods . . . . . . . . . . . . . . . . . . . 34

4.3.2 Approximation Methods . . . . . . . . . . . . . . . . . 41

5 APPLICATION 48

5.1 Foreign Exchange Market . . . . . . . . . . . . . . . . . . . . 48

5.1.1 Quotation Style . . . . . . . . . . . . . . . . . . . . . . 51

5.2 Foreign Exchange Basket Option . . . . . . . . . . . . . . . . 52

5.2.1 Correlation in foreign exchange . . . . . . . . . . . . . 53

5.3 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6

## CHAPTER ONE

General Introduction

In this chapter we give some denitions in probability theory needed for our

thesis and provide some introduction to the work.

1.1 Preliminaries:

We begin by introducing a number of probabilistic concepts.

1.1.1 -algebra

Let

be a non-empty set and B a non-empty collection of subset of

, B is

called a -algebra if the following properties hold:

i

2 B

ii A 2 ) A0 2

iii fAj : j 2 Jg B )

S

j2J

Aj 2 B for any nite or innite countable

subset J of N

7

1.1.2 Probability Space

1. Let

be a nonempty set and B a – algebra of subsets of

. Then

the pair (

; B) is called a measurable space and a member of B is

called a measurable set.

2. Let (

; B) be a measurable space and : B ?! R be a real valued

map on . Then is called a probability Measure if the following

properties hold:

i (A) 0 8A 2

ii (

) = 1

iii For fAngn2N , with Aj Ak = ; for j 6= k

(

S

n2N

An) =

P

n2N

(An) i.e is -additive(or countably additive).

3. If (

; ) is a measurable Space and is a probability measure on

(

; ),then the triple (

; ; ) is called Probability Space.

1.1.3 Borel -algebra

If is a collection of subsets of

,then the smallest -algebra of subsets of

which contains , denoted by () is called the -algebra generated by

.

Let X be a nonempty set and a topology on X, i.e is the collection

of all open subsets of X.Then ( ) ia called the Borel -algebra of the

topological space (X; ).

8

1.1.4 A random variable:

Let (

; ; ) be an arbitrary probability space,B(Rd) be the Borel -algebra

of Rd and (Rd;B(Rd)) the d-dimension Borel measurable space. Then, a

measurable map X :

?! Rd is called a random vector. In the case

d=1, X is called a random variable.

1.1.5 Probability distribution

Let (

; ; ) be a probability space, (Rd; (Rd)) be the d-dimensional Borel

measurable space, and X :

?! Rd a random vector. Then the map

X : (Rd) ?! [0; 1] dened by X(A) = (X?1(A)), A 2 (Rd) is called

the probability distribution of X:

1.1.6 Normal distribution

A standard univariate normal distribution(i.e of mean zero and variance 1)

has density (x) = p1

2

e?x2

2 ,?1 < x < 1 and cumulative distribution

function

(x) =

R x

?1

p1

2

e?u2

2 du

In general a normal distribution with mean and variance 2; > 0

has density ;(x) = 1

p

2

e?(x?)2

22 and cumulative distribution function

;(x) = (x?

)

The notation X N(; 2) means the random variable X is normally

distributed with mean and variance 2.

If Y N(0; 1) (i.e Y has the standard normal distribution, then +Y

N(; 2). Thus given a method for generating the samples Y1; Y2; from

the standard normal distribution, we can generate samples X1;X2; from

N(; 2). It therefore suces to consider methods for sampling from N(0,1).

[2]

9

1.1.7 A d-dimensional Normal distribution

This is characterised by a d-vector and a dd covariance matrix ; and is

abbreviated as N(; ). If is positive denite (i.e xTx > 0,8x 6= 0 2 Rd),

then the normal distribution N(; ) has density

;(x) =

1

(2)

d

2 jj

1

2

exp(?

1

2

(x ? )T?1(x ? ));

x 2 Rd with jj the determinant of :

The standard d-dimensional normal distribution N(0; Id); with Id the dd

identity matrix, is the special case

1

(2)

d

2

exp(?1=2 xT x):

If X N(; ) (i.e the random vector X has a multivariate normal

distribution)then its ith component Xi has distribution N(i;2i

) with 2

i =

ii.The ith and jth component have covariances cov(Xi;Xj) = E[(Xi ?

i)(Xj ? j )] = ij which justies calling the covariance matrix. The

correlation between Xi and Xj is given by ij = ij

ij

.

If a d d symmetric matrix is positive semi-denite but not positive

denite then the rank of is less than d, fells to be invertible, and there

is no normal density with covariance matrix . In this case we can dene

the normal distribution N(; ) as the distribution of X = + AZ with

Y N(0; Id) for any d d matrix A: AAT = . The resulting distribution

is independent of which A is chosen. The random vector X does not have a

density in Rd, but if has rank then one can nd k component of X with

multivariate normal density in Rk.

Any linear transformation of a normal vector is again normal, X

N(; ) ) AX N(A;AAT ) for any d-vector and dd matrix and

any d k matrix A, for any k.[2]

10

1.1.8 Log-normal Distribution

In simple terms: A random variable X is said to have a lognormal distri-

bution if its logarithm has a normal distribution. I.e ln[X] N(; ). An

important property of this distribution is that it does not take values less

than 0.

A lognormal distribution is very much what the name suggest “lognor-

mal”. Imagine that you have a function that is the exponent of some input

variable X. The input variable itself is a normal distribution function . e.g.

y = k:eX

Now, if we take a natural log of this function gives a normal distribution.

1.1.9 Mathematical Expectation

Let (

,,) be a probability space. If X 2 L1(

; ; ), then

E(X) =

Z

X(!)d(!)

is called the mathematical expectation or expected value or mean of X:

The map X 7?! E(X) ,X 2 L1(

; ; ) has the following properties:

i E is linear: E(X + Y ) = E(X) + E(Y ), for all X; Y 2 L1(

; ;

and, ; 2 R

ii Markov’s inequality holds, i.e let X 2 L1(

; ; ) be R-valued. Then

(f! 2

: jX(!)j g) E(jXj)

) = kXk1

, where > 0.

iii E is positivity preserving i.e if X is real-valued and lies in L1(

; ; )

and X 0, then E(X) 0.

11

iv Chebychev’s inequality holds: Let X 2 L1(

; ; ) be a R-valued ran-

dom variable with mean E(X) = and variance 2X

.Then for > 0

(f! 2

: jX(!) ? j g) 2X

2 ).

v Jensen’s inequality holds i.e, if X is real-valued and lies in L1(

; ; ).

: R ?! R is convex and (X) 2 L1(

; ; ), then E((X))

(E(X)).

1.1.10 Variance and covariance of random variables:

Let (

; ; ) be a probability space and X an R-valued random variable on

,

such that X 2 L2(

; ; ). Then X is automatically in L1(

; ; ) (because

in general if p q, then Lq(

; ; ) Lp(

; ; ) for all p 2 [1;1) [ f1g:)

The variance of X is dened as

V ar(X) = E((X ? E(X))2):

The number X =

p

V ar(X) is called the standard deviation/error of

X. Now let X, Y 2 L2(

; ; ). Then the covariance of X and Y is given

by:

Cov(X; Y ) = E((X ? E(X))(Y ? E(Y )))

And the correlation is given by:

corr(X; Y ) = (X; Y ) =

Cov(X)Cov(Y ) p

V ar(X)V ar(Y )

Two random variables X,Y are called uncorrelated if cov(X; Y ) = 0.

1.1.11 Characteristic function

Let (

; ; ) be a probability space and X 2 L0(

;R). Dene the C-valued

function on R by:

(t) = E(eitx) = E(costx) + iE(sintx):

12

Then is called the characteristic function of X.

Note: 0(0)

i = E(X)

1.1.12 Stochastic process

A stochastic process X indexed by a set J is a family X = fX(t) : t 2 Jg of

members of L(

;Rd).The value of X(t) at ! 2

is written as X(t,!).

1.1.13 Sample Paths

If X is a stochastic process and w 2

then the map t 7?! X(t;w) 2 Rd is

called a sample path or trajectory of X.

1.1.14 Brownian Motion

Let Z = fZ(t) 2 L(

;Rd) : t 2 4g ,where 4 R+ = [0;1] be an Rd

Stochastic process on

with the following properties:

i Z(0) = 0, almost surely.

ii Z(t) ? Z(s) is an N(0,(t-s)I) random vector for all t s 0,where I

is the d d identity matrix.

iii Z has stochastically independent increments i.e for 0 < t1 < t2 < <

tn, the random vectors Z(t1);Z(t2) ? Z(t1); ;Z(tn) ? Z(tn?1) are

stochastically independent.

iv Z has continuous sample paths t 7?! Z(t;w) for xed w 2

Then Z is called the standard d-dimensional Brownian Motion or d-dimensional

Weiner process.

13

For a d-dimensional Brownian motion Z(t) = (Z1(t); ;Zd(t)) we have

the following:

i E(Zj(t)) = 0, j = 1; 2; ; d

ii E(Zj(t)2) = t, j = 1; 2; ; d

iii E(Zj(t)Zk(s)) = jkt^k = jkminft; sg, for t; s 2 4, j; k = 1; 2; ; d

1.1.15 Filtration:

Let (

; ; ) be a probability space and consider F() = ft : t 2 g a

family of -subalgebras of with the following properties:

i For each t 2 , t contains all the -null members of .

ii s t whenever t s, s; t 2

Then F() is called a ltration of and (

; ; F(); ) is called a ltered

probability space or stochastic basis.

We interpret t as the information available at time t and F() describe the

ow of information.

1.1.16 Adaptedness

A Stochastic process X = fX(t) 2 L(

;Rn) : t 2 Tg is said to be adapted

to the ltration F() = ft : t 2 Tg if X(t) is measurable with respect to

t for each t 2 T. It is plain that every stochastic process is adapted to its

natural ltration.

1.1.17 Conditional expectation

Let (

; ; ) be a probability space, X a real random variable in L1(

; ; )

and a -subalgebra of . Then the conditional expectation of X given

14

written E(X j ) is dened as any random variable Y such that:

(i) Y is measurable with respect to i.e. for any A 2 (R), the set Y ?1(A) 2

.

(ii)

R

B X(!)d(!) =

R

B Y (!)d(!) for arbitrary B 2 :

A random variable Y which satises (i) and (ii) is called a version of E(X j

):

1.1.18 Martingale

Let X = fX(t) 2 L1(

; ; ) : t 2 g be a real-valued stochastic process on

a ltered probability space (

; ; F(); ). Then X is a

1. submartingale, if E(X(t)=s) X(s) a.s whenever t s

2. Supermartingale, if E(X(t)=s) X(s) a.s whenever t s

3. Martingale, if X is both a submartingale and supermartingale i.e E(X(t)=s) =

X(s) a.s whenever t s

1.1.19 Quadratic variation

Let X be a stochastic process on a ltered probability space (

; ; F(); ):

Then the quadratic variation of X on [0; t], t > 0, is the stochastic process

hXi dened by

hXi(t) = limjPj!0

nX?1

j=0

jX(tj+1) ? X(tj)j2

where P = ft; t1; ; tng is any partition of [0; t] i.e. 0 = t1 < t2 < < tn = t

and jPj = max0jn?1jtj+1 ? tj j

Note: If X is a dierentiable stochastic process, then hXi=0.

15

1.1.20 Stochastic dierential equations

These are dierential equations in which one or more terms is a stochastic

process, resulting in a solution which is itself a stochastic process. SDE are

used to model diverse phenomena such as uctuating stock prices or physical

system subject to thermal uctuations. They are of the form

dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)

with initial condition X(t) = x, where W denotes a Wiener process (stan-

dard Brownian motion). These are equations of the form

dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)

with initial condition X(t) = x

1.1.21 Ito formula and lemma

Let (

; ; F(); ) be a ltered probability space, X an adapted stochastic

process on (

; ; F(); ) with quadratic variation hXi and U 2 C1;2([0; 1]

R): Then

U(t;X(t)) = U(s;X(s)) +

Z t

s

@U

@t

(;X( ))ds +

Z t

s

@U

@x

(;X( ))dX( )

+

1

2

Z t

s

@2U

@x2 (;X( ))dhXi( )

which may be written as

dU(t; x) =

@U

@t

(t;X(t))dt +

@U

@x

(t;X(t))dX(t) +

1

2

@2U

@x2 (t;X(t))dhXi(t)

The equation above is referred to as the Ito formula. If X satsies the

stochastic dierential equation (SDE)

dX(t) = g(t;X(t))dt + f(t;X(t))dW(t)

X(t) = x;

16

then

dU(t;X(t)) = gu(t;X(t))dt + fu(t;X(t))dW(t)

U(t;X(t)) = U(t; x)

where

gu(t; x) =

@U

@t

(t; x) + g(t; x)

@U

@x

(t; x) +

1

2

(f(t; x))2 @2U

@x2 (t; x);

fu(t; x) = f(t; x)

@U

@x

(t; x)

We obtain a particular case of the Ito formula called the Ito lemma, if we

take X = Z, by setting g 0 and f 1 on T R. Then

dU(t;Z(t)) = [

@U

@t

(t;Z(t)) +

1

2

@2U

@x2 (t;Z(t))]dt +

@U

@x

(t;Z(t))dZ(t)

The equation above is referred to as the Ito lemma.

Table 1.1: Ito Multiplication Table

x dt dZ(t)

dt 0 0

dZ(t) 0 dt

1.1.22 Gamma distribution

The probability density function g? of a gamma distributed variable is given

by g?(x; ; ) =

e

?x

( x

)?1

?() ,x , ; 0

The corresponding cumulative distribution function G? is dene as:

G?(x; ; ) =

Z x

0

g?(u; x; )du

=

R x

0 u?1eudu

?()

=

(; x

)

?()

;

17

where

?(z) =

Z 1

0

tz?1e?tdt

The ith moment of the gamma distribution is given by:

E[Y i] =

i?(i + )

?()

The ith moment of the inverse gamma distribution can be obtained for

? < i 0 for i ? the moments are 1

If Y is reciprocally gamma distributed then:

E[Y i] =

1

i( ? 1)( ? 2) ( ? i)

:

Let gR be the inverse gamma probability distribution function. Then

gR(x; ; ) =

g?( 1

x ;;)

x2 , x 0; ; > 0

1.1.23 Risk-neutral Probabilities

These are probabilities for future outcomes adjusted for risk, which are then

used to compute expected asset values. The benet of this risk-neutral pric-

ing approach is that once the risk-neutral probabilities are calculated, they

can be used to price every asset based on its expected payo. These theo-

retical risk-neutral probabilities dier from actual real world probabilities;

if the latter were used, expected values of each security would need to be

adjusted for its individual risk prole. A key assumption in computing risk-

neutral probabilities is the absence of arbitrage. The concept of risk-neutral

probabilities is widely used in pricing derivatives.

1.2 Overview

A nancial derivative is a contract whose price is dependent upon or derived

from one or more underlying assets. The underlying assets could be stocks,

18

commodities, currencies e.t.c. An option is a nancial derivative that gives

the holder the right but not the obligation to buy or sell an underlying asset

at a certain date and price. Options were rst traded on the Chicago Board

Options Exchange on April 26th, 1973. Basket option is a type of derivative

security where the underlying asset is a group of commodities, securities or

currencies. Since the early 1990s, basket options have been used as a tool

for reducing risks (Hedging).

The pricing and hedging of basket options is dicult, due to the number

of state variables. The usual methods employed in pricing options are not

used to price Basket options, like Black and Scholes(1973) model. A sin-

gle underlying asset is assumed to follow a geometric Brownian motion and

therefore log-normally distributed, the problem arises from the fact that sum

of correlated log-normally distributed random variables is not log-normal,

thereby making it dicult to price the basket options and have a closed form

pricing formula and hedging ratios. Some Practitioners sometimes take the

basket itself also as a log-normal distribution. However, it leads to an incon-

sistency in the basic assumption “The distribution of a weighted average of

correlated log-normals is anything but log-normal. Another diculty that

prevents the price of basket options from being exactly known the correla-

tion structure involved in the basket. Correlation is observed to be volatile

over time as is the volatility. A lot of research have been done to overcome

this diculty. Several methods has been proposed, comprising numerical

methods and analytical approximation.

Instead of buying an option on each underlying asset, one may buy a

single option on all the underlying assets “Basket options” as this will be

cheaper, since there is only one option to monitor and exercise.

In the second chapter we give a literature review in pricing of basket

options, highlighting some of the important contributions.

In the third chapter, we discuss nancial derivatives and basket options,

so as to have a clear idea of the nancial market. We provide the: Denition

19

of option, types of option, some examples of nancial derivatives, traders,

and some examples of basket options.

In the fourth chapter, we discuss the pricing of basket options and the

methods used in the pricing, which is the main work of this thesis. The seller

of a nancial derivative, in particular options, requires a compensation for

the risk he is bearing, by selling the option to the buyer. The buyer must

pay a certain amount called a premium, in order to get the right to buy or

sell the underlying asset and that is what is referred to as the price of the

option. Several factors aect the pricing of basket option which include the

initial prices, volatilities of the underlying asset, correlation e.t.c.

Various methods have been used in pricing of basket options, which

include Monte-Carlo simulation (by assuming that the assets follow corre-

lated geometric Brownian motion processes) rst suggested by Boyle(1977).

Monte-Carlo methods are suitable numerical methods used in pricing op-

tions that do not have an analytical closed form solution, especially basket

options, Cox and Ross (1976) noted that if a riskless hedge can be formed,

the option value is the risk-neutral and discounted expectation of its pay-

o, that is the price can be represented by an integral, therefore making

it possible to estimate the price of the option by Monte Carlo methods,

which is done by simulating many independent paths of the underlying as-

sets and taking the discounted mean of the generated pay-o’s. We also have

Tree based method (in the case of few state variables), analytical approx-

imations such as Taylor approximation, Reciprocal gamma approximation,

Log-normal approximation e.t.c.

In the last chapter, we give some applications of log-normal approxima-

tion by considering foreign exchange basket options, and give details on how

they are priced.

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