PRICING OF BASKET OPTIONS BY FAUZIYYA ADO YAKASAI A THESIS SUBMITTED TO THE AFRICAN UNIVERSITY OF SCIENCE AND TECHNOLOGY ABUJA-NIGERIA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF MASTER OF SCIENCE DEGREE IN PURE AND APPLIED MATHEMATICS Supervisor: Professor G.O.S EKHAGUERE PhD, DIC(london), FAAS African University of Science and Technology www.aust.edu.ng P.M.B 681, Garki, Abuja F.C.T Nigeria. May 2013 PRICING OF BASKET OPTIONS A THESIS APPROVED BY SUP ERV ISOR P rof: G:O:S Ekhaguere MEMBER P rof: C:E Chidume MEMBER Dr: Guy Degla 2 DEDICATION This work is dedicate to my beloved parents, Hajiya Binta Kilishi and Engr.Ado Umar Yakasai, a father like no other, who believes a daughter has equal right to education. They made me believe in myself and always feel I can do it. Thanks for your love and support. 3 ACKNOWLEDGEMENT My first gratitude goes to the Almighty Allah whose without guidance and protection I will not have been where I am today. My special thanks goes to my supervisor Professor G.O.S EKHAGUERE for his time and contribution to the progress of this work. I would also like to send my gratitude to my lecturers: Professor Charles Chidume, Dr.Guy Degla and Malam Ma'aruf.S.Minjibir, for the guidance and knowledge they gave me through out my stay at AUST, may God reward you abundantly. My warm hearted thanks goes to my family: Dr.Umar Ado Yakasai, Eng.Shamsu Ado Yakasai, Naja Ado Yakasai, Sadiq Ado Yakasai and My fiance Ibrahim Usman Danbauchi for their endless support and encourage- ment. I Love You All! Finally I would like to thank my friends: Farouk Tijjani Sa'ad, Hajara Abdulwahab Nnya, Olulano Olu-Seun Morenrkeji, Ismail Abdulrashid and Usman Abdulmalik for their caring and support for more than a year, I will never forget your kindness towards me. This is an opportunity for me to thank my best friend Maryam Ibrahim Tijjani for always being by my side. God bless you all and anybody who support me in one way or the other. 4 Contents 1 General Introduction 7 1.1 Preliminaries: . .7 1.1.1 σ-algebra . .7 1.1.2 Probability Space . .8 1.1.3 Borel σ-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 differential 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 Definition . 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 1 General Introduction In this chapter we give some definitions 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 σ-algebra Let Ω be a non-empty set and B a non-empty collection of subset of Ω, B is called a σ-algebra if the following properties hold: iΩ 2 B ii A 2 β ) A0 2 β S iii fAj : j 2 Jg ⊂ B ) Aj 2 B for any finite or infinite countable j2J subset J of N 7 1.1.2 Probability Space 1. Let Ω be a nonempty set and B a σ- algebra of subsets of Ω. Then the pair (Ω; B) is called a measurable space and a member of B is called a measurable set. 2. Let (Ω; B) be a measurable space and µ : B −! R be a real valued map on β. Then µ is called a probability Measure if the following properties hold: i µ(A) ≥ 0 8A 2 β ii µ(Ω) = 1 iii For fAngn2N ⊂ β, with Aj \ Ak = ; for j 6= k S P µ( An) = µ(An) i.e µ is σ-additive(or countably additive). n2N n2N 3. If (Ω; β) is a measurable Space and µ is a probability measure on (Ω; β),then the triple (Ω; β; µ) is called Probability Space. 1.1.3 Borel σ-algebra If Υ is a collection of subsets of Ω,then the smallest σ-algebra of subsets of Ω which contains Υ, denoted by σ(Υ) is called the σ-algebra generated by Υ. Let X be a nonempty set and τ a topology on X, i.e τ is the collection of all open subsets of X.Then σ(τ) ia called the Borel σ-algebra of the topological space (X; τ). 8 1.1.4 A random variable: d Let (Ω; β; µ) be an arbitrary probability space,B(R ) be the Borel σ-algebra d d d of R and (R ;B(R )) the d-dimension Borel measurable space. Then, a d measurable map X :Ω −! R is called a random vector. In the case d=1, X is called a random variable. 1.1.5 Probability distribution d d Let (Ω; β; µ) be a probability space, (R ; β(R )) be the d-dimensional Borel d measurable space, and X :Ω −! R a random vector. Then the map d −1 d µX : β(R ) −! [0; 1] defined by µX (A) = µ(X (A)), A 2 β(R ) 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) 2 1 − x has density φ(x) = p e 2 ,−∞ < x < 1 and cumulative distribution 2π function 2 R x 1 − u Φ(x) = p e 2 du −∞ 2π In general a normal distribution with mean µ and variance σ2; σ > 0 (x−µ)2 1 − has density φ = p e 2σ2 and cumulative distribution function µ,σ(x) σ 2π x−µ Φµ,σ(x) = Φ( σ ) The notation X ∼ N(µ, σ2) means the random variable X is normally distributed with mean µ and variance σ2. If Y ∼ N(0; 1) (i.e Y has the standard normal distribution, then µ+σY ∼ 2 N(µ, σ ). Thus given a method for generating the samples Y1;Y2; ··· from the standard normal distribution, we can generate samples X1;X2; ··· from N(µ, σ2). It therefore suffices 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 µ and a d×d covariance matrix Σ; and is T d abbreviated as N(µ, Σ). If Σ is positive definite (i.e x Σx > 0,8x 6= 0 2 R ), then the normal distribution N(µ, Σ) has density 1 1 T −1 φµ,Σ(x) = d 1 exp(− (x − µ) Σ (x − µ)); (2π) 2 jΣj 2 2 d x 2 R with jΣj the determinant of Σ: The standard d-dimensional normal distribution N(0;Id); with Id the d × d identity matrix, is the special case 1 T d exp(−1=2 x x): (2π) 2 If X ∼ N(µ, Σ) (i.e the random vector X has a multivariate normal 2 2 distribution)then its ith component Xi has distribution N(µi; Σi ) with σi = Σii.The ith and jth component have covariances cov(Xi;Xj) = E[(Xi − µi)(Xj − µj)] = Σij which justifies calling Σ the covariance matrix. The correlation between X and X is given by ρ = Σij . i j ij σiσj If a d × d symmetric matrix Σ is positive semi-definite but not positive definite then the rank of Σ is less than d, Σ fells to be invertible, and there is no normal density with covariance matrix Σ. In this case we can define the normal distribution N(µ, Σ) as the distribution of X = µ + AZ with T Y ∼ N(0;Id) for any d × d matrix A: AA = Σ. The resulting distribution is independent of which A is chosen. The random vector X does not have a d density in R , but if Σ has rank then one can find k component of X with k multivariate normal density in R . Any linear transformation of a normal vector is again normal, X ∼ N(µ, Σ) ) AX ∼ N(Aµ, AΣAT ) for any d-vector µ and d × d matrix Σ 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.