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Numerical Methods for the Valuation of Financial Derivatives

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Numerical Methods for the Valuation of Financial Derivatives NUMERICAL METHODS FOR THE VALUATION OF FINANCIAL DERIVATIVES DAVIS BUNDI NTWIGA A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science, in the Department of Mathematics and Applied Mathematics, University of Western Cape, South Africa. Supervisor: Prof. Jean Claude Ndogmo November 2005 NUMERICAL METHODS FOR THE VALUATION OF FINANCIAL DERIVATIVES. DAVIS BUNDI NTWIGA KEYWORDS Black Scholes. Financial Derivatives. Exotic Options. Futures. American and European Options. Stochastic Differential Equations. Multi Period Binomial Model. Finite Difference Method. Monte Carlo Simulation. Differential Equations. i ABSTRACT NUMERICAL METHODS FOR THE VALUATION OF FINANCIAL DERIVATIVES. DAVIS BUNDI NTWIGA MSc thesis, Department of Mathematics and Applied Mathematics, University of Western Cape. Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps. We discuss the Finite difference methods quite extensively with a focus on the Implicit and Crank-Nicolson methods, and apply these numerical techniques to the pricing of vanilla options. Finally, we compare the convergence of the multi-period binomial model, the Implicit and Crank Nicolson methods to the analytical Black Scholes price of the option. We conclude with the pricing of exotic options with special emphasis on path dependent options. Monte Carlo simulation technique is applied as this method is very versatile in cases where there is no closed form analytical formula. The method is slow and time consuming but very flexible even for multi dimensional problems. The data analysis is presented by means of tables and graphs. The sample programs used in generating the tables are based on Matlab programming language and are listed in the appendix. ii DECLARATION I declare that Numerical Methods for the Valuation of Financial Derivatives is my own work, that it has not been submitted for any degree or examination in any other University, and that all the sources I have or quoted have been indicated and acknowledged by complete references. Davis Bundi Ntwiga November 2005 Signed: .................... iii ACKNOWLEDGMENTS First and foremost, my special gratitude goes to the University of the Western Cape through the Dean, Faculty of Science, and the African Institute for Mathematical Sciences (AIMS) for their financial support toward my masters’ studies. Sincere thanks to Prof. Neil Turok, the founder of AIMS. I am also grateful to my supervisor Prof. Jean Claude Ndogmo for his guidance, sugges- tions and assistance during the course of this work. I thank Prof. Peter Witbooi for his gratifying support during my stay and in particular my arrival at the University of Western Cape, the effort and sacrifice shown by Justin, Elmerie, Paul and Ian for assisting me in proof reading the document, and not forgetting the staff and students in the Department for creating a warm and friendly environment that enabled me to undertake my research. Finally, I am highly indebted to my family and relatives back at home, and friends both at home and here in South Africa. The love, patience, encouragements and their support through constant communication have been an invaluable asset for me. I wish to mention a few names, Doris, Martin, Michael, Maina, Benta, Timothy, Edwin, James, Kendi and my Dad, Evasius as the list is endless. To my late mum, Hellen Kagendo. iv LIST OF ACRONYMS NSE Nairobi Stock Exchange. MTM Marking To Market. GBM Geometric Brownian Motion. SDE Stochastic Differential Equation. CRR Cox, Ross and Rubinstein. PDE Partial Differential Equation. FDE Finite Difference Equation. SOR Successive Over Relaxation. JSE Johannesburg Stock Exchange. ATS Automated Trading System. SSF Single Stock Futures. SAFEX South African Futures Exchange. MCS Monte Carlo Simulation. v Contents 1 Introduction 1 1.1 Background . 1 1.1.1 Aims of Study . 2 1.2 Financial Derivatives . 3 1.2.1 Futures . 4 1.2.2 Forwards . 5 1.2.3 Options . 5 1.2.4 Value of an Option . 7 1.3 The Dynamics of Derivative Prices . 8 1.3.1 Stochastic Process . 9 1.3.2 Markov Process . 9 1.3.3 Martingale . 9 1.3.4 Brownian Motion . 10 1.3.5 Brownian Motion as the Limit of a Random Walk . 11 1.3.6 Geometric Brownian Motion . 12 1.4 Arbitrage . 12 1.4.1 Arbitrage-Free Market . 13 1.4.2 Risk Neutral Valuation . 15 2 Black - Scholes Model 17 2.1 Stochastic Differential Equation . 17 2.1.1 Ito^ Process . 18 2.1.2 Ito^’s Lemma . 18 2.2 Lognormal Dynamics . 22 vi 2.2.1 Factors Affecting Option Value . 23 2.3 Black - Scholes Equation . 24 2.3.1 American and European Options . 26 2.3.2 Solution of the Black-Scholes Equation . 28 2.3.3 Volatility . 31 2.3.4 Dividend Paying Stock . 33 2.4 Options on Futures . 34 2.4.1 American Options on Futures . 35 2.4.2 European Options on Futures . 36 2.4.3 Black’s Model . 37 3 Binomial Model 40 3.1 CRR Model . 41 3.2 Numerical Implementation . 44 3.2.1 Dividend Paying Stock . 46 3.3 Single Stock Futures Contracts . 48 3.3.1 Options on SSFs . 49 3.3.2 Valuation of Options on Futures . 51 4 Finite Difference Methods 53 4.1 Discretization of the Equation . 53 4.1.1 Finite Difference Approximations . 55 4.1.2 Boundary and Initial Conditions . 56 4.1.3 Log Transform of the Black Scholes Equation . 57 4.2 The Explicit Finite Difference Method . 59 4.2.1 Stability Analysis . 61 4.2.2 Change of Variable - The Explicit Method . 65 4.3 The Implicit Finite Difference Method . 66 4.3.1 Solving Systems of Linear Equations . 68 4.3.2 Change of Variable - The Implicit Method . 71 4.4 The Crank Nicolson Method . 72 4.4.1 Options on Futures . 75 vii 4.4.2 Matlab Implementation . 77 5 Monte Carlo Simulation 79 5.1 Simulation . 79 5.1.1 Monte Carlo Method . 80 5.2 Variance Reduction Procedures . 83 5.2.1 Antithetic Variable Technique . 83 5.2.2 Control Variate Technique . 84 5.3 Exotic Options . 85 5.3.1 Path Dependent Option . 85 Bibliography 92 A Appendix 95 A.1 Binomial Model for a Non-Dividend Paying Stock . 95 A.2 Binomial Model for Options on Futures. 96 A.3 Binomial Model for a Known Dividend Paying Stock. 97 A.4 Fully Implicit Finite Difference Method for Options on Spot . 98 A.5 Crank Nicolson Finite Difference Method for Options on Spot . 99 A.6 Crank Nicolson Finite Difference Method for Options on Futures . 100 A.7 Standard Asian Options . 101 viii Chapter 1 Introduction 1.1 Background In 1973; the Chicago Board of Trade (CBOT) became the first organized exchange to start trading so-called options, as well as futures and other financial derivatives. The question that arose was what price was the buyer willing to pay for those financial instruments? Black, Scholes and Merton approached the problem of pricing an option in a physicist’s way by assuming a reasonable model for the price of a risky asset. This search had not started at this point but can be traced back to the time of a botanist Robert Brown (1827); who first described the unusual motion exhibited by a small particle that is totally immersed in a liquid or gas. Then, Albert Einstein (1905) expounded more on this motion. Further, Norbert Wiener gave in a mathematically concise definition of the theory on Brownian Motion in a series of papers originating in 1918: Before Einstein, a young French PhD student, called Louis Bachelier proposed in his 1900 thesis, Brownian motion as a model for speculative prices. In 1960’s the economist Samuelson propagated to his students at Massachusetts Institute of Technology (MIT) the exponential of Brownian motion (Geometric Brownian motion) for modeling prices which are subject to uncertainty. Brownian motion is a natural analogue of a random walk in continuous time. A random walk is defined as discrete time equidistant instants of time. In finance, we use it to model prices at every instant of time and the notion of continuous time becomes important. The use of Brownian motion has a major flaw when used to model stock prices since it assumes that the price of a stock follows a normal random variable. This leads amongst others, to negative stock 1 values which is not realistic. The geometric Brownian motion is the basic mathematical model for price movements and Black, Scholes and Merton used this principle. However, they realized that Brownian motion is closely related to stochastic or Ito^ calculus, named after the Japanese mathematician Kiyosi Ito^ who developed this theory in the 1940’s. They further used the notion that the derivative security can be exactly replicated by a dynamic trading strategy utilizing the stocks and the risk-less bond. A trading strategy which replicates the value of the option at maturity is called a hedge.
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