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Numerical Methods for Pricing American Put Options Numerical Methods for Pricing American Put Options Daniil Kolesnikov Master's Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam Faculty of Economics and Business Amsterdam School of Economics Author: Daniil Kolesnikov Student nr: 10604189 Email: [email protected] Date: April 23, 2015 Supervisor: Prof. Dr. Roger Laeven Second reader: Andrei Lalu Statement of Originality This document is written by Student Daniil Kolesnikov who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervi- sion of completion of the work, not for the contents. Pricing of American Put Options | Daniil Kolesnikov iii Abstract This thesis considers the models for pricing American put options. The models applied are the binomial tree model, the Monte-Carlo least squares simulation method and the decomposition method. The first two methods are widely known while the latter one is is not so commonly applied. It was first used by Kim (1990) and it allows to decompose the value of the American put option into a corresponding European put option price and the early exercise premium. The thesis presents the numerical solution of the Kim equation using quadrature for- mulas and involves Newton-Raphson iteration to compute the optimal exercise boundaries at each time step. All three models are fitted to the real data and the results are compared to determine the most efficient method. The sensitivity of the models to particular parameter choices is analyzed and examined. Keywords American put option, Binomial tree, Black-Scholes, Decomposition method, Early exercise bound- ary, Integral equations, Least-squares method, Monte-Carlo simulation, Volatility Contents Preface vii Introduction 1 1 Numerical Approaches for Pricing American Options 3 1.1 Binomial Tree Model . .3 1.2 Monte-Carlo model (LSM) . .5 1.3 Decomposition Method . .6 2 Practical Implementation 12 2.1 Data Description . 12 2.2 Binomial Tree Model . 14 2.3 Least-Square Monte-Carlo Model (LSM) . 15 2.4 Decomposition method . 15 3 Results 16 3.1 Review of Results. 16 3.2 Sensitivity analysis . 19 Conclusion 22 APPENDICES 23 A Derivation of equation (1.3) 23 B Proof of equation (1.30) 25 C Risk-free interest rates 26 D Code Listings 28 D.1 MATLAB Binomial Tree Method Code . 28 D.2 MATLAB Least-Square Method Code . 29 D.3 Wolfram Mathematica Decomposition Method Code . 31 Bibliography 33 iv List of Tables 2.1 Option contract specifications . 12 2.2 Option contracts average prices for moneyness/maturity categories (in $) . 13 3.1 Option contracts standard errors of estimate in binomial tree model . 16 3.2 Errors relative to real option prices in binomial tree model . 17 3.3 Option contracts standard errors of estimate in LSM . 17 3.4 Errors relative to real option prices in LSM . 18 3.5 Option contracts standard errors of estimate in Decomposition method . 18 3.6 Errors relative to real option prices in Decomposition method . 18 3.7 Overall relative error excluding out-of-the-money options for all methods . 19 3.8 Option contracts standard errors of estimate in LSM with a linear regression . 20 3.9 Relative change in errors between simple regression and Laguerre polynomials in the LSM pricing approach. 20 v List of Figures 2.1 Daily Yahoo! Inc. returns 20/12/13 - 19/12/14. Source: Yahoo! Finance. 14 3.1 Critical stock price B(t) for an option with T = 345, K = 55 and S0 = 44:05. 19 vi Preface I wish to express my sincere gratitude to my supervisors Professor Dr. Roger Laeven and Andrei Lalu. I am thankful to them for sharing their expertise and their valuable guidance. I am also grateful to my wife and my parents for help, moral support and encour- agement. vii viii Daniil Kolesnikov | Pricing of American Put Options Introduction An American put option is a derivative contract that gives its holder the right to sell an asset for a certain price prior to and including the maturity date. Most of the options traded on exchanges are American style (Hull, 2012). These types of derivatives can be found in all financial markets: individual equity, equity index, currency, energy, agriculture, precious metals, credit, insurance, etc. The option contract trading volume has increased by 3.1% from 2013 to 2014 resulting in 9.7 billion contracts traded. Furthermore, individual equity is the largest category with the volume of options traded, around 6.5 billion in 2014.1 While it is never optimal for an American call option contract on non-dividend paying stock to be exercised prior to maturity, hence can be priced as a European call, no closed form solution exists for American put, except for perpetual put on a non-dividend paying stock. Not only the valuation but the optimal exercise of the American options is one of the most puzzling problems in derivatives finance and it continues to be of a great interest to finance theorists. One of the most famous and influential works on option pricing was written by Fischer Black and Myron Scholes in 1973, which discusses the existence of unique and rational price for European options. The work is based on the assumption of a continuous change of the underlying asset following a geometric Brownian motion with a drift. The work was refined by Merton the same year to show that Black-Scholes equation can be applied to American style options on non-dividend paying stock, but the closed form solution for American put does not exist. Another common approach to pricing options was proposed by Cox, Ross and Rubinstein in 1978, known as a binomial tree model. The model relies on a discretization of the various price paths for the underlying asset to find the value of the option. It is widely used due to it simplicity and applicability. A paper concerning American put pricing was written by Brennan and Schwartz (1977) which applied a finite difference method to valuate the options. The method approximates the differential equations that describe the development of the option price by a set of difference equations. The price of the American put option is obtained after these equations are solved iteratively. One of the most recent methods which uses Monte-Carlo simulation for American-style options was developed by Longstaff and Schwartz (2001). The idea is to simulate the underlying asset's price paths and using backward induction at each discrete tilmestep to determine by using regressions the value of continuation and based on this, decide whether it is optimal to exercise or not by comparing value of continuation and the immediate exercise value. There are other models available to price American put options, but one is of a particular interest. It was first proposed by Kim (1990). He suggests the decomposition of the American put into a European put and early exercise premium. The European part is calculated using Black-Scholes model and the early exercise premium is computed as an integral containing the critical stock price (the price below 1According to FIA Annual Volume Survey. 1 2 Daniil Kolesnikov | Pricing of American Put Options which the option should be exercised). Only a few papers implemented a numerical solution to evaluate the integral and that is done in this paper. This method is of great interest because unlike many other methods it provides the boundary for optimal exercise during the life of the option. This thesis provides the solution to two problems: applying the appropriate models for the pricing of American put options and evaluating the performance of these models by analyzing and comparing the pricing results. Hence, answering to the main question of this thesis: \How well do the option pricing models reproduce the American put option market price and which of the methods is the most efficient?" In order to implement this research the performance of three models was examined using a sample of American put options on Yahoo!Inc. non-dividend paying stocks by fitting the prices of these options during a one-month period. The models applied are the binomial tree model, the least squares Monte-Carlo model and the decomposition model. While the first two are commonly used the latter is infrequently applied which makes this research relevant and up-to-date. The structure of this thesis is as follows: the first chapter contains the literature review of the existing option pricing models as well as theoretical specifications. The second chapter is dedicated to the practical implementation of the models and justification of the choices made. The third chapter contains the results and their analysis, including the analyses of the models' sensitivity to the choices made earlier. Finally, in the last chapter the conclusion is provided. Chapter 1 Numerical Approaches for Pricing American Options This chapter contains a literature review. It includes the detailed overview of the existing pricing models for American put options, different sections of this chapter describe different methods. 1.1 Binomial Tree Model One of the most popular and useful methods for computing option prices is the binomial option pricing model that was initially proposed by Cox, Ross and Rubinstein (1979). The basic idea of this model is constructing a binomial tree { that is a graph that represents various possible paths that the stock price can follow during the life of an option. The underlying assumption is that the stock price follows a geometric Brownian motion and that the valuation is done under risk neutrality.
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