Quasi-Monte Carlo Methods for Option Pricing

本篇論文主旨在於評估類蒙地卡羅法。傳統蒙地卡羅法已經被證明是估計封閉解不存在的資產價格的一件有價值的工具；而類蒙特卡羅法保留了傳統蒙地卡羅法的彈性，更增加了模擬速度快與收斂速度快的特性。 本篇論文比較此發法應用於3種類選擇權定價問題︰歐式選擇權，彩虹選擇權和亞式選擇權。Monte Carlo simulation has proved to be a valuable tool for estimating security prices for which closed form solutions do not exist. This thesis evaluate the Quasi-Monte Carlo method that has attractive properties for the numerical valuation of derivatives and examines the use of Monte Carlo simulation with low-discrepancy sequences for valuing derivatives versus the traditional Monte Carlo method using pseudo-random sequences. The relative performance of the methods is evaluated based on three financial securities pricing problems: European call options, rainbow options, and Asian options.Contents 1 Introduction 1.1 Introduction 1 1.2 Organization of This Thesis 2 2 Background 2.1 Monte Carlo Simulation 3 2.2 Estimating the Greeks Using Simulation 4 2.3 Antithetic Variates 5 3 Quasi-Monte Carlo Methods 3.1 Low Discrepancy Sequences 6 3.1.1 Halton Sequences 7 3.1.2 Faure Sequences 8 3.1.3 Sobol Sequences 9 3.2 Pseudo Random Uniform Sequences 10 3.2.1 rand() 10 3.2.2 The Mersenne Twister 10 3.3 Normal Inversion Methods 11 4 Numerical Results 4.1 Evaluating Vanilla Call Options 13 4.2 Evaluating Rainbow Options 15 4.3 Evaluating Asian Options 18 5 Conclusions 20 Bibliography 21 Appendix 2