| 诺贝尔经济学奖已经至少3次授予以数学为工具分析金融问题的经济学家。金融数学这门新兴的交叉学科已经成为国际金融界的一枝奇葩。金融数学的发展曾两次引发了“华尔街革命”。今天，金融数学家已经是华尔街最抢手的人才之一。目前国内金融行业最紧缺的就是掌握现代金融衍生工具的应用、能对金融风险做定量分析的、既然懂金融又懂数学的高级复合型人才。 本书是牛津大学金融数学教材，含有大量的习题和例子，面向有一定数学基础的读者。并被斯坦福大学、芝加哥大学、加州大学亚圣迭戈分校等名校选用。书中介绍了一些基本概念和二叉树、鞅、布朗运动、随机积分及Black-Scholes期权定价公式及一些复杂的金融模型和金融产品。 |
| Alison Etheridge牛津大学Madgalen学院教授。拥有牛津大学博士学位，并在剑桥大学做博士后研究。她曾先后任教于加州大学伯克利分校、爱丁堡大学和伦敦大学。主要研究兴趣是随机过程和偏微分方程及其应用。除本书外，她还著有Introduction to Superprocesses一书。 |
| 1 Single period models 1 Summary 1 1.1 Some definitions from finance 1 1.2 Pricing a forward 4 1.3 The one-step binary model 6 1.4 A ternary model 8 1.5 A characterisation of no arbitrage 9 1.6 The risk-neutral probability measure 13 Exercises 18 2 Binomial trees and discrete parameter martingales 21 Summary 21 2.1 The multiperiod binary model 21 2.2 American options 26 2.3 Discrete parameter martingales and Markov processes 28 2.4 Some important martingale theorems 38 2.5 The Binomial Representation Theorem 43 2.6 Overture to continuous models 45 Exercises 47 3 Brownian motion 51 Summary 51 3.1 Definition of the process 51 3.2 Lévy's construction of Brownian motion 56 3.3 The reflection principle and scaling 59 3.4 Martingales in continuous time 63 Exercises 67 4 Stochastic calculus 71 Summary 71 4.1 Stock prices are not differentiable 72 4.2 Stochastic integration 74 4.3 It?'s formula 85 4.4 Integration by parts and a stochastic Fubini Theorem 93 4.5 The Girsanov Theorem 96 4.6 The Brownian Martingale Representation Theorem 100 4.7 Why geometric Brownian motion? 102 4.8 The Feynman-Kac representation 102 Exercises 107 5 The Black-Scholes model 112 Summary 112 5.1 The basic Black-Scholes model 112 5.2 Black-Scholes price and hedge for European options 118 5.3 Foreign exchange 122 5.4 Dividends 126 5.5 Bonds 131 5.6 Market price of risk 132 Exercises 134 6 Oifferent payoffs 139 Summary 139 6.1 European options with discontinuous payoffs 139 6.2 Multistage options 141 6.3 Lookbacks and barriers 144 6.4 Asian options 149 6.5 American options 150 Exercises 154 7 Bigger models 159 Summary 159 7.1 General stock model 160 7.2 Multiple stock models 163 7.3 Asset prices with jumps 175 7.4 Model error 181 Exercises 185 Bibliography 189 Notation 191 Index 193 |
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