内容简介
《金融衍生品数学模型(第2版)》旨在运用金融工程方法讲述模型衍生品背后的理论,作为重点介绍了对大多数衍生证券很常用的鞅定价原理。书中还分析了固定收入市场中的大量金融衍生品,强调了定价、对冲及其风险策略。《金融衍生品数学模型(第2版)》从著名的期权定价模型的Black-Scholes-Merton公式开始,讲述衍生品定价模型和利率模型中的最新进展,解决各种形式衍生品定价问题的解析技巧和数值方法。目次:衍生品工具介绍;金融经济和随机计算;期权定价模型;路径依赖期权;美国期权;定价期权的数值方案;利率模型和债券计价;利率衍生品:债券期权、LIBOR和交换产品。
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目录
Preface
1 Introduction to Derivative Instruments
1.1 Financial Options and Their Trading Strategies
1.1.1 Trading Strategies Involving Options
1.2 Rational Boundaries for Option Values
1.2.1 Effects of Dividend Payments
1.2.2 Put-Call Parity Relations
1.2.3 Foreign Currency Options
1.3 Forward and Futures Contracts
1.3.1 Values and Prices of Forward Contracts
1.3.2 Relation between Forward and Futures Prices
1.4 Swap Contracts
1.4.1 Interest Rate Swaps
1.4.2 Currency Swaps
1.5 Problems
2 Financial Economics and Stochastic Calculus
2.1 Single Period Securities Models
2.1.1 Dominant Trading Strategies and Linear Pricing Measures
2.1.2 Arbitrage Opportunities and Risk Neutral Probability Measures
2.1.3 Valuation of Contingent Claims
2.1.4 Principles of Binomial Option Pricing Model
2.2 Filtrations, Martingales and Multiperiod Models
2.2.1 Information Structures and Filtrations
2.2.2 Conditional Expectations and Martingales
2.2.3 Stopping Times and Stopped Processes
2.2.4 Multiperiod Securities Models
2.2.5 Multiperiod Binomial Models
2.3 Asset Price Dynamics and Stochastic Processes
2.3.1 Random Walk Models
2.3.2 Brownian Processes
2.4 Stochastic Calculus: Itos Lemma and Girsanovs Theorem
2.4.1 Stochastic Integrals
2.4.2 Itos Lemma and Stochastic Differentials
2.4.3 Itos Processes and Feynman-Kac Representation Formula
2.4.4 Change of Measure: Radon-Nikodym Derivative and Girsanovs Theorem.
2.5 Problems
3 Option Pricing Models: Blaek-Scholes-Merton Formulation
3.1 Black-Scholes-Merton Formulation
3.1.1 Riskless Hedging Principle
3.1.2 Dynamic Replication Strategy
3.1.3 Risk Neutrality Argument
3.2 Martingale Pricing Theory
3.2.1 Equivalent Martingale Measure and Risk Neutral Valuation
3.2.2 Black-Scholes Model Revisited
3.3 Black-Scholes Pricing Formulas and Their Properties
3.3.1 Pricing Formulas for European Options
3.3.2 Comparative Statics
3.4 Extended Option Pricing Models
3.4.1 Options on a Dividend-Paying Asset
3.4.2 Futures Options
3.4.3 Chooser Options
3.4.4 Compound Options
3.4.5 Mertons Model of Risky Debts
3.4.6 Exchange Options
3.4.7 Equity Options with Exchange Rate Risk Exposure
3.5 Beyond the Black-Scholes Pricing Framework
3.5.1 Transaction Costs Models
3.5.2 Jump-Diffusion Models
3.5.3 Implied and Local Volatilities
3.5.4 Stochastic Volatility Models
3.6 Problems
4 Path Dependent Options
4.1 Barrier Options
4.1.1 European Down-and-Out Call Options
4.1.2 Transition Density Function and First Passage Time Density
4.1.3 Options with Double Barriers
4.1.4 Discretely Monitored Barrier Options
4.2 Lookback Options
4.2.1 European Fixed Strike Lookback Options
4.2.2 European Floating Strike Lookback Options
4.2.3 More Exotic Forms of European Lookback Options
4.2.4 Differential Equation Formulation
4.2.5 Discretely Monitored Lookback Options
4.3 Asian Options.
4.3.1 Partial Differential Equation Formulation
4.3.2 Continuously Monitored Geometric Averaging Options
4.3.3 Continuously Monitored Arithmetic Averaging Options
4.3.4 Put-Call Parity and Fixed-Floating Symmetry Relations
4.3.5 Fixed Strike Options with Discrete Geometric Averaging
4.3.6 Fixed Strike Options with Discrete Arithmetic Averaging
4.4 Problems
5 American Options
5.1 Characterization of the Optimal Exercise Boundaries
5.1.1 American Options on an Asset Paying Dividend Yield
5.1.2 Smooth Pasting Condition.
5.1.3 Optimal Exercise Boundary for an American Call
5.1.4 Put-Call Symmetry Relations.
5.1.5 American Call Options on an Asset Paying Single Dividend
5.1.6 One-Dividend and Multidividend American Put Options
5.2 Pricing Formulations of American Option Pricing Models
5.2.1 Linear Complementarity Formulation
5.2.2 Optimal Stopping Problem
5.2.3 Integral Representation of the Early Exercise Premium
5.2.4 American Barrier Options
5.2.5 American Lookback Options
5.3 Analytic Approximation Methods
5.3.1 Compound Option Approximation Method
5.3.2 Numerical Solution of the Integral Equation
5.3.3 Quadratic Approximation Method
5.4 Options with Voluntary Reset Rights
5.4.1 Valuation of the Shout Floor
5.4.2 Reset-Strike Put Options
5.5 Problems
6 Numerical Schemes for Pricing Options
6.1 Lattice Tree Methods
6.1.1 Binomial Model Revisited
6.1.2 Continuous Limits of the Binomial Model
6.1.3 Discrete Dividend Models
6.1.4 Early Exercise Feature and Callable Feature
6.1.5 Trinomial Schemes
6.1.6 Forward Shooting Grid Methods
6.2 Finite Difference Algorithms
6.2.1 Construction of Explicit Schemes
6.2.2 Implicit Schemes and Their Implementation Issues
6.2.3 Front Fixing Method and Point Relaxation Technique
6.2.4 Truncation Errors and Order of Convergence
6.2.5 Numerical Stability and Oscillation Phenomena
6.2.6 Numerical Approximation of Auxiliary Conditions
6.3 Monte Carlo Simulation
6.3.1 Variance Reduction Techniques
6.3.2 Low Discrepancy Sequences
6.3.3 Valuation of American Options
6.4 Problems
7 Interest Rate Models and Bond Pricing
7.1 Bond Prices and Interest Rates
7.1.1 Bond Prices and Yield Curves
7.1.2 Forward Rate Agreement, Bond Forward and Vanilla Swap
7.1.3 Forward Rates and Short Rates
7.1.4 Bond Prices under Deterministic Interest Rates
7.2 One-Factor Short Rate Models
7.2.1 Short Rate Models and Bond Prices
7.2.2 Vasicek Mean Reversion Model
7.2.3 Cox-Ingersoll-Ross Square Root Diffusion Model
7.2.4 Generalized One-Factor Short Rate Models
7.2.5 Calibration to Current Term Structures of Bond Prices
7.3 Multifactor Interest Rate Models
7.3.1 Short Rate/Long Rate Models
7.3.2 Stochastic Volatility Models
7.3.3 Affine Term Structure Models
7.4 Heath-Jarrow-Morton Framework
7.4.1 Forward Rate Drift Condition
7.4.2 Short Rate Processes and Theft Markovian Characterization
7.4.3 Forward LIBOR Processes under Ganssian HIM Framework
7.5 Problems
8 Interest Rate Derivatives: Bond Options, LIBOR and Swap Products
8.1 Forward Measure and Dynamics of Forward Prices
8.1.1 Forward Measure
8.1.2 Pricing of Equity Options under Stochastic Interest Rates
8.1.3 Futures Process and Futures-Forward Price Spreadi
8.2 Bond Options and Range Notes
8.2.1 Options on Discount Bonds and Coupon-Bearing Bonds
8.2.2 Range Notes
8.3 Caps and LIBOR Market Models
8.3.1 Pricing of Caps under Gaussian HJM Framework
8.3.2 Black Formulas and LIBOR Market Models
8.4 Swap Products and Swaptions
8.4.1Forward Swap Rates and Swap Measure
8.4.2 Approximate Pricing of Swaption under Lognormal LIBOR Market Model
8.4.3 Cross-Currency Swaps
8.5 Problems
References
Author Index
Subject Index
前言/序言
In the past three decades, we have witnessed the phenomenal growth in the trading of financial derivatives and structured products in the financial markets around the globe and the surge in research on derivative pricing theory,cading financial institutions are hiring graduates with a science background who can use advanced analyrical and numerical techniques to price financial derivatives and manage portfolio risks, a phenomenon coined as Rocket Science on Wall Street. There are now more than a hundred Master level degreed programs in Financial Engineering/Quantitative Finance/Computational Finance in different continents. This book is written as an introductory textbook on derivative pricing theory for students enrolled in these degree programs. Another audience of the book may include practitioners in quantitative teams in financial institutions who would like to acquire the knowledge of option pricing techniques and explore the new development in pricing models of exotic structured derivatives. The level of mathematics in this book is tailored to readers with preparation at the advanced undergraduate level of science and engineering majors, in particular, basic proficiencies in probability and statistics, differential equations, numerical methods, and mathematical analysis. Advance knowledge in stochastic processes that are relevant to the martingale pricing theory, like stochastic differential calculus and theory of martingale, are introduced in this book.
The cornerstones of derivative pricing theory are the Black-Scholes-Merton pricing model and the martingale pricing theory of financial derivatives. The renowned risk neutral valuation principle states that the price of a derivative is given by the expectation of the discounted terminal payoff under the risk neutral measure,in accordance with the property that discounted security prices are martingales under this measure in the financial world of absence of arbitrage opportunities. This second edition presents a substantial revision of the first edition. The new edition presents the theory behind modeling derivatives, with a strong focus on the martingale pricing principle. The continuous time martingale pricing theory is motivated through the analysis of the underlying financial economics principles within a discrete time framework. A wide range of financial derivatives commonly traded in the equity and fixed income markets are analyzed, emphasizing on the aspects of pricing, hedging, and their risk management. Starting from the Black-Scholes-Merton formulation of the option pricing model, readers are guided through the book on the new advances in the state-of-the-art derivative pricing models and interest rate models. Both analytic techniques and numerical methods for solving various types of derivative pricing models are emphasized. A large collection of closed form price formulas of various exotic path dependent equity options (like barrier options, lookback options, Asian options, and American options) and fixed income derivatives are documented.
金融市场的波动之舞:探索衍生品的数学世界 在现代金融体系中,金融衍生品扮演着至关重要的角色,它们如同金融市场的“调味剂”,既能对冲风险、规避不确定性,又能提供投机获利的机会。然而,这些复杂而精妙的金融工具背后,隐藏着一套严谨而深奥的数学原理。本书并非直接介绍金融衍生品的具体种类或交易策略,而是将目光投向支撑这些工具的核心——数学模型。 本书旨在为读者揭示金融衍生品在数学模型层面的奥秘。它将带领读者深入理解,为何特定的数学公式能够有效地描述和预测衍生品的价格走势,以及这些模型是如何被构建、验证和应用的。我们不在此详细列举各类衍生品(如期权、期货、掉期等)及其交易细则,而是聚焦于驱动它们定价与风险管理的数学逻辑。 理论的基石:从概率论到随机过程 金融市场的本质,在很大程度上是随机和不确定的。因此,理解金融衍生品离不开概率论和统计学的严谨工具。本书将首先回顾并深入探讨概率论中的关键概念,例如随机变量、概率分布、期望值、方差等,这些是理解不确定性事件的基础。在此基础上,我们将引入更高级的工具——随机过程。 随机过程是描述随时间演变的随机现象的数学框架。在金融领域,股票价格、利率、汇率等变量的变动往往表现出随机性,它们的未来走势无法精确预测,只能用概率的语言来描述。本书将重点介绍几类对金融衍生品建模至关重要的随机过程,例如: 布朗运动(Wiener过程): 这是描述随机游走最基本也是最重要的模型,它被广泛应用于股票价格的随机波动建模。我们将探讨其性质,如独立增量、平稳增量以及连续路径等。 几何布朗运动: 考虑到金融资产价格通常是正值且其变动率更具比例性,几何布朗运动模型被认为是更贴合实际的股票价格模型。我们将分析其在衍生品定价中的应用。 泊松过程: 用于描述离散事件的发生,例如市场中可能出现的突发性新闻或重大事件。在某些衍生品(如带障碍期的期权)的建模中,泊松过程也发挥着作用。 马尔可夫过程: 强调未来的状态只取决于当前的状态,而与过去的历史无关。许多金融模型都基于马尔可夫性质,这大大简化了模型的分析。 本书将循序渐进地介绍这些随机过程的数学特性,并阐述它们如何被用来刻画金融市场中各种资产价格的动态行为。 核心的桥梁:理解定价公式的数学渊源 一旦我们掌握了描述资产价格变动的数学工具,我们就可以开始构建金融衍生品的定价模型。本书将深入探讨这些模型的数学基础,而不是直接罗列各种复杂定价公式。我们将重点关注以下几个方面: 无套利定价原理: 这是现代金融衍生品定价的基石。本书将阐述,在没有套利机会的市场中,任何衍生品的价格都必须与其复制组合(由标的资产和无风险资产构成)的价格相等。这一原理将是理解许多定价模型的核心。 风险中性定价: 在风险中性世界里,所有资产的预期收益率都等于无风险利率。风险中性定价法将复杂的贴现过程转化为一个计算期望值的问题,极大地简化了衍生品定价。本书将详细解释为何风险中性定价在无套利市场中是有效的。 偏微分方程(PDEs): 许多著名的衍生品定价模型,如Black-Scholes模型,最终可以归结为求解一个偏微分方程。本书将介绍求解这些PDEs的数学方法,例如有限差分法,以及这些方法如何被应用于计算期权价格。 随机微分方程(SDEs): 描述随机过程演变的方程。本书将介绍如何利用SDEs来构建更复杂的金融模型,并探讨求解这些方程的数值方法。 通过对这些数学概念的深入剖析,读者将能够理解为何Black-Scholes模型能够产生我们熟悉的期权定价公式,以及如何将类似的思想推广到其他类型的衍生品。本书不会详细展开Black-Scholes公式的推导过程,而是侧重于其背后的数学逻辑和模型假设。 风险的量化:模型在风险管理中的应用 金融衍生品的高杠杆性和复杂性,使得风险管理成为其应用中不可或缺的一环。本书将探讨数学模型如何在量化和管理衍生品风险方面发挥作用: Greeks(希腊字母): 衍生品价格对各种市场因素(如标的资产价格、波动率、时间流逝等)敏感性的度量。我们将介绍Delta、Gamma、Theta、Vega等关键的Greeks,并阐述它们是如何从数学模型中推导出来的,以及在对冲和风险管理中的意义。 VaR(Value at Risk)和CVaR(Conditional Value at Risk): 用于衡量投资组合在一定置信水平下的最大潜在损失。本书将探讨如何利用历史模拟、参数法(基于模型假设)等方法来计算VaR和CVaR,以及模型在这些风险度量指标中的作用。 蒙特卡洛模拟: 对于一些复杂的衍生品,解析解难以获得,蒙特卡洛模拟成为一种强大的数值工具。本书将介绍如何利用蒙特卡洛方法来模拟资产价格的随机路径,并基于模型计算衍生品的价格和风险。 本书旨在为读者提供一个理解金融衍生品数学建模的坚实基础,帮助他们领会那些隐藏在复杂金融工具背后的数学智慧。通过掌握这些数学模型,读者将能够更深刻地理解金融市场的运行机制,并为进一步学习衍生品交易、投资组合管理和风险控制打下坚实的基础。本书不会涉及具体的衍生品交易技巧或实盘操作指导,而是专注于理论框架的构建和数学工具的应用。