內容簡介
《金融衍生品數學模型(第2版)》旨在運用金融工程方法講述模型衍生品背後的理論,作為重點介紹瞭對大多數衍生證券很常用的鞅定價原理。書中還分析瞭固定收入市場中的大量金融衍生品,強調瞭定價、對衝及其風險策略。《金融衍生品數學模型(第2版)》從著名的期權定價模型的Black-Scholes-Merton公式開始,講述衍生品定價模型和利率模型中的最新進展,解決各種形式衍生品定價問題的解析技巧和數值方法。目次:衍生品工具介紹;金融經濟和隨機計算;期權定價模型;路徑依賴期權;美國期權;定價期權的數值方案;利率模型和債券計價;利率衍生品:債券期權、LIBOR和交換産品。
內頁插圖
目錄
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.
金融市場的波動之舞:探索衍生品的數學世界 在現代金融體係中,金融衍生品扮演著至關重要的角色,它們如同金融市場的“調味劑”,既能對衝風險、規避不確定性,又能提供投機獲利的機會。然而,這些復雜而精妙的金融工具背後,隱藏著一套嚴謹而深奧的數學原理。本書並非直接介紹金融衍生品的具體種類或交易策略,而是將目光投嚮支撐這些工具的核心——數學模型。 本書旨在為讀者揭示金融衍生品在數學模型層麵的奧秘。它將帶領讀者深入理解,為何特定的數學公式能夠有效地描述和預測衍生品的價格走勢,以及這些模型是如何被構建、驗證和應用的。我們不在此詳細列舉各類衍生品(如期權、期貨、掉期等)及其交易細則,而是聚焦於驅動它們定價與風險管理的數學邏輯。 理論的基石:從概率論到隨機過程 金融市場的本質,在很大程度上是隨機和不確定的。因此,理解金融衍生品離不開概率論和統計學的嚴謹工具。本書將首先迴顧並深入探討概率論中的關鍵概念,例如隨機變量、概率分布、期望值、方差等,這些是理解不確定性事件的基礎。在此基礎上,我們將引入更高級的工具——隨機過程。 隨機過程是描述隨時間演變的隨機現象的數學框架。在金融領域,股票價格、利率、匯率等變量的變動往往錶現齣隨機性,它們的未來走勢無法精確預測,隻能用概率的語言來描述。本書將重點介紹幾類對金融衍生品建模至關重要的隨機過程,例如: 布朗運動(Wiener過程): 這是描述隨機遊走最基本也是最重要的模型,它被廣泛應用於股票價格的隨機波動建模。我們將探討其性質,如獨立增量、平穩增量以及連續路徑等。 幾何布朗運動: 考慮到金融資産價格通常是正值且其變動率更具比例性,幾何布朗運動模型被認為是更貼閤實際的股票價格模型。我們將分析其在衍生品定價中的應用。 泊鬆過程: 用於描述離散事件的發生,例如市場中可能齣現的突發性新聞或重大事件。在某些衍生品(如帶障礙期的期權)的建模中,泊鬆過程也發揮著作用。 馬爾可夫過程: 強調未來的狀態隻取決於當前的狀態,而與過去的曆史無關。許多金融模型都基於馬爾可夫性質,這大大簡化瞭模型的分析。 本書將循序漸進地介紹這些隨機過程的數學特性,並闡述它們如何被用來刻畫金融市場中各種資産價格的動態行為。 核心的橋梁:理解定價公式的數學淵源 一旦我們掌握瞭描述資産價格變動的數學工具,我們就可以開始構建金融衍生品的定價模型。本書將深入探討這些模型的數學基礎,而不是直接羅列各種復雜定價公式。我們將重點關注以下幾個方麵: 無套利定價原理: 這是現代金融衍生品定價的基石。本書將闡述,在沒有套利機會的市場中,任何衍生品的價格都必須與其復製組閤(由標的資産和無風險資産構成)的價格相等。這一原理將是理解許多定價模型的核心。 風險中性定價: 在風險中性世界裏,所有資産的預期收益率都等於無風險利率。風險中性定價法將復雜的貼現過程轉化為一個計算期望值的問題,極大地簡化瞭衍生品定價。本書將詳細解釋為何風險中性定價在無套利市場中是有效的。 偏微分方程(PDEs): 許多著名的衍生品定價模型,如Black-Scholes模型,最終可以歸結為求解一個偏微分方程。本書將介紹求解這些PDEs的數學方法,例如有限差分法,以及這些方法如何被應用於計算期權價格。 隨機微分方程(SDEs): 描述隨機過程演變的方程。本書將介紹如何利用SDEs來構建更復雜的金融模型,並探討求解這些方程的數值方法。 通過對這些數學概念的深入剖析,讀者將能夠理解為何Black-Scholes模型能夠産生我們熟悉的期權定價公式,以及如何將類似的思想推廣到其他類型的衍生品。本書不會詳細展開Black-Scholes公式的推導過程,而是側重於其背後的數學邏輯和模型假設。 風險的量化:模型在風險管理中的應用 金融衍生品的高杠杆性和復雜性,使得風險管理成為其應用中不可或缺的一環。本書將探討數學模型如何在量化和管理衍生品風險方麵發揮作用: Greeks(希臘字母): 衍生品價格對各種市場因素(如標的資産價格、波動率、時間流逝等)敏感性的度量。我們將介紹Delta、Gamma、Theta、Vega等關鍵的Greeks,並闡述它們是如何從數學模型中推導齣來的,以及在對衝和風險管理中的意義。 VaR(Value at Risk)和CVaR(Conditional Value at Risk): 用於衡量投資組閤在一定置信水平下的最大潛在損失。本書將探討如何利用曆史模擬、參數法(基於模型假設)等方法來計算VaR和CVaR,以及模型在這些風險度量指標中的作用。 濛特卡洛模擬: 對於一些復雜的衍生品,解析解難以獲得,濛特卡洛模擬成為一種強大的數值工具。本書將介紹如何利用濛特卡洛方法來模擬資産價格的隨機路徑,並基於模型計算衍生品的價格和風險。 本書旨在為讀者提供一個理解金融衍生品數學建模的堅實基礎,幫助他們領會那些隱藏在復雜金融工具背後的數學智慧。通過掌握這些數學模型,讀者將能夠更深刻地理解金融市場的運行機製,並為進一步學習衍生品交易、投資組閤管理和風險控製打下堅實的基礎。本書不會涉及具體的衍生品交易技巧或實盤操作指導,而是專注於理論框架的構建和數學工具的應用。