金融市場(chǎng)數(shù)學(xué)

前言

This work is aimed at an audience with a sound mathematical background wishing to learn about the rapidly expanding field of mathematical finance. Its content is suitable particularly for graduate students in mathematics who have a background in measure theory and probability.The emphasis throughout is on developing the mathematical concepts required for the theory within the context of their application. No attempt is made to cover the bewildering variety of novel （or exotic） financial in- struments that now appear on the derivatives markets; the focus through- out remains on a rigorous development of the more basic options that lie at the heart of the remarkable range of current applications of martingale theory to financial markets.The first five chapters present the theory in a discrete-time framework. Stochastic calculus is not required, and this material should be accessible to anyone familiar with elementary probability theory and linear algebra.The basic idea of pricing by arbitrage （or, rather, by non-arbitrage） is presented in Chapter 1. The unique price for a European option in a single-period binomial model is given and then extended to multi-period binomial models. Chapter 2 introduces the idea of a martingale measure for price processes. Following a discussion of the use of self-financing trad- ing strategies to hedge against trading risk, it is shown how options can be priced using an equivalent measure for which the discounted price pro- cess is a martingale. This is illustrated for the simple binomial Cox-Ross- Rubinstein pricing models, and the Black-Scholes formula is derived as the limit of the prices obtained for such models. Chapter 3 gives the funda- mental theorem of asset pricing, which states that if the market does not contain arbitrage opportunities there is an equivalent martingale measure. Explicit constructions of such measures are given in the setting of finite market models. Completeness of markets is investigated in Chapter 4; in a complete market, every contingent claim can be generated by an admissible self-financing strategy （and the martingale measure is unique）. Stopping times, martingale convergence results, and American options are discussed in a discrete-time framework in Chapter 5.

內(nèi)容概要

　　本書旨在講述研究現(xiàn)代金融市場(chǎng)衍生證券，如期權(quán)、期貨和交換業(yè)務(wù)等所需的數(shù)學(xué)知識(shí)。建立在著名的Black-Scholes理論基礎(chǔ)上的理想化連續(xù)時(shí)間模型需要對(duì)現(xiàn)代微積分有較深的了解。然而，書中許多潛在的知識(shí)點(diǎn)完全可以在離散時(shí)間的框架內(nèi)理解。本書是在第1版的基礎(chǔ)上做了較多增補(bǔ)，使得連續(xù)時(shí)間理論應(yīng)用范圍更加廣泛，更加詳細(xì)地介紹Black-Scholes模型及其推廣、期限結(jié)構(gòu)和消費(fèi)投資問(wèn)題。增加的內(nèi)容有：一致性風(fēng)險(xiǎn)測(cè)度及其在對(duì)沖中的應(yīng)用；一般離散市場(chǎng)模型中資產(chǎn)估價(jià)的第一基本定理；不完全離散市場(chǎng)的套利區(qū)間；完全離散市場(chǎng)的特征；Black-Scholes模型中的風(fēng)險(xiǎn)、回報(bào)和靈敏度。本書內(nèi)容安排相當(dāng)謹(jǐn)慎、詳細(xì)，而不是泛泛羅列所有盡可能多的內(nèi)容，對(duì)期權(quán)的處理相當(dāng)精辟。通過(guò)本書的學(xué)習(xí)，讀者也可以了解更多的科研動(dòng)態(tài)。目次：套利定價(jià)；鞅測(cè)度；第一基本定理；完全市場(chǎng)；離散時(shí)間美國(guó)期權(quán)；連續(xù)時(shí)間隨機(jī)計(jì)算；美國(guó)賣方期權(quán)；債券和期限結(jié)構(gòu)；消費(fèi)投資策略；風(fēng)險(xiǎn)度量?！　∽x者對(duì)象：數(shù)學(xué)專業(yè)的研究生、科研人員以及具有一定數(shù)學(xué)背景的金融愛(ài)好者。

書籍目錄

Preface Preface to the Second Edition 1 Prlcing by Arbitrage   1.1 Introduction: Pricing and Hedging   1.2 Single-Period Option Pricing Models 　1.3 A General Single-Period Model 　1.4 A Single-Period Binomial Model 　1.5 Multi-period Binomial Models 　1.6 Bounds on Option Prices 2 Martingale Measures 　2.1 A General Discrete-Time Market Model 　2.2 Trading Strategies 　2.3 Martingales and Risk-Neutral Pricing 　2.4 Arbitrage Pricing: Martingale Measures 　2.5 Strategies Using Contingent Claims 　2.6 Example: The Binomial Model 　2.7 From CRR to Blaek-Scholes 3 The First Fundamental Theorem 　3.1 The Separating Hyperplane Theorem in Rn 　3.2 Construction of Martingale Measures 　3.3 Pathwise Description 　3.4 Examples 　3.5 General Discrete Models 4 Complete Markets 　4.1 Completeness and Martingale Representation 　4.2 Completeness for Finite Market Models 　4.3 The CRR Model 　4.4 The Splitting Index and Completeness 　4.5 Incomplete Models: The Arbitrage Interval 　4.6 Characterisation of Complete Models 5 Discrete-time American Options 　5.1 Hedging American Claims 　5.2 Stopping Times and Stopped Processes 　5.3 Uniformly Integrable Martingales 　5.4 Optimal Stopping: The Snell Envelope 　5.5 Pricing and Hedging American Options 　5.6 Consumption-Investment Strategies 6 Continuous-Time Stochastic Calculus 　6.1 Continuous-Time Processes 　6.2 Martingales 　6.3 Stochastic Integrals 　6.4 The It8 Calculus 　6.5 Stochastic Differential Equations 　6.6 Markov Property of Solutions of SDEs 7 Continuous-Time European Options 　7.1 Dynamics 　7.2 Girsanov's Theorem 　7.3 Martingale Representation 　7.4 Self-Financing Strategies 　7.5 An Equivalent Martingale Measure 　7.6 Black-Scholes Prices 　7.7 Pricing in a Multifactor Model 　7.8 Barrier Options 　7.9 The Black-Scholes Equation 　7.10 The Greeks 8 The American Put Option 　8.1 Extended Trading Strategies 　8.2 Analysis of American Put Options 　8.3 The Perpetual Put Option 　8.4 Early Exercise Premium 　8.5 Relation to Free Boundary Problems 　8.6 An Approximate Solution 9 Bonds and Term Structure 　9.1 Market Dynamics 　9.2 Future Price and Futures Contracts 　9.3 Changing Numeraire 　9.4 A General Option Pricing Formula 　9.5 Term Structure Models 　9.6 Short-rate Diffusion Models 　9.7 The Heath-Jarrow-Morton Model 　9.8 A Markov Chain Model 10 Consumption-Investment Strategies 　10.1 Utility Functions 　10.2 Admissible Strategies 　10.3 Maximising Utility of Consumption 　10.4 Maximisation of Terminal Utility 　10.5 Consumption and Terminal Wealth 11 Measures of Risk 　11.1 Value at Risk 　11.2 Coherent Risk Measures 　11.3 Deviation Measures 　11.4 Hedging Strategies with Shortfall Risk Bibliography Index

章節(jié)摘錄

插圖：The unreasonable effectiveness of mathematics is evidenced by the fre- quency with which mathematical techniques that were developed without thought for practical applications find unexpected new domains of appli- cability in various spheres of life. This phenomenon has customarily been observed in the physical sciences; in the social sciences its impact has per- haps been less evident. One of the more remarkable examples of simulta- neous revolutions in economic theory and market practice is provided by the opening of the worlds first options exchange in Chicago in 1973, and the ground-breaking theoretical papers on preference-free option pricing by Black and Scholes [27] （quickly extended by Merton [222]） that appeared in the same year, thus providing a workable model for the rational market pricing of traded options.From these beginnings, financial derivatives markets worldwide have become one of the most remarkable growth industries and now constitute a major source of employment for graduates with high levels of mathemat- ical expertise. The principal reason for this phenomenon has its origins in the simultaneous stimuli just described, and the explosive growth of these secondary markets （whose levels of activity now frequently exceed the un- derlying markets on which their products are based） continues unabated, with total trading volume now measured in trillions of dollars. The vari- ety and complexity of new financial instruments is often bewildering, and much effort goes into the analysis of the （ever more complex） mathematical models on which their existence is predicated.

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