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This book is devoted to some mathematical problems encountered by the au-thor in his capacity as a mathematician turned financial engineer at BankersTrust and Deutsche Bank. The exposition is restricted mainly to problemsoccurring in the foreign exchange （forex） context not only due to the fact thatit is the authors current area of responsibility but also because mathematicalmethods of financial engineering can be described more vividly when the expo-sition is centered on a single topic.
内容简介
《外汇用的数学方法》是一部系统、综合讲述了金融中尤其是外汇运用中的数学模型，具有很强的实用性。它揭示了金融工程的各个相关方面，包括衍生工具定价。《外汇用的数学方法》自成体系，介绍了必需的数学、经济以及简单贸易背景。除了标准材料的处理，还增加了许多原始结果。
目录
Preface
Ⅰ Introduction
1 Foreign exchange markets
1.1 Introduction
1.2 Historical background
1.3 Forex as an asset class
1.4 Spot forex
1.5 Derivatives： forwards， futures， calls， puts， and all that
1.6 References and further reading
Ⅱ Mathematical preliminaries
2 Elements of probability theory
2.1 Introduction
2.2 Probability spaces
2.3 Random variables
2.4 Convergence of random variables and limit theorems
2.5 References and further reading
3 Discrete-time stochastic engines
3.1 Introduction
3.2 Time series
3.3 Binomial stochastic engines for single- and multi-period markets
3.4 Multinomial stochastic engines
3.5 References and further reading
4 Continuous-time stochastic engines
4.1 Introduction
4.2 Stochastic processes
4.3 Markov processes
4.4 Diffusions
4.5 Wiener processes
4.6 Poisson processes
4.7 SDE and Mappings
4.8 Linear SDEs
4.9 SDEs for jump-diffusions
4.10 Analytical solution of PDEs
4.10.1 Introduction
4.10.2 The reduction method
4.10.3 The Laplace transform method
4.10.4 The eigenfunction expansion method
4.11 Numerical solution of PDEs
4.11.1 Introduction
4.11.2 Explicit， implicit， and Crank-Nicolson schemes for solving oneimensional problems
4.11.3 ADI scheme for solving two-dimensional problems
4.12 Numerical solution of SDEs
4.12.1 Introduction
4.12.2 Formulation of the problem
4.12.3 The Euler-Maruyama scheme
4.12.4 The Milstein scheme
4.13 References and further reading
Ⅲ Discrete-time models
5 Single-period markets
5.1 Introduction
5.2 Binomial markets with nonrisky investments
5.3 Binomial markets without nonrisky investments
5.4 General single-period markets
5.5 Economic constraints
5.6 Pricing of contingent claims
5.7 Elementary portfolio theory
5.8 The optimal investment problem
5.9 Elements of equilibrium theory
5.10 References and further reading
6 Multi-period markets
6.1 Introduction
6.2 Stationary binomial markets
6.3 Non-stationary binomial markets
6.3.1 Introduction
6.3.2 The nonrecombining case
6.3.3 The recombining case
6.4 General multi-period markets
6.5 Contingent claims and their valuation and hedging
6.6 Portfolio theory
6.7 The optimal investment problem
6.8 References and further reading
Ⅳ Continuous-time models
7 Stochastic dynamics of forex
7.1 Introduction
7.2 Two-country markets with deterministic investments
7.3 Two-country markets without deterministic investments .
7.4 Multi-country markets
7.5 The nonlinear diffusion model.
7.6 The jump diffusion model
7.7 The stochastic volatility model
7.8 The general forex evolution model
7.9 References and further reading
8 European options： the group-theoretical approach
8.1 Introduction
8.2 The two-country homogeneous problem， Ⅰ
8.2.1 Formuiation of the problem
8.2.2 Reductions of the pricing problem
8.2.3 Continuous hedging and the Greeks
8.3 Forwards， calls and puts
8.3.1 Definitions
8.3.2 Pricing via the Feynman-Kac formula
8.3.3 A naive pricing attempt
8.3.4 Pricing via the Fourier transform method
8.3.5 Pricing via the Laplace transform method
8.3.6 The limiting behavior of calls and puts
8.4 Contingent claims with arbitrary payoffs
8.4.1 Introduction
8.4.2 The decomposition formula
8.4.3 Call and put bets
8.4.4 Log contracts and modified log contracts
8.5 Dynamic asset allocation
8.6 The two-country homogeneous problem， Ⅱ
8.7 The multi-country homogeneous problem
8.7.1 Introduction
8.7.2 The homogeneous pricing problem
8.7.3 Reductions
8.7.4 Probabilistic pricing and hedging
8.8 Some representative multi-factor options
8.8.1 Introduction
8.8.2 Outperformance options
8.8.3 Options on the maximum or minimum of several FXRs
8.8.4 Basket options
8.8.5 Index options
8.8.6 The multi-factor decomposition formula
8.9 References and further reading
9 European options， the classical approach
9.1 Introduction
9.2 The classical two-country pricing problem， Ⅰ
9.2.1 The projection method
9.2.2 The classical method .
9.2.3 The impact of the actual drift
9.3 Solution of the classmal pricing problem
9.3.1 Nondimensionalization
9.3.2 Reductions
9.3.3 The pricing and hedging formulas for forwards， calls and puts
9.3.4 European options with exotic payoffs
9.4 The classical two-country pricing problem， Ⅱ
9.5 The multi-country classical pricing problem
9.5.1 Introduction
……
Bilbiography
Index
前言/序言
This book is devoted to some mathematical problems encountered by the au-thor in his capacity as a mathematician turned financial engineer at BankersTrust and Deutsche Bank. The exposition is restricted mainly to problemsoccurring in the foreign exchange （forex） context not only due to the fact thatit is the authors current area of responsibility but also because mathematicalmethods of financial engineering can be described more vividly when the expo-sition is centered on a single topic. Studying forex is interesting and importantbecause it is the grease on the wheels of the world economy. Besides， whilethe meaning of some financial instruments is difficult to comprehend withoutprior experience， everyone who has ever travelled abroad has had to exchangecurrencies thus acquiring direct experience of such concepts as spot forex rate，bid-ask spread， transaction costs （in the form of commissions）， etc. At thesame time， the reader who acquires working knowledge of the material pre-sentedin this book should be able to handle efficiently most of the problemsoccurring in equity markets and some of the problems relevant for fixed incomemarkets.
If one were to choose just one word in order to characterize financial mar-kets， that word would be uncertainty since it is their dominant feature. Someinvestors consider uncertainty a blessing， while others think it a curse， yetboth groups participate in the intricate inner workings of the markets. Thefact that foreign exchange rates （relative prices of different currencies）， as wellas prices of bonds （government or corporate obligations to repay debts） andstocks （claims on future cash flows generated by companies） are random andfinancial investments are risky was realized long ago and has been a sourceof fascination for economists， mathematicians， speculators， philosophers， andmoralists， not to mention laymen.
