本书由剑桥大学出版社出版，原书名为：Financial Engineering and Computation: Principles, Mathematics, and Algorithms，是一本非常优秀的有关金融计算的图书。

如今打算在金融领域工作的学生和专家不仅要掌握先进的概念和数学模型，还要学会如何在计算上实现这些模型。本书内容广泛，不仅介绍了金融工程背后的理论和数学，并把重点放在了计算上，以便和金融工程在今天资本市场的实际运作保持一致。本书不同于大多数的有关投资、金融工程或者衍生证券方面的书，而是从金融的基本想法开始，逐步建立理论。作者提供了很多定价、风险评估以及项目组合管理的算法和理论。本书的重点是有关金融产品和衍生证券、期权、期货、远期、利率衍生产品、抵押证券等等的定价问题。每个工具都有简要的介绍，每章都可以独立被引用。本书的算法均使用Java算法编程实现的，并可以在相关的网站上下载。

本书适合经济学、金融数学专业的学生、教师，以及从事金融事务领域的相关人员使用参考。

Foreword

Preface

1 Probability theory: basic notions

 1.1 Introduction

 1.2 Probability distributions

 1.3 Typical values and deviations

 1.4 Moments and characteristic function

 1.5 Divergence of moments-asymptotic behaviour

 1.6 Gaussian distribution

 1.7 Log-normal distribution

 1.8 Levy distributions and Paretian tails

 1.9 Other distributions (*)

 1.10 Summary

2 Maximum and addition of random variables

 2.1 Maximum of random variables

 2.2 Sums of random variables

2.2.1 Convolutions

2.2.2 Additivity of cumulants and of tail amplitudes

2.2.3 Stable distributions and self-similarity

 2.3 Central limit theorem

2.3.1 Convergence to a Gaussian

2.3.2 Convergence to a Levy distribution

2.3.3 Large deviations

2.3.4 Steepest descent method and Cram~r function (*)

2.3.5 The CLT at work on simple cases

2.3.6 Truncated L6vy distributions

2.3.7 Conclusion: survival and vanishing of tails

2.4 From sum to max: progressive dominance of extremes (*)

2.5 Linear correlations and fractional Brownian motion

 2.6 Summary

3 Continuous time limit, Ito calculus and path integrals

 3. I Divisibility and the continuous time limit

3.1.1 Divisibility

3.1.2 Infinite divisibility

3.1.3 Poisson jump processes

 3.2 Functions of the Brownian motion and Ito calculus

3.2.1 Ito's lemma

3.2.2 Novikov's formula

3.2.3 Stratonovich's prescription

 3.3 Other techniques

3.3.1 Path integrals

3.3.2 Girsanov's formula and the Martin-Siggia-Rose trick

 3.4 Summary

4 Analysis of empirical data

 4.1 Estimating probability distributions

4.1.1 Cumulative distribution and densities - rank histogram

4.1.2 Kolmogorov-Smirnov test

4.1.3 Maximum likelihood

4.1.4 Relative likelihood

4.1.5 A general caveat

 4.2 Empirical moments: estimation and error

4.2.1 Empirical mean

4.2.2 Empirical variance and MAD

4.2.3 Empirical kurtosis

4.2.4 Error on the volatility

 4.3 Correlograms and variograms

4.3.1 Variogram

4.3.2 Correlogram

4.3.3 Hurst exponent

4.3.4 Correlations across different time zones

 4.4 Data with heterogeneous volatilities

 4.5 Summary

5 Financial products and financial markets

 5.1 Introduction

 5.2 Financial products

5.2.1 Cash (Interbank market)

5.2.2 Stocks

5.2.3 Stock indices

5.2.4 Bonds

5.2.5 Commodities

5.2.6 Derivatives

 5.3 Financial markets

5.3.1 Market participants

5.3.2 Market mechanisms

5.3.3 Discreteness

5.3.4 The order book

5.3.5 The bid-ask spread

5.3.6 Transaction costs

5.3.7 Time zones, overnight, seasonalities

 5.4 Summary

6 Statistics of real prices: basic results

 6.1 Aim of the chapter

 6.2 Second-order statistics

6.2.1 Price increments vs. returns

6.2.2 Autocorrelation and power spectrum

 6.3 Distribution of returns over different time scales

6.3.1 Presentation of the data

6.3.2 The distribution of returns

6.3.3 Convolutions

 6.4 Tails, what tails？

 6.5 Extreme markets

 6.6 Discussion

 6.7 Summary

7 Non-linear correlations and volatility fluctuations

 7.1 Non-linear correlations and dependence

7.1.1 Non identical variables

7.1.2 A stochastic volatility model

7.1.3 GARCH(I,I)

7.1.4 Anomalous kurtosis

7.1.5 The case of infinite kurtosis

 7.2 Non-linear correlations in financial markets: empirical results

7.2.1 Anomalous decay of the cumulants

7.2.2 Volatility correlations and variogram

 7.3 Models and mechanisms

7.3.1 Multifractality and multifractal models (*)

7.3.2 The microstructure of volatility

 7.4 Summary

8 Skewness and price-volatility correlations

 8.1 Theoretical considerations

8.1.1 Anomalous skewness of sums of random variables

8.1.2 Absolute vs. relative price changes

8.1.3 The additive-multiplicative crossover and the q-transformation

 8.2 A retarded model

8.2.1 Definition and basic properties

8.2.2 Skewness in the retarded model

 8.3 Price-volatility correlations: empirical evidence

8.3.1 Leverage effect for stocks and the retarded model

8.3.2 Leverage effect for indices

8.3.3 Return-volume correlations

 8.4 The Heston model: a model with volatility fluctuations and skew

 8.5 Summary

9 Cross-correlations

 9.1 Correlation matrices and principal component analysis

9.1.1 Introduction

9.1.2 Gaussian correlated variables

9.1.3 Empirical correlation matrices

 9.2 Non-Gaussian correlated variables

9.2.1 Sums of non Gaussian variables

9.2.2 Non-linear transformation of correlated Gaussian variables

9.2.3 Copulas

9.2.4 Comparison of the two models

9.2.5 Multivariate Student distributions

9.2.6 Multivariate L~vy variables (*)

9.2.7 Weakly non Gaussian correlated variables (*)

 9.3 Factors and clusters

9.3.1 One factor models

9.3.2 Multi-factor models

9.3.3 Partition around medoids

9.3.4 Eigenvector clustering

9.3.5 Maximum spanning tree

 9.4 Summary

 9.5 Appendix A: central limit theorem for random matrices

 9.6 Appendix B: density of eigenvalues for random correlation matrices

10 Risk measures

 10.1 Risk measurement and diversification

 10.2 Risk and volatility

 10.3 Risk of loss, 'value at risk' (VaR) and expected shortfall

10.3.1 Introduction

10.3.2 Value-at-risk

10.3.3 Expected shortfall

 10.4 Temporal aspects: drawdown and cumulated loss

 10.5 Diversification and utility-satisfaction thresholds

 10.6 Summary

11 Extreme correlations and variety

 11.1 Extreme event correlations .

11.1.1 Correlations conditioned on large market moves

11.1.2 Real data and surrogate data

11.1.3 Conditioning on large individual stock returns: exceedance correlations

11.1.4 Tail dependence

11.1.5 Tail covariance (*)

 11.2 Variety and conditional statistics of the residuals

11.2.1 The variety

11.2.2 The variety in the one-factor model

11.2.3 Conditional variety of the residuals

11.2.4 Conditional skewness of the residuals

 11.3 Summary

 11.4 Appendix C: some useful results on power-law variables

12 Optimal portfolios

 12.1 Portfolios of uncorrelated assets

12.1.1 Uncorrelated Gaussian assets

12.1.2 Uncorrelated 'power-law' assets

12.1.3 Exponential' assets

12.1.4 General case: optimal portfolio and VaR (*)

 12.2 Portfolios of correlated assets

12.2.1 Correlated Gaussian fluctuations

12.2.2 Optimal portfolios with non-linear constraints (*)

12.2.3 'Power-law' fluctuations - linear model (*)

 12.2.4 'Power-law' fluctuations - Student model (*)

 12.3 Optimized trading

 12.4 Value-at-risk- general non-linear portfolios (*)

12.4.1 Outline of the method: identifying worst cases

12.4.2 Numerical test of the method

 12.5 Summary

13 Futures and options: fundamental concepts

 13.1 Introduction

13.1.1 Aim of the chapter

13.1.2 Strategies in uncertain conditions

13.1.3 Trading strategies and efficient markets

 13.2 Futures and forwards

13.2.1 Setting the stage

13.2.2 Global financial balance

13.2.3 Riskless hedge

13.2.4 Conclusion: global balance and arbitrage

 13.3 Options: definition and valuation

13.3.1 Setting the stage

13.3.2 Orders of magnitude

13.3.3 Quantitative analysis-option price

13.3.4 Real option prices, volatility smile and 'implied' kurtosis

13.3.5 The case of an infinite kurtosls

 13.4 Summary

14 Options: hedging and residual risk

 14.1 Introduction

 14.2 Optimal hedging strategies

14.2.1 A simple case: static hedging

14.2.2 The general case and 'A' hedging

 14.2.3 Global hedging vs. instantaneous hedging

14.3 Residual risk

14.3.1 The Black-Scholes miracle

14.3.2 The 'stop-loss' strategy does not work

14.3.3 Instantaneous residual risk and kurtosis risk

  14.3.4 Stochastic volatility models

 14.4 Hedging errors. A variational point of view

 14.5 Other measures of risk-hedging and VaR (*)

 14.6 Conclusion of the chapter

 14.7 Summary

 14.8 Appendix D

15 Options: the role of drift and correlations

 15.1 Influence of drift on optimally hedged option

15.1.1 A perturbative expansion

15.1.2 'Risk neutral' probability and martingales

 15.2 Drift risk and delta-hedged options

15.2.1 Hedging the drift risk

15.2.2 The price of delta-hedged options

15.2.3 A general option pricing formula

 15.3 Pricing and hedging in the presence of temporal correlations (*)

15.3.1 A general model of correlations

15.3.2 Derivative pricing with small correlations

  15.3.3 The case of delta-hedging

 15.4 Conclusion

15.4.1 Is the price of an option unique？

15.4.2 Should one always optimally hedge？

 15.5 Summary

 15.6 Appendix E

16 Options: the Black and Scholes model

 16.1 Ito calculus and the Black-Scholes equation

16.1.1 The Gaussian Bachelier model

16.1.2 Solution and Martingale

16.1.3 Time value and the cost of hedging

16.1.4 The Log-normal Black-Scholes model

16.1.5 General pricing and hedging in a Brownian world

16.1.6 The Greeks

 16.2 Drift and hedge in the Gaussian model (*)

16.2.1 Constant drift

16.2.2 Price dependent drift and the Omstein-Uhlenbeck paradox

 16.3 The binomial model

 16.4 Summary

17 Options: some more specific problems

 17.1 Other elements of the balance sheet

17.1.1 Interest rate and continuous dividends

17.1.2 Interest rate corrections to the hedging strategy

17.1.3 Discrete dividends

17.1.4 Transaction costs

 17.2 Other types of options

17.2.1 'Put-call' parity

17.2.2 'Digital' options

17.2.3 'Asian' options

17.2.4 'American' options

17.2.5 'Barrier' options (*)

17.2.6 Other types of options

 17.3 The 'Greeks' and risk control

 17.4 Risk diversification (*)

 17.5 Summary

18 Options: minimum variance Monte-Carlo

 18.1 Plain Monte-Carlo

18.1.1 Motivation and basic principle

18.1.2 Pricing the forward exactly

18.1.3 Calculating the Greeks

18.1.4 Drawbacks of the method

 18.2 An 'hedged' Monte-Carlo method

18.2.1 Basic principle of the method

18.2.2 A linear parameterization of the price and hedge

18.2.3 The Black-Scholes limit

 18.3 Non Gaussian models and purely historical option pricing

 18.4 Discussion and extensions. Calibration

 18.5 Summary

 18.6 Appendix F: generating some random variables

19 The yield curve

 19.1 Introduction

 19.2 The bond market

 19.3 Hedging bonds with other bonds

19.3.1 The general problem

19.3.2 The continuous time Ganssian limit

 19.4 The equation for bond pricing

19.4.1 A general solution

19.4.2 The Vasicek model

19.4.3 Forward rates

19.4.4 More general models

 19.5 Empirical study of the forward rate curve

19.5.1 Data and notations

19.5.2 Quantities of interest and data analysis

 19.6 Theoretical considerations (*)

19.6.1 Comparison with the Vasicek model

19.6.2 Market price of risk

19.6.3 Risk-premium and the law

 19.7 Summary

 19.8 Appendix G: optimal portfolio of bonds

20 Simple mechanisms for anomalous price statistics

 20.1 Introduction

 20.2 Simple models for herding and mimicry

20.2.1 Herding and percolation

20.2.2 Avalanches of opinion changes

 20.3 Models of feedback effects on price fluctuations

20.3.1 Risk-aversion induced crashes

20.3.2 A simple model with volatility correlations and tails

20.3.3 Mechanisms for long ranged volatility correlations

 20.4 The Minority Game

 20.5 Summary

Index of most important symbols

Index