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Theory of Financial Risk and Derivative Pricing, Second Edition

Book Description

Risk control and derivative pricing have become of major concern to financial institutions, and there is a real need for adequate statistical tools to measure and anticipate the amplitude of the potential moves of the financial markets. Summarising recent theoretical developments in the field, this second edition has been substantially expanded. Additional chapters now cover stochastic processes, Monte-Carlo methods, Black-Scholes theory, the theory of the yield curve, and Minority Game. There are discussions on aspects of data analysis, financial products, non-linear correlations, and herding, feedback and agent based models. This book has become a classic reference for graduate students and researchers working in econophysics and mathematical finance, and for quantitative analysts working on risk management, derivative pricing and quantitative trading strategies.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Foreword
  6. Preface
  7. 1 Probability theory: basic notions
    1. 1.1 Introduction
    2. 1.2 Probability distributions
    3. 1.3 Typical values and deviations
    4. 1.4 Moments and characteristic function
    5. 1.5 Divergence of moments – asymptotic behaviour
    6. 1.6 Gaussian distribution
    7. 1.7 Log-normal distribution
    8. 1.8 Lévy distributions and Paretian tails
    9. 1.9 Other distributions (*)
    10. 1.10 Summary
  8. 2 Maximum and addition of random variables
    1. 2.1 Maximum of random variables
    2. 2.2 Sums of random variables
      1. 2.2.1 Convolutions
      2. 2.2.2 Additivity of cumulants and of tail amplitudes
      3. 2.2.3 Stable distributions and self-similarity
    3. 2.3 Central limit theorem
      1. 2.3.1 Convergence to a Gaussian
      2. 2.3.2 Convergence to a Lévy distribution
      3. 2.3.3 Large deviations
      4. 2.3.4 Steepest descent method and Cramèr function (*)
      5. 2.3.5 The CLT at work on simple cases
      6. 2.3.6 Truncated Lévy distributions
      7. 2.3.7 Conclusion: survival and vanishing of tails
    4. 2.4 From sum to max: progressive dominance of extremes (*)
    5. 2.5 Linear correlations and fractional Brownian motion
    6. 2.6 Summary
  9. 3 Continuous time limit, Ito calculus and path integrals
    1. 3.1 Divisibility and the continuous time limit
      1. 3.1.1 Divisibility
      2. 3.1.2 Infinite divisibility
      3. 3.1.3 Poisson jump processes
    2. 3.2 Functions of the Brownian motion and Ito calculus
      1. 3.2.1 Ito’s lemma
      2. 3.2.2 Novikov’s formula
      3. 3.2.3 Stratonovich’s prescription
    3. 3.3 Other techniques
      1. 3.3.1 Path integrals
      2. 3.3.2 Girsanov’s formula and the Martin–Siggia–Rose trick (*)
    4. 3.4 Summary
  10. 4 Analysis of empirical data
    1. 4.1 Estimating probability distributions
      1. 4.1.1 Cumulative distribution and densities – rank histogram
      2. 4.1.2 Kolmogorov–Smirnov test
      3. 4.1.3 Maximum likelihood
      4. 4.1.4 Relative likelihood
      5. 4.1.5 A general caveat
    2. 4.2 Empirical moments: estimation and error
      1. 4.2.1 Empirical mean
      2. 4.2.2 Empirical variance and MAD
      3. 4.2.3 Empirical kurtosis
      4. 4.2.4 Error on the volatility
    3. 4.3 Correlograms and variograms
      1. 4.3.1 Variogram
      2. 4.3.2 Correlogram
      3. 4.3.3 Hurst exponent
      4. 4.3.4 Correlations across different time zones
    4. 4.4 Data with heterogeneous volatilities
    5. 4.5 Summary
  11. 5 Financial products and financial markets
    1. 5.1 Introduction
    2. 5.2 Financial products
      1. 5.2.1 Cash (Interbank market)
      2. 5.2.2 Stocks
      3. 5.2.3 Stock indices
      4. 5.2.4 Bonds
      5. 5.2.5 Commodities
      6. 5.2.6 Derivatives
    3. 5.3 Financial markets
      1. 5.3.1 Market participants
      2. 5.3.2 Market mechanisms
      3. 5.3.3 Discreteness
      4. 5.3.4 The order book
      5. 5.3.5 The bid-ask spread
      6. 5.3.6 Transaction costs
      7. 5.3.7 Time zones, overnight, seasonalities
    4. 5.4 Summary
  12. 6 Statistics of real prices: basic results
    1. 6.1 Aim of the chapter
    2. 6.2 Second-order statistics
      1. 6.2.1 Price increments vs. returns
      2. 6.2.2 Autocorrelation and power spectrum
    3. 6.3 Distribution of returns over different time scales
      1. 6.3.1 Presentation of the data
      2. 6.3.2 The distribution of returns
      3. 6.3.3 Convolutions
    4. 6.4 Tails, what tails?
    5. 6.5 Extreme markets
    6. 6.6 Discussion
    7. 6.7 Summary
  13. 7 Non-linear correlations and volatility fluctuations
    1. 7.1 Non-linear correlations and dependence
      1. 7.1.1 Non identical variables
      2. 7.1.2 A stochastic volatility model
      3. 7.1.3 GARCH(1,1)
      4. 7.1.4 Anomalous kurtosis
      5. 7.1.5 The case of infinite kurtosis
    2. 7.2 Non-linear correlations in financial markets: empirical results
      1. 7.2.1 Anomalous decay of the cumulants
      2. 7.2.2 Volatility correlations and variogram
    3. 7.3 Models and mechanisms
      1. 7.3.1 Multifractality and multifractal models (*)
      2. 7.3.2 The microstructure of volatility
    4. 7.4 Summary
  14. 8 Skewness and price-volatility correlations
    1. 8.1 Theoretical considerations
      1. 8.1.1 Anomalous skewness of sums of random variables
      2. 8.1.2 Absolute vs. relative price changes
      3. 8.1.3 The additive-multiplicative crossover and the q-transformation
    2. 8.2 A retarded model
      1. 8.2.1 Definition and basic properties
      2. 8.2.2 Skewness in the retarded model
    3. 8.3 Price-volatility correlations: empirical evidence
      1. 8.3.1 Leverage effect for stocks and the retarded model
      2. 8.3.2 Leverage effect for indices
      3. 8.3.3 Return-volume correlations
    4. 8.4 The Heston model: a model with volatility fluctuations and skew
    5. 8.5 Summary
  15. 9 Cross-correlations
    1. 9.1 Correlation matrices and principal component analysis
      1. 9.1.1 Introduction
      2. 9.1.2 Gaussian correlated variables
      3. 9.1.3 Empirical correlation matrices
    2. 9.2 Non-Gaussian correlated variables
      1. 9.2.1 Sums of non Gaussian variables
      2. 9.2.2 Non-linear transformation of correlated Gaussian variables
      3. 9.2.3 Copulas
      4. 9.2.4 Comparison of the two models
      5. 9.2.5 Multivariate Student distributions
      6. 9.2.6 Multivariate Lévy variables (*)
      7. 9.2.7 Weakly non Gaussian correlated variables (*)
    3. 9.3 Factors and clusters
      1. 9.3.1 One factor models
      2. 9.3.2 Multi-factor models
      3. 9.3.3 Partition around medoids
      4. 9.3.4 Eigenvector clustering
      5. 9.3.5 Maximum spanning tree
    4. 9.4 Summary
    5. 9.5 Appendix A: central limit theorem for random matrices
    6. 9.6 Appendix B: density of eigenvalues for random correlation matrices
  16. 10 Risk measures
    1. 10.1 Risk measurement and diversification
    2. 10.2 Risk and volatility
    3. 10.3 Risk of loss, ‘value at risk’ (VaR) and expected shortfall
      1. 10.3.1 Introduction
      2. 10.3.2 Value-at-risk
      3. 10.3.3 Expected shortfall
    4. 10.4 Temporal aspects: drawdown and cumulated loss
    5. 10.5 Diversification and utility – satisfaction thresholds
    6. 10.6 Summary
  17. 11 Extreme correlations and variety
    1. 11.1 Extreme event correlations
      1. 11.1.1 Correlations conditioned on large market moves
      2. 11.1.2 Real data and surrogate data
      3. 11.1.3 Conditioning on large individual stock returns: exceedance correlations
      4. 11.1.4 Tail dependence
      5. 11.1.5 Tail covariance (*)
    2. 11.2 Variety and conditional statistics of the residuals
      1. 11.2.1 The variety
      2. 11.2.2 The variety in the one-factor model
      3. 11.2.3 Conditional variety of the residuals
      4. 11.2.4 Conditional skewness of the residuals
    3. 11.3 Summary
    4. 11.4 Appendix C: some useful results on power-law variables
  18. 12 Optimal portfolios
    1. 12.1 Portfolios of uncorrelated assets
      1. 12.1.1 Uncorrelated Gaussian assets
      2. 12.1.2 Uncorrelated ‘power-law’ assets
      3. 12.1.3 ‘Exponential’ assets
      4. 12.1.4 General case: optimal portfolio and VaR (*)
    2. 12.2 Portfolios of correlated assets
      1. 12.2.1 Correlated Gaussian fluctuations
      2. 12.2.2 Optimal portfolios with non-linear constraints (*)
      3. 12.2.3 ‘Power-law’ fluctuations – linear model (*)
      4. 12.2.4 ‘Power-law’ fluctuations – Student model (*)
    3. 12.3 Optimized trading
    4. 12.4 Value-at-risk – general non-linear portfolios (*)
      1. 12.4.1 Outline of the method: identifying worst cases
      2. 12.4.2 Numerical test of the method
    5. 12.5 Summary
  19. 13 Futures and options: fundamental concepts
    1. 13.1 Introduction
      1. 13.1.1 Aim of the chapter
      2. 13.1.2 Strategies in uncertain conditions
      3. 13.1.3 Trading strategies and efficient markets
    2. 13.2 Futures and forwards
      1. 13.2.1 Setting the stage
      2. 13.2.2 Global financial balance
      3. 13.2.3 Riskless hedge
      4. 13.2.4 Conclusion: global balance and arbitrage
    3. 13.3 Options: definition and valuation
      1. 13.3.1 Setting the stage
      2. 13.3.2 Orders of magnitude
      3. 13.3.3 Quantitative analysis – option price
      4. 13.3.4 Real option prices, volatility smile and ‘implied’ kurtosis
      5. 13.3.5 The case of an infinite kurtosis
    4. 13.4 Summary
  20. 14 Options: hedging and residual risk
    1. 14.1 Introduction
    2. 14.2 Optimal hedging strategies
      1. 14.2.1 A simple case: static hedging
      2. 14.2.2 The general case and ‘Δ’ hedging
      3. 14.2.3 Global hedging vs. instantaneous hedging
    3. 14.3 Residual risk
      1. 14.3.1 The Black–Scholes miracle
      2. 14.3.2 The ‘stop-loss’ strategy does not work
      3. 14.3.3 Instantaneous residual risk and kurtosis risk
      4. 14.3.4 Stochastic volatility models
    4. 14.4 Hedging errors. A variational point of view
    5. 14.5 Other measures of risk – hedging and VaR (*)
    6. 14.6 Conclusion of the chapter
    7. 14.7 Summary
    8. 14.8 Appendix D
  21. 15 Options: the role of drift and correlations
    1. 15.1 Influence of drift on optimally hedged option
      1. 15.1.1 A perturbative expansion
      2. 15.1.2 ‘Risk neutral’ probability and martingales
    2. 15.2 Drift risk and delta-hedged options
      1. 15.2.1 Hedging the drift risk
      2. 15.2.2 The price of delta-hedged options
      3. 15.2.3 A general option pricing formula
    3. 15.3 Pricing and hedging in the presence of temporal correlations (*)
      1. 15.3.1 A general model of correlations
      2. 15.3.2 Derivative pricing with small correlations
      3. 15.3.3 The case of delta-hedging
    4. 15.4 Conclusion
      1. 15.4.1 Is the price of an option unique?
      2. 15.4.2 Should one always optimally hedge?
    5. 15.5 Summary
    6. 15.6 Appendix E
  22. 16 Options: the Black and Scholes model
    1. 16.1 Ito calculus and the Black-Scholes equation
      1. 16.1.1 The Gaussian Bachelier model
      2. 16.1.2 Solution and Martingale
      3. 16.1.3 Time value and the cost of hedging
      4. 16.1.4 The Log-normal Black–Scholes model
      5. 16.1.5 General pricing and hedging in a Brownian world
      6. 16.1.6 The Greeks
    2. 16.2 Drift and hedge in the Gaussian model (*)
      1. 16.2.1 Constant drift
      2. 16.2.2 Price dependent drift and the Ornstein–Uhlenbeck paradox
    3. 16.3 The binomial model
    4. 16.4 Summary
  23. 17 Options: some more specific problems
    1. 17.1 Other elements of the balance sheet
      1. 17.1.1 Interest rate and continuous dividends
      2. 17.1.2 Interest rate corrections to the hedging strategy
      3. 17.1.3 Discrete dividends
      4. 17.1.4 Transaction costs
    2. 17.2 Other types of options
      1. 17.2.1 ‘Put-call’ parity
      2. 17.2.2 ‘Digital’ options
      3. 17.2.3 ‘Asian’ options
      4. 17.2.4 ‘American’ options
      5. 17.2.5 ‘Barrier’ options (*)
      6. 17.2.6 Other types of options
    3. 17.3 The ‘Greeks’ and risk control
    4. 17.4 Risk diversification (*)
    5. 17.5 Summary
  24. 18 Options: minimum variance Monte–Carlo
    1. 18.1 Plain Monte-Carlo
      1. 18.1.1 Motivation and basic principle
      2. 18.1.2 Pricing the forward exactly
      3. 18.1.3 Calculating the Greeks
      4. 18.1.4 Drawbacks of the method
    2. 18.2 An ‘hedged’ Monte-Carlo method
      1. 18.2.1 Basic principle of the method
      2. 18.2.2 A linear parameterization of the price and hedge
      3. 18.2.3 The Black-Scholes limit
    3. 18.3 Non Gaussian models and purely historical option pricing
    4. 18.4 Discussion and extensions. Calibration
    5. 18.5 Summary
    6. 18.6 Appendix F: generating some random variables
  25. 19 The yield curve
    1. 19.1 Introduction
    2. 19.2 The bond market
    3. 19.3 Hedging bonds with other bonds
      1. 19.3.1 The general problem
      2. 19.3.2 The continuous time Gaussian limit
    4. 19.4 The equation for bond pricing
      1. 19.4.1 A general solution
      2. 19.4.2 The Vasicek model
      3. 19.4.3 Forward rates
      4. 19.4.4 More general models
    5. 19.5 Empirical study of the forward rate curve
      1. 19.5.1 Data and notations
      2. 19.5.2 Quantities of interest and data analysis
    6. 19.6 Theoretical considerations (*)
      1. 19.6.1 Comparison with the Vasicek model
      2. 19.6.2 Market price of risk
      3. 19.6.3 Risk-premium and the θ law
    7. 19.7 Summary
    8. 19.8 Appendix G: optimal portfolio of bonds
  26. 20 Simple mechanisms for anomalous price statistics
    1. 20.1 Introduction
    2. 20.2 Simple models for herding and mimicry
      1. 20.2.1 Herding and percolation
      2. 20.2.2 Avalanches of opinion changes
    3. 20.3 Models of feedback effects on price fluctuations
      1. 20.3.1 Risk-aversion induced crashes
      2. 20.3.2 A simple model with volatility correlations and tails
      3. 20.3.3 Mechanisms for long ranged volatility correlations
    4. 20.4 The Minority Game
    5. 20.5 Summary
  27. Index of most important symbols
  28. Index