Table of contents
Table of contents v
Preface xi
Chapter 1 Introduction to probability 1
1 Intuitive explanation . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Number of favorable cases over the total number of cases 2
2 Axiomatic definition . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Random experiment . . . . . . . . . . . . . . . . . . . 2
2.2 Event . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Algebra of events . . . . . . . . . . . . . . . . . . . . 3
2.4 Probability axioms . . . . . . . . . . . . . . . . . . . . 4
2.5 Conditional probabilities . . . . . . . . . . . . . . . . . 5
2.6 Independent events . . . . . . . . . . . . . . . . . . . . 7
Chapter 2 Introduction to random variables 9
1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 Cumulative distribution function . . . . . . . . . . . . 10
1.2 Probability density function . . . . . . . . . . . . . . . 10
1.3 Mean, variance and higher moments of a random variable
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Characteristic function of a random variable . . . . . . 20
2 Random vectors . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Cumulative distribution function of a random vector . . 22
2.2 Probability density function of a random vector . . . . . 22
2.3 Marginal distribution of a random vector . . . . . . . . 23
2.4 Conditional distribution of a random vector . . . . . . . 24
2.5 Mean, variance and higher moments of a random vector 27
2.6 Characteristic function of a random vector . . . . . . . 29
3 Transformation of random variables . . . . . . . . . . . . . . . 30
4 Transformation of random vectors . . . . . . . . . . . . . . . . 34
5 Approximation of the standard normal cumulative distribution
function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
vi Stochastic Simulation and Applications in Finance
Chapter 3 Random sequences 39
1 Sum of independent random variables . . . . . . . . . . . . . . 39
2 Law of large numbers . . . . . . . . . . . . . . . . . . . . . . . 41
3 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . 42
4 Convergence of sequences of random variables . . . . . . . . . 44
4.1 Sure convergence . . . . . . . . . . . . . . . . . . . . 45
4.2 Almost sure convergence . . . . . . . . . . . . . . . . . 45
4.3 Convergence in probability . . . . . . . . . . . . . . . . 45
4.4 Convergence in quadratic mean . . . . . . . . . . . . . 45
Chapter 4 Introduction to computer simulation of random variables 47
1 Uniform random variable generator . . . . . . . . . . . . . . . 48
2 Generating discrete random variables . . . . . . . . . . . . . . 49
2.1 Finite discrete random variables . . . . . . . . . . . . . 49
2.2 Infinite discrete random variables: Poisson distribution . 51
3 Simulation of continuous random variables . . . . . . . . . . . 51
3.1 Cauchy distribution . . . . . . . . . . . . . . . . . . . . 52
3.2 Exponential law . . . . . . . . . . . . . . . . . . . . . 52
3.3 Rayleigh random variable . . . . . . . . . . . . . . . . 53
3.4 Gaussian distribution . . . . . . . . . . . . . . . . . . . 53
4 Simulation of random vectors . . . . . . . . . . . . . . . . . . . 55
4.1 Case of a two-dimensional random vector . . . . . . . . 56
4.2 Cholesky decomposition of the variance-covariance matrix 57
4.3 Eigenvalue decomposition of the variance-covariance matrix
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Simulation of a Gaussian random vector with MATLAB 61
5 Acceptance-rejection method . . . . . . . . . . . . . . . . . . . 61
6 Markov Chain Monte Carlo Method (MCMC) . . . . . . . . . . 64
6.1 Definition of a Markov process . . . . . . . . . . . . . . 64
6.2 Description of the MCMC technique . . . . . . . . . . . 64
Chapter 5 Foundations of Monte Carlo simulations 67
1 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2 Introduction to the concept of precision . . . . . . . . . . . . . 69
3 Quality of Monte Carlo simulations results . . . . . . . . . . . . 72
4 Improvement of the quality of Monte Carlo simulations or variance
reduction techniques . . . . . . . . . . . . . . . . . . . . . 75
4.1 Quadratic resampling . . . . . . . . . . . . . . . . . . . 75
4.2 Reduction of the number of simulations using antithetic
variables . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3 Reduction of the number of simulations using control
variates . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Importance sampling . . . . . . . . . . . . . . . . . . . 81
5 Application cases of random variables simulations . . . . . . . . 86
5.1 Application case: Generation of random variables as a
function of the number of simulations . . . . . . . . . . 86
Front matter vii
5.2 Application case: Simulations and improvement of the
simulations? quality . . . . . . . . . . . . . . . . . . . . 88
Chapter 6 Fundamentals of Quasi Monte Carlo (QMC) simulations 93
1 Van Der Corput sequence (basic sequence) . . . . . . . . . . . . 94
2 Halton sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3 Faure sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Sobol sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Latin Hypercube sampling . . . . . . . . . . . . . . . . . . . . 103
6 Comparison of the different sequences . . . . . . . . . . . . . . 105
Chapter 7 Introduction to random processes 111
1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 111
1.1 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 111
1.2 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . 113
1.3 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . 114
2 Notion of continuity, differentiability and integrability . . . . . . 114
2.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . 114
2.2 Differentiability . . . . . . . . . . . . . . . . . . . . . . 115
2.3 Integrability . . . . . . . . . . . . . . . . . . . . . . . . 115
3 Examples of random processes . . . . . . . . . . . . . . . . . . 116
3.1 Gaussian process . . . . . . . . . . . . . . . . . . . . . 116
3.2 Random walk . . . . . . . . . . . . . . . . . . . . . . . 117
3.3 Wiener process . . . . . . . . . . . . . . . . . . . . . . 118
3.4 Brownian bridge . . . . . . . . . . . . . . . . . . . . . 120
3.5 Fourier transform of a Brownian bridge . . . . . . . . . 122
3.6 Example of a Brownian bridge . . . . . . . . . . . . . . 122
Chapter 8 Solution of stochastic differential equations 125
1 Introduction to stochastic calculus . . . . . . . . . . . . . . . . 126
2 Introduction to stochastic differential equations . . . . . . . . . 128
2.1 Ito?s integral . . . . . . . . . . . . . . . . . . . . . . . 128
2.2 Ito?s lemma . . . . . . . . . . . . . . . . . . . . . . . 129
2.3 Ito?s lemma in the multi-dimensional case . . . . . . . . 132
2.4 Solutions of some stochastic differential equations . . . 133
3 Introduction to stochastic processes with jump . . . . . . . . . . 134
4 Numerical solutions of some stochastic differential equations
(SDE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.1 Ordinary differential equations . . . . . . . . . . . . . . 136
4.2 Stochastic differential equations . . . . . . . . . . . . . 138
5 Application case: Generation of a stochastic differential
equation using the Euler and Milstein schemes . . . . . . . . . . 141
5.1 Sensitivity with respect to the number of simulated series
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.2 Sensitivity with respect to the confidence interval . . . . 144
5.3 Sensitivity with respect to the number of simulations . . 144
5.4 Sensitivity with respect to the time step . . . . . . . . . 144
viii Stochastic Simulation and Applications in Finance
6 Application case: Simulation of a stochastic differential equation
with control and antithetic variables . . . . . . . . . . . . . 145
6.1 Simple simulations . . . . . . . . . . . . . . . . . . . . 145
6.2 Simulations with control variables . . . . . . . . . . . . 146
6.3 Simulations with antithetic variables . . . . . . . . . . . 147
7 Application case: Generation of a stochastic differential equation
with jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Chapter 9 General approach to the valuation of contingent claims 153
1 The Cox, Ross and Rubinstein (1979) binomial model of option
pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 154
1.2 Price of a call option . . . . . . . . . . . . . . . . . . . 155
1.3 Extension to N periods . . . . . . . . . . . . . . . . . . 157
2 Black-Scholes-Merton (1973) option pricing model . . . . . . . 160
2.1 Fundamental equation for the valuation of contingent
claims . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
2.2 Exact analytical value of European call and put options . 162
2.3 Hedging ratios and the sensitivity coefficients . . . . . . 165
3 Derivation of the Black-Scholes formula using the risk-neutral
valuation principle . . . . . . . . . . . . . . . . . . . . . . . . 168
3.1 The Girsanov theorem and the risk-neutral probability . 168
3.2 Derivation of the Black and Scholes formula under the
risk neutralized or equivalent martingale principle . . . . 169
Chapter 10 Pricing options using Monte Carlo simulations 173
1 Plain vanilla options: European put and call . . . . . . . . . . . 173
1.1 Simple simulations . . . . . . . . . . . . . . . . . . . . 173
1.2 Simulations with antithetic variables . . . . . . . . . . 174
1.3 Simulations with control variates . . . . . . . . . . . . . 176
1.4 Simulations with stochastic interest rate . . . . . . . . . 180
1.5 Simulations with stochastic interest rate and stochastic
volatility . . . . . . . . . . . . . . . . . . . . . . . . . 183
2 American options . . . . . . . . . . . . . . . . . . . . . . . . . 186
2.1 Simulations using the Least-Squares Method of Longstaff
and Schwartz (2001) . . . . . . . . . . . . . . . . . . . 186
2.2 Simulations using the Dynamic Programming Technique
of Barraquand and Martineau (1995) . . . . . . . . . . . 195
3 Asian options . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.1 Asian options on arithmetic mean . . . . . . . . . . . . 202
3.2 Asian options on geometric mean . . . . . . . . . . . . 203
4 Barrier options . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5 Estimation methods for the sensitivity coefficients or Greeks . . 207
5.1 Pathwise derivative estimates . . . . . . . . . . . . . . . 207
5.2 Likelihood ratio method . . . . . . . . . . . . . . . . . 210
5.3 Retrieval of volatility method . . . . . . . . . . . . . . 213
Front matter ix
Chapter 11 Term structure of interest rates and interest rate derivatives
219
1 General approach and the Vasicek (1977) model . . . . . . . . 220
1.1 General formulation . . . . . . . . . . . . . . . . . . . 220
1.2 Risk neutral approach . . . . . . . . . . . . . . . . . . 222
1.3 Particular case: One factor Vasicek model . . . . . . . . 223
2 The general equilibrium approach: The Cox, Ingersoll and Ross
(CIR, 1985) model . . . . . . . . . . . . . . . . . . . . . . . . 225
3 The affine model of the term structure . . . . . . . . . . . . . . 227
4 Market models . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.1 The Heath, Jarrow and Morton (HJM, 1992) model . . . 228
4.2 The Brace, Gatarek and Musiela (BGM, 1997) model . . 235
Chapter 12 Credit risk and the valuation of corporate securities 245
1 Valuation of corporate risky debts: The Merton (1974) model . . 246
1.1 The Black and Scholes (1973) model revisited . . . . . 246
1.2 Application of the model to the valuation of a risky debt 248
1.3 Analysis of the debt risk . . . . . . . . . . . . . . . . . 251
1.4 Relation between the firm?s asset volatility and its equity
volatility . . . . . . . . . . . . . . . . . . . . . . . . . 254
2 Insuring debt against default risk . . . . . . . . . . . . . . . . . 256
2.1 Isomorphism between a put option and a financial guarantee
. . . . . . . . . . . . . . . . . . . . . . . . . . . 256
2.2 Insuring the default risk of a risky debt . . . . . . . . . 258
2.3 Establishing a lower bound for the price of the insurance
strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 260
3 Valuation of a risky debt: The reduced-form approach . . . . . . 261
3.1 The discrete case with a zero-coupon bond . . . . . . . 261
3.2 General case in continuous time . . . . . . . . . . . . . 262
Chapter 13 Valuation of portfolios of financial guarantees 265
1 Valuation of a portfolio of loan guarantees . . . . . . . . . . . . 265
1.1 Firms? and guarantor?s dynamics . . . . . . . . . . . . . 266
1.2 Value of loss per unit of debt . . . . . . . . . . . . . . . 267
1.3 Value of guarantee per unit of debt . . . . . . . . . . . . 269
2 Valuation of credit insurance portfolios using Monte Carlo simulations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
2.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . 272
2.2 Expected shortfall and credit insurance valuation . . . . 273
2.3 MATLAB Program . . . . . . . . . . . . . . . . . . . . 275
Chapter 14 Risk management and Value at Risk (VaR) 281
1 Types of financial risks . . . . . . . . . . . . . . . . . . . . . . 282
1.1 Market risk . . . . . . . . . . . . . . . . . . . . . . . . 282
1.2 Liquidity risk . . . . . . . . . . . . . . . . . . . . . . . 282
1.3 Credit risk . . . . . . . . . . . . . . . . . . . . . . . . 282
1.4 Operational risk . . . . . . . . . . . . . . . . . . . . . . 283
x Stochastic Simulation and Applications in Finance
2 Definition of the Value at Risk (VaR) . . . . . . . . . . . . . . . 283
3 The regulatory environment of Basle . . . . . . . . . . . . . . . 283
3.1 Stress testing . . . . . . . . . . . . . . . . . . . . . . . 285
3.2 Back testing . . . . . . . . . . . . . . . . . . . . . . . . 285
4 Approaches to compute VaR . . . . . . . . . . . . . . . . . . . 286
4.1 Non-parametric approach: Historical simulations . . . . 286
4.2 Parametric approaches . . . . . . . . . . . . . . . . . . 286
5 Computing VaR by Monte Carlo simulations . . . . . . . . . . . 287
5.1 Description of the procedure . . . . . . . . . . . . . . . 287
5.2 Application: VaR of a simple bank account . . . . . . . 288
5.3 Application: VaR of a portfolio composed of one domestic
stock and one foreign stock . . . . . . . . . . . . 291
Chapter 15 VaR and Principal Components Analysis (PCA) 295
1 Introduction to the Principal Components Analysis . . . . . . . 296
1.1 Graphical illustration . . . . . . . . . . . . . . . . . . 296
1.2 Analytical illustration . . . . . . . . . . . . . . . . . . 296
1.3 Illustrative example of the PCA . . . . . . . . . . . . . 299
2 Computing the VaR of a bond portfolio . . . . . . . . . . . . . 302
2.1 Sample description and methodology . . . . . . . . . . 302
2.2 Principal components analysis (PCA) . . . . . . . . . . 304
2.3 Linear interpolation or bootstrapping for the intermediate
spot rates . . . . . . . . . . . . . . . . . . . . . . . 306
2.4 Computing VaR by MC and QMC simulations . . . . . 307
Appendix A Review of mathematics 313
1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
1.1 Elementary operations on matrices . . . . . . . . . . . . 314
1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 315
1.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 315
1.4 Determinants of matrices . . . . . . . . . . . . . . . . . 316
2 Solution of a system of linear equations . . . . . . . . . . . . . 318
3 Matrix decomposition . . . . . . . . . . . . . . . . . . . . . . . 319
4 Polynomial and linear approximation . . . . . . . . . . . . . . . 320
5 Eigenvectors and eigenvalues of a matrix . . . . . . . . . . . . . 320
Appendix B MATLAB°R Functions 323
Bibliography 327
Index 338