Numerical probability. An introduction with applications to finance
This textbook deals with the different aspects of the Monte Carlo method. The book is divided into two parts, one is devoted to exact simulation and the other to approximate simulation. Chapter 1 deals with the simulation of random numbers. The inverse distribution function method and the acceptance-rejection method are presented. The simulation of Poisson distributions and Gaussian random vectors are considered. Chapter 2 is devoted to the Monte Carlo method and applications to option pricing. It is considered the rate of convergence, confidence intervals and vanilla option pricing in the Black-Scholes model. The sensitivity of the option parameters is studied. Variance reduction is considered in Chapter 3. The Monte Carlo method revisited: static control variate, regression-based control variates and an application to option pricing are presented. A pre-condition method, stratified sampling, and importance sampling are considered. The quasi-Monte Carlo method, its theoretical foundation, and its limitations are developed and analyzed in Chapter 4. A lot of practical advices to practitioners are presented. Chapter 5 is devoted to optimal vector quantization-based cubature formulas for the fast computation of expectations in medium dimensions with possible applications to a universal stratified sampling variance reduction method. Stochastic approximation with applications to finance is presented in Chapter 6. An application to recursive variance reduction by importance sampling, an application to implicit correlation search, the paradigm of model calibration by simulation, recursive computation of the VaR and the CVaR, and stochastic optimization methods for optimal quantization are presented. Chapter 7 deals with approximate simulation through the presentation and analysis of discretization schemes of Brownian diffusions, especially the Euler-Maruyama and Milstein scheme. The analysis of their convergence in both the strong and the weak sense are presented with their consequences on the Monte Carlo simulation of ``vanilla'' payoffs in local volatility models. Chapter 8 deals with the diffusion bridge method and an application to path-dependent options. The Brownian bridge method and the diffusion bridge method are considered. An application to look-back style path-dependent options, applications to regular barrier options, and Asian style options are presented. Chapter 9 is devoted to biased Monte Carlo simulation and the multilevel paradigm. An abstract framework for biased Monte Carlo simulation, crude Monte Carlo simulation, and Richardson-Romberg extrapolation are presented. It is considered the multilevel paradigm, namely: the weighted multilevel setting and a regular multilevel estimator. The antithetic scheme for Brownian diffusion and the antithetic scheme for nested Monte Carlo are studied. As examples of simulation, the Clark-Cameron system, option pricing, nested Monte Carlo, and multilevel Monte Carlo are presented. The randomized multilevel Monte Carlo is studied. The computation of sensitivities is revisited in Chapter 10. The shock method, tangent flow, the log-likelihood method for the Euler scheme, and Malliavin-Monte Carlo are considered. In Chapter 11, the author considers the optimal stopping for multi-asset American/Bermudian options. It is studied the optimal stopping for discrete-time \(\mathbb R^{d}\)-valued Markov chains, the backward dynamic programming principle, and time discretization for Snell envelopes based on a diffusion dynamics. The numerical methods, namely: the regression methods, and quantization methods are presented. In Chapter 12 it is presented a nonlinear problem and optimal stopping theory, mostly in a Markovian framework. Time discretization schemes of the Snell envelope are considered. The author presents a detailed analysis of regression methods and of the quantization tree approach.
scientific article; zbMATH DE number 6875545