Numerical probability. An introduction with applications to finance
type: Item
instance of: scholarly article
title: Numerical probability. An introduction with applications to finance
zbMATH Open document ID: 1418.91016
DOI: 10.1007/978-3-319-90276-0
author: Gilles Pagès
published in: Universitext
publication date: 2018-05-28T00:00:00Z
review text: 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 the…
reviewed by: Aleksandr D. Borisenko
Mathematics Subject Classification ID: 91-02
Mathematics Subject Classification ID: 91G60
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 91G20
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 65C10
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 6875545
zbMATH Keywords: Monte Carlo method
zbMATH Keywords: stochastic approximation
zbMATH Keywords: quantization-based cubature formulas
zbMATH Keywords: sensitivity computation
zbMATH Keywords: variance reduction
zbMATH Keywords: discretization schemes
zbMATH Keywords: path-dependent functionals
zbMATH Keywords: diffusion bridge
zbMATH Keywords: quasi-Monte Carlo method
zbMATH Keywords: optimal stopping
zbMATH Keywords: spatial discretization
zbMATH Keywords: multilevel paradigm
zbMATH Keywords: application to path-dependent options
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-90276-0
OpenAlex ID: W4241975315
description: scientific article; zbMATH DE number 6875545