Numerical Probability: An Introduction with Applications to Finance

by Gilles Pagès

Overview

Now in a thoroughly revised and expanded second edition, this textbook offers a comprehensive and self-contained introduction to numerical methods in probability, with particular emphasis on shastic optimization and its applications in financial mathematics.

The volume covers a broad range of topics, including Monte Carlo simulation techniques—such as the simulation of random variables, variance reduction strategies, quasi-Monte Carlo methods—and recent advancements like the multilevel Monte Carlo paradigm. It further discusses discretization schemes for shastic differential equations and optimal quantization methods. A rigorous treatment of shastic optimization is provided, encompassing shastic gradient descent, including Langevin-based gradient descent algorithms, new to this edition. Detailed applications are presented in the context of numerical methods for pricing and hedging financial derivatives, the computation of risk measures (including value-at-risk and conditional value-at-risk), parameter implicitation, and model calibration.

Intended for graduate students and advanced undergraduates, the textbook includes numerous illustrative examples and over 200 exercises, rendering it well-suited for both classroom use and independent study.

Product Details

ISBN-13:	9783032100917
Publisher:	Springer Nature Switzerland
Publication date:	12/29/2025
Series:	Universitext
Edition description:	Second Edition 2026
Pages:	623
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Gilles Pagès is a Professor of Mathematics at Sorbonne Université specializing in probability theory, numerical probability and mathematical finance. He has published over 130 research articles in probability theory, numerical probability and financial modelling, and is also the author of several graduate and undergraduate textbooks in statistics, applied probability and mathematical finance. He has supervised over 20 doctoral theses.

Table of Contents

1 Simulation of Random Variables.- 2 The Monte Carlo Method and Applications to Option Pricing.- 3 Variance Reduction.- 4 The Quasi-Monte Carlo Method.- 5 Optimal Quantization Methods I: Cubatures.- 6 Shastic Optimization with Applications to Finance.- 7 Discretization Scheme(s) of a Brownian Diffusion.- 8 The Diffusion Bridge Method: Application to Path-Dependent Options (II).- 9 Biased Monte Carlo Simulation, Multilevel Paradigm.- 10 Back to Sensitivity Computation.- 11 Optimal Stopping, Multi-Asset American/Bermudan Options.- 12 Langevin Gradient Descent Algorithms.- 13 Miscellany.