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Full Description
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 stochastic 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 stochastic differential equations and optimal quantization methods. A rigorous treatment of stochastic optimization is provided, encompassing stochastic 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.
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 Stochastic 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.
