Numerical Probability

An Introduction with Applications to Finance

2026
Buch

Verfasst von: Gilles Pagès
Buchreihe: Universitext
Verlag: Springer Nature Switzerland
Enthalten in: Springer Professional "Wirtschaft+Technik"; Springer Professional "Wirtschaft"

Über dieses Buch

In einer gründlich überarbeiteten und erweiterten zweiten Auflage bietet dieses Lehrbuch eine umfassende und in sich geschlossene Einführung in numerische Wahrscheinlichkeitsmethoden, insbesondere in die stochastische Optimierung und ihre Anwendung in der Finanzmathematik. Der Band deckt ein breites Spektrum an Themen ab, darunter Monte-Carlo-Simulationstechniken - wie die Simulation von Zufallsvariablen, Varianzreduktionsstrategien, Quasi-Monte-Carlo-Methoden - und jüngste Fortschritte wie das Monte-Carlo-Paradigma auf mehreren Ebenen. Außerdem werden Diskretisierungsschemata für stochastische Differentialgleichungen und optimale Quantisierungsmethoden diskutiert. Eine rigorose Behandlung stochastischer Optimierung wird angeboten, einschließlich stochastischer Gradienten-Deszendenz, einschließlich Langevin-basierter Gradienten-Deszensus-Algorithmen, die in dieser Ausgabe neu sind. Detaillierte Anwendungen werden im Kontext numerischer Methoden zur Preisgestaltung und Absicherung von Finanzderivaten, der Berechnung von Risikomaßnahmen (einschließlich Value-at-Risk und bedingtem Value-at-Risk), der Implikation von Parametern und der Modellkalibrierung dargestellt. Das Lehrbuch richtet sich an Studenten und fortgeschrittene Studenten und enthält zahlreiche anschauliche Beispiele und über 200 Übungen, wodurch es sowohl für den Einsatz im Klassenzimmer als auch für unabhängige Studien gut geeignet ist.

Mit KI übersetzt

Inhaltsverzeichnis

Frontmatter

Chapter 1. Simulation of Random Variables

Abstract

This chapter presents the basic principles of the simulation of random variables and random vectors, including a short introduction to the simulation of pseudo-random numbers, the inverse distribution function method and von Neumann's acceptance-rejection method, with applications to the simulation of Gaussian vectors, (fractional) Brownian motion and Poisson process paths.

Gilles Pagès

Chapter 2. The Monte Carlo Method and Applications to Option Pricing

Abstract

This chapter is devoted to the Monte Carlo method for the computation of expectation of simulable random variables. After a presentation of the Strong Law of Large Numbers (without proof) and the different ways to measure its rate of convergence (quadratic mean, Central Limit theorem, Law of the Iterated Logarithm), we introduce the notion of confidence interval at a given confidence level, illustrated by a simple application to the pricing of a multi-asset European option. Then, we propose a first approach to sensitivities computation - known as the Greeks when dealing with derivative financial products - by Monte Carlo simulation.

Gilles Pagès

Chapter 3. Variance Reduction

Abstract

The rate of convergence of the Monte Carlo method being universal, the only way to shrink the confidence interval is to reduce its asymptotic variance by replacing the random variable of interest by another one with the same expectation but lower variance. This chapter presents the main methods to do so: static and dynamic (regression) control variate, convexity methods (Jensen's inequality), antithetic method, pre-conditioning (Blackwell-Rao), stratification and importance sampling. Various applications to the pricing of derivatives are proposed. In particular, the use of parity equations - such as the call-put parity equations -- to produce synthetic call or put payoffs with a very low variance.

Gilles Pagès

Chapter 4. The Quasi-Monte Carlo Method

Abstract

In this chapter we present the so-called Quasi-Monte Carlo (QMC) method, which can be seen as a deterministic alternative to the standard Monte Carlo method: the pseudo-random numbers are replaced by deterministic computable sequences of vectors which, once substituted in place of pseudo-random numbers in the Monte Carlo method, may significantly speed up its rate of convergence, making it almost independent of the structural dimension of the simulation.

Gilles Pagès

Chapter 5. Optimal Quantization Methods I: Cubatures

Abstract

This chapter is a first introduction to optimal vector quantization and its application to numerical probability. Optimal quantization produces the best approximation of probability distribution by finitely supported distributions in the sues of the Wasserstein distance. It naturally yields cubature formulas to compute. An appropriate Richardson–Romberg extrapolation based on the combination of such cubature formulas turn out to be competitive with regular Monte Carlo simulation up to at least 5 dimensions. A first approach to the computation of (quadratic) optimal quantizers of a given distribution is developed. Quantization is also investigated in chapter 6 from an algorithmic view point and 11 as a numerical method to price American and Bermudan options.

Gilles Pagès

Chapter 6. Stochastic Optimization and Applications to Finance

Abstract

This chapter provides an introduction to stochastic approximation theory, a part of stochastic optimization now widely used in Machine Learning and Deep Learning applications. We show when and how to design a stochastic gradient, a stochastic pseudo-gardient or a zero search recursive stochastic algorithm. We prove the main convergence theorems as well as their rates of convergence (Central Limit theorem). The Ruppert & Polyak averaging procedure, which allows to minimize the asymptotic variance of such procedures, is also analyzed. Various applications to finance are developed: computation of implicit parameters (volatility, correlation, etc), calibration and the computation of value-at-risk and conditional value-at-risk (expected shortfall).

Gilles Pagès

Chapter 7. Discretization Scheme(s) of a Brownian Diffusion

Abstract

This chapter is devoted to the discretization schemes of the solutions of a stochastic differential equation driven by a Brownian motion (diffusion): the (discrete time and continuous) Euler scheme and the Milstein scheme. The existence of moments, the strong (or pathwise) convergence rate of both schemes are established under Lipschitz assumptions of the diffusion coefficients (Euler scheme) or of their partial derivatives (Milstein scheme). Several other important properties of these schemes are established (e.g., conditions for the simulability of the Milstein scheme in higher dimension, Lipschitz property of the flow, etc). The main weak error results for the Euler scheme, either under smoothness (Talay–Tubaro) or ellipticity (Bally–Talay) assumptions, are stated, with a detailed proof in the first setting. Applications to the Richardson-Romberg extrapolation to reduce the bias in Monte Carlo simulations is presented and illustrated in an example.

Gilles Pagès

Chapter 8. The Diffusion Bridge Method: Application to Path-Dependent Options (II)

Abstract

This chapter provides a (partial) answer to the following question: can we simulate the continuous—or genuine—Euler scheme? To this end, we first investigate the Brownian bridge and its avatar for diffusion processes which allows to simulate in an exact way some functionals of the genuine Euler scheme involving its maximum or its minimum over a given time interval and provide sharper approximations of functionals involving time integrals. Several first order weak error are stated with precise references. Applications to several families of path-dependent European options (Asian, lookback, barrier) are given, including some variance reduction methods for barrier options.

Gilles Pagès

Chapter 9. Biased Monte Carlo Simulation, Multilevel Paradigm

Abstract

We introduce in this chapter the paradigm of multilevel simulation, whose aim is to dramatically reduce the bias in a Monte Carlo simulation when the (probability distribution of the) random variable under consideration cannot be simulated at a reasonable cost but can be approximated by simulable random variables with a controlled complexity. As typical examples let us cite the discretization scheme of a stochastic process or nested Monte Carlo simulations. The paradigm relies on the existence of both a strong rate and an expansion of the weak error convergence of approximating random variables. We propose an in-depth analysis of both weighted and regular multilevel methods in an abstract framework, in presence of a higher or first order expansion of the weak error. Various applications are detailed like the pricing of path-dependent or forward start options, quantile computations in actuarial sciences (SCR). When the strong convergence rate is fast enough (like the Milstein scheme for diffusion), multilevel simulation behaves like an unbiased simulation. We conclude with a section about randomized multilevel quantization.

Gilles Pagès

Chapter 10. Back to Sensitivity Computation

Abstract

We present and analyze three types of sensitivity computation for a general diffusion process with a focus on the sensitivity with respect to the initial value. First, an in-depth analysis of the (elementary but popular) finite difference method—also known as the shock or bump method—is carried out with both a constant and decreasing step settings. Then the tangent process method, based on pathwise differentiation, is described for smooth enough payoffs. As a third method, we revisit the log-likelihood method applied to the Euler scheme of the diffusion. As a conclusion, we provide some first stakes on the way to Malliavin calculus by proving the formulas of both Haussmann-Clark-Ocone and Bismut. We show how to use them for the computations of various sensitivities. Numerical experiments illustrate the typical behavior of these methods depending on the regularity of the payoffs under consideration.

Gilles Pagès

Chapter 11. Optimal Stopping, Multi-Asset American/Bermudan Options

Abstract

The main aim of this chapter is to present and analyze two methods for the pricing of multi-asset American—in practice Bermudan—options: the regression methods “à la Longstaff-Schwarz” and the quantization methods. The pricing of American options is a typical example of an optimal stopping problem. So, we start from such a general optimal stopping problem but for a diffusion model, we introduce the Snell envelope of a payoff or reward process and its value function in ou Markov framework. We give some rate results for various “levels” of time discretization of such Snell envelope. Then we consider a discrete time optimal stopping problem in a Markov framework. We introduce the backward dynamic programming principle and its variant based on optimal stopping times. We describe in detail the algorithmic aspects of both the regression and the quantization method. The Monte Carlo error (Central Limit Theorem) induced by the regression method is described. Finally, a theoretical analysis of the convergence rate of the quantization method is carried out.

Gilles Pagès

Chapter 12. Langevin Gradient Descent Algorithms

Abstract

We introduce the so-called Langevin dynamics associated to a recursive optimization procedure, whether stochastic or not. We focus on the Langevin version of gradient descent and stochastic gradient descent induced by a smooth coercive potential function but this could also be applied to the Adam procedure among others. We first provide, in a nutshell, some basics about invariant distributions of (mean-reverting) Brownian diffusions with a focus on the Langevin diffusion. We show in particular that its invariant distribution concentrates on the argmin of the potential function when the Brownian noise fades. The Langevin dynamics applied to a gradient descent consist in adding an exogenous noise in the appropriate scale in order to improve the exploration of the state space. Convergence rates are established for the Langevin version of both regular and stochastic gradient descents.

Gilles Pagès

Chapter 13. Miscellany

Abstract

This chapter gathers various tools and results from measure and probability theory, martingale theory, uniform integrability, essential extrema and stochastic calculus (Itô’s formula)—some with a complete proof, others simply with precise references—which are used throughout the book. We also included the proofs of two specific mathematical results (discrepancy of the Halton sequence and the Pitman–Yor identity), which are not essential in the context of numerical applications but give the mathematical flavor of the underlying theories we use at several places in the book.

Gilles Pagès

Titel: Numerical Probability
Verfasst von: Gilles Pagès
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-032-10092-4
Print ISBN: 978-3-032-10091-7
DOI: https://doi.org/10.1007/978-3-032-10092-4

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Numerical Probability

An Introduction with Applications to Finance

Über dieses Buch

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Simulation of Random Variables

Chapter 2. The Monte Carlo Method and Applications to Option Pricing

Chapter 3. Variance Reduction

Chapter 4. The Quasi-Monte Carlo Method

Chapter 5. Optimal Quantization Methods I: Cubatures

Chapter 6. Stochastic Optimization and Applications to Finance

Chapter 7. Discretization Scheme(s) of a Brownian Diffusion

Chapter 8. The Diffusion Bridge Method: Application to Path-Dependent Options (II)

Chapter 9. Biased Monte Carlo Simulation, Multilevel Paradigm

Chapter 10. Back to Sensitivity Computation

Chapter 11. Optimal Stopping, Multi-Asset American/Bermudan Options

Chapter 12. Langevin Gradient Descent Algorithms

Chapter 13. Miscellany

Backmatter

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