Numerical Probability: An Introduction With Applications to Finance

✍ Scribed by Gilles Pagès

• Publisher: Springer Nature
• Year: 2018
• Tongue: English
• Leaves: 591
• Edition: 1st ed. 2018
• Category: Library

✦ Synopsis

This textbook provides a self-contained introduction to numerical methods in probability with a focus on applications to finance.

Topics covered include the Monte Carlo simulation (including simulation of random variables, variance reduction, quasi-Monte Carlo simulation, and more recent developments such as the multilevel paradigm), stochastic optimization and approximation, discretization schemes of stochastic differential equations, as well as optimal quantization methods. The author further presents detailed applications to numerical aspects of pricing and hedging of financial derivatives, risk measures (such as value-at-risk and conditional value-at-risk), implicitation of parameters, and calibration.

Aimed at graduate students and advanced undergraduate students, this book contains useful examples and over 150 exercises, making it suitable for self-study.

✦ Table of Contents

Preface
Contents
Notation
⊳ General Notation
⊳ Probability (and Integration) Theory Notation
1 Simulation of Random Variables
1.1 Pseudo-random Numbers
1.2 The Fundamental Principle of Simulation
1.3 The (Inverse) Distribution Function Method
1.4 The Acceptance-Rejection Method
1.5 Simulation of Poisson Distributions (and Poisson Processes)
1.6 Simulation of Gaussian Random Vectors
1.6.1 d-dimensional Standard Normal Vectors
1.6.2 Correlated d-dimensional Gaussian Vectors, Gaussian Processes
2 The Monte Carlo Method and Applications to Option Pricing
2.1 The Monte Carlo Method
2.1.1 Rate(s) of Convergence
2.1.2 Data Driven Control of the Error: Confidence Level and Confidence Interval
2.1.3 Vanilla Option Pricing in a Black--Scholes Model: The Premium
2.1.4 Practitioner's Corner
2.2 Greeks (Sensitivity to the Option Parameters): A First Approach
2.2.1 Background on Differentiation of Functions Defined by an Integral
2.2.2 Working on the Scenarii Space (Black--Scholes Model)
2.2.3 Direct Differentiation on the State Space: The Log-Likelihood Method
2.2.4 The Tangent Process Method
3 Variance Reduction
3.1 The Monte Carlo Method Revisited: Static Control Variate
3.1.1 Jensen's Inequality and Variance Reduction
3.1.2 Negatively Correlated Variables, Antithetic Method
3.2 Regression-Based Control Variates
3.2.1 Optimal Mean Square Control Variates
3.2.2 Implementation of the Variance Reduction: Batch versus Adaptive
3.3 Application to Option Pricing: Using Parity Equations to Produce Control Variates
3.3.1 Complexity Aspects in the General Case
3.3.2 Examples of Numerical Simulations
3.3.3 The Multi-dimensional Case
3.4 Pre-conditioning
3.5 Stratified Sampling
3.6 Importance Sampling
3.6.1 The Abstract Paradigm of Important Sampling
3.6.2 How to Design and Implement Importance Sampling
3.6.3 Parametric Importance Sampling
3.6.4 Computing the Value-At-Risk by Monte Carlo Simulation: First Approach
4 The Quasi-Monte Carlo Method
4.1 Motivation and Definitions
4.2 Application to Numerical Integration: Functions with Finite Variation
4.3 Sequences with Low Discrepancy: Definition(s) and Examples
4.3.1 Back Again to the Monte Carlo Method on [0,1]d
4.3.2 Roth's Lower Bounds for the Star Discrepancy
4.3.3 Examples of Sequences
4.3.4 The Hammersley Procedure
4.3.5 Pros and Cons of Sequences with Low Discrepancy
4.3.6 Practitioner's Corner
4.4 Randomized QMC
4.4.1 Randomization by Shifting
4.4.2 Scrambled (Randomized) QMC
4.5 QMC in Unbounded Dimension: The Acceptance-Rejection Method
4.6 Quasi-stochastic Approximation I
5 Optimal Quantization Methods I: Cubatures
5.1 Theoretical Background on Vector Quantization
5.2 Cubature Formulas
5.2.1 Lipschitz Continuous Functions
5.2.2 Convex Functions
5.2.3 Differentiable Functions With Lipschitz Continuous Gradients (calC1Lip)
5.2.4 Quantization-Based Cubature Formulas for mathbbE(F(X)|Y)
5.3 How to Get Optimal Quantization?
5.3.1 Dimension 1…
5.3.2 The Case of the Normal Distribution calN(0;Id) on mathbbRd, dge2
5.3.3 Other Multivariate Distributions
5.4 Numerical Integration (II): Quantization-Based Richardson--Romberg Extrapolation
5.5 Hybrid Quantization-Monte Carlo Methods
5.5.1 Optimal Quantization as a Control Variate
5.5.2 Universal Stratified Sampling
5.5.3 A(n Optimal) Quantization-Based Universal Stratification: A Minimax Approach
6 Stochastic Approximation with Applications to Finance
6.1 Motivation
6.2 Typical a.s. Convergence Results
6.3 Applications to Finance
6.3.1 Application to Recursive Variance Reduction by Importance Sampling
6.3.2 Application to Implicit Correlation Search
6.3.3 The Paradigm of Model Calibration by Simulation
6.3.4 Recursive Computation of the VaR and the CVaR (I)
6.3.5 Stochastic Optimization Methods for Optimal Quantization
6.4 Further Results on Stochastic Approximation
6.4.1 The Ordinary Differential Equation (ODE) Method
6.4.2 L2-Rate of Convergence and Application to Convex Optimization
6.4.3 Weak Rate of Convergence: CLT
6.4.4 The Averaging Principle for Stochastic Approximation
6.4.5 Traps (A Few Words About)
6.4.6 (Back to) VaRα and CVaRα Computation (II): Weak Rate
6.4.7 VaRα and CVaRα Computation (III)
6.5 From Quasi-Monte Carlo to Quasi-Stochastic Approximation
6.6 Concluding Remarks
7 Discretization Scheme(s) of a Brownian Diffusion
7.1 Euler--Maruyama Schemes
7.1.1 The Discrete Time and Stepwise Constant Euler Schemes
7.1.2 The Genuine (Continuous) Euler Scheme
7.2 The Strong Error Rate and Polynomial Moments (I)
7.2.1 Main Results and Comments
7.2.2 Uniform Convergence Rate in Lp(¶)
7.2.3 Proofs in the Quadratic Lipschitz Case for Autonomous Diffusions
7.3 Non-asymptotic Deviation Inequalities for the Euler Scheme
7.4 Pricing Path-Dependent Options (I) (Lookback, Asian, etc)
7.5 The Milstein Scheme (Looking for Better Strong Rates…)
7.5.1 The One Dimensional Setting
7.5.2 Higher-Dimensional Milstein Scheme
7.6 Weak Error for the Discrete Time Euler Scheme (I)
7.6.1 Main Results for mathbbEf(XT): the Talay--Tubaro and Bally--Talay Theorems
7.7 Bias Reduction by Richardson--Romberg Extrapolation (First Approach)
7.7.1 Richardson--Romberg Extrapolation with Consistent Brownian Increments
7.8 Further Proofs and Results
7.8.1 Some Useful Inequalities
7.8.2 Polynomial Moments (II)
7.8.3 Lp-Pathwise Regularity
7.8.4 Lp-Convergence Rate (II): Proof of Theorem7.2
7.8.5 The Stepwise Constant Euler Scheme
7.8.6 Application to the a.s.-Convergence of the Euler Schemes and its Rate
7.8.7 The Flow of an SDE, Lipschitz Continuous Regularity
7.8.8 The Strong Error Rate for the Milstein Scheme: Proof of Theorem7.5
7.8.9 The Feynman--Kac Formula and Application to the Weak Error Expansion by the PDE Method
7.9 The Non-globally Lipschitz Case (A Few Words On)
8 The Diffusion Bridge Method: Application to Path-Dependent Options (II)
8.1 Theoretical Results About Time Discretization of Path-Dependent Functionals
8.2 From Brownian to Diffusion Bridge: How to Simulate Functionals of the Genuine Euler Scheme
8.2.1 The Brownian Bridge Method
8.2.2 The Diffusion Bridge (Bridge of the Genuine Euler Scheme)
8.2.3 Application to Lookback Style Path-Dependent Options
8.2.4 Application to Regular Barrier Options: Variance Reduction by Pre-conditioning
8.2.5 Asian Style Options
9 Biased Monte Carlo Simulation, Multilevel Paradigm
9.1 Introduction
9.2 An Abstract Framework for Biased Monte Carlo Simulation
9.3 Crude Monte Carlo Simulation
9.4 Richardson--Romberg Extrapolation (II)
9.4.1 General Framework
9.4.2 Practitioner's Corner
9.4.3 Going Further in Killing the Bias: The Multistep Approach
9.5 The Multilevel Paradigm
9.5.1 Weighted Multilevel Setting
9.5.2 Regular Multilevel Estimator (Under First Order Weak Error Expansion)
9.5.3 Additional Comments and Provisional Remarks
9.6 Antithetic Schemes (a Quest for β>1)
9.6.1 The Antithetic Scheme for Brownian Diffusions: Definition and Results
9.6.2 Antithetic Scheme for Nested Monte Carlo (Smooth Case)
9.7 Examples of Simulation
9.7.1 The Clark--Cameron System
9.7.2 Option Pricing
9.7.3 Nested Monte Carlo
9.7.4 Multilevel Monte Carlo Research Worldwide
9.8 Randomized Multilevel Monte Carlo (Unbiased Simulation)
9.8.1 General Paradigm of Unbiased Simulation
9.8.2 Connection with Former Multilevel Frameworks
9.8.3 Numerical Illustration
10 Back to Sensitivity Computation
10.1 Finite Difference Method(s)
10.1.1 The Constant Step Approach
10.1.2 A Recursive Approach: Finite Difference with Decreasing Step
10.2 Pathwise Differentiation Method
10.2.1 (Temporary) Abstract Point of View
10.2.2 The Tangent Process of a Diffusion and Application to Sensitivity Computation
10.3 Sensitivity Computation for Non-smooth Payoffs: The Log-Likelihood Approach (II)
10.3.1 A General Abstract Result
10.3.2 The log-Likelihood Method for the Discrete Time Euler Scheme
10.4 Flavors of Stochastic Variational Calculus
10.4.1 Bismut's Formula
10.4.2 The Haussman--Clark--Occone Formula: Toward Malliavin Calculus
10.4.3 Toward Practical Implementation: The Paradigm of Localization
10.4.4 Numerical Illustration: What is Localization Useful for?
11 Optimal Stopping, Multi-asset American/Bermudan Options
11.1 Introduction
11.1.1 Optimal Stopping in a Brownian Diffusion Framework
11.1.2 Interpretation in Terms of American Options (Sketch)
11.2 Optimal Stopping for Discrete Time mathbbRd-Valued Markov Chains
11.2.1 General Theory, the Backward Dynamic Programming Principle
11.2.2 Time Discretization for Snell Envelopes Based on a Diffusion Dynamics
11.3 Numerical Methods
11.3.1 The Regression Methods
11.3.2 Quantization Methods II: Non-linear Problems (Quantization Tree)
11.4 Dual Form of the Snell Envelope (Discrete Time)
12 Miscellany
12.1 More on the Normal Distribution
12.1.1 Characteristic Function
12.1.2 Numerical Approximation of the Cumulative Distribution Function Φ0
12.1.3 Table of the Distribution Function of the Normal Distribution
12.2 Black--Scholes Formula(s) (To Compute Reference Prices)
12.3 Measure Theory
12.4 Uniform Integrability as a Domination Property
12.5 Interchanging…
12.6 Weak Convergence of Probability Measures on a Polish Space
12.7 Martingale Theory
12.8 Itô Formula for Itô Processes
12.8.1 Itô Processes
12.8.2 The Itô Formula
12.9 Essential Supremum (and Infimum)
12.10 Halton Sequence Discrepancy (Proof of an Upper-Bound)
12.11 A Pitman--Yor Identity as a Benchmark
Bibliography
Index