Numerical Probability: An Introduction With Applications to Finance

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