Numerical Solution of Stochastic Differential Equations with Jumps in Finance

Citation

Platen, E., & Bruti-Liberati, N. (2010). Numerical Solution of Stochastic Differential Equations with Jumps in Finance. Springer-Verlag.

Chapter Summary

The book is a research monograph of about 850 pages. Its subject is the design and analysis of discrete-time approximations to stochastic differential equations (SDEs) driven by Wiener processes and Poisson jump measures. The preface divides it into four parts: a theoretical basis with stochastic expansions, strong approximations together with estimation and filtering, weak approximations, and a final part on numerical stability, martingale representations and variance reduction. Chapter 18 contains solutions to the exercises that close every chapter. The preface distinguishes regular schemes, whose time grid ignores the jump times, from jump-adapted schemes, whose grid includes them, and strong approximations, which suit scenario analysis, filtering and hedge simulation, from weak ones, which suit expectations such as derivative prices and risk measures.

Chapter 1: Stochastic Differential Equations with Jumps

A self-contained introduction to the theory: stochastic processes, supermartingales and martingales, quadratic variation and covariation, the Itô integral and Itô formula, SDEs, linear SDEs, SDEs with jumps, and the existence and uniqueness of solutions.

Chapter 2: Exact Simulation of Solutions of SDEs

When the solution of an SDE can be sampled exactly, no discretization error arises. The chapter covers sampling from transition distributions, exact solutions of multi-dimensional SDEs, functions of exact solutions, almost exact solutions by conditioning and by time change, and functionals of solutions.

Chapter 3: Benchmark Approach to Finance and Insurance

This chapter presents the benchmark approach of the first author as a framework for modelling risk beyond the standard risk-neutral setting: the market model, the best performing portfolio, the supermartingale property and pricing, diversification, real world pricing under some models and under the minimal market model, and binomial option pricing.

Chapter 4: Stochastic Expansions

The mathematical engine behind high-order schemes: an introduction to Wagner-Platen expansions, multiple stochastic integrals, coefficient functions, the expansions themselves, and moment estimates for multiple stochastic integrals.

Chapter 5: Introduction to Scenario Simulation

The chapter starts from approximating solutions of ordinary differential equations, moves to scenario simulation, and introduces strong Taylor schemes and derivative-free strong schemes.

Chapter 6: Regular Strong Taylor Approximations with Jumps

Discrete-time approximation with jumps is set up, followed by the strong order 1.0 Taylor scheme, commutativity conditions, convergence results, a lemma on multiple Itô integrals and a proof of the convergence theorem.

Chapter 7: Regular Strong Itô Approximations

Explicit regular strong schemes, drift-implicit schemes, balanced implicit methods and predictor-corrector schemes, with convergence results.

Chapter 8: Jump-Adapted Strong Approximations

The jump-adapted counterparts of the earlier schemes (Taylor, derivative-free, drift-implicit, predictor-corrector, exact simulation), convergence results, numerical results on strong schemes and the approximation of pure jump processes.

Chapter 9: Estimating Discretely Observed Diffusions

Maximum likelihood estimation, discretization of estimators, transform functions for diffusions, estimation of affine diffusions, asymptotics of estimating functions and the estimation of jump diffusions.

Chapter 10: Filtering

The Kalman-Bucy filter, hidden Markov chain filters, filtering a mean reverting process, the balanced method in filtering, and a benchmark approach to filtering in finance.

Chapter 11: Monte Carlo Simulation of SDEs

The weak-approximation part opens with an introduction to Monte Carlo simulation, weak Taylor schemes, derivative-free weak approximations, extrapolation methods, and implicit and predictor-corrector methods.

Chapter 12: Regular Weak Taylor Approximations

Weak Taylor schemes, commutativity conditions and convergence results.

Chapter 13: Jump-Adapted Weak Approximations

Jump-adapted weak schemes, derivative-free and predictor-corrector schemes, some exact weak schemes, convergence of jump-adapted weak Taylor schemes and of other jump-adapted weak schemes, and numerical results on weak schemes.

Chapter 14: Numerical Stability

Asymptotic p-stability, and the stability of predictor-corrector methods, of some implicit methods and of simplified schemes.

Chapter 15: Martingale Representations and Hedge Ratios

General contingent claim pricing, hedge ratios for one-dimensional processes, explicit hedge ratios, martingale representation for non-smooth payoffs, absolutely continuous payoff functions, the maximum of several assets, and hedge ratios for lookback options.

Chapter 16: Variance Reduction Techniques

Various variance reduction methods, measure transformation techniques, discrete-time variance reduced estimators, control variates and the HP variance reduction method.

Chapter 17: Trees and Markov Chain Approximations

Numerical effects of tree methods, the efficiency of simplified schemes, higher order Markov chain approximations, finite difference methods and a convergence theorem for Markov chains.

Chapter 18: Solutions for Exercises

Worked solutions to the exercises that end each chapter, about 25 pages.

Key Concepts

Chapters 1–4: Theory and framework

• SDE with jumps: an equation driven by both a Wiener process and a Poisson jump measure, the object all later schemes approximate.
• Exact and almost exact simulation: sampling solutions without discretization error, or nearly so, by conditioning or time change.
• Benchmark approach: the first author's framework for pricing and risk, centred on the best performing portfolio.
• Wagner-Platen expansion: the stochastic Taylor-type expansion whose truncations generate the schemes of later chapters.

Chapters 5–10: Strong approximation, estimation, filtering

• Strong order of convergence: pathwise accuracy of a scheme, used to grade strong Taylor and Itô schemes.
• Regular versus jump-adapted schemes: whether the time grid ignores or includes the jump times.
• Drift-implicit, balanced implicit and predictor-corrector schemes: methods that trade extra computation per step for better stability.
• Maximum likelihood estimation and filtering: parameter estimation for discretely observed diffusions, and the Kalman-Bucy and hidden Markov chain filters.

Chapters 11–17: Weak approximation and applications

• Weak order of convergence: accuracy of expectations rather than paths, the criterion for Monte Carlo pricing.
• Commutativity conditions: conditions under which simpler schemes keep their order.
• Numerical stability: whether errors stay controlled as the step grows, studied through p-stability.
• Hedge ratios and martingale representation: simulation of derivatives of prices with respect to model inputs.
• Variance reduction: control variates, measure transformations and HP variance reduction.

Critical Analysis

Strengths

• A systematic treatment of jumps. The book builds the whole pipeline for SDEs with jumps, from theory and expansions through strong and weak schemes, which the preface says goes significantly beyond Kloeden and Platen's earlier book on SDEs without jumps.
• Clear organization by purpose. The split into strong and weak schemes, and into regular and jump-adapted ones, is reflected in the contents, so the reader can go straight to the family needed for scenario work or for Monte Carlo pricing.
• Explicit reading routes. The "Suggestions for the Reader" front section gives four flowcharts, for readers wanting applications without deep mathematics, for engineers and quantitative analysts, for mathematicians, and for financial professionals. Each lists the chapters and sections to read and which to skip.
• Exercises with solutions. Every chapter ends with exercises, and Chapter 18 provides solutions.
• Breadth beyond simulation. Exact simulation, parameter estimation, filtering, hedge ratios and variance reduction are included alongside the approximation schemes.

Limitations

• Size and density. At over 850 pages with proofs of convergence theorems (for example Sections 6.5 and 6.6, and the convergence sections of Chapters 7, 8 and 13), the book is a monograph, not a short course. The preface says it does not claim to be a complete account of the state of the art.
• Finance used to motivate, not to teach. The preface says finance was chosen to motivate the methods. Readers who need derivative pricing, term structure models or risk management as subjects will need other books.
• Benchmark approach may be unfamiliar. The preface says the benchmark approach "may be new to most readers", and Chapter 3 is the main introduction to it.
• Written before later developments. It is a 2010 text and does not cover later work; the author's website is named in the preface as a place for updates.

Where It Fits in the Course

Within this course, the book is the specialised numerical companion to the pricing theory books. Björk, Musiela and Rutkowski, Bingham and Kiesel and Lamberton and Lapeyre explain what a derivative is worth in continuous-time models; Platen and Bruti-Liberati ask how to compute such values when the model has no closed-form solution and includes jumps. Glasserman's Monte Carlo Methods in Financial Engineering is the broader, more finance-oriented simulation text, covering random number generation, variance reduction, quasi-Monte Carlo, Greeks, American options and risk management; this book goes deeper into discretization schemes, convergence orders and numerical stability, and adds jumps.

For a first look at derivatives, Hull's Options, Futures, and Other Derivatives and the shorter Fundamentals of Futures and Options Markets come first. The book also builds on Kloeden and Platen's 1992 text on SDEs without jumps, which the preface names as its predecessor.

How to Use This Book

The preface says the book is written for a reader with engineer-level undergraduate training in mathematical methods and is accessible even to those who need only numerical recipes. That is the prerequisite for the application routes; the mathematical route assumes more.

Choose one of the four flowcharts in "Suggestions for the Reader". Those wanting to apply jump SDEs outside finance follow the first, which reads Sections 1.1 to 1.8, then the introductory scenario simulation chapter, the regular and jump-adapted strong schemes, and the weak schemes in Chapters 11 to 13, ending with Chapter 14 and Section 16.1. Engineers and quantitative analysts add Chapters 2 and 4 and all of Chapter 16. Mathematicians skip Chapter 1 and go directly to Chapter 2 and then to the convergence-oriented chapters. Financial practitioners add Chapters 2, 3, 9, 10, 15, 16 and 17 to the application route.

Whatever the route, do the exercises at the end of each chapter and check them against Chapter 18. For jump-adapted schemes, read the numerical results sections (8.8 and 13.7) as a guide to what accuracy to expect. Readers who also use Glasserman's book can treat it as the guide to simulation practice and this one as the reference for the numerical analysis of the schemes.