Introduction to Stochastic Calculus Applied to Finance

Citation

Lamberton, D., & Lapeyre, B. (2007). Introduction to Stochastic Calculus Applied to Finance (2nd ed.). Chapman & Hall/CRC.

Chapter Summary

Chapter 1: Discrete-Time Models

The book begins in discrete time, where the central ideas can be seen without stochastic integrals. It sets up the formalism of assets, trading strategies and self-financing portfolios, links martingales to the absence of arbitrage, and discusses complete markets and option pricing. The second edition adds complements on Rogers' approach to the fundamental theorem of asset pricing and on super-replication in incomplete markets. The chapter ends with a problem on the Cox, Ross and Rubinstein model, a discrete-time counterpart of Black-Scholes.

Chapter 2: Optimal Stopping Problem and American Options

Defines stopping times and introduces the Snell envelope, the decomposition of supermartingales, and their application to the pricing and hedging of American options in discrete time.

Chapter 3: Brownian Motion and Stochastic Differential Equations

The continuous-time toolkit: Brownian motion and its properties, the stochastic integral and Ito calculus (including Ito's formula), and stochastic differential equations.

Chapter 4: The Black-Scholes Model

Describes the model, uses a change of probability to price and hedge options, and treats American options. The second edition adds implied volatility and local volatility models, including Dupire's formula.

Chapter 5: Option Pricing and Partial Differential Equations

Connects European option prices with diffusions and the partial differential equations they satisfy, covers numerical solution of parabolic equations, and discusses the PDE approach to American options.

Chapter 6: Interest Rate Models

The principles of interest rate modelling, followed by the classical models (Vasicek, Cox-Ingersoll-Ross and Heath-Jarrow-Morton). The second edition adds change-of-numeraire techniques, forward measures and the forward Libor model.

Chapter 7: Asset Models with Jumps

Introduces the Poisson process, jump-diffusion models for asset prices, and option pricing when prices jump.

Chapter 8: Credit Risk Models

A chapter new to the second edition, covering structural models based on a firm's asset value, intensity-based models, and copulas for the joint distribution of default times.

Chapter 9: Simulation and Algorithms for Financial Models

Simulation of financial models, variance reduction methods, and computer experiments. The second edition extends the chapter with numerical experiments that illustrate variance reduction and hedging strategies.

Appendix

Reviews normal random variables, conditional expectation and the separation of convex sets, followed by a bibliography and index.

Key Concepts

Chapters 1–2: Discrete Time and Optimal Stopping

• Self-financing strategy and arbitrage: a trading strategy that needs no outside cash, and the possibility of risk-free profit with no net investment.
• Martingale and complete market: the characterization of arbitrage-free prices, and markets in which every claim can be replicated.
• Snell envelope and stopping time: the tools for pricing American options as an optimal stopping problem.

Chapters 3–5: Continuous Time and Black-Scholes

• Brownian motion and Ito calculus: the stochastic integral and Ito's formula.
• Change of probability: the technique that moves pricing to a risk-neutral measure.
• Implied and local volatility: extensions that let volatility vary with strike and maturity.
• Link between pricing and PDEs: option prices as solutions of parabolic equations, which can be solved numerically.

Chapters 6–9: Extensions

• Short-rate and forward-rate models: Vasicek, Cox-Ingersoll-Ross and Heath-Jarrow-Morton.
• Jump-diffusion models: asset prices with discontinuities driven by a Poisson process.
• Structural and intensity-based credit models, copulas: ways of modelling default and dependence among defaults.
• Monte Carlo simulation and variance reduction: numerical valuation and the techniques that make it efficient.

Critical Analysis

Strengths

• Concise and compact. At about 250 pages, the book supplies the stochastic calculus it needs and applies it to the main topics of derivative pricing, rather than treating stochastic analysis for its own sake. The publisher describes it as a solid introduction to the stochastic approaches used in finance.
• A discrete-time start. Martingales, arbitrage, completeness and American options are first met in discrete time, so the economic ideas are separate from the technicalities of continuous time.
• Breadth for its size. Besides the Black-Scholes model, the book reaches interest rate models, jump models, credit risk and simulation.
• Exercises and problems. The publisher notes additional exercises and problems in the second edition, and the first chapter ends with a worked modelling problem.
• A substantially updated second edition. The new material includes local volatility and Dupire's formula, change of numeraire, forward measures, the forward Libor model and the credit risk chapter.

Limitations

• Probability background is assumed. The book provides stochastic calculus but expects the reader to be at ease with measure-theoretic probability, conditional expectation and martingales; only a short appendix reviews selected results.
• Brief later chapters. Interest rate, jump and credit models each occupy a single chapter in a book of this length, so they are introductions rather than full treatments.
• Limited computational guidance. The simulation chapter is the only one on computation, and the book does not offer extended implementation material.

Where It Fits in the Course

In the asset pricing and derivatives list, this is one of the shorter, mathematics-first introductions. Björk's Arbitrage Theory in Continuous Time, Bingham and Kiesel's Risk-Neutral Valuation and Musiela and Rutkowski's Martingale Methods in Financial Modelling cover the same arc at greater length, with the last of these much larger and especially deep on interest rates. Hunt and Kennedy's Financial Derivatives in Theory and Practice is another route into pricing theory.

Hull's Options, Futures, and Other Derivatives gives the market and product background that Lamberton and Lapeyre do not, and Hull's shorter Fundamentals of Futures and Options Markets is a gentler start still. Glasserman's Monte Carlo Methods in Financial Engineering and Platen and Bruti-Liberati's book on numerical solution of stochastic differential equations with jumps go much further than the book's final chapter on simulation.

How to Use This Book

The book is for students with a solid grounding in probability and calculus who want a compact, rigorous route from discrete-time models to Black-Scholes and its extensions. It suits a course in mathematical finance or self-study alongside a market-oriented text.

Read Chapters 1 and 2 first, then Chapters 3 and 4, which form the core. Chapter 5 is needed by readers who will price numerically through PDEs. Chapters 6 to 8 are largely independent of one another and can be chosen by interest. Chapter 9 is the natural companion to any programming practice. Use the appendix to refresh conditional expectation, and work the exercises, particularly the problem at the end of Chapter 1, before moving to continuous time.