Martingale Methods in Financial Modelling

Citation

Musiela, M., & Rutkowski, M. (2005). Martingale Methods in Financial Modelling (2nd ed.). Springer. Stochastic Modelling and Applied Probability, vol. 36.

Chapter Summary

Part I: Spot and Futures Markets

Chapter 1: An Introduction to Financial Derivatives

Introduces options, futures and forward contracts, replicating portfolios, martingale measures and the absence of arbitrage, setting out the vocabulary used throughout.

Chapter 2: Discrete-time Security Markets

The Cox-Ross-Rubinstein model, the martingale properties of discrete-time prices, the Black-Scholes formula as a limit, and American options in discrete time. Later printings extend the chapter with sections on optimal stopping and Dynkin games.

Chapter 3: Benchmark Models in Continuous Time

The Black-Scholes model and its relatives: dividend-paying stocks, the Bachelier and Black models, robustness of the Black-Scholes approach, market imperfections, and numerical methods.

Chapter 4: Foreign Market Derivatives

Cross-currency market models, currency forward contracts and options, foreign equity forwards and options, and futures in foreign markets.

Chapter 5: American Options

Valuation of American claims, American call and put options, the early exercise representation, analytical approaches and approximations, including options on dividend-paying stocks.

Chapter 6: Exotic Options

A catalogue with pricing methods for packages, forward-start, chooser, compound, digital, barrier, lookback, Asian, basket and quantile options.

Chapter 7: Volatility Risk

The chapter new in the second edition. It covers implied and historical volatilities, vega hedging, extensions of the Black-Scholes model, local and stochastic volatility models, and the dynamics of volatility surfaces.

Chapter 8: Continuous-time Security Markets

General standard market models, multidimensional Black-Scholes models, variance-minimizing and risk-minimizing hedging, and market imperfections.

Part II: Fixed-income Markets

Chapter 9: Interest Rates and Related Contracts

Zero-coupon and coupon bonds, interest rate futures, swaps and forward rate agreements, and stochastic models of bond prices.

Chapters 10–11: Short-Term Rate Models; Models of Instantaneous Forward Rates

Single-factor and multi-factor short-rate models, the extended CIR model and bond options, followed by the Heath-Jarrow-Morton methodology, Gaussian HJM models, European spot options and the specification of volatilities and correlations.

Chapters 12–13: Market LIBOR Models; Alternative Market Models

Forward and futures LIBORs, caps and floors, LIBOR market models and valuation in the Gaussian HJM model, then swap models for swaptions, with co-terminal, co-initial and co-sliding forward swap rates and Markov-functional models.

Chapter 14: Cross-currency Derivatives

Arbitrage-free cross-currency markets, Gaussian models, and models of forward LIBOR rates across currencies.

Appendix

A reference summary of Ito stochastic calculus: conditional expectation, filtrations, martingales, Brownian motion, stopping times, stochastic integrals, semimartingales, Ito's lemma and Girsanov's theorem.

Key Concepts

Part I: Spot and Futures Markets

• Martingale measure and absence of arbitrage: the link between no-arbitrage prices and expectations under an equivalent probability.
• Cox-Ross-Rubinstein and Black-Scholes models: the discrete and continuous benchmark models.
• Volatility risk, local and stochastic volatility: how the benchmark models are extended to fit observed option prices.
• Hedging in incomplete markets: variance- and risk-minimizing strategies when claims cannot be replicated.

Part II: Fixed-income Markets

• Heath-Jarrow-Morton methodology: modelling the whole forward curve.
• LIBOR and swap market models: models whose inputs are the quoted rates behind caps, floors and swaptions.
• Cross-currency derivatives: arbitrage-free modelling of interest rates in several currencies.

Critical Analysis

Strengths

• A unified method. The same martingale approach runs through equity, currency, exotic, interest rate and cross-currency products, which lets the reader see one theory behind many contracts.
• Breadth and depth. At over 600 pages and 14 chapters, it includes topics rarely collected in one place, such as the catalogue of exotic options, volatility surfaces, Markov-functional models and cross-currency LIBOR models.
• A reference appendix on stochastic calculus. The appendix collects the Ito calculus results used in the book, which makes it easier to check an unfamiliar step.
• A revised second edition. The Springer description and the preface report that some sections of Part I were omitted for readability, Part II was extensively revised, and a new chapter on volatility risk was added.

Limitations

• Demanding prerequisites. The book is written in the language of filtrations, martingales and stochastic integrals, and it assumes the reader can handle measure-theoretic probability.
• Dense and long. It is better suited as a reference or a source for selected courses than as a text to be read cover to cover.
• Edition differences. The chapter structure described here is that of the second edition; the first edition is organised differently, and later printings of the second edition add some sections.

Where It Fits in the Course

In the asset pricing and derivatives list, this is the encyclopaedic mathematical reference. Lamberton and Lapeyre's Introduction to Stochastic Calculus Applied to Finance is a much shorter route through the first part of the story. Björk's Arbitrage Theory in Continuous Time and Bingham and Kiesel's Risk-Neutral Valuation are comparable rigorous texts of more moderate length. Rebonato's Modern Pricing of Interest-Rate Derivatives develops the LIBOR market model, which Musiela and Rutkowski treat in a single chapter, with a focus on calibration.

For practitioners' background, Hull's Options, Futures, and Other Derivatives is the place to learn products and market conventions first. Glasserman's book on Monte Carlo methods supplies numerical methods for pricing.

How to Use This Book

The book is for graduate students and researchers in mathematical finance who already know probability and stochastic calculus, and for quantitative analysts who need to look up how a model is derived.

Do not read it linearly. For a course on equity derivatives, use Chapters 1 to 3, then 5 to 7 as needed; Chapter 4 is for currency products. For interest rates, move to Part II after Chapter 3 or Chapter 8, with Chapters 9 to 12 as the core and Chapters 13 and 14 as extensions. Keep the appendix beside you for the stochastic calculus results. Work through the derivations by hand and compare each model with the Black-Scholes benchmark of Chapter 3.