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Arbitrage Theory in Continuous Time cover

Economics

Economics

Arbitrage Theory in Continuous Time

Tomas Björk

A rigorous graduate-level guide to arbitrage theory, spanning discrete and continuous time models, complete and incomplete markets, and practical applications in derivative pricing, portfolio optimization, and risk management.

Difficulty Level
Advanced
Academic Level
Graduate
financial mathematicsderivatives pricingstochastic calculusarbitrage theory

01 / Reading Guide

Reading Guide

Citation

Björk, T. (2019). Arbitrage Theory in Continuous Time (4th ed.). Oxford University Press. Oxford Finance Series.

Chapter Summary

The 38 chapters are grouped in six parts and followed by appendices on measure and integration, probability theory, martingales and stopping times, and convex duality. The publisher's description says the book offers two full treatments of arbitrage theory, the classical delta-hedging approach and the modern martingale approach, and that they can be studied independently of each other. The fourth edition added the chapters on incomplete markets (Part V) and the part on dynamic equilibrium theory (Part VI).

Chapter 1: Introduction

Sets out the aims of arbitrage theory and the problem of pricing financial derivatives, using European call options and a currency-risk example to motivate hedging.

Part I, Chapters 2–3: Discrete Time Models (The Binomial Model; A More General One Period Model)

The binomial model introduces replicating portfolios and arbitrage-free pricing in one and several periods. Chapter 3 generalizes the one-period model, bringing in martingale measures and completeness.

Part II, Chapters 4–5: Stochastic Calculus (Stochastic Integrals; Stochastic Differential Equations)

Wiener processes, stochastic integrals, Itô's formula and stochastic differential equations, with the link between SDEs and partial differential equations. This is the mathematical toolkit for everything that follows.

Part III, Chapters 6–10: Portfolio Dynamics to Delta Hedging

Chapters 6 to 8 (Portfolio Dynamics; Arbitrage Pricing; Completeness and Hedging) develop self-financing portfolios, the Black–Scholes pricing equation, and the conditions for completeness. Chapter 9, A Primer on Incomplete Markets, previews the issues taken up in Part V, and Chapter 10, Parity Relations and Delta Hedging, covers parity relations and delta hedging.

Part III, Chapters 11–15: The Martingale Approach

Chapters 11 and 12 (The Martingale Approach to Arbitrage Theory; The Mathematics of the Martingale Approach) recast pricing as expectation under a martingale measure and supply the required mathematics, including the Girsanov theorem. Chapter 13 derives Black–Scholes from this viewpoint, Chapter 14 extends it to several dimensions, and Chapter 15 introduces change of numeraire.

Part III, Chapters 16–18: Dividends; Forward and Futures Contracts; Currency Derivatives

Applications of the martingale machinery to stocks paying dividends, forwards and futures, and options on foreign exchange.

Part III, Chapters 19–24: Interest rates

Bonds and Interest Rates (19), Short Rate Models (20), Martingale Models for the Short Rate (21), Forward Rate Models (22), LIBOR Market Models (23) and Potentials and Positive Interest (24). Term structure modelling is developed from short rates to forward rate and market models, and the part closes with a potential-theoretic approach to positive interest rates.

Part IV, Chapters 25–28: Optimal Control and Investment Theory

Stochastic Optimal Control (25), Optimal Consumption and Investment (26), The Martingale Approach to Optimal Investment (27), and Optimal Stopping Theory and American Options (28). Dynamic programming and the martingale method are applied to portfolio choice, and optimal stopping gives the pricing of American options.

Part V, Chapters 29–34: Incomplete Markets

Incomplete Markets (29), The Esscher Transform and the Minimal Martingale Measure (30), Minimizing f-divergence (31), Portfolio Optimization in Incomplete Markets (32), Utility Indifference Pricing and Other Topics (33) and Good Deal Bounds (34). When perfect hedging fails, a specific martingale measure or price bounds must be chosen.

Part VI, Chapters 35–38: Dynamic Equilibrium Theory

Equilibrium Theory: A Simple Production Model (35), The Cox–Ingersoll–Ross Factor Model (36), The Cox–Ingersoll–Ross Interest Rate Model (37) and Endowment Equilibrium: Unit Net Supply (38). Prices and interest rates are derived from agents' optimization rather than by arbitrage alone.

Key Concepts

Parts I–II: Discrete models and stochastic calculus

  • Replicating portfolio: a trading strategy that reproduces a derivative's payoff and so fixes its arbitrage-free price.
  • Itô's formula: the chain rule for functions of a stochastic process, used to derive pricing equations.

Part III: Arbitrage theory

  • Self-financing portfolio and completeness: a strategy needing no external cash flow, and a market in which every claim can be replicated.
  • Martingale measure: a probability measure under which discounted prices are martingales and derivative prices are expectations.
  • Change of numeraire: choosing a different price as the unit of account to simplify valuation.
  • Term structure models: short rate, forward rate and LIBOR market models.

Parts IV–VI: Control, incompleteness and equilibrium

  • Stochastic optimal control and optimal stopping: tools for consumption-investment problems and American options.
  • Minimal martingale measure and good deal bounds: ways to select a price when a market is incomplete.
  • Equilibrium models: prices determined by supply and demand and agents' optimization.

Critical Analysis

Strengths

  • Two parallel routes to the theory. According to the publisher, the delta-hedging and martingale approaches are treated in full and can be studied independently, so a less mathematical reader can follow the first while a more advanced reader gets the second.
  • Economic interpretation alongside mathematics. The description stresses intuitive economic arguments for all concepts, a solved example for every new technique, numerous exercises and further reading in each chapter.
  • Wide scope in one volume. The book spans option pricing, interest-rate models, American options, optimal investment, incomplete markets and equilibrium, with appendices that collect the required measure theory, probability, martingale and convex duality background.
  • Substantially extended fourth edition. New chapters cover incomplete markets topics such as the Esscher transform, f-divergences and good deal bounds, and a new part treats dynamic equilibrium.

Limitations

  • Size and level. At 592 pages and aimed at graduate students, the book is a long read, and the appendices on measure theory, probability and convex duality indicate how much mathematics the later parts rely on.
  • No chapters on numerical implementation. The chapter titles show a focus on theory; calibration, simulation and numerical methods are not topics of their own, so a numerical text must supplement the book.
  • No credit risk. The table of contents has no chapters on default or credit derivatives, which a separate credit-risk text has to supply.
  • A very long Part III. Part III runs from Chapter 6 to Chapter 24, half of all chapters, and combines the delta-hedging route, the martingale route and the interest-rate chapters, so a reader needs a selection plan.

Where It Fits in the Course

In the asset-pricing and derivatives course list, Björk is a graduate theory text, a middle path between market-oriented introductions and technical monographs. The books by Hull are institution- and application-oriented: Fundamentals of Futures and Options Markets introduces the instruments without calculus, and Options, Futures and Other Derivatives gives the standard market and valuation account with accessible mathematics. Björk is the place to go afterwards for derivations of the pricing results.

For the mathematics, Lamberton and Lapeyre's introduction to stochastic calculus applied to finance is a separate, finance-oriented route to the same tools. Musiela and Rutkowski's Martingale Methods in Financial Modelling and Bingham and Kiesel's Risk-Neutral Valuation are comparable graduate treatments. Specialist books deepen individual topics: Rebonato's on the LIBOR market model, Glasserman's on Monte Carlo methods, and Platen and Bruti-Liberati's on numerical solution of SDEs with jumps. Fixed-income texts by Tuckman, Veronesi and Sundaresan are market-side companions to Chapters 19 to 23, and Duffie and Singleton's book supplies the credit risk that Björk does not cover.

How to Use This Book

The book is meant for graduate students in economics and mathematics and for advanced undergraduates and professionals in mathematical finance, according to the publisher. The practical minimum is calculus and probability at an advanced undergraduate level; the appendices supply the measure-theoretic background.

A first course: Chapters 1 to 3 for intuition in discrete time, Chapters 4 and 5 for stochastic calculus, then Chapters 6 to 10 for the delta-hedging route to Black–Scholes. Those who want the modern approach can continue with Chapters 11 to 15 and then pick topics: Chapters 16 to 18 for dividends, futures and currencies, Chapters 19 to 24 for interest rates. Chapters 25 to 28 suit a course on investment theory and American options, while Parts V and VI are for advanced readers.

Work the exercises, since each technique comes with a solved example and the exercises extend it, and consult the appendices when measure-theoretic points arise.