Stochastic Calculus for Finance I: The Binomial Asset Pricing Model

Citation

Shreve, S. E. (2005). Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Springer. Springer Finance.

Chapter Summary

This is the first of two volumes that grew out of the first ten years of Carnegie Mellon's professional master's programme in computational finance. It is short, under 190 pages: six chapters and an appendix proving the fundamental properties of conditional expectations. The publisher states the volume's purpose plainly: the binomial asset-pricing model is presented mainly as a vehicle for introducing, in a simple setting, the concepts needed for the continuous-time theory of Volume II. Every chapter ends with a summary, notes and classroom-tested exercises, some extending the theory and some drawn from practical problems in quantitative finance.

Chapter 1: The Binomial No-Arbitrage Pricing Model

The chapter builds the binomial model of a stock price and shows how a derivative security is priced by no-arbitrage, building up to the multiperiod binomial model. A section on computational considerations follows.

Chapter 2: Probability Theory on Coin Toss Space

The probability needed for the rest of the book, developed on the finite space of coin-toss outcomes that drive the binomial model: random variables, distributions and expectations, conditional expectations, martingales and Markov processes. Because the space is finite, these ideas can be defined and used without measure theory; the appendix proves the fundamental properties of conditional expectations.

Chapter 3: State Prices

The chapter introduces the Radon–Nikodým derivative process that connects the actual probability measure with the risk-neutral measure, and develops state prices, with a section on the capital asset pricing model.

Chapter 4: American Derivative Securities

Derivatives that may be exercised early. The chapter treats non-path-dependent American derivatives first, then introduces stopping times and handles general American derivatives, and ends with American call options.

Chapter 5: Random Walk

The symmetric random walk as a discrete forerunner of Brownian motion: first passage times, the reflection principle and a section devoted to an extended example.

Chapter 6: Interest-Rate-Dependent Assets

The binomial framework applied to interest rates: a binomial model for interest rates, fixed-income derivatives, forward measures and futures.

Key Concepts

Chapters 1–2: Pricing and probability on coin-toss space

• No-arbitrage pricing: the price of a derivative is the initial capital of a portfolio that replicates its payoff.
• Risk-neutral probability measure: the measure under which discounted asset prices are martingales, giving the risk-neutral pricing formula.
• Conditional expectation: the best estimate of a random quantity given the coin tosses observed so far, the central tool of the book.
• Martingales and Markov processes: processes with no expected drift, and processes whose future depends only on the present state.

Chapters 3–4: Change of measure and early exercise

• Radon–Nikodým derivative process: the process that converts expectations under the actual measure into expectations under the risk-neutral measure.
• State prices: the value today of a payment made in a single future outcome.
• Stopping times: random times decided only by information available so far, used to describe when an American option is exercised.

Chapters 5–6: Random walk and interest rates

• First passage time and reflection principle: when a random walk first reaches a level, and the symmetry argument used to compute its distribution.
• Forward measure: a change of measure adapted to pricing payments at a fixed future date.
• Futures and forward contracts: interest-rate-dependent contracts priced within the binomial model.

Critical Analysis

Strengths

• Every key idea in its simplest setting. Conditional expectation, martingales, Markov processes, change of measure and stopping times all appear here on a finite coin-toss space, where they can be computed by hand. Readers meet them again in continuous time in Volume II already knowing what they mean.
• Modest prerequisites. The publisher states that the material has been used with students whose background is calculus and calculus-based probability; the finite setting means measure theory is not needed.
• Built for teaching. The text grew out of a professional master's programme and offers, according to the publisher, precise statements of results, plausibility arguments and some proofs, with intuitive explanations refined in class. Each chapter has a summary, notes and exercises.
• Short and focused. Fewer than 190 pages cover pricing, hedging, American options, random walk and interest-rate models, so the whole volume can be worked through in part of a semester.

Limitations

• Not a stand-alone account of financial mathematics. By design the volume prepares for Volume II; Brownian motion and the continuous-time theory belong to the second volume, not this one.
• Only discrete time. Every model is a binomial tree, so readers learn nothing here about continuous-time processes beyond the random walk of Chapter 5.

Where It Fits in the Course

Within this course, the book is the bridge between general stochastic processes and continuous-time finance. Robert Dobrow's Introduction to Stochastic Processes with R and Sheldon Ross's Introduction to Probability Models give the broader background in Markov chains, Poisson processes and Brownian motion; Shreve can be read alongside them by any student with a probability course, or straight after the martingale material in Dobrow's Chapter 8.

After Shreve, the continuous-time books take over. Lamberton and Lapeyre's Introduction to Stochastic Calculus Applied to Finance and Tomas Björk's Arbitrage Theory in Continuous Time develop option pricing with stochastic calculus, Björk also starting from a discrete-time model; Bernt Øksendal's Stochastic Differential Equations treats the underlying mathematics of Itô calculus rigorously. Sidney Resnick's Adventures in Stochastic Processes is a rigorous general treatment of stochastic processes for those who want more probability before or after the finance. Shreve's own Volume II is the most direct continuation, but it is not part of this course list.

How to Use This Book

The book suits advanced undergraduates and master's students in mathematical finance, financial engineering, mathematics or economics who have taken calculus and a first course in probability. The publisher names calculus and calculus-based probability as the background, and the finance is built up from the binomial model itself.

A reading plan:

• Core: Chapters 1 and 2 in full. Chapter 1 sets out the replication argument on which everything rests; Chapter 2 supplies the probability language, and its sections on conditional expectations and martingales are the most important in the book.
• Next: Chapter 3 on the Radon–Nikodým derivative process, which anticipates the change of measure used constantly in continuous time, and Chapter 4 on American derivatives and stopping times.
• Then: Chapter 5 as preparation for Brownian motion, and Chapter 6 for students interested in fixed income.
• As needed: the appendix on the properties of conditional expectations, for readers who want the proofs.

Work small binomial trees by hand until pricing by replication is routine, then write a short program to price options in many-period trees. Use the chapter summaries to check understanding before attempting the exercises, and keep notes on each concept as it will reappear in continuous time.