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Unlike much of the existing literature, Stochastic Finance: A Numeraire Approach treats price as a number of units of one asset needed for an acquisition of a unit of another asset instead of expressing prices in dollar terms exclusively. This numeraire approach leads to simpler pricing options for complex products, such as barrier, lookback, quanto, and Asian options. Most of the ideas presented rely on intuition and basic principles, rather than technical computations.

The first chapter of the book introduces basic concepts of finance, including price, no arbitrage, portfolio, financial contracts, the First Fundamental Theorem of Asset Pricing, and the change of numeraire formula. Subsequent chapters apply these general principles to three kinds of models: binomial, diffusion, and jump models. The author uses the binomial model to illustrate the relativity of the reference asset. In continuous time, he covers both diffusion and jump models in the evolution of price processes. The book also describes term structure models and numerous options, including European, barrier, lookback, quanto, American, and Asian.

Classroom-tested at Columbia University to graduate students, Wall Street professionals, and aspiring quants, this text provides a deep understanding of derivative contracts. It will help a variety of readers from the dynamic world of finance, from practitioners who want to expand their knowledge of stochastic finance, to students who want to succeed as professionals in the field, to academics who want to explore relatively advanced techniques of the numeraire change.

Jan Vecer is a professor of finance and has taught courses on stochastic finance at Columbia University, the University of Michigan, Kyoto University, and the Frankfurt School of Finance and Management. His research interests encompass areas within financial statistics, financial engineering, and applied probability, including option pricing, optimal trading strategies, stochastic optimal control, and stochastic processes. He earned a Ph.D. in mathematical finance from Carnegie Mellon University.

Table of Contents

Introduction

Elements of Finance
Price Arbitrage Time Value of Assets, Arbitrage and No-Arbitrage Assets Money Market, Bonds, and Discounting Dividends Portfolio Evolution of a Self-Financing Portfolio Fundamental Theorems of Asset Pricing Change of Measure via Radon–Nikodým Derivative Leverage: Forwards and Futures

Binomial Models
Binomial Model for No-Arbitrage Assets Binomial Model with an Arbitrage Asset

Diffusion Models
Geometric Brownian Motion General European Contracts Price as an Expectation Connections with Partial Differential Equations Money as a Reference Asset Hedging Properties of European Call and Put Options Stochastic Volatility Models Foreign Exchange Market

Interest Rate Contracts
Forward LIBOR Swaps and Swaptions Term Structure Models

Barrier Options
Types of Barrier Options Barrier Option Pricing via Power Options Price of a Down-and-In Call Option Connections with the Partial Differential Equations

Lookback Options
Connections of Lookbacks with Barrier Options Partial Differential Equation Approach for Lookbacks Maximum Drawdown

American Options
American Options on No-Arbitrage Assets American Call and Puts on Arbitrage Assets Perpetual American Put Partial Differential Equation Approach

Contracts on Three or More Assets: Quantos, Rainbows and "Friends"
Pricing in the Geometric Brownian Motion Model Hedging

Asian Options
Pricing in the Geometric Brownian Motion Model Hedging of Asian Options Reduction of the Pricing Equations

Jump Models
Poisson Process Geometric Poisson Process Pricing Equations European Call Option in Geometric Poisson Model Lévy Models with Multiple Jump Sizes

Appendix: Elements of Probability Theory

Solutions to Selected Exercises

References

Index

342 pages, Hardcover