Stochastic Finance: A Numeraire Approach / Edition 1

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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.

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Editorial Reviews

From the Publisher
… a nice book for researchers and practitioners. … this book can be regarded as a wonderful application of stochastic analysis, as it includes not only detailed theoretical proofs but also practical illustrative examples. With the systematic and feasible numeraire techniques, the book can serve as an everyday reference book for practitioners, but also as a powerful tool to deal with pricing and hedging for complicated financial assets. Most importantly, the representation of prices as a pairwise relationship of two assets is the most novel characteristic of this book, which could lead to deeper understanding of derivative contracts.
—Jian Ping Wan, Mathematical Reviews, 2012f

Although the importance of the choice of numeraire has been recognized for quite some time, this is the first book to stress the fundamental role that numeraires play in modern asset pricing theory. The author is the leading expert on the subject so it is a pleasure to highly recommend this book.
—Peter Carr, Ph.D., Managing Director of Morgan Stanley and Executive Director of NYU’s Master of Science Program in Mathematics in Finance

Finally, we have a full volume with a systematic treatment of the change of numeraire techniques. Jan Vecer has taken years of teaching experience and practitioners’ feedback to unify a previously complicated topic into the most elegant and easily accessible numeraire textbook to come down the pike. Now it has become fun to learn about parity and duality relationships among exotic options in a whole variety of models. The practitioners will be happy about the dimension reduction methods. There should be more such books.
—Uwe Wystup, Ph.D., Managing Director of MathFinance AG

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Product Details

Meet the Author

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.

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Table of Contents


Elements of Finance
Time Value of Assets, Arbitrage and No-Arbitrage Assets
Money Market, Bonds, and Discounting
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
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

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



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