Stochastic Finance: A Numeraire Approach

Focuses on fundamental principles of pricing
Shows how to identify the basic assets of a given contract
Explains how to compute the prices of contingent claims in terms of various reference assets
Presents examples of a model independent formula for European call options; a simple method for pricing barrier options, lookback options, and Asian options; and a formula for options on LIBOR
Provides prerequisite material on probability and stochastic calculus in the appendix
Includes solutions to odd-numbered exercises at the back of the book
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.

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?ikod羸m 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
L矇vy Models with Multiple Jump Sizes

Appendix: Elements of Probability Theory
Solutions to Selected Exercises
References
Index