Stochastic Finance: A Numeraire Approach

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