Chapter 1. Financial markets and derivatives
Chapter 2. Binomial model
Chapter 3. Finite market model
Chapter 4. Black-Scholes model
Chapter 5. Multi-dimensional Black-Scholes model
Appendix A. Conditional expectation and Lp
-spaces
Appendix B. Discrete time stochastic processes
Appendix C. Continuous time stochastic processes
Appendix D. Brownian motion and stochastic integration

The book begins with the development of the basic ideas of hedging and pricing of European and American derivatives in the discrete (i.e., discrete time and discrete state) setting of binomial tree models. Then a general discrete finite market model is introduced, and the fundamental theorems of asset pricing are proved in this setting. Tools from probability such as conditional expectation, filtration, (super)martingale, equivalent martingale measure, and martingale representation are all used first in this simple discrete framework. This provides a bridge to the continuous (time and state) setting, which requires the additional concepts of Brownian motion and stochastic calculus. The simplest model in the continuous setting is the famous Black-Scholes model, for which pricing and hedging of European and American derivatives are developed. The book concludes with a description of the fundamental theorems for a continuous market model that generalizes the simple Black-Scholes model in several directions.