ABSTRACT
Chapter 3 begins the book’s mathematical foundation. Three primary derivative pricing approaches (tree models, simulation models, and analytic models) are introduced through a simple dice game. After that, the chapter details the crucial concept of risk-neutral pricing through a binomial tree model. P-measures, Q-measures, and risk aversion are introduced in a tree setting. Next, geometric Brownian motion and simulation are introduced as a foundation to generating random paths. Ito’s lemma is detailed next. Then an instantaneous risk-free portfolio is constructed to provide an intuitive derivation of the Black-Scholes PDE. The risk-neutral principal in a continuous model is implied by the Black-Scholes PDE. Finally, applying the risk-neutral principal, the analytic Black-Scholes call and put price formulas are derived through integration. At the end of the chapter, Monte Carlo simulation is used to value calls and puts. Sample codes of simulation in R are offered so that readers can practice and experiment. Sample spreadsheets with a small numbers of steps are offered so that readers can visualize the underlying economics and mathematics and extend them. Finally, a project compares the three general derivative valuation methods (analytic, binomial tree by spreadsheet, and simulation by R) and analyzing convergence, a potentially important issue.
