Math 194
The Mathematics of Finance

Winter 2008

In this course we examine the mathematics of some of the basic derivative securities encountered in financial markets. A prototype for such a derivative is the European put option, which is a contract giving its owner the right (but not the obligation) to sell a share of a specific stock at a fixed price (the strike price) on a fixed date. The buying of such an option provides the owner with a hedge against some of the risk associated with owning the stock---if on the fixed date the price of the stock exceeds the strike price, the option is worthless, but if the stock price falls below the strike price, the owner can exercise the option and sell the stock at a better-than-market price. Economists R.~Merton and M.~Scholes won the Nobel Prize for their work on pricing such European put (and call) options, when the stock price is modeled by an exponential Brownian motion. (Scholes' collaborator F.~Black died before the Nobel was awarded; Merton refined and extended the early work of Black and Scholes.) In this course we will study a discrete version of the Black-Scholes-Merton (BSM) model, the Cox-Ross-Rubinstein (CRR) model. The mathematics of the CRR model is simpler than that needed for the BSM model, but all of the key ideas necessary for the analysis are present in our analysis of the CCR model. The model and its analysis are based on probability theory learned in Math 180A, and will also make use of linear algebra (Math 20F) and differential equations (Math 20D) learned in prerequisite courses.

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January 14, 2008