Valuing American style Asian and look back options using state space methods

Abstract

Numerical procedures are popular techniques used to value derivatives when analytical formulas are not available. For example, in Binomial Tree we represent different possible paths that might be followed by the stock price over the life of the option. However, Binomial Tree Model is not an precise model of reality. Even in a multiple step binomial model, at each time interval the stock price is assumed to take only two possible states. This assumption is not correct. We know that if the a stock price S, follows a geometric Brownian motion, then the stock's price at time t is lognormally distributed. The standard deviation of the logarithm of the stock price is aVt. It is proportional to the square root of how far ahead we are looking. In our entire project, we assume that stock's price is log-normally distributed. State Space Methods is an approach in which we divide the option maturity time (T) into small intervals of length At = T/N. In each time interval the stock price is assumed to follow a log normal distribution. The possible values the stock price are represented in form of a grid. The option values at maturity (or end of Nth time interval) are known. We then work backwards. Because a risk neutral world is being assumed, the value at each point at time T-At can be calculated as the expected value at time T discounted at rate r for a period At. If we consider a point (i,j) at time iAt , the probability that stock shall reach point (i+l,k) in time in (i+l)At can be calculated from the fact that stock price is log-normally distributed. The first objective of this project was to use State Space Methods in valuing options. In order to compare the results from this approach, we started with valuing those options for which analytical formulas are available. Having achieved encouraging results (less than 0.04% error from exact analytical solution), we extended this approach to options for which no analytical formulas are available. Results were compared with the best available solutions to measure such options.