# The optimal exercising problem from American options: a comparison of solution methods

Traditionally, the stock options are simulated using Monte-Carlo methods and all stock prices along the path are stored, and then the optimal exercise time is determined starting at the final time period and continuing backward in time. Also, as the number of paths simulated increases, the number of simultaneous equations that need to be solved at each time step grow proportionally. Currently, two theoretical methods have emerged in determining the optimal exercise problem. The first method uses the concept of least-squares approach in linear regression to estimate the value of continuing to hold on to the option via a set of randomly generated future stock prices. Then, the value of continuing can be compared to the payoff at current time from exercising the option and a decision can be reached, which gives the investor a higher value. The second method uses the finite difference approach to establish an exercise boundary for the American option via an artificially generated mesh on both possible stock prices and decision times. Then, the stock price is simulated and the method checks to see if it is inside the exercise boundary.

In this research, these two solution approaches are evaluated and compared using discrete event simulation. This allows complex methods to be simulated with minimal coding efforts. Finally, the results from each method are compared. Although a more conservative method cannot be determined, the least-squares method is faster, more concise, easier to implement, and requires less memory than the mesh method.

The motivation for this research stems from interest in simulating and evaluating complicated solution methods to the optimal exercise problem, yet requiring little programming effort to produce accurate and efficient estimation results.

Advisor:

School:Kansas State University

School Location:USA - Kansas

Source Type:Master's Thesis

Keywords:american options least squares method finite difference economics finance 0508 engineering industrial 0546 operations research 0796

ISBN:

Date of Publication:01/01/2007