University of Houston
Department of Computer Science
In partial fulfillment of the Requirements for the Degree of
Master of
Science
KaiLong Cai
will defend his thesis
NUMERICAL METHODS FOR OPTIONS PRICING
Abstract
The purpose of this paper is to apply numerical integration techniques to
options pricing in financial derivatives markets. After reviewing the
concept of options and Black-Scholes options pricing differential equation,
I apply numerical methods of
nonlinear
equations in order to use the Black-Scholes options pricing model to
estimate securities'
implied standard deviation. I then approximate the Black-Scholes options
pricing model
numerically by solving the related partial differential equations by the
Euler's explicit
method, the Euler's implicit method, the Crank-Nicolson method and the
Fourth-Order
Runge-Kutta method. I then compare those methods with respect to stability,
accuracy
and computational complexity.
My conclusions are as follows: First, among three numerical methods of
nonlinear
equations to estimate securities' implied standard deviation from options
prices, the
Secant method with polynomial approximation is the one with least
computational
complexity and least execution time to provide the same accuracy. Second, to
approximate the Black-Scholes options pricing model, the Crank-Nicolson
method
generally has the overall advantage over the other methods. This is because
the Crank-
Nicolson method can guarantee its stability without any restriction and can
run relatively
fast but produce relatively small error. The Fourth-Order Runge-Kutta method
may win
over the Crank-Nicolson method with respect to speed and accuracy in the
situation that
the stability limitation on the Fourth-Order Runge-Kutta method does not
significantly
slow down its computing speed.
Date: Tuesday, November 11, 2003
Time: 01:00 PM
Place:
550-PGH
Faculty, students, and the general public are
invited.
Thesis Advisor: Dr. Olin Johnson