Valuation of Standard Option with Dividend Paying Stock Using Finite Difference Method

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Numerical methods form a significant part of the pricing of financial derivatives, especially in cases where there is no closed form analytical solution. The evaluation of American options using the Black-Scholes Model where early exercise is possible and a general closed-form solution does not exist leads to a free boundary value problem. A common way to deal with this problem is to apply numerical methods. In this thesis we price American options with dividend paying stock on a single asset. We start from the Black-Scholes equation with a free boundary value, the free boundary value problem is then transformed into a Linear Complementarity Problem, and an Obstacle Problem. We solve the Linear Complementarity Problem by introducing the method of Finite Difference method. Finite difference methods is discussed quite extensively with a focus on the Crank-Nicolson scheme. This leads to a constraint linear system of equations which is solved on a discrete domain by applying the PSOR method. The simulation results showed that the price of the American option exceeds the analytical solution. The payoff function intersects the European option at lower prices relative to the American option; this gives us the early exercise value. We conclude that the American option with dividend paying stock is preferred to the European option.
A Thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology in partial fulfillment of the requirements for the degree of Master of Philosophy in Applied Mathematics, June-2013