Solving partial differential equations related to option pricing with numerical method
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Date
2014-10-15
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Abstract
In this thesis, we solve the special differential partial equation, Black-Scholes Equation for valuing option pricing through numerical methods. Options considered include European style, American style, and the exotic option with major reference to the European style option. The aim is to find accurately the value of the various option styles by determining whether a grid point not greater than 60 can be used to determine the value of options with reliable accuracy, by setting a higher order discretization in space and time as well as grid stretching around the interesting region. Whether a highly accurate scheme will also work for the exotic options and finally whether implied volatility can be calculated using iterative methods in less iteration. The fourth order difference scheme and the grid stretching in space by means of an analytic coordinate transformation are employed. By experiment, we showed numerically that a grid size or space of 20 to 40 is all that is needed to achieve accuracy in the value of option. A fourth order Backward Difference Scheme is sufficient enough to yield accuracy in the exotic options and also it is possible to use few iteration to obtain implied volatility. The numerical experiment thus confirms the proposed methods.
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A thesis submitted to the Department of Mathematics,
Kwame Nkrumah University of Science and Technology
in partial fulfillment of the requirement for the degree of
Masters of Science, 2014