22 January 2016Abstract:
The price of an option can under some assumptions be determined by the solution of the Black-Scholes partial differential equation. Often options are issued on more than one asset. In this case it turns out that the option price is governed by the multi-dimensional version of the Black-Scholes equation. Options issued on a large number of underlying assets, such as index options, are of particular interest, but pricing such options is a challenge due to the "curse of dimensionality". The multi-dimensional PDE turn out to be computationally expensive to solve accurately even in quite a low number of dimensions.
In this thesis we develop a radial basis function partition of unity method for pricing multi-asset options up to moderately high dimensions. Our approach requires the use of a lower number of node points per dimension than other standard PDE methods, such as finite differences or finite elements, thanks to a high order convergence rate. Our method shows good results for both European style options and American style options, which allow early exercise. For the options which do not allow early exercise, the method exhibits an exponential convergence rate under node refinement. For options that allow early exercise the option pricing problem becomes a free boundary problem. We incorporate two different approaches for handling the free boundary into the radial basis function partition of unity method: a penalty method, which leads to a nonlinear problem, and an operator splitting method, which leads to a splitting scheme. We show that both methods allow for locally high algebraic convergence rates, but it turns out that the operator splitting method is computationally more efficient than the penalty method. The main reason is that there is no need to solve a nonlinear problem, which is the case in the penalty formulation.
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