Abstract
We discuss the issue of how to improve the efficiency of standard spectral methods for pricing options. As a prototype example, we consider American options which are frequently used in the market as they enjoyed the early exercise opportunities. Such options are often modeled by nonlinear partial differential equations, solutions of which are not obtainable in a closed form and thus necessitates robust numerical techniques. For the discretisation in space (asset) direction, we use a rational spectral methods. For time integration involved in the process, we looked at comparing the efficiency of the method by using contour integral method based on the inversion of the Laplace transform, and the Exponential Time Differencing Runge–Kutta method of order 4. It is known that when these spectral methods are applied to option pricing problems, they no longer retain their higher order convergence. This order reduction is attributed due to the non-smooth initial conditions. To overcome this, we propose a domain decomposition method based on our rational spectral method. For the problem under consideration, we divide the domain into two sub-domains, with the transition point as the strike price. To represent the solution in each sub-domains, we choose to approximate the solution by linear rational interpolants due to their improved stability properties as compared to the polynomial interpolant. Comparative results are also presented.
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Acknowledgments
We would like to acknowledge anonymous referees for their valuable suggestions. E. Ngounda would like to acknowledge the Office of Research and Development, University of the Western Cape, for the financial support. KCP’s research was also supported by the South African National Research Foundation.
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Ngounda, E., Patidar, K.C. Limitations and improvements of standard spectral methods for pricing standard options. Int J Adv Eng Sci Appl Math 7, 106–113 (2015). https://doi.org/10.1007/s12572-015-0140-3
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DOI: https://doi.org/10.1007/s12572-015-0140-3