A Note on Relative Efficiency of Some Numerical Methods for Pricing of American Options Under Levy Processes
40 Pages Posted: 28 Oct 2004
Date Written: October 11, 2004
We analyze properties of prices of American options under Levy processes, and the related difficulties for design of accurate and efficient numerical methods for pricing of American options. The case of Levy processes with insignificant diffusion component and jump part of infinite activity but finite variation (the case most relevant to practice according to the empirical study in Carr et. al., Journ. of Business (2002)) appears to be the most difficult. Several numerical methods suggested for this case are discussed and compared. It is shown that approximations by diffusions with embedded jumps may be too inaccurate unless time to expiry is large, but two methods: the fitting by a diffusion with embedded exponentially distributed jumps and a new finite difference scheme suggested in the paper can be used as good complements, which ensure accurate and fast calculation of the option prices both close to expiry and far from it. We demonstrate that if the time to expiry is 2 months or more, and the relative error 1-2% is admissible then the fitting by a diffusion with embedded exponentially distributed jumps and the calculation of prices using the semi-explicit pricing procedure in Levendorskii, IJTAF (2004), is the best choice.
Keywords: American options, Levy processes, numerical methods
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