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Zbl 1160.91346
Almendral, Ariel; Oosterlee, Cornelis W.
On American options under the variance gamma process. (English)
[J] Appl. Math. Finance 14, No. 2, 131-152 (2007). ISSN 1350-486X

Summary: American options are considered in a market where the underlying asset follows a Variance Gamma process. A sufficient condition is given for the failure of the smooth fit principle for finite horizon call options. A second-order accurate finite-difference method is proposed to find the American option price and the exercise boundary. The problem is formulated as a Linear Complementarity Problem and solved numerically by a convenient splitting. Computations have been accelerated with the help of the Fast Fourier Transform. A stability analysis shows that the scheme is conditionally stable, with a mild stability condition of the form $k = O(|\log(h)|^{-1})$. The theoretical results are verified numerically throughout a series of numerical experiments.

MSC 2000:: *91B28 Finance etc.
90C33 Complementarity problems
45J05 Integro-ordinary differential equations
65M06 Finite difference methods (IVP of PDE)
65T50 Discrete and fast Fourier transforms

Keywords: integro-differential equations; variance gamma; finite differences; FFT

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Zentralblatt MATH Berlin [Germany]