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Two-dimensional Shannon wavelet inverse Fourier technique for pricing European options. (English) Zbl 1414.91409

Summary: The SWIFT method for pricing European-style options on one underlying asset was recently published and presented as an accurate, robust and highly efficient technique. The purpose of this paper is to extend the method to higher dimensions by pricing exotic option contracts, called rainbow options, whose payoff depends on multiple assets. The multidimensional extension inherits the properties of the one-dimensional method, being the exponential convergence one of them. Thanks to the nature of local Shannon wavelets basis, we do not need to rely on a-priori truncation of the integration range, we have an error bound estimate and we use fast Fourier transform (FFT) algorithms to speed up computations. We test the method for similar examples with state-of-the-art methods found in the literature, and we compare our results with analytical expressions when available.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65T50 Numerical methods for discrete and fast Fourier transforms
91G20 Derivative securities (option pricing, hedging, etc.)
60G51 Processes with independent increments; Lévy processes

Software:

Sinc-Pack; BENCHOP
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Full Text: DOI Link

References:

[1] Daubechies, I., Ten Lectures on Wavelets (1992), Society for Industrial and Applied Mathematics · Zbl 0776.42018
[2] Fang, F.; Oosterlee, C. W., A novel option pricing method based on Fourier-cosine series expansions, SIAM J. Sci. Comput., 31, 2, 826-848 (2008) · Zbl 1186.91214
[3] Kirkby, J. L., Efficient option pricing by frame duality with the fast Fourier transform, SIAM J. Financ. Math., 6, 1, 713-747 (2015) · Zbl 1320.91155
[4] Mallat, S., A Wavelet Tour of Signal Processing (2009), Academic Press · Zbl 1170.94003
[5] Maree, S. C.; Ortiz-Gracia, L.; Oosterlee, C. W., Pricing early-exercise and discrete barrier options by Shannon wavelet expansions, Numer. Math. (2016), in press · Zbl 1378.91124
[6] Margrabe, W., The value of an option to exchange one asset for another, J. Finance, 33, 177-186 (1978)
[7] Meyer, Y., Wavelets and Operators (1993), Cambridge University Press · Zbl 0810.42015
[8] Moller, T., On valuation and risk management at the interface of insurance and finance, Br. Actuar. J, 8, IV, 787-827 (2002)
[9] Ortiz-Gracia, L.; Oosterlee, C. W., Robust pricing of European options with wavelets and the characteristic function, SIAM J. Sci. Comput., 35, 5, B1055-B1084 (2013) · Zbl 1281.62227
[10] Ortiz-Gracia, L.; Oosterlee, C. W., A highly efficient Shannon wavelet inverse Fourier technique for pricing European options, SIAM J. Sci. Comput., 38, 1, B18-B143 (2016) · Zbl 1330.91184
[11] Ruiter, M. J.; Oosterlee, C. W., Two-dimensional Fourier cosine series expansion method for pricing financial options, SIAM J. Sci. Comput., 34, 5, B642-B671 (2012) · Zbl 1258.91222
[12] Stenger, F., Handbook of Sinc Numerical Methods, Chapman & Hall/CRC Numerical Analysis and Scientific Computing (2011), CRC Press: CRC Press Boca Raton, FL · Zbl 1208.65143
[13] von Sydow, L., BENCHOP - the benchmarking project in option pricing, Int. J. Comput. Math., 92, 2361-2379 (2015) · Zbl 1335.91113
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