Abstract
In this chapter we describe the pricing of Bermudan options by means of Fourier cosine expansions. We further propose a technique to price early-exercise call options with the help of the (European) put-call parity n. Direct pricing of the call options with cosine expansions may give rise to some sensitivity regarding the size of the domain in which the Fourier expansion is applied. By employing the put-call parity relation, this can be avoided so that call options governed by fat-tailed asset price distributions can be priced as easily as put options.
MSC code: 65C30, 60H35, 65T50
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Notes
- 1.
For example so that \(\vert \int_{\mathbb{R}}f(y\vert x)dy -\int_{a}^{b}f(y\vert x)dy\vert < TOL.\)
- 2.
This is mainly the case when we consider real options or insurance products with a long life time.
- 3.
Here we have a long list of arguments, as they are important for the use of the put-call duality.
- 4.
Without any dividend payments, of course, the American call option value is equal to the European call option value.
References
A. Almendral and C. W. Oosterlee. On American options under the Variance Gamma process. Appl. Math. Finance, 14(2):131–152, 2007
A. D. Andricopoulos, M. Widdicks, P. W. Duck, and D. P. Newton. Universal option valuation using quadrature methods. J. Fin. Economics, 67:447–471, 2003
A. D. Andricopoulos, M. Widdicks, P. W. Duck, and D. P. Newton. Extending quadrature methods to value multi-asset and complex path dependent options. J. Fin. Economics, 83(2):471–500, 2007
O. E. Barndorff-Nielsen. Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist., 24(1–13), 1997
M. Broadie and Y. Yamamoto. Application of the fast Gauss transform to option pricing. Management Sci., 49:1071–1008, 2003
M. Broadie and Y. Yamamoto. A double-exponential fast Gauss transform for pricing discrete path-dependent options. Operations Research, 53(5):764–779, 2005
P. P. Carr, H. Geman, D. B. Madan, and M. Yor. The fine structure of asset returns: An empirical investigation. J. Business, 75:305–332, 2002
P. P. Carr and D. B. Madan. Option valuation using the fast Fourier transform. J. Comp. Finance, 2:61–73, 1999
C-C Chang, S-L Chung, and R. C. Stapleton. Richardson extrapolation technique for pricing American-style options. J. Futures Markets, 27(8):791–817, 2007
R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman and Hall, Boca Raton, FL, 2004
A. Eydeland. A fast algorithm for computing integrals in function spaces: financial applications. Computational Economics, 7(4):277–285, 1994
J. Fajardo, E. Mordecki. Symmetry and duality in Leacutevy markets. Quantitative Finance, 6(3): 219–227, 2006
F. Fang and C. W. Oosterlee. A novel option pricing method based on Fourier-cosine series expansions. SIAM J. Sci. Comput., 31(2):826–848, 2008
F. Fang and C. W. Oosterlee. Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions. Numerische Mathematik, 114(1):27–62, 2009
L. Feng and V. Linetsky. Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Heston transform approach. Math. Finance, 18(3):337–384, 2008
R. Geske and H. Johnson. The American put valued analytically. J. of Finance, 39:1511–1542, 1984
K. Jackson, S. Jaimungal, and V. Surkov. Option pricing with regime switching Lévy processes using Fourier space time-stepping. Proc. 4th IASTED Intern. Conf. Financial Engin. Applic., pages 92–97, 2007
S. G. Kou. A jump diffusion model for option pricing. Management Science, 48(8):1086–1101, 2002
R. Lord, F. Fang, F. Bervoets, and C.W. Oosterlee. A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes. SIAM J. Sci. Comput., 30:1678–1705, 2008
R. Lord and C. Kahl. Optimal Fourier inversion in semi-analytical option pricing. J. Comp. Finance, 10:1–30, 2007
D. B. Madan, P. R. Carr, and E. C. Chang. The Variance Gamma process and option pricing. European Finance Review, 2:79–105, 1998
R. Merton. Option pricing when underlying stock returns are discontinuous. J. Financial Economics, 3:125–144, 1976
W. Schoutens. Lévy processes in finance: Pricing financial derivatives. Wiley, 2003
Y. Yamamoto. Double-exponential fast Gauss transform algorithms for pricing discrete lookback options. Publ. Res. Inst. Math. Sci., 41:989–1006, 2005
B. Zhang, C.W. Oosterlee. An efficient pricing algorithm for swing options based on Fourier Cosine expansions. To appear in the Journal of Computational Finance
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Zhang, B., Oosterlee, C.W. (2012). Fourier Cosine Expansions and Put–Call Relations for Bermudan Options. In: Carmona, R., Del Moral, P., Hu, P., Oudjane, N. (eds) Numerical Methods in Finance. Springer Proceedings in Mathematics, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25746-9_10
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