Efficient Pricing of European Options with Stochastic Interest Rate Using Fourier Transform Method
American Journal of Applied Mathematics
Volume 4, Issue 4, August 2016, Pages: 181-185
Received: May 6, 2016; Accepted: May 20, 2016; Published: Jul. 13, 2016
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Authors
Michael Chimezie Anyanwu, Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria
Roseline Ngozi Okereke, Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria
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Abstract
The inclusion of stochastic interest rate is an essential element of any realistic option pricing method. Therefore, the purpose of this paper is to incorporate interest rates in the Fourier transform method for pricing European options in exponential Levy models. With the assumption of stochastic independence between the underlying log asset price and the stochastic interest rate, we obtain a pricing of pure discount bond available in the market. Our method of valuation is to apply eigenfunction expansion to the variable that describes the evolution of the interest rate, and Fourier transform to the variable that describes the log asset price.
Keywords
Fourier Transform, Eigenfunction Expansion, Stochastic Interest Rate, Levy Process, Simple Harmonic Oscillator
To cite this article
Michael Chimezie Anyanwu, Roseline Ngozi Okereke, Efficient Pricing of European Options with Stochastic Interest Rate Using Fourier Transform Method, American Journal of Applied Mathematics. Vol. 4, No. 4, 2016, pp. 181-185. doi: 10.11648/j.ajam.20160404.13
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Black, F. and M. S. Scholes (1973). The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 637-654.
[2]
Hoston, S. L. (1993) A closed-form Solution for Options with Stochastic Volatility with Application to Bond and Currency Options. Review of Financial Studies 6 (2), 327-343.
[3]
Stein, E. and Stein, J. (1991) Stock Price Distributions with Stochastic Volatility: An Analytic Approach. Review of Financial Studies 4, 727-752.
[4]
Carr, P. P. and Madan, D. B. (1999) Option valuation using the fast Fourier transform, Journal of Computational Finance 2 (4), 61-73.
[5]
Bailey, D. and Swarztrauber, P. (1991). The fractional Fourier transform and applications, SIAM Review 33, 389-404.
[6]
Attari, M. (2004) Option Pricing using Fourier Transforms: A Numerically Efficient Simplification. Working Paper, Charles River Associates. Available at SSRN: http://ssrn.com/abstract=520042.
[7]
Benhamou, E. (2000). Fast Fourier Transform for Discrete Asian Options, EFMA 2001 Lugano Meetings. Available at SSRN: http://ssrn.com/abstract=269491
[8]
Hurd, T. R. and Zhou, Z. (2009). A Fourier transform method for spread option pricing, Preprint, arXiv: 0902.3643v1.
[9]
Lewis, A. (2001). A simple option Formula for General Jump-Diffusion and other Exponential Levy Processes, Envision Financial Systems and Option City. net, California. Available at http://optioncity.net/pubs/ExpLevy.pdf.
[10]
Borak, S., Detlefsen, K. and Hardle, W. (2005) FFT Based Option Pricing. Discussion Paper, Sonderforschungsbereich (SFB) 649, Humboldt Universität Berlin.
[11]
Dempster, M. A. and Hong, S. S. (2002) Spread Option Valuation and the Fast Fourier Transform. Technical Report WP26/2000, University of Cambridge.
[12]
Itkin, A. (2005) Pricing options with VG model using FFT. arXiv: physics/0503137v1
[13]
O’Sullivan, C. (2005) Path Dependent Option Pricing under Levy Processes. EFA 2005 Moscow Meetings Paper. Available at SSRN: http://ssrn.com/abstract=673424
[14]
Mckean, Henry P. J. (1956). Elementary solution for certain parabolic differential equations. Transaction of the American mathematical society, 82 (2), 519-548.
[15]
Linetsky, V. (2004). The spectral decomposition of the option value. International Journal of Theoretical and Applied Finance, 7 (3), 337-384.
[16]
Davydov, V. and Linetsky, V. (2000). Pricing Options on scalar diffusions: An Eigenfunction Expansion Approach, Operations Research 512, 185-209.
[17]
Boyarchenko, N. and Levendorskii, S. Z. (2004). Eigenfunction Expansion method in multi-factor quadratic term-structure models. Working paper. Available at SSRN: http://ssrn.com/abstract=627642
[18]
Cox, J. C., Ingersol, J. E and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 55, 349-383.
[19]
Vasicek, O. A. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177-188.
[20]
Gorovoi, V. and Linetsky, V. (2004) Black’s model of interest rates as options, eigenfunction expansion and Japanese interest rates. Mathematical Finance, 14 (1), 48-78.
[21]
Cont, R. and Tankov, P. (2004). Financial Modeling with Jump Processes. (Financial Mathematics Series), Chapman and Hall/CRC, New York.
[22]
Merton, R. C. (1976). Option pricing when the underlying stock returns are discontinuous. Journal of Financial Economics 3, 125-144.
[23]
Kou, S. G. (2002). A jump – diffusion model for option pricing. Management Science 48, 1086-1101.
[24]
Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The Variance Gamma Process and Option Pricing. European Finance Review 2, 79-105.
[25]
Barndorff-Nielson, O. Processes of Normal Inverse Gaussian Type. Finance and Stochastics, 2, (1998), 41-68.
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