On the Efficacy of Fourier Series Approximations for Pricing European Options

A. S. Hurn¹, K. A. Lindsay¹, A. J. McClelland²

¹School of Economics and Finance, Queensland University of Technology, Brisbane, Australia.
²Numerix Pty Ltd, Sydney, Australia.

DOI: 10.4236/am.2014.517267                

Abstract

This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.

Keywords

Fourier Transform, Fourier Series, Characteristic Function, Option Price

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Johannes, M.S., Polson, N.G. and Stroud, J.R. (2009) Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices. Review of Financial Studies, 22, 2759-2799. http://dx.doi.org/10.1093/rfs/hhn110
[2]	Broadie, M., Chernov, M. and Johannes, M. (2007) Model Specification and Risk Premia: Evidence from Futures Options. Journal of Finance, 62, 1453-1490. http://dx.doi.org/10.1111/j.1540-6261.2007.01241.x
[3]	Christoffersen, P., Jacobs, K. and Mimouni, K. (2010) Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns and Option Prices. Review of Financial Studies, 23, 3141-3189. http://dx.doi.org/10.1093/rfs/hhq032
[4]	Hurn, A.S., Lindsay, K.A. and McClelland, A.J. (2012) Estimating the Parameters of Stochastic Volatility Models Using Option Price Data. Unpublished Working Paper, NCER.
[5]	Andersen, T.G., Fusari, N. and Todorov, V. (2012) Parametric Inference and Dynamic State Recovery from Option Panels. NBER Working Paper Series.
[6]	Carr, P.P. and Madan, D.B. (1999) Option Evaluation Using the Fast Fourier Transform. Journal of Computational Finance, 2, 61-73.
[7]	Borak, S., Detlefsen, K. and Hardle, W. (2005) FFT Based Option Pricing. SFB Discussion Paper 649.
[8]	Lord, R., Fang, F. Bervoets, F. and Oosterlee, C.W. (2007) A Fast and Accurate FFT-Based Methodology for Pricing Early-Exercise Options under Levy Processes. SIAM Journal of Scientific Computing, 20, 1678-1705.
[9]	Kwok, Y.K., Leung, K.S. and Wong, H.Y. (2012) Efficient Options Pricing Using the Fast Fourier Transform. In: Duan, J.C., Ed., Handbook of Computational Finance, Springer, Berlin, 579-604. http://dx.doi.org/10.1007/978-3-642-17254-0_21
[10]	Fang, F. and Oosterlee, C.W. (2008) A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions. SIAM Journal on Scientific Computing, 31, 826-848. http://dx.doi.org/10.1137/080718061
[11]	Zhang, B., Grzelak, L.A. and Oosterlee, C.W. (2012) Efficient Pricing of Commodity Options with Earlyexercise under the Ornstein-Uhlenbeck Process. Applied Numerical Mathematics, 62, 91-111. http://dx.doi.org/10.1016/j.apnum.2011.10.005
[12]	Heston, S.L. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6, 327-343.
[13]	Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654. http://dx.doi.org/10.1086/260062
[14]	Johannes, M.S., Polson, N.G. and Stroud, J.R. (2009) Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices. Review of Financial Studies, 22, 2759-2799. http://dx.doi.org/10.1093/rfs/hhn110
[15]	Broadie, M., Chernov, M. and Johannes, M. (2007) Model Specification and Risk Premia: Evidence from Futures Options. Journal of Finance, 62, 1453-1490. http://dx.doi.org/10.1111/j.1540-6261.2007.01241.x
[16]	Christoffersen, P., Jacobs, K. and Mimouni, K. (2010) Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns and Option Prices. Review of Financial Studies, 23, 3141-3189. http://dx.doi.org/10.1093/rfs/hhq032
[17]	Hurn, A.S., Lindsay, K.A. and McClelland, A.J. (2012) Estimating the Parameters of Stochastic Volatility Models Using Option Price Data. Unpublished Working Paper, NCER.
[18]	Andersen, T.G., Fusari, N. and Todorov, V. (2012) Parametric Inference and Dynamic State Recovery from Option Panels. NBER Working Paper Series.
[19]	Carr, P.P. and Madan, D.B. (1999) Option Evaluation Using the Fast Fourier Transform. Journal of Computational Finance, 2, 61-73.
[20]	Borak, S., Detlefsen, K. and Hardle, W. (2005) FFT Based Option Pricing. SFB Discussion Paper 649.
[21]	Lord, R., Fang, F. Bervoets, F. and Oosterlee, C.W. (2007) A Fast and Accurate FFT-Based Methodology for Pricing Early-Exercise Options under Levy Processes. SIAM Journal of Scientific Computing, 20, 1678-1705.
[22]	Kwok, Y.K., Leung, K.S. and Wong, H.Y. (2012) Efficient Options Pricing Using the Fast Fourier Transform. In: Duan, J.C., Ed., Handbook of Computational Finance, Springer, Berlin, 579-604. http://dx.doi.org/10.1007/978-3-642-17254-0_21
[23]	Fang, F. and Oosterlee, C.W. (2008) A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions. SIAM Journal on Scientific Computing, 31, 826-848. http://dx.doi.org/10.1137/080718061
[24]	Zhang, B., Grzelak, L.A. and Oosterlee, C.W. (2012) Efficient Pricing of Commodity Options with Earlyexercise under the Ornstein-Uhlenbeck Process. Applied Numerical Mathematics, 62, 91-111. http://dx.doi.org/10.1016/j.apnum.2011.10.005
[25]	Heston, S.L. (1993) A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6, 327-343.
[26]	Black, F. and Scholes, M. (1973) The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81, 637-654. http://dx.doi.org/10.1086/260062