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 AM  Vol.5 No.17 , October 2014
On the Efficacy of Fourier Series Approximations for Pricing European Options
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.
Cite this paper: Hurn, A. , Lindsay, K. and McClelland, A. (2014) On the Efficacy of Fourier Series Approximations for Pricing European Options. Applied Mathematics, 5, 2786-2807. doi: 10.4236/am.2014.517267.
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

 
 
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