On the Efficacy of Fourier Series Approximations for Pricing European Options

Affiliation(s)

^{1}
School of Economics and Finance, Queensland University of Technology, Brisbane, Australia.

^{2}
Numerix Pty Ltd, Sydney, Australia.

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.

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.

Hurn, A. , Lindsay, K. and McClelland, A. (2014) On the Efficacy of Fourier Series Approximations for Pricing European Options.

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

[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