Fast Fourier Transform Option Pricing: Efficient Approximation Methods Under Multi-Factor Stochastic Volatility and Jumps
Fourier option-pricing methods are popular due to the dual benefits of wide applicability and computational efficiency. The literature tends to focus on a limited subset of models with analytic conditional characteristic functions (CCFs). Models that require numerical solutions of the CCF undermine the efficiency of Fourier methods. To tackle this problem, an ad hoc approximate numerical method was developed that provide CCF values accurately much faster than traditional methods. This approximation, based on averaging, Taylor expansions and asymptotic behaviour of the CCFs, is presented and tested for a range of affine models, with multi-factor stochastic volatility and jumps. The approximation leads to average run-time accelerations up to 50 times those of other numerical implementations, with very low absolute and relative errors reported.
KeywordsFast Fourier Transform Option Price Stochastic Volatility Strike Price Saddlepoint Approximation
Jean Charpin acknowledges the support of the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie) funded by the Science Foundation Ireland Mathematics Initiative grant 06/MI/005.
- 4.Benhamou, E.: Fast Fourier transform for discrete Asian options. J. Comput. Financ. 6, 49–61 (2002)Google Scholar
- 5.Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Financ. 3, 463–520 (1999)Google Scholar
- 7.Carr, P., Géman, H., Madan D., et al.: Option pricing using integral transforms. New York University. Available online. http://www.math.nyu.edu/research/carrp/papers/pdf/integtransform.pdf (2011). Last time accessed May 2012
- 10.Dempster, M.A.H., Hong, S.S.G.: Spread option valuation and the fast Fourier transform. In: German, H., Madan, D., Pliska, S.R., Vorst, T. (ed.) Mathematical Finance – Bachelier Congress 2000: Selected Papers form the World Congress of the Bachelier Finance Society. Springer-Verlag, Berlin (2001)Google Scholar
- 12.Eraker, B.: Do stock prices and volatility jump? Reconciling evidence from spot and option prices. J. Financ. 59, 1367–1404 (2004)Google Scholar
- 13.Filipovic, D., Mayerhoffer, E., Schneider, P.: Density approximations for multivariate affine jump-diffusion processes [Working Paper]. Available via arXiv. http://arxiv.org/abs/1104.5326 (2011). Last time accessed May 2012
- 14.Glasserman, P., Kim, K-K.: Saddlepoint approximations for affine jump-diffusion models. J. Econ. Dyn. Control. 33, 15–36 (2009)Google Scholar
- 16.Hurd, T.R., Zhou, Z.: A Fourier transform method for spread option pricing [Working Paper]. McMaster University. Available via arXiv. http://arxiv.org/PS\_cache/arxiv/pdf/0902/0902.3643v1.pdf (2009). Last time accessed May 2012
- 18.Jaimungal, S., Surkov, V.: Fourier space time-stepping for option pricing with Levy models [Working Paper]. University of Toronto. Available via Social Science Research Network. http://papers.ssrn.com/sol3/papers.cfm?abstract\_id=1302887 (2009). Last time accessed May 2012
- 19.Lee, R.W.: Option pricing by transform methods: Extensions, unification, and error control. J. Comput. Financ. 7, 51–86 (2004)Google Scholar
- 21.Lord, R., Kahl, C.: Optimal Fourier inversion in semi-analytical option pricing. J. Comput. Financ. 10, 1–30 (2007)Google Scholar
- 24.Takahashi, A., Takehara, K.: Fourier transform method with an asymptotic expansion approach: An application to currency options [Working Paper]. University of Tokyo. Available online. http://www.cirje.e.u-tokyo.ac.jp/research/dp/2008/2008cf538.pdf (2009). Last time accessed May 2012