Abstract
In Leitao et al. (Appl Math Comput 293:461–479, 2017), we have presented a one time-step Monte Carlo simulation of the SABR model (Hagan et al. Wilmott Mag 84–108, 2002). The technique is based on an efficient simulation of SABR’s time-integrated variance process. We base our approach on the derivation of the cumulative distribution function of the integrated variance by means of Fourier techniques and the use of a copula to approximate the conditional distribution (integrated variance conditional on the SABR volatility process). Resulting is a fast simulation algorithm which can be employed to price European options under the SABR dynamics. This converts our approach into an alternative to Hagan analytic formula for short maturities, where some known issues of the implied volatility expression for small strike values are overcome. A generalization of this technique to the multiple time-step case has been presented in Leitao et al. (On an efficient multiple time-step Monte Carlo simulation of the SABR model 2016, submitted for publication. Available at SSRN: http://ssrn.com/abstract=2764908).
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Notes
- 1.
These time points are not to be confused with the Monte Carlo time steps. We will have only one Monte Carlo time-step. M is the number of points for the discrete approximation of Y (T).
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Acknowledgements
The first author is supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE—Novel Methods in Computational Finance).
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Leitao, Á., Grzelak, L.A., Oosterlee, C.W. (2017). A Highly Efficient Numerical Method for the SABR Model. In: Ehrhardt, M., Günther, M., ter Maten, E. (eds) Novel Methods in Computational Finance. Mathematics in Industry(), vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-61282-9_14
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DOI: https://doi.org/10.1007/978-3-319-61282-9_14
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