Accelerating Closed-Form Heston Pricers for Calibration
Calibrating models against the markets is a crucial step to obtain meaningful results in the subsequent pricing processes. In general, calibration can be seen as a minimization problem that tries to fit modeled product prices to the observed ones on the market (compare Chap. 2 by Sayer and Wenzel). This means that during the calibration process the modeled prices need to be calculated many times, and therefore the run time of the product pricers have the highest impact on the overall calibration run time. Therefore, in general, only products are used for calibration for which closed-form mathematical pricing formulas are known.
While for the Heston model (semi) closed-form solutions exist for simple products, their evaluation involves complex functions and infinite integrals. So far these integrals can only be solved with time-consuming numerical methods. However, over the time, more and more theoretical and practical subtleties have been revealed for doing this and today a large number of possible approaches are known. Examples are different formulations of closed-formulas and various integration algorithms like quadrature or Fourier methods. Nevertheless, all options only work under specific conditions and depend on the Heston model parameters and the input setting.
In this chapter we present a methodology how to determine the most appropriate calibration method at run time. For a practical setup we study the available popular closed-form solutions and integration algorithms from literature. In total we compare 14 pricing methods, including adaptive quadrature and Fourier methods. For a target accuracy of 10−3 we show that static Gauss-Legendre are best on Central Processing Units (CPUs) for the unrestricted parameter set. Further we show that for restricted Carr-Madan formulation the methods are 3.6× faster. We also show that Fourier methods are even better when pricing at least 10 options with the same maturity but different strikes.
We gratefully acknowledge the partial financial support from the Center of Mathematical and Computational Modelling (CM)2 of the University of Kaiserslautern, from the German Federal Ministry of Education and Research under grant number 01LY1202D, and from the Deutsche Forschungsgemeinschaft (DFG) within the RTG GRK 1932 “Stochastic Models for Innovations in the Engineering Sciences”, project area P2. The authors alone are responsible for the content of this work.
- 1.Aichinger, M., Binder, A., Fürst, J., Kletzmayr, C.: A fast and stable Heston model calibration on the GPU. In: Guarracino, M., Vivien, F., Träff, J., Cannatoro, M., Danelutto, M., Hast, A., Perla, F., Knüpfer, A., Di Martino, B., Alexander, M. (eds.) Euro-Par 2010 Parallel Processing Workshops, Ischia. Volume 6586 of Lecture Notes in Computer Science, pp. 431–438. Springer, Berlin/Heidelberg (2011)Google Scholar
- 2.Albrecher, H., Mayer, P., Schoutens, W., Tistaert, J.: The little Heston trap. Technical report 06, K.U. Leuven Section of Statistics, Sept 2006. Issue 05Google Scholar
- 3.Bailey, D.H., Swarztrauber, P.N.: The fractional Fourier transform and applications. SIAM Rev. 33(3), 389–404 (1991)Google Scholar
- 4.Bernemann, A., Schreyer, R., Spanderen, K.: Accelerating exotic Option pricing and model calibration using GPUs. Available at SSRN 1753596 (2011)Google Scholar
- 5.Brugger, C., Liu, G., de Schryver, C., Wehn, N.: A systematic methodology for analyzing closed-form Heston pricer regarding their accuracy and runtime. In: Proceedings of the 7th Workshop on High Performance Computational Finance, WHPCF ’14, Piscataway, pp. 9–16 (2014). IEEEGoogle Scholar
- 6.Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Finance 2(4), 61–73 (1999)Google Scholar
- 7.Chourdakis, K.: Option pricing using the fractional FFT. J. Comput. Finance 8(2), 1–18 (2004)Google Scholar
- 8.Gatheral, J.: The Volatility Surface: A Practitioner’s Guide, vol. 357. Wiley, Hoboken (2006)Google Scholar
- 9.Gatheral, J., Jacquier, A.: Convergence of Heston to SVI. Quant. Finance 11(8), 1129–1132 (2011)Google Scholar
- 10.Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327 (1993)Google Scholar
- 11.Janek, A., Kluge, T., Weron, R., Wystup, U.: FX smile in the Heston model. In: Čižek, P., Hardle, W., Weron, R. (eds.) Statistical Tools for Finance and Insurance, pp. 133–162. Springer, Berlin/Heidelberg/New York (2011)Google Scholar
- 12.Korn, R., Korn, E., Kroisandt, G.: Monte Carlo Methods and Models in Finance and Insurance. CRC, Boca Raton (2010)Google Scholar
- 13.Lee, R.W.: Option pricing by transform methods: extensions, unification and error control. J. Comput. Finance 7(3), 51–86 (2004)Google Scholar
- 14.Lord, R., Kahl, C.: Optimal Fourier inversion in semi-analytical option pricing. Technical report, Tinbergen Institute Discussion Paper (2006)Google Scholar
- 15.Lord, R., Koekkoek, R., van Dijk, D.: A comparison of biased simulation schemes for stochastic volatility models. Quant. Finance 10(2), 177–194 (2010)Google Scholar
- 16.Ng, M.W.: Option pricing via the FFT and its application to calibration. PhD thesis, Master Thesis, Applied Institute of Mathematics, TU Delft (2005). [Online]. Available: http://ta.twi.tudelft.nl/mf/users/oosterle/oosterlee/ng.pdf. Last access: 31 Dec 2014
- 17.Schmelzle, M.: Option pricing formulae using Fourier transform: theory and application. Technical report (2010)Google Scholar
- 18.Schoutens, W., Simons, E., Tistaert, J.: A perfect calibration! Now what? Wilmott Mag. March. 576, 66–78 (2004)Google Scholar