Reduced Models in Option Pricing
We consider the computational efficiency of the backward vs. forward approaches and compare these with the respective ones resulting from a parametric reduced order model, whose speed-up can be put to good use in the calibration of the underlying dynamics. We apply a global Proper Orthogonal Decomposition in the time domain to obtain the reduced basis and the Modified Craig-Sneyd ADI and Chang-Cooper schemes to numerically solve the partial differential equations. The numerical results are presented for the Black-Scholes and Heston models.
The work of the authors was partially 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, http://www.itn-strike.eu). The authors would like to thank Prof. Karel in ’t Hout for providing the ADI MCS code for the Heston Model.
- 4.Dupire, B.: Pricing with a smile. Risk 7(1), 18–20 (1994)Google Scholar
- 6.Haentjens, T.: ADI schemes for the efficient and stable numerical pricing of financial options via multidimensional partial differential equations. Ph.D. thesis, Universiteit Antwerpen (2013)Google Scholar
- 7.in ’t Hout, K., Toivanen, J.: Application of operator splitting methods in finance. In: Glowinski, R., Osher, S.J., Yin, W. (Eds): Splitting Methods in Communication, Imaging, Science, and Engineering. Series Scientific Computation, Springer International Publishing Switzerland, 541–575 (2017)Google Scholar
- 8.Le Floc’h, F.: Positive second order finite difference methods on Fokker-Planck equations with Dirac initial data–application in finance. SSRN-id 2605160 (2015)Google Scholar
- 11.Seydel, R.U.: Tools for Computational Finance. Universtext 5, Springer, London (2012)Google Scholar
- 12.Silva, J., ter Maten, E.J.W., Günther, M., Ehrhardt, M.: Proper orthogonal decomposition in option pricing: basket options and Heston model. In: G. Russo, V. Capasso, G. Nicosia, V. Romano (eds.) Progress in Industrial Mathematics at ECMI 2014. Mathematics in Industry Series, vol. 22. Springer, Berlin (2016)Google Scholar
- 13.Tavella, D., Randall, C.: Pricing Financial Instruments: The Finite Difference Method. John Wiley & Sons Inc., New York (2000)Google Scholar