Abstract
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.
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References
Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. SIAM–Society for Industrial and Applied Mathematics, Philadelphia (2005)
Balajewiecz, M., Toivanen, J.: Reduced order models for pricing American options under stochastic volatility and jump-diffusion models. Procedia Comput. Sci. 80, 734–743 (2016)
Chang, J., Cooper, G.: A practical difference scheme for Fokker-Planck equations. J. Comput. Phys. 6(1), 1–16 (1970)
Dupire, B.: Pricing with a smile. Risk 7(1), 18–20 (1994)
Günther, M., Jüngel, A.: Finanzderivate mit MATLAB, 2nd edn. Vieweg+ Teubner Verlag and Springer Fachmedien Wiesbaden GmbH, Wiesbaden (2010)
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)
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)
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)
Mohammadi, M., Borzì, A.: Analysis of the Chang–Cooper discretization scheme for a class of Fokker–Planck equations. J. Numer. Math. 23(3), 271–288 (2015)
Paul-Dubois-Taine, A., Amsallem, D.: An adaptive and efficient greedy procedure for the optimal training of parametric reduced-order models. Int. J. Numer. Methods Eng. 102(5), 1262–1292 (2014)
Seydel, R.U.: Tools for Computational Finance. Universtext 5, Springer, London (2012)
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)
Tavella, D., Randall, C.: Pricing Financial Instruments: The Finite Difference Method. John Wiley & Sons Inc., New York (2000)
Acknowledgements
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.
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Silva, J.P., Maten, E.J.W.t., Günther, M., Ehrhardt, M. (2017). Reduced Models in Option Pricing. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_23
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DOI: https://doi.org/10.1007/978-3-319-63082-3_23
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