A priori error estimates for reduced order models in finance
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, p. 449-469

Mathematical models for option pricing often result in partial differential equations. Recent enhancements are models driven by Lévy processes, which lead to a partial differential equation with an additional integral term. In the context of model calibration, these partial integro differential equations need to be solved quite frequently. To reduce the computational cost the implementation of a reduced order model has shown to be very successful numerically. In this paper we give a priori error estimates for the use of the proper orthogonal decomposition technique in the context of option pricing models.

DOI : https://doi.org/10.1051/m2an/2012039
Classification:  35K15,  65M15,  91G80
Keywords: option pricing models, proper orthogonal decomposition, a priori error estimate
@article{M2AN_2013__47_2_449_0,
author = {Sachs, Ekkehard W. and Schu, Matthias},
title = {A priori error estimates for reduced order models in finance},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
pages = {449-469},
doi = {10.1051/m2an/2012039},
zbl = {1268.91182},
language = {en},
url = {http://www.numdam.org/item/M2AN_2013__47_2_449_0}
}

Sachs, Ekkehard W.; Schu, Matthias. A priori error estimates for reduced order models in finance. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, pp. 449-469. doi : 10.1051/m2an/2012039. http://www.numdam.org/item/M2AN_2013__47_2_449_0/

[1] Y. Achdou and O. Pironneau, Computational Methods for Option Pricing. SIAM (2005). | MR 2159611 | Zbl 1078.91008

[2] A. Almendral and C. Oosterlee, Numerical valuation of options with jumps in the underlying. Appl. Numer. Math. 53 (2005) 1-18. | MR 2122957 | Zbl 1117.91028

[3] L. Andersen and J. Andreasen, Jump-diffusion processes : Volatility smile fitting and numerical methods for option pricing. Rev. Deriv. Res. 4 (2000) 231-262. | Zbl 1274.91398

[4] H. Antil, M. Heinkenschloss and R. Hoppe, Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system. Optim. Methods Soft. 26 (2011) 643-669. | MR 2837792 | Zbl 1227.49046

[5] H. Antil, M. Heinkenschloss, R. Hoppe and D. Sorensen, Domain decomposition and model reduction for the numerical solution of pde constrained optimization problems with localized optimization variables. Comput. Vis. Sci. 13 (2010) 249-264. | MR 2748450 | Zbl 1220.65074

[6] N.J. Armstrong, K.J. Painter and J.A. Sherratt, A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 243 (2006) 98-113. | MR 2279324

[7] F. Black and M. Scholes, The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973) 637-654. | Zbl 1092.91524

[8] R. Cont, N. Lantos and O. Pironneau, A reduced basis for option pricing. SIAM J. Financ. Math. 2 (2011) 287-316. | MR 2781177 | Zbl 1227.91033

[9] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall (2004). | MR 2042661 | Zbl 1052.91043

[10] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, in Evolution Problems I 5, Springer (1992). | MR 1156075 | Zbl 0755.35001

[11] B. Dupire, Pricing with a smile. Risk 7 (1994) 18-20.

[12] A. Gerisch, On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion. J. Numer. Anal. 30 (2010) 173-194. | MR 2580553 | Zbl 1185.65166

[13] A. Gerisch and M. Chaplain, Mathematical modelling of cancer cell invasion of tissue : Local and non-local models and the effect of adhesion. J. Theoret. Biol. 250 (2008) 684-704. | MR 2945039 | Zbl 1230.92022

[14] M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM : M2AN 39 (2005) 157-181. | Numdam | MR 2136204 | Zbl 1079.65096

[15] P. Hepperger, Option pricing in Hilbert space-valued jump-diffusion models using partial integro-differential equations. SIAM J. Financ. Math. 1 (2008) 454-489. | MR 2669401 | Zbl 1198.91230

[16] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319-345. | MR 2396870 | Zbl 1191.49040

[17] P. Holmes, J. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press (1996). | MR 1422658 | Zbl 0923.76002

[18] J.C. Hull, Options, Futures and Other Derivatives, Prentice-Hall, Upper Saddle River, N.J., 6th edition (2006). | Zbl 1087.91025

[19] S.G. Kou, A jump-diffusion model for option pricing. Manage. Sci. 48 (2002) 1086-1101. | Zbl 1216.91039

[20] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148. | MR 1868765 | Zbl 1005.65112

[21] A.-M. Matache, T. Von Petersdorff and C. Schwab, Fast deterministic pricing of options on Lévy driven assets. ESAIM : M2AN 38 (2004) 37-72. | Numdam | MR 2073930 | Zbl 1072.60052

[22] R.C. Merton, Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 (1976) 125-144. | Zbl 1131.91344

[23] O. Pironneau, Calibration of options on a reduced basis. J. Comput. Appl. Math. 232 (2009) 139-147. | MR 2554229 | Zbl 1173.91398

[24] E.W. Sachs and M. Schu, Reduced order models (POD) for calibration problems in finance, edited by K. Kunisch, G. Of and O. Steinbach. ENUMATH 2007, Numer. Math. Adv. Appl. (2008) 735-742. | Zbl 1154.91476

[25] E.W. Sachs and M. Schu, Reduced order models in PIDE constrained optimization. Control and Cybernetics 39 (2010) 661-675. | MR 2791366 | Zbl 1282.49022

[26] E.W. Sachs and A. Strauss, Efficient solution of a partial integro-differential equation in finance. Appl. Numer. Math. 58 (2008) 1687-1703. | MR 2458476 | Zbl 1155.65109

[27] E.W. Sachs and S. Volkwein, POD-Galerkin approximations in PDE-constrained optimization. GAMM Reports 33 (2010) 194-208. | MR 2843959 | Zbl 1205.35011

[28] W. Schoutens, Lévy-Processes in Finance, Wiley (2003).

[29] S. Volkwein, Optimal control of a phase-field model using proper orthogonal decomposition. Z. Angew. Math. Mech. 81 (2001) 83-97. | MR 1818724 | Zbl 1007.49019

[30] S. Volkwein, Model reduction using proper orthogonal decomposition. Lecture Notes, University of Constance (2011).