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.
Mots-clés : 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 = {\protect\emph{A priori }error estimates for reduced order models in finance},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {449--469},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
doi = {10.1051/m2an/2012039},
zbl = {1268.91182},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012039/}
}
TY - JOUR AU - Sachs, Ekkehard W. AU - Schu, Matthias TI - A priori error estimates for reduced order models in finance JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 449 EP - 469 VL - 47 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012039/ DO - 10.1051/m2an/2012039 LA - en ID - M2AN_2013__47_2_449_0 ER -
%0 Journal Article %A Sachs, Ekkehard W. %A Schu, Matthias %T A priori error estimates for reduced order models in finance %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 449-469 %V 47 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012039/ %R 10.1051/m2an/2012039 %G en %F 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 , Tome 47 (2013) no. 2, pp. 449-469. doi : 10.1051/m2an/2012039. http://archive.numdam.org/articles/10.1051/m2an/2012039/
[1] and , Computational Methods for Option Pricing. SIAM (2005). | MR | Zbl
[2] and , Numerical valuation of options with jumps in the underlying. Appl. Numer. Math. 53 (2005) 1-18. | MR | Zbl
[3] and , Jump-diffusion processes : Volatility smile fitting and numerical methods for option pricing. Rev. Deriv. Res. 4 (2000) 231-262. | Zbl
[4] , and , Domain decomposition and balanced truncation model reduction for shape optimization of the Stokes system. Optim. Methods Soft. 26 (2011) 643-669. | MR | Zbl
[5] , , and , 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 | Zbl
[6] , and , A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 243 (2006) 98-113. | MR
[7] and , The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973) 637-654. | Zbl
[8] , and , A reduced basis for option pricing. SIAM J. Financ. Math. 2 (2011) 287-316. | MR | Zbl
[9] and , Financial Modelling with Jump Processes, Chapman and Hall (2004). | MR | Zbl
[10] and , Mathematical Analysis and Numerical Methods for Science and Technology, in Evolution Problems I 5, Springer (1992). | MR | Zbl
[11] , Pricing with a smile. Risk 7 (1994) 18-20.
[12] , On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion. J. Numer. Anal. 30 (2010) 173-194. | MR | Zbl
[13] and , 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 | Zbl
[14] and , A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM : M2AN 39 (2005) 157-181. | Numdam | MR | Zbl
[15] , Option pricing in Hilbert space-valued jump-diffusion models using partial integro-differential equations. SIAM J. Financ. Math. 1 (2008) 454-489. | MR | Zbl
[16] and , Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319-345. | MR | Zbl
[17] , and , Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press (1996). | MR | Zbl
[18] , Options, Futures and Other Derivatives, Prentice-Hall, Upper Saddle River, N.J., 6th edition (2006). | Zbl
[19] , A jump-diffusion model for option pricing. Manage. Sci. 48 (2002) 1086-1101. | Zbl
[20] and , Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148. | MR | Zbl
[21] , and , Fast deterministic pricing of options on Lévy driven assets. ESAIM : M2AN 38 (2004) 37-72. | Numdam | MR | Zbl
[22] , Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 (1976) 125-144. | Zbl
[23] , Calibration of options on a reduced basis. J. Comput. Appl. Math. 232 (2009) 139-147. | MR | Zbl
[24] and , 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
[25] and , Reduced order models in PIDE constrained optimization. Control and Cybernetics 39 (2010) 661-675. | MR | Zbl
[26] and , Efficient solution of a partial integro-differential equation in finance. Appl. Numer. Math. 58 (2008) 1687-1703. | MR | Zbl
[27] and , POD-Galerkin approximations in PDE-constrained optimization. GAMM Reports 33 (2010) 194-208. | MR | Zbl
[28] , Lévy-Processes in Finance, Wiley (2003).
[29] , Optimal control of a phase-field model using proper orthogonal decomposition. Z. Angew. Math. Mech. 81 (2001) 83-97. | MR | Zbl
[30] , Model reduction using proper orthogonal decomposition. Lecture Notes, University of Constance (2011).
Cité par Sources :
