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Calibration of Lévy processes using optimal control of Kolmogorov equations with periodic boundary conditions

    Mario Annunziato Affiliation
    ; Hanno Gottschalk Affiliation

Abstract

We present an optimal control approach to the problem of model calibration for Lévy processes based on an non-parametric estimation procedure of the measure. The optimization problem is related to the maximum likelihood theory of sieves [25] and is formulated with the Fokker-Planck-Kolmogorov approach [3, 4].


We use a generic spline discretization of the Lévy jump measure and select an adequate size of the spline basis using the Akaike Information Criterion (AIC) [12]. The first order necessary optimality conditions are derived based on the Lagrange multiplier technique in a functional space. The resulting Partial Integral-Differential Equations (PIDE) are discretized, numerically solved using a scheme composed of Chang-Cooper, BDF2 and direct quadrature methods, jointly to a non-linear conjugate gradient method. For the numerical solver of the Kolmogorov's forward equation we prove conditions for non-negativity and stability in the L1 norm of the discrete solution.

Keyword : optimal control of PIDE, Kolmogorov-Fokker-Planck equation, Lévy processes, non-parametric maximum likelihood method, IMEX numerical method

How to Cite
Annunziato, M., & Gottschalk, H. (2018). Calibration of Lévy processes using optimal control of Kolmogorov equations with periodic boundary conditions. Mathematical Modelling and Analysis, 23(3), 390-413. https://doi.org/10.3846/mma.2018.024
Published in Issue
Jun 14, 2018
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References

[1] Y. Aït-Sahalia and A. W. Lo. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53(2):499-547, 1998. https://doi.org/10.1111/0022-1082.215228.

[2] D.N. Allen and R.V. Southwell. Relaxation methods applied to determine the motion, in 2-D, of a viscous fluid past a fixed cylinder. Quart-J. Mech. Appl., VIII(2):129-145, 1955. https://doi.org/10.1093/qjmam/8.2.129.

[3] M. Annunziato and A. Borzì. Optimal control of probability density functions of stochastic processes. Mathematical Modelling and Analysis, 15(4):393-407, 2010. https://doi.org/10.3846/1392-6292.2010.15.393-407.

[4] M. Annunziato and A. Borzì. A Fokker-Planck control framework for multidimensional stochastic processes. Journal of Computational and Applied Mathematics, 237(1):487-507, 2013. ISSN 0377-0427. https://doi.org/10.1016/j.cam.2012.06.019.

[5] M. Annunziato, A. Borzì, F. Nobile and R. Tempone. On the connection between the Hamilton-Jacobi-Bellman and the Fokker-Planck control frameworks. Appl. Mathematics, 5:2476-2484, 2014. https://doi.org/10.4236/am.2014.516239.

[6] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2 edition, 2009. https://doi.org/10.1017/CBO9780511809781.

[7] D. Belomestny and M. Reiß. Spectral calibration of exponential Lévy models. Finance Stoch., 10:449-474, 2006. https://doi.org/10.1007/s00780-006-0021-5.

[8] D. Belomestny and J. Schoenmakers. A jump-diffusion Libor model and its robust calibration. Quantitative Finance, 11(4):529-546, 2011. https://doi.org/10.1080/14697680903295176.

[9] C. Berg and G. Forst. Potential theory on locally compact Abelian groups, volume 87. Springer, Berlin - Heidelberg - New York, 1975. https://doi.org/10.1007/978-3-642-66128-0.

[10] A. Borzì and V. Schulz. Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, Philadephia, 2012. https://doi.org/10.1137/1.9781611972054.

[11] M. Briani, R. Natalini and G. Russo. Implicit{explicit numerical schemes for jump-diffusion processes. Calcolo, 44(1):33-57, 2007. https://doi.org/10.1007/s10092-007-0128-x.

[12] K. P. Burnham and D. R. Anderson. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer, New York, 2002. https://doi.org/10.1007/b97636.

[13] J.S. Chang and G. Cooper. A practical difference scheme for Fokker-Planck equations. J. Comput. Phys., 6:1-16, 1970. https://doi.org/10.1016/0021-9991(70)90001-X.

[14] F. Comte and V. Genon-Catalot. Nonparametric adaptive estimation for pure jump Lévy processes. Annales de l'Institut Henri Poincaré, 46(3), 2010. https://doi.org/10.1214/09-AIHP323.

[15] R. Cont and P. Tankov. Financial Modeling with Jump Processes. Chapman & Hall, Boca Raton - London - New York - Washington D.C., 2004.

[16] R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM Journal on Numerical Analysis, 43(4):1596-1626, 2005. https://doi.org/10.1137/S0036142903436186.

[17] S. Csörgö and V. Totil. On how long intervals is the empirical characteristic function uniformly consistent? Acta Sci. Math., 45:141-149, 1983.

[18] Y.-H. Dai. Nonlinear conjugate gradient methods. In Wiley Encyclopedia of Operations Research and Management Science. John Wiley & Sons, Inc., 2010. ISBN 9780470400531. https://doi.org/10.1002/9780470400531.eorms0183.

[19] Y.-H. Dai and Y. Yuan. A nonlinear conjugate gradient with a strong global convergence property. SIAM J. Optim., 10:177-182, 1999. https://doi.org/10.1137/S1052623497318992.

[20] D. J. Duffy. Numerical analysis of jump diffusion models: A partial differential equation approach. Wilmott Magazine, pp. 68-73, 2009.

[21] T. S. Fergusson. A Course in Large Sample Theory. Chapman & Hall, Boca Raton - London - New York - Washington D.C., 1996.

[22] T. L. Friesz. Dynamic Optimization and Differential Games. Springer, New York - Dortrecht - Heidelberg - London, 2010. https://doi.org/10.1007/978-0-387-72778-3.

[23] B. Gaviraghi, M. Annunziato and A. Borzì. Analysis of splitting methods for solving a partial integro-differential Fokker-Planck equation. Applied Mathematics and Computation, 294:1-17, 2017. https://doi.org/10.1016/j.amc.2016.08.050.

[24] B. Gaviraghi, A. Schindele, M. Annunziato and A. Borzì. On optimal sparsecontrol problems governed by jump-diffusion processes. Applied Mathematics, 7:1978-2004, 2016. https://doi.org/10.4236/am.2016.716162.

[25] S. Geman and C.-H. Hwang. Non-parametric maximum likelihood estimation by the method of sieves. Ann. of Statistics, 10(2):401-414, 1982. https://doi.org/10.1214/aos/1176345782.

[26] J.C. Gilbert and J. Nocedal. Global convergence properties of conjugate gradient methods for optimization. SIAM J. Optim., 2:21-42, 1992. https://doi.org/10.1137/0802003.

[27] H. Gottschalk, B. Smii and H. Thaler. The Feynman graph representation for the convolution semigroup and its applications to Lévy statistics. Bernoulli, 14(2):322-351, 2008. https://doi.org/10.3150/07-BEJ106.

[28] Z. Grbac, A. Papapantoleon, J. Schoenmakers and D. Skovmand. Affine LIBOR models with multiple curves: Theory examples and calibration. SIAM J. Finan. Math., 6:984-1025, 2015. https://doi.org/10.1137/15M1011731.

[29] S. Iacus. Option Pricing and Estimation of Financial Models with R. Wiley, Chichester, 2011. https://doi.org/10.1002/9781119990079.

[30] J. Kappus and M. Reiß. Estimation of the characteristics of a Lévy process observed at arbitrary frequency. Statistica Neerlandica, 64(3):314-328, 2010. https://doi.org/10.1111/j.1467-9574.2010.00461.x.

[31] K. Knight. Mathematical Statistics. Chapman & Hall, 1999. https://doi.org/10.1201/9781584888567.

[32] D. Marazzina, O. Reichmann and C. Schwab. hp-DGFEM for Kolmogorov-Fokker-Planck equations of multivariate Lévy processes. Math. Mod. and Meth. in Appl. Sci., 22:1150005-1-37, 2012.

[33] M. Mohammadi and A. Borzì. Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations. Journal of Numerical Mathematics, 23:271-288, 2015. https://doi.org/10.1515/jnma-2015-0018.

[34] J. Nocedal and S.J. Wright. Numerical Optimization. Springer, New York, 1999. https://doi.org/10.1007/b98874.

[35] R.J. Plemmons. M-matrix characterizations. I-nonsingular Mmatrices. Linear Algebra and its Applications, 18(2):175-188, 1977. https://doi.org/10.1016/0024-3795(77)90073-8.

[36] O. Reichmann and C. Schwab. Numerical analysis of additive, Lévy and Feller processes with applications to option pricing Lévy matters. In Lévy Matters I: Recent Progress in Theory and Applications: Foundations, Trees and Numerical Issues in Finance, Lecture Notes in Mathematics 2001, pp. 137-196. Springer-Verlag, Berlin - Heidelberg, 2010.

[37] L. A. Sakhonovich. Lévy Processes, Integral Equations, Statistical Physics: Connections and Interactions. Birkhäuser, Basel, 2012. https://doi.org/10.1007/978-3-0348-0356-4.

[38] D. L. Scharfetter and H. K. Gummel. Large signal analysis of a silicon Read diode oscillator. IEEE Trans. Electron. Dev., 16:64-77, 1969. https://doi.org/10.1109/T-ED.1969.16566.

[39] D.F. Shanno. Conjugate gradient methods with inexact searches. Math. Oper. Res., 3:244-256, 1978. https://doi.org/moor.3.3.244.