Calibration of Lévy processes using optimal control of Kolmogorov equations with periodic boundary conditions
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
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[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.
[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.