The Analysis of Real Data Using a Stochastic Dynamical System Able to Model Spiky Prices

Affiliation(s)

Dipartimento di Matematica e Informatica, Università di Camerino, Camerino, Italy.

CERI—Centro di Ricerca “Previsione, Prevenzione e Controllo dei Rischi Geologici”, Università di Roma “La Sapienza”,.

Dipartimento di Scienze Sociali “D. Serrani”, Università Politecnica delle Marche, Ancona, Italy.

Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Roma, Italy.

Dipartimento di Matematica e Informatica, Università di Camerino, Camerino, Italy.

CERI—Centro di Ricerca “Previsione, Prevenzione e Controllo dei Rischi Geologici”, Università di Roma “La Sapienza”,.

Dipartimento di Scienze Sociali “D. Serrani”, Università Politecnica delle Marche, Ancona, Italy.

Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Roma, Italy.

ABSTRACT

In this paper we use filtering and maximum likelihood methods to solve a calibration problem for a stochastic dynamical system used to model spiky asset prices. The data used in the calibration problem are the observations at discrete times of the asset price. The model considered has been introduced by V.A. Kholodnyi in [1,2] and describes spiky as-set prices as the product of two independent stochastic processes: the spike process and the process that represents the asset prices in absence of spikes. A Markov chain is used to regulate the transitions between presence and absence of spikes. As suggested in [3] in a different context the calibration problem for this model is translated in a maximum likelihood problem with the likelihood function defined through the solution of a filtering problem. The estimated values of the model parameters are the coordinates of a constrained maximizer of the likelihood function. Furthermore, given the calibrated model, we develop a sort of tracking procedure able to forecast forward asset prices. Numerical examples using synthetic and real data of the solution of the calibration problem and of the performance of the tracking procedure are presented. The real data studied are electric power price data taken from the UK electricity market in the years 2004-2009. After calibrating the model using the spot prices, the forward prices forecasted with the tracking procedure and the observed forward prices are compared. This comparison can be seen as a way to validate the model, the formulation and the solution of the calibration problem and the forecasting procedure. The result of the comparison is satisfactory. In the website: http://www.econ.univpm.it/recchioni/finance/w10 some auxiliary material including animations that helps the understanding of this paper is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/ recchioni/finance.

In this paper we use filtering and maximum likelihood methods to solve a calibration problem for a stochastic dynamical system used to model spiky asset prices. The data used in the calibration problem are the observations at discrete times of the asset price. The model considered has been introduced by V.A. Kholodnyi in [1,2] and describes spiky as-set prices as the product of two independent stochastic processes: the spike process and the process that represents the asset prices in absence of spikes. A Markov chain is used to regulate the transitions between presence and absence of spikes. As suggested in [3] in a different context the calibration problem for this model is translated in a maximum likelihood problem with the likelihood function defined through the solution of a filtering problem. The estimated values of the model parameters are the coordinates of a constrained maximizer of the likelihood function. Furthermore, given the calibrated model, we develop a sort of tracking procedure able to forecast forward asset prices. Numerical examples using synthetic and real data of the solution of the calibration problem and of the performance of the tracking procedure are presented. The real data studied are electric power price data taken from the UK electricity market in the years 2004-2009. After calibrating the model using the spot prices, the forward prices forecasted with the tracking procedure and the observed forward prices are compared. This comparison can be seen as a way to validate the model, the formulation and the solution of the calibration problem and the forecasting procedure. The result of the comparison is satisfactory. In the website: http://www.econ.univpm.it/recchioni/finance/w10 some auxiliary material including animations that helps the understanding of this paper is shown. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/ recchioni/finance.

Cite this paper

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "The Analysis of Real Data Using a Stochastic Dynamical System Able to Model Spiky Prices,"*Journal of Mathematical Finance*, Vol. 2 No. 1, 2012, pp. 1-12. doi: 10.4236/jmf.2012.21001.

L. Fatone, F. Mariani, M. Recchioni and F. Zirilli, "The Analysis of Real Data Using a Stochastic Dynamical System Able to Model Spiky Prices,"

References

[1] V. A. Kholodnyi, “Valuation and Hedging of European Contingent Claims on Power with Spikes: A Non-Markovian Approach,” Journal of Engineering Mathematics, Vol. 49, No. 3, 2004, pp. 233-252. doi:10.1023/B:ENGI.0000031203.43548.b6

[2] V. A. Kholodnyi, “The Non-Markovian Approach to the Valua-tion and Hedging of European Contingent Claims on Power with Scaling Spikes,” Nonlinear Analysis: Hybrid Systems, Vol. 2, No. 2, 2008, pp. 285-304. doi:10.1016/j.nahs.2006.05.002

[3] F. Mariani, G. Pacelli and F. Zirilli, “Maximum Likelihood Estimation of the Heston Stochastic Volatility Mo- del Using Asset and Option Prices: An Application of Nonlinear Filtering Theory,” Optimization Letters, Vol. 2, No. 2, 2008, pp. 177-222. doi:10.1007/s11590-007-0052-7

[4] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Maximum Like-lihood Estimation of the Parameters of a System of Stochastic Differential Equations that Models the Returns of the Index of Some Classes of Hedge Funds,” Journal of Inverse and Ill Posed Problems, Vol. 15, No. 5, 2007, pp. 493-526. doi:10.1515/jiip.2007.028

[5] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Calibration of the Heston Stochastic Volatility Model Using Filtering and Maximum Likelihood Methods,” G. S. Ladde, N. G. Medhin, C. Peng and M. Sambandham Eds., Proceedings of Dynamic Systems and Applications, Dynamic Publishers, Atlanta, Vol. 5, 2008, pp. 170-181.

[6] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Calibration of a Multiscale Stochastic Volatility Model Using European Option Prices,” Mathematical Methods in Ecomics and Finance, Vol. 3, No. 1, 2008, pp. 49-61.

[7] L. Fatone, F. Mariani, M. C. Recchioni and F .Zirilli, “An Explicitly Solvable Multi-Scale Stochastic Volatility Model: Option Pricing and Calibration,” Journal of Futures Markets, Vol. 29, No. 9, 2009, pp. 862-893. doi:10.1002/fut.20390

[8] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Analysis of Real Data Using a Multiscale Stochastic Volatility Model,” European Financial Management, 2012, in press.

[9] P. Capelli, F. Mariani, M. C. Recchioni, F. Spinelli and F. Zirilli, “Determining a Stable Relationship between Hedge Fund Index HFRI-Equity and S&P 500 Behaviour, Using Filtering and Maximum Likelihood,” Inverse Problems in Science and Engineering, Vol. 18, No. 1, 2010, pp. 93-109.

[10] C. R. Knittel and M. R. Roberts, “An Empirical Examination of Restructured Electricity Prices,” Energy Economics, Vol. 27, No. 5, 2005, pp. 791-817. doi:10.1016/j.eneco.2004.11.005

[11] J. H. Hull, “Options, Futures, and Other Derivatives,” 7th Edition, Pearson Prentice Hall, London, 2008

[1] V. A. Kholodnyi, “Valuation and Hedging of European Contingent Claims on Power with Spikes: A Non-Markovian Approach,” Journal of Engineering Mathematics, Vol. 49, No. 3, 2004, pp. 233-252. doi:10.1023/B:ENGI.0000031203.43548.b6

[2] V. A. Kholodnyi, “The Non-Markovian Approach to the Valua-tion and Hedging of European Contingent Claims on Power with Scaling Spikes,” Nonlinear Analysis: Hybrid Systems, Vol. 2, No. 2, 2008, pp. 285-304. doi:10.1016/j.nahs.2006.05.002

[3] F. Mariani, G. Pacelli and F. Zirilli, “Maximum Likelihood Estimation of the Heston Stochastic Volatility Mo- del Using Asset and Option Prices: An Application of Nonlinear Filtering Theory,” Optimization Letters, Vol. 2, No. 2, 2008, pp. 177-222. doi:10.1007/s11590-007-0052-7

[4] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Maximum Like-lihood Estimation of the Parameters of a System of Stochastic Differential Equations that Models the Returns of the Index of Some Classes of Hedge Funds,” Journal of Inverse and Ill Posed Problems, Vol. 15, No. 5, 2007, pp. 493-526. doi:10.1515/jiip.2007.028

[5] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Calibration of the Heston Stochastic Volatility Model Using Filtering and Maximum Likelihood Methods,” G. S. Ladde, N. G. Medhin, C. Peng and M. Sambandham Eds., Proceedings of Dynamic Systems and Applications, Dynamic Publishers, Atlanta, Vol. 5, 2008, pp. 170-181.

[6] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “Calibration of a Multiscale Stochastic Volatility Model Using European Option Prices,” Mathematical Methods in Ecomics and Finance, Vol. 3, No. 1, 2008, pp. 49-61.

[7] L. Fatone, F. Mariani, M. C. Recchioni and F .Zirilli, “An Explicitly Solvable Multi-Scale Stochastic Volatility Model: Option Pricing and Calibration,” Journal of Futures Markets, Vol. 29, No. 9, 2009, pp. 862-893. doi:10.1002/fut.20390

[8] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli, “The Analysis of Real Data Using a Multiscale Stochastic Volatility Model,” European Financial Management, 2012, in press.

[9] P. Capelli, F. Mariani, M. C. Recchioni, F. Spinelli and F. Zirilli, “Determining a Stable Relationship between Hedge Fund Index HFRI-Equity and S&P 500 Behaviour, Using Filtering and Maximum Likelihood,” Inverse Problems in Science and Engineering, Vol. 18, No. 1, 2010, pp. 93-109.

[10] C. R. Knittel and M. R. Roberts, “An Empirical Examination of Restructured Electricity Prices,” Energy Economics, Vol. 27, No. 5, 2005, pp. 791-817. doi:10.1016/j.eneco.2004.11.005

[11] J. H. Hull, “Options, Futures, and Other Derivatives,” 7th Edition, Pearson Prentice Hall, London, 2008