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Journal of Probability and Statistics
Volume 2012, Article ID 931609, 20 pages
http://dx.doi.org/10.1155/2012/931609
Research Article

The Use of Statistical Tests to Calibrate the Black-Scholes Asset Dynamics Model Applied to Pricing Options with Uncertain Volatility

1Dipartimento di Matematica e Informatica, Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy
2CERI-Centro di Ricerca “Previsione, Prevenzione e Controllo dei Rischi Geologici”, Università di Roma “La Sapienza”, Palazzo Doria Pamphilj, Piazza Umberto Pilozzi 9, Valmontone 00038 Roma, Italy
3Dipartimento di Scienze Sociali “D. Serrani”, Università Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy
4Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Roma, Italy

Received 28 October 2011; Revised 28 February 2012; Accepted 13 March 2012

Academic Editor: A. Thavaneswaran

Copyright © 2012 Lorella Fatone et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A new method for calibrating the Black-Scholes asset price dynamics model is proposed. The data used to test the calibration problem included observations of asset prices over a finite set of (known) equispaced discrete time values. Statistical tests were used to estimate the statistical significance of the two parameters of the Black-Scholes model: the volatility and the drift. The effects of these estimates on the option pricing problem were investigated. In particular, the pricing of an option with uncertain volatility in the Black-Scholes framework was revisited, and a statistical significance was associated with the price intervals determined using the Black-Scholes-Barenblatt equations. Numerical experiments involving synthetic and real data were presented. The real data considered were the daily closing values of the S&P500 index and the associated European call and put option prices in the year 2005. The method proposed here for calibrating the Black-Scholes dynamics model could be extended to other science and engineering models that may be expressed in terms of stochastic dynamical systems.