A New Tempered Stable Distribution and Its Application to Finance
In this paper, we will discuss a parametric approach to risk-neutral density extraction from option prices based on the knowledge of the estimated historical density. A flexible distribution is needed in order to find an equivalent change of measure and, at the same time, take into account the historical estimates. To this end, we introduce a new tempered stable distribution that we refer to as the KR distribution.
Some properties of this distribution will be discussed in this paper, along with the advantages in applying it to financial modeling. Since the KR distribution is infinitely divisible, a Lélvy process can be induced from it. Furthermore, we can develop an exponential Lévy model, called the exponential KR model, and prove that it is an extension of the Carr, Geman, Madan, and Yor (CGMY) model.
The risk-neutral process is fitted by matching model prices to market prices of options using nonlinear least squares. The easy form of the characteristic function of the KR distribution allows one to obtain a suitable solution to the calibration problem. To demonstrate the advantages of the exponential KR model, we present the results of the parameter estimation for the S&P 500 Index and option prices.
KeywordsOption Price Stable Distribution Equivalent Martingale Measure Canonical Process Market Measure
Unable to display preview. Download preview PDF.
- 3.Andrews, L. D. (1998). Special Functions of Mathematics for Engineers, 2nd Edn, Oxford University Press, OxfordGoogle Scholar
- 8.Carr, P. and Madan, D. B. (1999). Option Valuation Using the Fast Fourier Transform, Journal of Computational Finance, 2, 4, 61–73Google Scholar
- 9.Cont, R. and Tankov P. (2004). Financial Modelling with Jump Processes, Chapman & Hall/CRC, LondonGoogle Scholar
- 10.D'Agostino, R. B. and Stephens, M. A. (1986). Goodness of Fit Techniques, Dekker, New YorkGoogle Scholar
- 11.Kawai, R. (2004). Contributions to Infinite Divisibility for Financial modeling, Ph.D. thesis, http://hdl.handle.net/1853/4888
- 12.Kim, Y. S., Rachev, S. T., Chung, D. M., and Bianchi. M. L. The Modified Tempered Stable Distribution, GARCH-Models and Option Pricing, Probability and Mathematical Statistics, to appearGoogle Scholar
- 13.Kim, Y. S. and Lee, J. H. (2007). The Relative Entropy in CGMY Processes and Its Applications to Finance, to appear in Mathematical Methods of Operations Research Google Scholar
- 15.Lewis, A. L. (2001). A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes, avaible from http:// www.optioncity.net.
- 16.Lukacs, E. (1970). Characteristic Functions, 2nd Ed, Griffin, LondonGoogle Scholar
- 18.Marsaglia, G., Tsang, W. W. and Wang, G. (2003). Evaluating Kolmogorov's Distribution, Journal of Statistical Software, 8, 18Google Scholar
- 19.Marsaglia, G. and Marsaglia, J. (2004). Evaluating the Anderson-Darling Distribution, Journal of Statistical Software, 9, 2Google Scholar
- 21.Rachev, S. and Mitnik S. (2000). Stable Paretian Models in Finance, Wiley, New YorkGoogle Scholar
- 22.Rachev, S., Menn C., and Fabozzi F. J. (2005). Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio selection, and Option Pricing, Wiley, New YorkGoogle Scholar
- 23.Rosiński, J. (2006). Tempering Stable Processes, Working Paper, http://www.math.utk.edu/̃rosinski/Manuscripts/tstableF.pdf.
- 24.Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, CambridgeGoogle Scholar
- 25.Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, WileyGoogle Scholar
- 26.Shao, J. (2003). Mathematical Statistics, 2nd Ed, Springer, Berlin Heidelberg New YorkGoogle Scholar