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.
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Kim, Y.S., Rachev, S.T., Bianchi, M.L., Fabozzi, F.J. (2009). A New Tempered Stable Distribution and Its Application to Finance. In: Bol, G., Rachev, S.T., Würth, R. (eds) Risk Assessment. Contributions to Economics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2050-8_5
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