Computation of the Stochastic Volatility and Levy Index using the Kolmogorov-Feller Equation with Applications to Carbon Price Data Analysis.

We derive new algorithms for computing time variations in the Stochastic Volatility and the L´evy index using a standard financial price model and a Green’s function solution to the Kolmogorov-Feller equation. A principal condition upon which the algorithms are based is the Phase Only Condition which allows the Power Spectral Density Function of a financial time series (specifically the log price differences) to be taken to be a constant. The paper is composed of four component parts: (i) the Stochastic Volatility is derived and studied numerically; (ii) the Kolmogorov-Feller equation is studied and solved to provide a model for the stochastic characteristics of a financial time series using the Levy Characteristic Function; (iii) a method for computing the L´evy index is proposed given price data and the Stochastic Volatility of the data; (iv) numerical algorithms are designed and example results presented. Although the models proposed and the algorithms developed are applicable to financial time series in general, in this paper, we consider a study of the Stochastic Volatility and L´evy index for Carbon price data. This is because of the increasing importance of ‘Carbon trading’ with regard to climatic control and the emission of Carbon Dioxide and other green-house gases. The results presented therefore represent a study of a financial indicator (in particular the Levy index) that may be of value for future energy commodities trading, and, in particular, Carbon price risk assessment modelling.