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

Abstract

We derive new algorithms for computing time variations in the Stochastic Volatility and the Levy 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 Levy 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 Levy 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.