Option Pricing with Log-stable Lévy Processes

Abstract

We model the logarithm of the price (log-price) of a financial asset as a random variable obtained by projecting an operator stable random vector with a scaling index matrix $\underline{\underline E}$ onto a non-random vector. The scaling index $\underline{\underline E}$ models prices of the individual financial assets (stocks, mutual funds, etc.). We find the functional form of the characteristic function of real powers of the price returns and we compute the expectation value of these real powers and we speculate on the utility of these results for statistical inference. Finally we consider a portfolio composed of an asset and an option on that asset. We derive the characteristic function of the deviation of the portfolio, $\mathfrak{D}_t^{(\mathfrak{t})}$, defined as a temporal change of the portfolio diminished by the the compound interest earned. We derive pseudo-differential equations for the option as a function of the log-stock-price and time and we find exact closed-form solutions to that equation. These results were not known before. Finally we discuss how our solutions correspond to other approximate results known from literature,in particular to the well known Black & Scholes equation.