pledPriceTime International Journal of Theoretical and Applied Finance c ○ World Scientific Publishing Company

Pricing of options on stocks that are driven by multi-dimensional coupled price-temporal infinitely divisible fluctuations. We model the price of a stock via a Langévin equation with multi-dimensional fluctuations coupled both in the price in time. We generalise previous models in that we assume that the fluctuations conditioned on the time step are assumed to be compound Poisson processes with operator stable jump intensities. We derive exact relations for Fourier transforms of the jump intensity in case of different scaling indices E of the process. We express the Fourier transform of the joint probability density of the process to attain given values at several different times and to attain a given maximal value in a given time period through Fourier transforms of the jump intensity. Then we consider a portfolio composed of stocks and of options on stocks and we derive the Fourier transform of a random variable Dt (deviation of the portfolio) that is defined as a small temporal change of the portfolio diminished by the the compound interest earned. We derive a functional equation for the price of the option on the stock as a function of the stock subject to the condition that the deviation of the portfolio has a zero mean E [Dt] = 0 Therefore the option pricing problem may have a solution.