Calculating the Price for Derivative Financial Assets of Bessel Processes Using the Sturm-Liouville Theory

Abstract

In the paper we apply the spectral theory to find the price for derivatives of financial assets assuming that the processes described are Markov processes and such that can be considered in the Hilbert space L^2 using the Sturm-Liouville theory. Bessel diffusion processes are used in studying Asian options. We consider the financial flows generated by the Bessel diffusions by expressing them in terms of the system of Bessel functions of the first kind, provided that they take into account the linear combination of the flow and its spatial derivative. Such expression enables calculating the size of the market portfolio and provides a measure of the amount of internal volatility in the market at any given moment, allows investigating the dynamics of the equity market. The expansion of the Green function in terms of the system of Bessel functions is expressed by an analytic formula that is convenient in calculating the volume of financial flows. All assumptions are natural, result in analytic formulas that are consistent with the empirical data and, when applied in practice, adequately reflect the processes in equity markets.