Financial

Saint-Petersburg State University Faculty of Physics and DATAART company

Experimental Program of Supplementary Education "Information Technologies, Econophysics, and Complex Systems Management"

Internship Project

"Analytical Results for Evaluation of Stock Options with Stochastic Volatility"

Accomplished by P.B. Goldin
Supervised by Y.A. Kuperin

This work is dedicated to the method of stock options evaluation which accounts for volatility stochastic dynamics. This approach is the generalization of the well-known Black-Sholes model, in which volatility is considered as constant. The result of this work is the elaboration of the compact analytical formula for stock option prices calculation which generalizes the Black-Sholes formula and approaches it at the constant volatility limit.

To implement this project the theoretical physics methods were used so this project can be considered as a true example of the econophysics application.

In this paper the problem of stochastic volatility is scrutinized from the point of view of the relatively new theory of Feynman path integrals. The most important results of the research are as follows:

The investigated problem has the strongly nonlinear nature of very high order and it was possible to present this non-linearity using the functional integration technique in a very elegant and convenient way.

While solving this problem the general approach for solving such non-linearity and getting compact analytical solutions was found.

In this work the new method was elaborated and on its basis the analytical model was built. The model allows to evaluate the cost of European put and call options accurately, effectively, and easily as well as evaluate the cost of some more exotic options with stochastic volatility.

The technique described in this paper can be helpful for traders for more accurate and realistic evaluations of the European options as well as for building new exotic options with stochastic volatility and their precise evaluations.