On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model

Abstract

This paper discusses the generalized Black-Scholes-Merton model, where the volatility coefficient, the drift coefficient of stocks, and the interest rate are time-dependent deterministic functions. Together with it, we make the assumption that the volatility, the drift, and the interest rate depend on a gamma or inverse-gamma random variable. This model includes the models of skew Student’s t- and variance-gamma-distributed stock log-returns. The price of the European forward-start call option is derived from the considered models in closed form. The obtained formulas are compared with the Black-Scholes formula through examples.

Keywords

• Black-Scholes formula
• time-dependence
• gamma distribution
• inverse-gamma distribution
• special function