Study of the dynamics of a jump-like change in price in the generalized Black-Scholes model
In this paper, we examine the stability of the generalized Black-Scholes model. We assume that the dynamics of the option price depends on random spasmodic changes in the volatility. We prove that the study of the mean-square stability of the Black-Scholes stochastic system can be reduced to the study of the stability of a deterministic system of second-order moments for the underlying asset. Using the example of an option with two structural volatility states, we obtain conditions of mean-square stability.