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.