Analytically pricing vulnerable options under the stochastic volatility model with stochastic long-term mean and stochastic liquidity

This paper investigates the analytical pricing of vulnerable European options under a two-factor stochastic volatility model that incorporates both a stochastic long-term mean and stochastic liquidity risk. We develop the framework by integrating the stochastic long-term mean model and the mean-reverting liquidity discount factor, addressing the limitation that the long-term volatility mean is conatant in existing models. Under the structural approach to counterparty credit risk, we derive the joint characteristic function for the log-prices of the underlying and the option writer’s assets by solving a system of Riccati equations and obtain an explicit analytical pricing formula of a vulnerable option using the characteristic function. Numerical experiments show that the pricing formula is accurate and more computationally efficient than Monte Carlo simulations. Moreover, from the numerical examples, we find that the drift of stochastic long-term means, initial variance levels, and liquidity parameters significantly impact vulnerable option prices.