Analytical valuation of vulnerable options under a stochastic volatility model with a stochastic long-term mean

We derive the explicit pricing formulas for vulnerable options under a stochastic volatility model with stochastic long-term mean. We extend the He and Chen model to incorporate counterparty default risk and derive explicit solutions for option prices using the characteristic function of the underlying asset's log-price. The option writer defaults when their asset value falls below a predetermined boundary, reducing the option payoff. Our numerical examples show that option prices are highly sensitive to default boundaries and exhibit asymmetric responses to volatility parameters.