In this paper, we investigate analytical approximations for foreign equity options under a stochastic volatility model with fast mean reversion. Foreign equity options are financial derivatives whose payoffs are determined by the price of an underlying asset and the foreign exchange rate, and they involve two currencies where the currency of the actual payoff differs from the currency of the underlying asset. We consider a stochastic volatility model to capture key stylized facts observed in financial markets, such as volatility clustering and the volatility smile, by modeling the volatilities of both the foreign asset and the exchange rate as rapidly mean-reverting processes. Specifically, we employ asymptotic expansion techniques to derive analytical pricing formulas for two types of options: Option struck in foreign currency and option struck in the domestic currency. This methodology provides accurate price approximations without complex numerical methods. Additionally, we provide some examples to show the importance of model parameters in foreign equity option pricing.

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.