In this study, we examine the pricing of vulnerable options under a stochastic volatility model based on the partial differential equation approach. Specifically, we consider a multiscale stochastic volatility model that is assumed to be driven by two diffusions (fast-scale and slow-scale) and use an asymptotic expansion approach to drive the approximate pricing formulas of vulnerable options, which allows the counterparty credit risk at maturity. Furthermore, we provide the Greek Delta of vulnerable options for the dynamic hedge and present the numerical results to examine the effect of the multiscale stochastic volatility model and to show the accuracy of our formula.

在这项研究中，我们研究了基于偏微分方程方法的随机波动率模型下脆弱期权的定价。具体来说，我们考虑一个假设为由两个扩散（快速扩展和慢速扩展）驱动的多尺度随机波动率模型，并使用渐近展开法来驱动脆弱期权的近似定价公式，这使得交易对手信用风险位于到期。此外，我们为动态套期保值提供了易受害期权的希腊三角洲，并给出了数值结果以检验多尺度随机波动率模型的影响并证明我们公式的准确性。