An asymptotic expansion approach to the valuation of vulnerable options under a multiscale stochastic volatility model

•This paper studies the vulnerable option under the multiscale stochastic volatility model.•An analytic pricing formula of option is derived using the asymptotic expansion method.•The Greek Delta for the hedge is derived.•Numerical examples are provided to illustrate the properties of the option under the proposed model. 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.