||The stochastic elasticity of variance model introduced by Kim, Lee, Zhu & Yu (2014) is a useful model for forecasting extraordinary volatility behavior which would take place in a financial crisis. High volatility of market could be linked to default risk of option contracts. So, the model is a desirable choice for the pricing of vulnerable option. This paper is an extended study of a previous work by Lee, Yang & Kim (2017) but obtains an explicit closed-form analytic expression of the price of vulnerable option under a framework with two separate scales that could minimize the number of necessary parameters for calibration but reflect the essential character of the underlying risky asset of the option and the firm value of the option writer. Using singular perturbation and double Mellin transform techniques, we obtain the formula which is given by the Black-Scholes formula for the vulnerable option plus correction terms that consist of simple derivatives of the Black-Scholes solution. This formula provides us to compute the vulnerableoption price easily and to perform the sensitivity analysis for the price with respect to default risk eectively.