A Mellin Transform Approach to the Pricing of Options with Default Risk

The stochastic elasticity of variance model introduced by Kim et al. (Appl Stoch Models Bus Ind 30(6):753-765, 2014) is a useful model for forecasting extraordinary volatility behavior which would take place in a financial crisis and high volatility of a market could be linked to default risk of option contracts. So, it is natural to study the pricing of options with default risk under the stochastic elasticity of variance. Based on a framework with two separate scales that could minimize the number of necessary parameters for calibration but reflect the essential characteristics of the underlying asset and the firm value of the option writer, we obtain a closed form approximation formula for the option price via double Mellin transform with singular perturbation. Our formula is explicitly expressed as the Black-Scholes formula plus correction terms. The correction terms are given by the simple derivatives of the Black-Scholes solution so that the model calibration can be done very fast and effectively.