Extended CreditGrades Model with Stochastic Volatility and Jumps

We present two robust extensions of the CreditGrades model: the first one assumes that the variance of returns on the firm's assets is stochastic, and the second one assumes that the firm's asset value process follows a double-exponential jump-diffusion. We derive closed-form formulas for pricing equity options on a reference firm in this setting and for calculating the survival probability of this firm during a finite time horizon. We apply these models for modeling credit default swap (CDS) and equity default swap (EDS) spreads. We calibrate our models to General Motors options data and discuss the results. It follows that both models provide a good fit to the data and lead to non-zero short-term CDS spreads. The contribution of this paper is threefold. First, we incorporate jumps into the CreditGrades model. Although the Merton's model with jump risk has already been considered in a number of studies, there was no reference on how to connect the default risk with equity risk, i.e. how to estimate default probabilities using equity options. Secondly, we consider the stochastic variance of the firm's value in the CreditGrades model and make a connection to equity options. This model seems to be new. Finally, we consider incorporating random default barriers and provide an alternative to the CreditGrades approach on how to deal with random default barriers by computing survival probabilities and option prices. This approach is based on the convexity adjustment and it can applied to diffusions with stochastic variance and jumps