Pricing of vulnerable options with early counterparty credit risk

The counterparty credit risk should be considered when valuing options traded in the over-the-counter (OTC) markets because the OTC markets have rapidly grown, and the credit risk in the OTC markets has become an important issue since the global financial crisis. In this paper, we propose two types of vulnerable options whose payoffs allow for the counterparty credit risk prior to maturity of the options. We use Mellin transforms to solve the partial differential equations for the vulnerable option prices with early counterparty credit risk. In addition, we present numerical experiment results with the Monte Carlo simulations to show the accuracy of the pricing formulas and provide graphs to illustrate the properties of options.