Explicit expressions to counterparty credit exposures for Forward and European Option

Abstract

With the fast development of over the counter (OTC) derivatives market, counterparty credit risk (CCR) has become one of the main risks that can even affect the survival of banks. As a result, it is essential to measure and manage CCR exposures. Giant financial institutions apply numerical methods such as the Monte Carlo to calculate exposures. However, the numerical methods usually cost a significant amount of time, and some advanced algorithms like distributed and parallel processing are usually used to accelerate the calculation. Nevertheless, for small banks, they cannot afford the calculation cost. In order to make small banks manage CCR more efficiently, this paper puts forward analytic models to measure CCR exposures and derives the explicit expressions to exposures for Forward contract and European Option, which are represented in the OTC market. The explicit expressions to credit exposures for Forward contract are derived under the assumption that the underlying market risk factors follow Geometric Brown Motion. For European Option case, the problem turns to be difficult since the analytic formula involves double-definite integral of Gaussian function that cannot be simplified into elementary functions. An approximating normal distribution function with integrability is proposed, then the analytic approximations of European Call Option’s credit exposures and maximum errors are presented.