Risk management of stock portfolios with jumps at exogenous default events

In this paper we study equity risk management of stock portfolios where the individual stock prices have downward jumps at the defaults of an exogenous group of defaultable entities. The default times can come from any type of credit portfolio model. In this setting we derive computational tractable formulas for several stock-related quantizes, such as loss distributions of equity portfolios and apply it to Value-at-Risk computations. We start with individual stock prices and then extend the setting to a portfolio framework. In the portfolio case our studies considers both small-time expansions of the loss-distribution for a heterogeneous portfolio via a linearization of the loss, but also for general time points when the stock portfolio is large and homogeneous and where we use a conditional version of the law of large numbers. Most of the derived formulas will heavily rely on the ability to efficiently compute the number of defaults distribution of the entities in the exogenous group of corporates negative affecting the stock prices in our equity portfolio. If the stock prices are unaffected by the exogenous defaults then our framework collapses into the traditional Black-Scholes model under the real probability measure. Finally, we give several numerical applications. For example, in a setting where the jumps in the stock prices are at default times which are generated by a one-factor Gaussian copula model, we study the time evolution of Value-at-Risk (i.e. VaR as function of time) for stock portfolios, both for a 20-day period and for a two-year period. We also perform similar numerical VaR-studies in a setting where the individual default intensities follow a CIR process. Our results are compared with the corresponding VaR-values in the Black-Scholes case with same drift and volatilises as in the jump models. Not surprisingly, we show that the VaR-values in stock portfolios with downward jumps at defaults of external entities, will have substantially higher VaR-values compared to the corresponding Black-Scholes cases. The numerical computations of the number of default distribution will in all our studies use fast and efficient saddlepoint methods.