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. In the portfolio case, our study 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 negatively affecting the stock prices in our equity portfolio. 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 both the Black-Scholes case and Kou model with the same drift and volatility as in the jump-at-default models. Unsurprisingly, 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, but also compared with VaR-numbers from the Kou model restricted to having only negative jumps.