Optimal Investment in Equity and Credit Default Swaps in the Presence of Default

Abstract

We consider an equity market subject to risk from both unhedgeable shocks and default. To partially offset default risk, investors may also dynamically trade in a rolling credit default swap (CDS) market. Assuming investment opportunities are driven by functions of an underlying diffusive factor process, we identify the certainty equivalent for a constant absolute risk aversion inve stor with a semi‐linear partial differential equation (PDE) that has quadratic growth in both the function and gradient coefficients. For general model specifications, we prove the existence of a solution to the PDE, which is also the certainty equivalent. We show the optimal policy in the CDS market covers not only equity losses upon default (as one would expect), but also losses due to restricted future trading opportunities. We use our results to price default‐dependent claims through the principle of utility indifference, and we show that provided the underlying equity market is complete absent the possibility of default, the equity‐CDS market is complete accounting for default. Lastly, through a numerical application, we show the optimal CDS policies are essentially static (and hence easily implementable) and that investing in CDS dramatically increases investors' indirect utility.