# Optimal Investment and Pricing in Models where the Underlying Asset May Default

he need for the pricing and hedging of credit events has increased since the financial crisis. For example, large banks are now mandated to compute prices of credit risk for all over-the-counter contracts. Such prices are known by the acronym CVA (Credit Valuation Adjustment), or more generally, XVA. Industry practitioners typically use risk-neutral pricing for such computations, the validity of which is questioned in incomplete markets. In our research, we consider an incomplete market where investment returns and variances are driven by a partially hedgeable factor process, modelled by a multi-dimensional diffusion. Additionally, the issuer of the stock may default, with the default intensity also driven by the factor process. Investors can freely trade the stock to hedge their positions in this market, and do so to maximize their utility. However, in the event of default, the investors lose their position in the stock. In this setting, we price defaultable claims using utility indifference pricing for an exponential investor. Due to the Markovian structure of the problem, we rely on PDE theory rather than BSDE theory to solve the utility maximization problem. This leads to explicit candidate solutions which we verify using the well-developed duality theory. As an application of our optimal investment result, we define, and compute, the dynamic utility indifference price for insurance against the defaultable stock.