Optimal investment with counterparty risk: a default-density model approach

We consider a financial market with a stock exposed to a counterparty risk inducing a jump in the price, and which can still be traded after this default time. The jump represents a loss or gain of the asset value at the default of the counterparty. We use a default-density modelling approach, and address in this incomplete market context the problem of expected utility maximization from terminal wealth. We show how this problem can be suitably decomposed in two optimization problems in a default-free framework: an after-default utility maximization and a global before-default optimization problem involving the former one. These two optimization problems are solved explicitly, respectively, by duality and dynamic programming approaches, and provide a detailed description of the optimal strategy. We give some numerical results illustrating the impact of counterparty risk and the loss or gain given default on optimal trading strategies, in particular with respect to the Merton portfolio selection problem. For example, this explains how an investor can take advantage of a large loss of the asset value at default in extreme situations as observed during the financial crisis.