Optimal Investment and Pricing in the Presence of Defaults

Abstract

We consider the optimal investment problem when the traded asset may default, causing a jump in its price. For an investor with constant absolute risk aversion, we compute indifference prices for defaultable bonds, as well as a price for dynamic protection against default. For the latter problem, our work complements [30], where it is implicitly assumed the investor is protected against default. We consider a factor model where the asset's instantaneous return, variance, correlation and default intensity are driven by a time-homogenous diffusion X taking values in an arbitrary region E ⊆ R d . We identify the certainty equivalent with a semi-linear degenerate elliptic partial differential equation with quadratic growth in both function and gradient. Under a minimal integrability assumption on the market price of risk, we show the certainty equivalent is a classical solution. In particular, our results cover when X is a one-dimensional affine diffusion and when returns, variances and default intensities are also affine. Numerical examples highlight the relationship between the factor process and both the indifference price and default insurance. Lastly, we show the insurance protection price is not the default intensity under the dual optimal measure. TETSUYA ISHIKAWA AND SCOTT ROBERTSON factors. Here, the certainty equivalent is identified with a degenerate elliptic semi-linear partial differential equation (PDE). Using powerful PDE existence results from [19,9], in conjunction with well-known duality results for exponential utility (see, for example, [16,4,25]), we show the certainty equivalent is a classical solution to the PDE under minimal assumptions on the model coefficients. In particular, our results can handle when the factor process takes values in an arbitrary region in R d , and when the asset return, volatility and default intensity are unbounded functions of the factor process.The utility maximization problem taking into account default has been widely studied. Through a complete literature review is too lengthy to give here, we wish to highlight where our work fits in regards to prior studies. First and foremost, our work fills a gap in the literature by considering the Markovian setting where one may solve the optimal investment problem using PDE techniques. In the PDE setting it is possible to obtain explicit solutions which highlight the dependency of optimal policies and bond prices upon broader economic factors. Though the Markovian case has been treated in some form dating at least back to [30,23], to the best of our knowledge the problem has not been studied when one assumes the investor loses her dollar position in the stock upon default. Indeed, in contrast to [30], we do not assume the investor is fully protected against losses due to default.The non-Markovian case has been much more thoroughly studied. For dynamics governed by Brownian adapted processes, the problem traces to [24,20,2] and the subsequent extensions in [15,14,21,12]. In fact, our setting is closest to [20, Sections 2,3] w...