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We consider the optimal investment problem when the traded asset may default,causing a jump in its price. For an investor with constant absolute riskaversion, we compute indifference prices for defaultable bonds, as well as aprice for dynamic protection against default. For the latter problem, our workcomplements Sircar & Zariphopoulou (2007), where it is implicitly assumed theinvestor is protected against default. We consider a factor model where theasset's instantaneous return, variance, correlation and default intensity aredriven by a time-homogenous diffusion X taking values in an arbitrary region E.We identify the certainty equivalent with a semi-linear degenerate ellipticpartial differential equation with quadratic growth in both function andgradient. Under a minimal integrability assumption on the market price of risk,we show the certainty equivalent is a classical solution. In particular, ourresults cover when X is a one-dimensional affine diffusion and when returns,variances and default intensities are also affine. Numerical examples highlightthe relationship between the factor process and both the indifference price anddefault insurance. Lastly, we show the insurance protection price is not thedefault intensity under the dual optimal measure.

Optimal Investment and Pricing in the Presence of Defaults