  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 Sircar & Zariphopoulou (2007), 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.
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.
