  Motivated by the interplay between structural and reduced form credit models,
we propose to model the firm value process as a time-changed Brownian motion
that may include jumps and stochastic volatility effects, and to study the
first passage problem for such processes. We are lead to consider modifying the
standard first passage problem for stochastic processes to capitalize on this
time change structure and find that the distribution functions of such "first
passage times of the second kind" are efficiently computable in a wide range of
useful examples. Thus this new notion of first passage can be used to define
the time of default in generalized structural credit models. Formulas for
defaultable bonds and credit default swaps are given that are both efficiently
computable and lead to realistic spread curves. Finally, we show that by
treating joint firm value processes as dependent time changes of independent
Brownian motions, one can obtain multifirm credit models with rich and
plausible dynamics and enjoying the possibility of efficient valuation of
portfolio credit derivatives.
