  We address the so-called calibration problem which consists of fitting in a
tractable way a given model to a specified term structure like, e.g., yield or
default probability curves. Time-homogeneous jump-diffusions like Vasicek or
Cox-Ingersoll-Ross (possibly coupled with compounded Poisson jumps, JCIR), are
tractable processes but have limited flexibility; they fail to replicate actual
market curves. The deterministic shift extension of the latter (Hull-White or
JCIR++) is a simple but yet efficient solution that is widely used by both
academics and practitioners. However, the shift approach is often not
appropriate when positivity is required, which is a common constraint when
dealing with credit spreads or default intensities. In this paper, we tackle
this problem by adopting a time change approach. On the top of providing an
elegant solution to the calibration problem under positivity constraint, our
model features additional interesting properties in terms of implied
volatilities. It is compared to the shift extension on various credit risk
applications such as credit default swap, credit default swaption and credit
valuation adjustment under wrong-way risk. The time change approach is able to
generate much larger volatility and covariance effects under the positivity
constraint. Our model offers an appealing alternative to the shift in such
cases.
