Calibration of portfolio credit risk model: solution of an inverse problem via intensity control
Rama Cont (Columbia University)
Pricing models for portfolio credit derivatives such as
CDOs involves the construction of a stochastic process for the
losses due to defaults which is compatible with a set of
observations of market spreads for CDO tranches. We propose an
efficient and stable algorithm to solve this inverse problem by
transforming it into a stochastic control probem. We formalize the
problem in terms of minimization of relative entropy with respect to
a prior jump process under calibration constraints and use convex
duality techniques to solve the problem. The dual problem is shown
to be an intensity control problem. We show that the corresponding
nonlinear Hamilton Jacobi system of differential equations can be
represented in terms of a nonlinear transform of a linear system of
ODEs and thus easily solved. Our method allows to construct a
Markovian jump process for defaults which leads to CDO tranche
spreads consistent with the observations. We illustrate our method
ITRAXX index data: our results reveal strong evidence for the dependence of loss
transitions rates on the past number of defaults, thus offering
quantitative evidence for ``contagion effects" in the riskneutral
loss process. Joint work with Andreea Minca.
