Raphael Douady, Monique Jeanblanc
We present a model in which a bond issuer subject to possible default is assigned a "continuous" rating R(t) in [0,1] that follows a jump-diffusion process. Default occurs when the rating reaches 0, which is an absorbing state. An issuer that never defaults has rating 1 (unreachable). The value of a bond is the sum of "default-zero-coupon" bonds (DZC), priced as follows:
D(t,x,R)=exp( L(t,x)-psi(t,x,R)) x = T-t
The default-free yield y(t,x,1)=L(t,x)/x follows a traditional interest rate model (e.g. HJM, BGM, "string", etc.). The "spread field" psi(t,x,R) is a positive random function of two variables R and x, decreasing with respect to R and such that psi(t,0,R)=0. The value psi(t,x,0) is given by the bond recovery value upon default. The dynamics of psi is represented as the solution of a finite dimensional SDE. Given psi such that dpsi/dR <0 a.s., we compute what should be the drift of the rating process R(t) under the risk-neutral probability, assuming its volatility and possible jumps are also given.
For several bonds, ratings are driven by correlated Brownian motions and jumps are produced by a combination of economic events.
Credit derivatives are priced by Monte-Carlo simulation. Hedge ratios are computed with respect to underlying bonds and CDS's.
Most other credit models (Merton, Jarrow-Turnbull, Duffie-Singleton, Hull-White, etc.) can be seen either as particular cases or as limit cases of this model, which has been specially designed to ease calibration.
Long-term statistics on yield spreads in each rating and seniority category provide the diffusion factors of psi. The rating process is, in a first step, statistically estimated, thanks to agency rating migration statistics from rating agencies (each agency rating is associated with a range for the continuous rating). Then its drift is replaced by the risk-neutral value, while the historical volatility and the jumps are left untouched.