BFS 2002 

Contributed Talk 
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 jumpdiffusion 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 "defaultzerocoupon" bonds (DZC), priced as follows:
D(t,x,R)=exp( L(t,x)psi(t,x,R)) x = Tt
The defaultfree 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 riskneutral 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 MonteCarlo simulation. Hedge ratios are computed with respect to underlying bonds and CDS's.
Most other credit models (Merton, JarrowTurnbull, DuffieSingleton, HullWhite, etc.) can be seen either as particular cases or as limit cases of this model, which has been specially designed to ease calibration.
Longterm 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 riskneutral value, while the historical volatility and the jumps are left untouched.