@MISC{Douady_arating-based, author = {Raphael Douady and Monique Jeanblanc}, title = {A RATING-BASED MODEL FOR CREDIT DERIVATIVES Authors:}, year = {} }

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Abstract

We present a model in which a bond issuer subject to possible default is assigned a “continuous” rating R t ∈ [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 “defaultzero-coupon” bonds (DZC), priced as follows: D (t, x, R) = exp (–l (t, x) – ψ(t, x, R) 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” ψ(t, x, R) is a positive random function of two variables R and x, decreasing with respect to R and such that ψ(t, 0, R) = 0. The value ψ(t, x, 0) is given by the bond recovery value upon default. The dynamics of ψ is represented as the solution of a finite dimensional SDE. Given ψ such that ∂Rψ < 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.