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lecture7 - CREDIT RISK Lecture V Lina El-Jahel 2006 1...

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CREDIT RISK: Lecture V Lina El-Jahel 2006 1 Reduced Form Models Reduced form models are often advocated on the basis that they are easier to implement than structural models. In fact, one can always implement structural model, by inferring hard to observe information on say capital structure form market prices so this arguments is not very convincing. A more significant argument in favor of reduced-form or structural models is that they may allow the user to link the default event to more variables than firm value (using Cox processes) and this approach can be linked to the entire set of tools usually used in term structure modelling. This means that tricks used for pricing derivatives can be transferred to defaultable claims. In ordinary term structure modelling our ignorance of what truly governs the dynamics of interest rates is often hidden by our modelling of the short rate for example. This idea is transferred to intensity based models where most models use exogenously given specifications of the default intensity. The mathematical structure of ordinary term structure models easily allows however, for the short rate to depend on multidimensional state variable processes and even to depend on observable state variable processes such as yields and volatilities. The reduced form approach is surely moving in this direction and many researchers are trying to isolate fundamental variables to explain the default event. Once these have been convincingly identified one can then incorporate them in an intensity model. In our presentation of reduced form models we will focus on models using doubly stochastic Poisson processes or Cox processes. This specification is more general than using a constant hazard rate and also allows the intensity process to be linked to other variables. 1
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2 The Cox Process Construction of a Single Jump Time Consider a zero-coupon issued by a risky firm at time 0. Assume that the maturity of the bond is T and the default time τ of the issuing firm has an intensity λ ( X t ) under the risk-neutral probability measure Q . Also, assume there is a short rate process r ( X t ) such that the zero coupon default free bond can be computed as p (0 , T ) = E " exp - Z T 0 r ( X s ) ds !# (1) The price of the risky bond at time 0 is then, assuming zero recovery , F (0 , T ) = E " exp - Z T 0 r ( X s ) ds ! 1 { τ>T } # = E " E " exp - Z T 0 r ( X s ) ds ! 1 { τ>T } | λ ( t ) t ## = E " exp - Z T 0 r ( X s ) ds ! E h 1 { τ>T } | λ ( t ) t i # = E " exp - Z T 0 r ( X s ) ds ! exp - Z T 0 λ ( X s ) ds !# = E " exp - Z T 0 r ( X s ) + λ ( X s ) ds !# The short rate has been replaced by the intensity-adjusted short rate ( r + λ ) This results can be easily modified to cover a contingent claim with a promised payment f ( X T ) and an actual payment f ( X T )1 τ>T . Duffie-Singleton (1999) examine this issue and introduce the concept of a discount rate higher than the risk free rate under the risk neutral measure to take into consideration the risk of default.
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