This Document is the property of Forbin Systems and is protected by copyright.  Permission to reproduce may be gramted by contacting ********

Implied Default Probabilties

Abstract

This paper describes a semi analytic approach to implying default probabilites from  vanilla CDS spread rates for CDS's having a range of maturities.A substantail part of the analysis presented in the paper will be placed in the context of the reference paper 1. In particular we will maintain the same termonology
as described on page 13 of refernce 1.

Analysis

The relationship between the CDS  rate and the default proabability curve is given by the following :

(1)

where  ⅆt is the probability of a bond with maturity T defaulting between t and t+ⅆt as seen at time zero and w[T] is anualized CDS rate to be payed as protection
against default . It is assumed that the default density for different bonds with different maturities is the same since if a company defaults on an  bond issued with a
particular maturity   it will default on  bonds of all maturities,

ie  === ect .
this property can be represented as
   
                  =H[t,T]
                  
                 where    The Heaviside function, H[t,T], is defined as:
                 
                              H[t,T]=1, 0<t<=T
                              H[t,T]=0, T<t
                                 
   (1) the becomes
   
                             (2)
   
        U[t] =u[t]+e[t], where u and e are defined in ref 1. ,and describe the CDS payment times.
        U[t] can be conviently  expressed as
       
           
         where          w[T] m[t] ⅆt  are the CDS payments made between t and t+ⅆt on a CDS maturing at time T to protect the purchase of a bond also maturing at time T.
         
         The probability of default and probability of survival before time t are given respectively by:
         
                                                                                                                                                              (3)
         
         S[t]=1-Q[t]                                                                                                                                                                    (4)                                                                                                                                                       
         
       From the above definitions and Integration by parts we get that
       
                                                                                       (5)
       
       Using (5) and (2) we obtain :
       
                                                                                                                 (6)
       
       where   .
       
       if  =0    then (6) can be integrated to give :
       
                                                                                                                                                        (7)
       
       if the CDS payments are made in arrears then m[t]=1 and (7) becomes :                                                                      (8)
       
                  
       
       For a non constant  w[T,] (6) can be rearranged to be read as Voltera eqaution of the 2 nd Kind , for which there are a multitude of solution tecniques                                                      
        
  

REFERENCES

        1. Hull, J. C. and A. White, "Valuing Credit Default Swaps I:  No Counterparty default risk"