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Hi,I'm trying to calibrate the intensity process in the standard CDS pricing formula. Assuming - independence between recovery payment and default,- recovery as being randomly distributed in the default leg, yields.Pan; Singleton (2005), "Default and Recovery Implicit in the Term Structure of Sovereign CDS Spreads" then assume independence between interest rates and default times which in turn yields,with D(t,u) being the price of a zero bond issued at t and expiring in u.They then state they "[...] construct a discrete approximation in terms of D(t,u) [..] and the risk-neutral survival probabilities [...]" to the above expression. The risk-neutral survival probabilities are .I can't really get to that approximation. Discretizing by itself isn't the problem, but the survival probs part is. You obviously can't just drop the \lambda from the expectancy. I've thought about replacing it with (1 - survival probability), but I'm not sure if that's correct.Any help is greatly appreciated!

OK, by now I've found that,see Duffie (1998), "First-to-Default Valuation".Unfortunately I'm not sure how to incorporate the zero bond...

Hello, As an advice, I would suggest to make all reasonnings in terms of effective cash-flows. Then the discount factors (DF) will appear naturally.The default leg of a CDS of maturity N years is supposed to pay 1-Recovery (1-'R') at time of default (called 'thau') => DefaultLeg=Expectation( (1-R)*DF(0,thau)*1_thau<N_) ), where 1_condition_ returns 1 if condition if true, 0 otherwiseTo be useful in practice, this formula must be discretized, eg. with a monthly time step, so that the integral form is seen as a discrete summation :=>DefaultLeg=(1-R)*Integral(t=0->t=N of DF(0,t)*Proba(thau=t)*dt) ~= Sum(m=1->m=12*N of DF(0,m/12)*Proba(thau=m/12))In the discrete case, default can only occurs at times m/12 and Proba(thau=m/12) is expressed in terms of survival probability Q(s,t) (probability of survival until t knowing that there was survival until s) : Q(0,(m-1)/12)-Q(0,m/12)Then one just have to replace in the above expression and find a suitable model for Q, eg. with a piecewise constant default intensity lambda:Q(s,t)=Exponential(-Integral(u=s->u=t of lambda(u)*du))Hope this helps