how to value Credit Default Swap default leg and default probabilities

how to value CDS (credit default swap) default leg with following time structure:


Suppose that default (at time \tau ) can only occur at discrete times t1,t2,t3,..
and Qi=survival probability until time t_i


  \tau  - time of default {=t_1,=t_2,=t_3 ...}

 = 1 - ( 1-Q_2 + Q_3)= Q_2-Q_3

default leg of Credit Default Swap pays:

 discounted payoff = (1-R) (  df_1 E{\bf1}(\tau=t_1) + df_2 E{\bf1}(\tau=t_2)+ ...)
 =(1-R) (  df_1 P(\tau=t_1)+df_2 P(\tau=t_2)+ ...)
 =(1-R) (  df_1(1-Q_1)+df_2(Q_1-Q_2)+df_3(Q_2-Q_3) ... )

where df_1 is discount factor for t_1 and R – recovery rate (normally assumed ot be 40%)

to use continuous time one normally use intensity:
 Q(t)=exp( -\int_{0}^{T}\lambda(s)ds)
in practice \lambda is assumed to be piecewise-constant chaning value on market CDS maturities

useful approximations:
Credit Default Swap spread \approx p(1-R) \approx \lambda (1-R)
where \lambda is intensity and approx. equal to probability of default in first year.

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