how to value Credit Default Swap default leg and default probabilities

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

0—-t1—–t2—-t3—–t4—–….—T

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

then

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

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

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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