# CDS protection/contingent leg pricing, taking expectation of interest and hazard rates

In the screenshot above, I am having trouble understanding the maths between equation 13 and equation 14.

Notation:

• $N$ = notational payment, e.g., £100
• $RR$ = recovery rate, the percentage of the $N$ recovered upon default, e.g., you get back 40%
• $\tau$ = time of default
• $t_v$ = valuation date
• $T$ = maturity date
• $\mathbb{I}_A$ = indicator function for event $A$
• $r(s)$ = instantaneous short rate at time $s$
• $P(t)$ = discount factor from time $t > 0 =$start date
• $Q(t)$ = survival probability at time $t$

What I have tried:

$$\mathbb{E}\left[e^{-\int_{t_v}^{T}r(s)ds}\mathbb{I}_{\tau<T}\right] = \int_{-\infty}^{\infty} \tau e^{-\int_{t_v}^{T}r(s)ds}\mathbb{I}_{\tau<T} d\tau = \int_{0}^{T} \tau \frac{P(\tau)}{P(t_v)}d\tau$$

From here, I cannot see how equation 14 is derived.

• Ciao! why a minus appears in the third line? (in the same line I think you should put $s$ instad of $\tau$ in the characteristic function). – clarkmaio Dec 14 '17 at 14:22