The Pricing and Risk Management of Credit Default Swaps, with a Focus on the ISDA Model Screenshot: Pricing protection leg of a CDS, by OpenGamma
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.