I'm trying to understand the functioning of "recovery of face-value" approach.

Let $V_t$ the fair-value, that is the price that the holder of a defaultable bond must pay for hedging of default of his counterparty.

Let $P_0(t,T):=e^{-\int_{t}^{T}r_sds}$, with $r_s$ deterministic rate, the price of a no-defaultable bond. Assuming the existence of an equivalente martingale under neutrality measure $\mathbb{Q}$, this price can be written $P_0(t,T):=\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{T}r_sds}|F_t]$ with $F_t$ a filtration of $F$.

This approach says that considering a time range $[0,T]$ and a stopping time $t\leq \tau \leq T$, the investor will receive at the time of default $\tau$ of his counterparty a sum of $(1-\delta_{\tau})P_0(t,T)$, sum that will be quantified in proportion to the value of a no-defaultable bond evaluated in that moment. In particular, that quantity is equal to the product of the price that the stock would have had if it had not been subject to default $P_0(t,T)$ and the Loss Given Default $(1-\delta_{\tau})$, that describes the loss suffered on projected incomes. Note that $\delta_{\tau}$ represents the recovery rate, and the $\tau$ placed as subscript means that also the recovery rate could be a stochastic process, not only the fair-value. However we assuming, for convenience, that this rate is constant: $\delta_{\tau}\equiv \delta$. Now, as I understand it, my text says that fair-value is equal to differential between the prospective gain $(1-\delta_{\tau})P_0(t,T)$ in case of default of counterparty and a fraction of gain that the investor would have obtained if it had not been default in the range of interest (i.e. for $\tau>T$). Formally:

$V_t=(1-\delta)P_0(t,T)-\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{T}}P_0(\tau,T)\mathbb{I}_{\tau>T}|\nu _t]$

with $\nu_t$ the smallest $\sigma$-algebra generated by random variables which constitutes a generic counting process $\begin{Bmatrix}N_t\end{Bmatrix}_{t\in[0,T]}:=k, \forall t\in [\tau_{k},\tau_{k+1})\sim \mathrm{Po}(\lambda_t:=\int_{0}^{t}\lambda_sds)$.

The problem is in the next step. In fact, professor writes that


as you can see here enter image description heresaying that the quantity highlighted is the "countervalue of recovery rate".

1) What it means?

2) What justifies the affixing of $P_0(\tau,T)$ in the denominator?

Thanks in advance!


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