I'm trying to take a closer look to option pricing in a risky environment.
Let's say a firm $A$ sells me an (European) option on an underlying $S$ (which of course can be any other financial product than the firm $A$) with payoff $h(S_T)$.
Let's take a r.v $\tau:\Omega \rightarrow \mathbb{R}_+$ to model the time at which the firm $A$ default (without really going into details of what "making default" means, we can say that it's the time the payoff of the option goes to $0$). A way to characterize the distribution of $\tau$ is to express it in function of the "hazard rate" $\lambda_t$ cf. Survival analysis. We can show that:
$$ \mathbb{P}\left(\tau > t\right) = e^{-\int_0^t\lambda_sds}. $$
In a second time, due to Feynman-Kac theorem, we can now (do we?) write the option price as:
$$P_t(T,K)=e^{-\int_t^Tr_sds}\mathbb{E}\left[h(S_T)\mathbb{1}_{\{\tau>T\}}|\mathcal{F}_t\right].$$
Before going any further, are $S_t$ and $\{\tau>t\}$ defined on the same filtered probability space? I cannot get an answer for this, because for me, intuitively (and I'm not saying my intuitions are good), we're most likely to have two filtrations $\mathcal{F}_t:=\sigma(S_s,s\leq t)$ and $\mathcal{G}_t:=\sigma(A_s,s\leq t)$ (with $A$ being the firm who sells the option) such that the option price is more likely to be something like:
$$P_t(T,K)=e^{-\int_t^Tr_sds}\mathbb{E}\left[h(S_T)\mathbb{1}_{\{\tau>T\}}|\mathcal{F}_t\vee\mathcal{G}_t\right],$$
if it does mean something...
I'm seeing it like something is horribly wrong in what I'm writing, but there is not so much literature for this, and I don't want to go any further before being sure of the basics in such an environment.
Thanks.
