First of all, I realise this question might border on `meta-finance', so I'd be totally OK if it gets closed.
Having said that, the question itself:
Given a stock $S$, in the absence of default it is quite clear that the probability the stock price at time $T$ takes values in $(0,\infty)$ is $1$ for all $T$.
In the presence of default, however, the price at $T$ isn't even defined if the company defaulted at $t<T$. Can we express this mathematically as $P(0<S_T < \infty) = f(T) \leq 1$?
To illustrate what I mean with a simple example, consider the probability density of the stock price (zero interest rate and dividend yield for now) given by $$ q(S_T,T | S_t,t) = e^{-c(T,t)} p(S_T,T | S_t,t) $$ where $p$ is the usual lognormal density with constant volatility $\sigma$ and $c(t,T)$ is a positive deterministic function of $t$ and $T$. In this simple example let's assume that we only observe a term structure of IVs and no skew/smile.
Then, $$ E\left[(S_T-K)_+ \right] = e^{-c(T,t)} C^{BS} (S_t,K,\sigma,T) $$ We can calibrate $c(t,T)$ to the IV TS through $$ C^{BS} (S_t,K,IV(T),T) = e^{-c(T,t)} C^{BS} (S_t,K,\sigma,T) $$
I am working on generalising this to stochastic $c$ and/or a local $c$ to get skew/smile.