finance mathematics - Finding the extrinsic value of an option with conditions

Background: Consider a spread option with the payoff $\max (P_{T} - HR\times G_T, 0)$, where $P$, $G$ are underlying prices and $HR$ is a constant.

Let's also assume, that the correlation between assets is $\text{corr}(\ln(P_t), \ln(G_t)) = 1$.

Let's additionally assume that the underlying variables are jointly elliptical.

Question: Characterize the conditions under which the extrinsic value of the option is equal to zero. That is, find the conditions under which: $E_{0}^{*}[\max (P_{T} - HR\times G_T, 0)] = \max (P_{0} - HR\times G_0, 0)$.

Answer

Find the conditions under which:

$E_{0}^{*}[\max (P_{T} - HR\times G_T, 0)] = \max (P_{0} - HR\times G_0, 0)$

We have a no-brainer solution - the condition that the drift and volatility of both $P$ and $G$ is zero, which means $P$ and $G$ are constants in time.

Second valid condition - the option is deep in the money or deep out of the money, such that chance of moneyness changing sign is remote (i.e. the volatility of $P$ and $G$ are not large enough to provide a meaningful chance of moneyness changing sign). Essentially, the payoff behaves as a forward, rather than an option.

The drifts of the two assets also need to cancel out.So either both the drifts should be zero, or the drift of $P$ should be $HR$ times the drift of $G$.

That's pretty much it, as far as I can see.