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

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