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) $ .
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.