When a Spread Option Has Zero Extrinsic Value

Summary

The document considers a spread option with payoff equal to the positive part of the difference between one asset price and a fixed hedge ratio times another asset price. It asks when the option’s expected payoff equals its current intrinsic value, assuming perfectly correlated log prices and a joint elliptical distribution.

The response gives two intuitive cases: both assets remain constant, or the option is so far in or out of the money that a change in moneyness is highly unlikely. It also says the assets’ drifts must cancel in the spread, either because both are zero or because their movements offset according to the hedge ratio. These are informal claims rather than a derivation. In particular, describing a small chance of crossing the strike as a condition for equality only suggests an approximation; the post does not establish exact conditions under the stated distributional assumptions or specify a pricing measure and dynamics. Treat the answer as intuition about time value, not a complete characterization.

Key ideas

• The question concerns a call-like option on the difference between two asset prices adjusted by a fixed hedge ratio.
• Zero extrinsic value means expected payoff equals current intrinsic value.
• The response identifies constant underlying prices as a sufficient intuitive case.
• It suggests deep in-the-money or out-of-the-money positions may have little time value when crossing moneyness is unlikely.
• The response says asset drifts should offset, but does not derive exact conditions under the stated assumptions.

Tags

# Finding the extrinsic value of an option with conditions


# 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 by bhutes (score 2, accepted)

https://quant.stackexchange.com/a/45654

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.

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