I'm currently reading Iain Clark's book Foreign Exchange Option Pricing and I got stuck at one sentence in the beginning of Section 3.3 that I feel is important to understand. He writes:

FX volatility smiles are characterized by providing volatilities, not as a function of strike, but as a function of delta. The choice of delta as the parameter describing the volatility smile is sensible, as otherwise a strike that might correspond to a considerably out-of-the-money option for small $T$ would be very close to at-the-money for large $T$.

where by $T$ he refers to the time left until expiry of the option. My question is: how do you know (or argue) that just because there is an option with expiry in 1 week that is out-of-the-money a similar option (with bigger $T$) will be very close to at-the-money?

Thanks in advance!


To a FX trader, "considerably out-of-the-money" means "low delta" and "close to at-the-money" means "close to 50% delta." That is, moneyness is measured in terms of delta.

A useful way to understand this is that delta measures the probability of finishing in the money[1]. A 10% delta option has 10% chance of finishing in the money; a 50% delta option has 50% chance of finishing in the money. But over larger maturities, larger spot moves are likely so a 10% delta option at long maturity has strike much farther from the forward than a 10% delta option at short maturity.

[1]Mathematically, the simple delta is a discount factor times the foreign-numeraire risk-neutral probability of finishing in the money. Psychologically, just think of it as probability.


It depends on how close your strike is to the forward (at expiry). Lets say you have an option which is expiring in a week, the forward will be close to the spot. Hence an out of the money strike for such as option will be closer to the at the money for an option expiring in 6 months (where the forward is pretty far from the spot).


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