Assuming for simplicity deterministic interest rate and dividend yield, then the forward price of an asset is $$ F = Se^{(r-q)T} $$ where $T$ is maturity date.

In studying IV skew asymptotics, the behaviour of the quantitites $$ d_{\pm} = \frac{\log F/K}{I\sqrt T} \pm \frac{I\sqrt T}{2} $$ is important. For example, in an answer to this question I argued, using these quantities, that the smile flattens as $T\to\infty$. However, I silently assumed that $r=q=0$.

For $r,q\neq 0$ I am not sure it is very clearcut what $\lim_{T\to\infty} d_{\pm}$ are, and hence how the skew behaves as $T\to\infty$.

Does anyone have any ideas about this and/or can refer to a paper that explicitly discusses this?

An easy way out would be to assume that there are constants $C_1,C_2 > 0$ such that for all $T$ we have $C_1 \leq F \leq C_2$. But would such a condition be realistic/possible?


For what it's worth, after working on this a bit more I found the solution to be quite simple:

It's possible to follow the argument in the question and answer mentioned above, but using normalized call/put prices where the normalization is to divide call/put prices by $Se^{-qT}$, and instead of looking at the slope of the IV for fixed strike to look at the slope of the IV for fixed moneyness $k$ where $k := \log Ke^{-rT} / Se^{-qT}$.

So there is no need to assume anything about boundedness of forward prices.



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