# Forward price of dividend paying asset and IV skew asymptotics as $T\to\infty$

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?

EDIT:

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.