Long-Maturity Implied Volatility Skew with Dividends and Rates

Summary

The document examines how deterministic interest rates and dividend yields affect the asymptotic behavior of implied-volatility skew as option maturity grows. It starts from the forward price of a dividend-paying asset and notes that the Black–Scholes quantities used to analyze skew depend on the forward-to-strike relationship and implied volatility. An earlier argument that the smile flattens at long maturities had assumed zero rates and dividends, leaving the more general case uncertain.

The proposed resolution is to normalize call and put prices by the dividend-adjusted spot value and study implied-volatility slope at fixed log-moneyness, rather than at fixed strike. Under this framing, the analysis does not need a separate assumption that forward prices remain bounded. The text reports this as a simplification of the prior argument, but supplies no full derivation, cited paper, or empirical test. Its conclusion is therefore tied to the stated asymptotic setup and does not establish behavior under stochastic rates or dividends.

Key ideas

• Forward prices for dividend-paying assets depend on rates, dividend yield, and time to maturity.
• Long-maturity implied-volatility skew analysis can change when nonzero rates and dividends are included.
• Normalize option prices by the dividend-adjusted spot value to formulate the asymptotic comparison.
• Evaluate the implied-volatility slope at fixed log-moneyness rather than fixed strike.
• The proposed method removes the need to assume bounded forward prices within the stated setup.

Tags

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


# 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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.