Why Heston Dynamics Can Make ATM Implied Volatility Non-Monotonic
Summary
The document explains why at-the-money implied volatility can rise or fall across option maturities. It challenges the assumption that the term structure should be monotonic and identifies several possible drivers, including return autocorrelation, dependence between price and volatility moves, volatility of volatility, and changes in instantaneous volatility. These forces affect the evolving moments of the underlying’s price distribution.
A Heston-model illustration holds initial variance, long-run variance, and mean-reversion speed fixed, then varies correlation and volatility of volatility while pricing ATM options at different maturities. The example shows that volatility of volatility can produce either monotonic or non-monotonic curves even with interest rates set to zero, so rate uncertainty is not required to explain the pattern. A second answer points to fat-tailed stock returns and volatility premiums in out-of-the-money options as a broad explanation. The discussion is conceptual and model-based; it does not establish which mechanism dominates in any particular market or dataset.
Key ideas
- ATM implied volatility need not move monotonically with option maturity.
- Price-volatility correlation, volatility of volatility, and changing instantaneous volatility can shape the term structure.
- A Heston model can generate non-monotonic ATM volatility even when interest rates are fixed at zero.
- Fat-tailed return distributions and wing premiums can also contribute to volatility patterns across maturities.
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Full text
# Why is the term structure of the implied volatility surface non-monotonic?
# Why is the term structure of the implied volatility surface non-monotonic?
Does this reflect expectations & uncertainty about interest rates (exposure to rho?), event driven concerns about the underlying, or something else?
## Answer by Quantuple (score 1, accepted)
https://quant.stackexchange.com/a/24966
On many occasions may the ATM volatility term structure implied from option prices exhibit non monotonicity. You could actually turn the question on its head and ask yourself why should it be monotonic?
> Does this reflect expectations & uncertainty about interest rates (exposure to rho?), event driven concerns about the underlying, or something else?
It could, but not necessarily. Several factors have a role to play: log-returns auto-correlation (or correlation between underlying moves and volatility), volatility of volatility, or simply time-varying instantaneous volatility (even if we assume the latter is deterministic) are just examples.
As a matter of fact, any factor which can influence the time evolution of the second and higher moments of the price distribution will have a role to play (e.g. deterministic time-varying instantaneous volatility = time-varying second moment, correlation and volatility of volatility = time-varying skewness and kurtosis etc.).
What I mean is that, even without considering stochastic interest rates, a simple stochastic volatility model such as Heston can already give rise to non-monotonic ATM term structures. This is what I illustrate below.
Let $v_0$ represent the initial variance, $\theta$ the long run variance, $\kappa$ the mean reversion speed, $\rho$ the correlation between vol/spot moves, $\xi$ the volatility of volatility. The interest rate is set to zero to show that this is not necessarily a determining factor. Without loss of generality let us pick $v_0=0.625$ ($\sqrt{v_0}\approx 25\%$), $\theta=0.0650$ ($\sqrt{\theta}\approx 25.5\%$), $\kappa=3$ and finally $\rho=0$ and $\xi=0.5$ (by default).
The idea is then to keep $v_0$, $\theta$ and $\kappa$ fixed (so that the term structure of expected realised variance, i.e. the price of variance swaps under Heston $K_{var}(T)=E_0[1/T \int_0^T v_t dt]$, remains unchanged) and let (1) the correlation parameter vary, (2) the vol-of-vol parameter vary. We then price ATM options of growing maturities under Heston, imply their BS volatility and plot the resulting curves.
As you can see from the figure below, both parameters shape the ATM implied volatility terms tructure. In the bottom subplot, I notably show how different values of volatility of volatility lead to either monotonic, or non-monotonic term structures ceteris paribus.
## Answer by dm63 (score 0)
https://quant.stackexchange.com/a/24984
Since equity option prices on the "wings" (i.e. deep out of the money puts and calls) often trade at significant volatility premiums to ATM, it's highly unlikely implied vol will be monotonic. The basic reason for this is that stock price distributions have fat tails relative to a lognormal distribution. One can get much more complex than that, but that's the basic situation.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.