# 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?

• Can we have more info on this IV surface? What equity (single name or stock index)? How was it built? That being, said the ATMF term structure can sometimes indeed be non monotonic (decreases on the short term and reincreases afterwards), but it is also important to know if it is an artefact of the method used to build it, or if its really the case in the market. – Quantuple Mar 18 '16 at 20:48
• Source: blog.nag.com/2013/10/… – Andrew Maliska Mar 19 '16 at 6:14
• Apologies for confusion about the particulars from the picture, the example was meant to be illustrative but your point about understanding the method used to generate the IV surface is well taken. I'm interested in what rationale(s) would lead to a consistently non-monotonic term structure across different strike prices as in the example (as opposed to a mere dislocation), especially if it's the case in the market as opposed to an artifact of a given method. – Andrew Maliska Mar 19 '16 at 6:36
• I see. I have provided you with a partial answer + example below. Hope this will help. – Quantuple Mar 19 '16 at 15:50

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.

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.

• I think we're talking about monotonicity along different axes - what you're describing is (I think) the relationship between implied volatilities for a single expiration date, as described by skew (the above graphic displaying a volatility smile). I was asking about monotonicity of the term structure for a given strike price. Please let me know if I've misunderstood your answer. – Andrew Maliska Mar 23 '16 at 2:17
• Andrew Maliska, I think that what @dm63 meant is that as soon as you let skewness or fat tails kick in, then you mechanically alter the implied volatility levels. And this is perfectly true: if you know how skewness and leptokurticity of your underlying's distribution evolve through time, you should be able to work out the shape of the ATM volatility term structure. Relating that to my post, correlation and volatility of volatility are clearly levers on which you can play to tune smiles' skewness and convexity in Heston, but also, incidentally, the ATM volatility term structure. – Quantuple Mar 23 '16 at 23:40
• I've edited my original answer to make that appear. – Quantuple Mar 23 '16 at 23:45
• Ah yes I see I did misunderstand the question. As a practitioner I'd say that the variation of implied vol versus time to expiration is driven by many factors, the most important being: (a) if the market is currently extremely volatile, the term structure is usually downward sloping; and if it is currently very quiet, it is usually upward sloping - reflecting the expectation that the future will likely revert to a 'normal' environment (b) volatile events whose date is known (Fed meetings, earnings announcements etc) will cause a local spike in vol around those dates (c) supply and demand – dm63 Apr 4 '16 at 0:49