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I’m doing some work around the variance risk premium currently, and I’m interested in understanding the situations when realized volatility is > implied volatility in practice.

I know in generally up-trending markets ("good times") implied vol tends to be higher than realized vol, in which case a delta hedger profiting from the vol spread will generally make money. But when might this relationship breakdown in real life; are there such situations in practice?

Appreciate the help on this

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  • $\begingroup$ Generally this occurs when realized vol is extremely high by historical standards, and the IV is lower than this because option market participants believe RV will come down. Generally in this situation the VIX Term Structure will be decreasing (backwardation). The situation around March 25, 2020 shortly after the "bounce" in S&P from its 3/23 low is typical. Time will tell if this was a good time to write/buy 3 month options or not ;) $\endgroup$ – noob2 May 3 '20 at 13:56
  • $\begingroup$ Yes this makes sense, so more or less during some stock market crash. What about for individual stocks @noob2, bar a general market crash, are there other situations in practice when RV > IV? I can only think of a stock takeover potentially, are there other types of events? $\endgroup$ – AShortSqueeze May 4 '20 at 3:52
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Since you talk about a variance risk premium, a word about some of the details involved might be important. Technically, your variance premium is the difference between the expected volatility under the physical and risk-neutral measures. In general those quantities aren't equal.

For example, in a stochastic volatility model, market incompleteness implies some premium must be attached to volatility risk. Here, you'd expect the variance risk premium to be negative because of the so-called "leverage effect" (Black, 1976) whereby measures of volatility tend to be negatively correlated with returns (at least for stock market indexes) -- intuitively, going long on volatility is an insurance because it pays during crises. There are other contexts where this would appear. In all GARCH-based option pricing models, the conditional variance process is perfectly anticipated one step ahead. However, starting from two steps ahead, this is no longer the case. As the conditional variance process under P and Q revert to different means (assuming a covariance stationnary process), you get a variance risk premium here too (but the risk comes from equity risk). Again, you'd expect this to be negative due to the leverage effect. Yet more generally still, if you introduce non-gaussian risk in the equity process, you would have a variance premium even on step ahead.

Now, what about the data? Several people have generated more or less sophisticated evidence that options require a negative variance risk premium. A simple way to go about it is to use the Breeden and Litzenberger (1978) trick (you know, you can get the conditional risk-neutral density of the underlying using the second derivative of the option price with respect to the strike). Then, you need some means of estimating a conditional density under the physical measure. Engle and Rosenberg (2002) used a GARCH model and a simulation procedure based on realized errors to get such an estimate. Then, you just remember that your Radon-Nikodym derivatives really like a ratio of densities -- i.e., if you divide one by the other, you should get a sense of what kind of pricing kernel can reconcile the time series properties with the cross-section properties of the data. When you do this, you usually find oddities: your pricing kernel is not monotonic in the space of returns. This is very odd because it means you have segments which sloppe upwards and, in a consumption-based model, that directly would translate into risk-loving rather than risk-aversion. If you take the difference of logarithms of both densities, you usually find a U-shape, suggesting your pricing kernel needs to be quadratic. Christoffersen, Heston and Jacobs (2013) proposed to introduce a term explicitly capturing changes in conditional variance in the pricing kernels used in GARCH models. (It works because the variance is related to the square of your returns, so you get that U-shape if the variance risk premium is negative).

Back to the implied volatility

I personally expect the variance risk premium to be positive at all times, but the difference between realized volatility and implied volatility is not exactly like measuring volatility under both measures. For starters, it's pretty darn obvious there are non-gaussian patterns in the return data and that immediately implies your realized volatility is polluted by higher order effects -- see, for example, Ian Martin (2017, QJE) on this. Moreover, your implied volatility isn't the same thing as Q volatility. Specifically, think about the Black-Scholes-Merton model you are inverting: it has one free parameter (volatility). Compensation for tail risk will be expressed as compensation for high volatility because it's the only mechanism BSM has to produce more "expansive" options. In a sense, a suitably weighted IV^2 index probably could be shown to be a noisy proxy of conditional Q entropy because it's going to look a hell of a lot like the VIX -- and the VIX is a consistent estimator of conditional Q entropy. So, that's the first point: those things aren't measuring what you think it measures. With that being said, I'll proceed bellow by using "volatility" a lot more loosely.

Second point is that your options have a well defined maturity, whereas the equity on which they are written do not. Your realized volatility captures the amount of movement in trading over the recent past and while it probably contains some information about the future, it's not obvious what period is concerned. On the other hand, you know exactly over which time span your IV measurements apply. Because high current volatility doesn't prevent people from betting it might fall in the next months, you can have situations where IV is lower than RV.

Over the last 30 years, using logarithmic returns on the S&P500 and a sample estimator at daily frequency, annualized volatility over 252 trading days would be somewhere around 20% on average. If you see RV up in the 50% annualized range, you might bet that it will fall in the next quarter. In that case, you'd like to be short volatility and that will end up pressuring option prices downward for some region of the volatility surface -- and, with them, IV will go down. So, if I had to bet, I'd say that around peaks of RV, you will find periods where many weighted IV indexes will be lower than RV.

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  • $\begingroup$ @ stephane, very interesting read. When you say peaks of RV you are more or less referring to stock market crashes/crisis periods, and other exogenous shocks yes? $\endgroup$ – AShortSqueeze May 5 '20 at 0:35
  • $\begingroup$ @AShortSqueeze I mean tp say local maxima. In practice, that probably happens mostly in bear markets. Most peaks you can spot with a good ol' eye ball test are indeed market crashes. $\endgroup$ – Stéphane May 5 '20 at 2:48
  • $\begingroup$ yes this makes sense $\endgroup$ – AShortSqueeze May 5 '20 at 3:31

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