*“Fear tends to manifest itself much more quickly than greed, so volatile markets tend to be on the downside. In up markets, volatility tends to gradually decline.”*

*Philip Roth*

**Introduction **

As options and other products whose price depends on volatility have become widespread, studying its behaviour is now a key skill required to every market participant.

The first distinction has to be made between implied and realized volatility. The latter is, in first approximation, the standard deviation of the returns of a certain asset over a specified time horizon. It can be easily calculated from time series. The former, namely implied, can be found by observing the prices of options traded on financial markets. It is the market best guess about how volatile the asset will be in the future.

Volatility obviously depends on the asset considered. Every publicly traded security has a volatility. The most famous volatility index is the VIX, which is the 30 days implied volatilities of ATM (at the money) options on the S&P 500. Its negative correlation with the index granted it its fame as “fear gauge”. For what concerns the S&P, VXST (9 days ATM implied volatility) and VXV (3 months ATM implied volatility) are also quoted.

We will begin by investigating the relationship between these 3 ATM gauges of implied volatilities.

**VXST, VIX and VXV**

In the chart above (please note axis values have been rescaled) you can find VXST, VIX and VXV plotted against the S&P500. As could be expected, the three volatility indexes exhibit strong positive correlation among themselves. However, their correlation with the S&P 500 varies widely depending on two factors: the index daily return and the time horizon examined. The first factor helps explaining the second.

In the above table, we find some statistical measures of correlation and dispersion. The lookback period has been set equal to 2 years, in order to capture the normalisation of markets after the “Eurozone Crisis”. In this time horizon, there is a decent negative correlation among the volatility indexes.

In particular, by analysing the historical correlations between the S&P 500 and the VIX we see a strong negative correlation only in late 2011 and early 2012, where it was close to -0.7 for all three indexes, and this did not happen by chance. It was a very stressful period for equities, and the S&P 500 lost about 15% in three months. How does this change the correlation between volatility indexes and the S&P 500?

Taking a closer look at the correlations between the S&P and the volatility indexes in those days in which losses amount to more than 1%, we see a much stronger relationship, displayed in the following table.

Therefore, we can argue that correlation was higher in 2011 and 2012 because the S&P was losing more often than not. When the situation came back to “normality”, the relative number of “black” days for the index was reduced and so was the correlation with the volatility indexes.

The shorter the maturity, the weaker the correlation with the S&P500 and the greater the variability. As a matter of fact, the VXST climbs higher than the VIX and VXV when there is a spike in the volatility, while in normal times the order is the reverse.

**The VXV-VIX Spread**

The VXV-VIX spread is particularly interesting. It is nearly always small and positive. However, when the S&P 500 suffers sharp losses, it gets large and negative.

A simple statistical analysis confirms our intuition. The distribution is negatively skewed, with a positive expected value and a good degree of variability. It is important to note that the VXV-VIX spread behaviour is inherently different from that of the volatility indexes. In normal times (S&P gaining or losing less than 1%), we have a slight negative correlation. When losses are non-negligible, we have almost no correlation between the spread and the S&P500.

**VXV: How precise is it?**

Taking a glance to Chart 1, we observe that the VXV is nearly always higher than the VIX. This would make sense if and only if the VIX were to be increasing over time. Yet, it has decreased steadily as we have seen over the past few years. This seems quite similar to what happens to the yield curve: longer maturities are riskier and demand a higher risk premium. This is assessed as yield for bonds, while it is implied volatility for options.

As a rough measure of how big and consistent the overestimation of 3 months volatility is, we compare the VXV to the average of the next three months VIX.

We can observe not only that the expected value is positive, but it is also quite consistently so.

Even if we did not compute the VIX-VXST spread directly, it is likely that it displays the same behaviour: VIX will on average overestimate volatility with respect to VXST.

**Trade Idea – Volatility (basic)**

As of December 3^{rd}, we have that the VIX is 12.07 and the VXX is 15.32. Our trade idea will be to go long on 1-month volatility (VIX) while going short on 3-month volatility (VXV).

From a mere volatility standpoint, there are two reasons. The first one is that the current VXV-VIX spread is above the expected value and will most likely revert to it. Moreover, the distribution is left-skewed and does not exhibit particularly fat tails. The second one is that, on average, it is profitable to go long on 1-month term volatility and to go short 3-month volatility.

**Trade Idea – Options (advanced)**

Options allow us to take a view not only on volatility, but also on the underlying. Our trade idea will be to do a short calendar spread: buy an ATM (strike price 2,075 or 2,050) 1-month call option and sell an ATM (with the same strike price) 3-month call option, as the latter is relatively more expensive.

From the point of view of the underlying, we support our position by being slightly bearish on the S&P 500 for two reasons.

The first one is that the put call ratio on the index has rebounded from 0.56 to more than 0.6, which might signal an imminent correction (as explained in https://bsic.it/putcall-ratio-guide-market-sentiment-sp-500-case/).

Secondly, the FOMC, which is the committee setting Fed monetary policy, will announce its interest rate target for December in two weeks. Even if we do not expect the Fed to raise interest rates, it is possible that the pledge to keep interest rates low “for a considerable amount of time” will be dropped. Evidence supporting this view can be found in last month minutes, which saw a long discussion about the topic, and in extremely strong non-farm payrolls figures published on December 5^{th}.

The above image shows the payoff of a short (or reverse) calendar spread at maturity. Even if it is just a rough approximation of the goals of the aforementioned trade idea, it is useful to visualize that the position tends to have negative profits when the underlying at maturity is close to the strike price, while it makes profits when the underlying moves in either directions. We wish to stress the fact that it tends to, but will not necessarily behave as the image suggests. This is because we are planning to close out the position before the expiration of the 1-month call option, while the payoff in the image is represented at expiration.

Even if an option position should not be taken and “left unmanaged” for weeks and a precise time horizon is quite arbitrary, we could target about 2 weeks for our trade. A shorter time horizon might not allow us to witness substantial change in the S&P 500, while the time decay of the position could become too big in a longer one.

We would have a lower payoff than in the standard case (i.e. the one represented in the above image) if the S&P 500 drops and a higher one if it raises, assuming the volatility term structure has a parallel shift. This is because our position is short parallel shifts in the ATM volatility term structure. However, it also profits from a flattening of the ATM term structure (i.e. VIX increasing more than the VXV). When the S&P500 drops the volatility usually undergoes both a parallel shift upwards and a flattening (short term implied volatility raises more than long term). These effects act in opposite ways, and partially offset each other.

As the S&P 500 goes down, both the calls will move out of the money. A decrease in the “moneyness” (i.e. the ratio between the underlying price and the strike price), which should be close to 100% at inception, results in higher implied volatility. As our position is Vega negative, (at least at inception) our profits will be partially hit by a decrease in moneyness. The reverse would happen in case the S&P 500, and consequently moneyness, increases.

Setting up such a position entails a hidden cost: the time decay. Our position loses value as time passes by, and it does so at an increasing pace. The closer we get to the maturity of the first option, the faster the time decay will be, unless a change in the underlying has occurred in the meanwhile. For this reason, it is extremely important to time the trade properly, and to keep track of the position closely.

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## 2 Comments

## leonardo · 16 December 2014 at 15:34

Very interesting article.. but id like to know your thinking about the gamma. How does the gamma change or contribute to the trade considering the vol position? Thanks

## jen · 11 March 2016 at 20:49

concise and good ideas on vol