During the early days of 2018 we have seen a consistent and huge move upwards in price in the SPX index, it has been a relentless move that is pushing the index to all time highs on a daily basis for almost all of the sessions in January so far. At the same we have seen a consistent increase in implied volatility, in particular VIX has been trending up at the same time the market is up and this is confusing a lot of market participants, however there is a valid explanation on why implied volatility can be higher with the market trading higher.
To understand this we need to remember what is the purpose of the VIX index. Recall that VIX is the expected variance in the SPX for the next 30 calendar days quoted in annual volatility points (as it is easier to read than pure variance). For today for instance is around 10.92%, which means that SPX options indicate an expected variance from now until Feb 23 of an equivalent 10.92% annual volatility. However the question is how do option dealers come up with those expectations ? Why not 9%? Or 12%? Or 15%? As you can imagine the 10.92% is a forecast and in fact it is a complex forecast that uses very complex mathematical and statistical models from each option dealer, and through price discovery settles on that market consensus value. The funny thing is that despite the complexity and secrecy of the mathematical models the most important factor in the forecast of future variance is past variance. In fact it is such an important factor that we can almost ignore the mathematical mambo-jumbo and focus only on past variance to understand the forecast.
The good thing is that compute past variance is tremendously simple, and in fact all of the information is publicly available, we don’t need any secret formulas or complex algorithms. There are two ways of computing realized variance, there is the pure statistical sample variance formula that is present in all mathematical packages, including something like Excel. So the only thing we need if to compute the last 20 log returns (as they cover about 30 calendar days) and get the sample standard deviation and scale it to annual volatility points like this:
Realized vol = StandardDeviationLogReturs*sqrt(252)*100
Where Excel can compute the standard deviation. If we do that for the last 20 sessions so far we get the following value:
Realized Vol = 6.73%
That is very interesting at first sight it seems that the forecast of future variance is overpricing a lot the past realized variance. However we need to remember that Wall Street tends to do things in a very different manner than normal statistics, and this is no different. In the real trading world, more in particular in the world of Variance Swaps which are financial instruments that trade all of the time, variance is defined slightly different than the statistical counterpart. The main difference is that the usual definition of sample variance is this one:
Sample_Variance = 1/(n-1) * Sum( (LogReturn-mean)^2)
However Wall Street drops the mean as it is a very small value most of the time (the mean daily move is very small most of the time). So here is the Wall Street version:
WS_variance = 1/(n-1) *Sum(LogReturn^2)
Most of the time the difference between those two computations is very small, however due to the incredible strong trending market we have seen, the difference now is actually pretty big. If we use the last formula to compute realized volatility for the last 20 sessions we get:
WS Realized Vol = 8.12%
So as you can see due to the omission of the mean we are getting very high values of realized volatility so a VIX of 10.92% it is only a tad higher than past variance and the premium is the usual variance risk premium we always see on the market (2.8% of VRP is very low actually). The conclusion here is that current values of VIX seem fairly close to recent variance (as compute by Wall Street) making the VRP fairly priced and therefore we can’t assume that the high levels of IV are due to market makers being pessimistic about the future at all, they are just pricing VIX as business as usual at this time.