A Journey to the Center of the VIX

We can start by looking at the VIX formula to determine what are the points of influence that option trading activity can have on the calculation. The VIX is calculated using the following formula:

where:

T is the time to expiration (in years). F is the option implied forward price, given by

r is the risk-free rate,K0 is the first strike price equal to or otherwise immediately below F, Ki is the strike price of the kth out-of-the-money (OTM) option (if Ki > K0 a call, but if the opposite a put, and if Ki = K0 both a call and put are used). Delta Ki is the difference between two consecutive strikes, and Q(Ki) is the price of the OTM option with strike i.

The variables that change due to option prices are F, K0, Delta Ki and Q(Ki). F is affected by the changes of the bid-ask spread of options near the current price of the S&P 500. K0 is affected by changes in the bid-ask spread of options with strike close to F. Delta Ki will change if an option is added or included in the calculation. The VIX methodology excludes all other put (calls) whose strike is above (below) the strikes of two consecutive options of the same type that have zero bids. Q(Ki) is affected by order flow.

In theory, it is possible but impractical to manipulate the VIX calculation by influencing Delta Ki and Q(Ki). To influence Delta Ki, you would need to reduce the number of options with zero bids. Since you don't have to execute a trade and use actual money, an investor can place orders on more options to make this happens. Q(Ki) depends on the quotes themselves and this can be influenced by making large orders especially on deep OTM options which receive a higher weight in the calculation.

Anyway, we can see from this that the number of options included in the calculation and the bid-ask spread (especially it's skew) affects the VIX figure. The time to expiry T, and risk-free rate have very small effects on the figure. The S&P 500 price also affects the calculation by affecting F and  K0.

I'll tell you another reason the VIX is called a 'fear' gauge—it’s because no one knows what to do when it goes up.

You can't go long or short vol right then and there. Firstly, because VIX futures (and therefore VIX ETPs) have a no-arbitrage relationship to the VIX, so they don't have to do what the VIX is doing, and secondly, because you can't tell how high it will go, or if and when it will pull back. You can't go long or short equities either because by the time the VIX is going up, the corresponding market drawdown is well in play, and the VIX won’t help you figure out if it is going to continue or snap back.

So, what is it good for, then? There are three possibilities: an estimate of volatility going forward, a measure of the amount of hedging and a measure of investor sentiment. I tried to answer that in a previous post. The VIX isn’t a great estimate of future volatility because it overestimates volatility in normal times and underestimates it in crisis times. I showed a strategy that used this stylized fact to take contrarian trades when an underlying asset breaches the range implied by its vol index. The strategy did very well in normal times, but when a crisis hit, profits were wiped out.

Because the VIX underestimates volatility in a downturn, it is not good for hedging. Some say the VIX is actually a measure of how much hedging is going on. The argument is that if it’s going up, it means that people are buying more puts than calls to hedge their portfolios. The rebuttal against this idea is that by the same theory, a rising VIX could be signaling greed as more people buy calls than puts. The problem is that VIX doesn’t distinguish fear and greed for you; just like the variance or standard deviation, it estimates both upside and downside volatility. There have been academic attempts, such as Serur et al. (2021), to make this distinction by for example calculating two sets of VIXs: one based on calls for the upside volatility and one based on puts for the downside volatility,  like so:

Serur et al. (2021) later come to the same conclusion I had noted in an earlier post: it is better to look at the contribution of puts to calls in the VIX calculation to gauge hedging. However, this is also not quite satisfactory. In said post, I showed that this PCR declines in a market downturn. This happens because as the market drops, a lot more puts go in-the-money, and calls start to go OTM, hence there are more calls than puts in the VIX calculation, and that would wrongly be interpreted as the market becoming greedier. One can analogously reason that a higher contribution of puts than calls in a bull market would erroneously be interpreted as an increase in fear.

Put simply, the PCR doesn’t distinguish between hedging and speculation/gambling. A related idea is that of Son & Robert (2012) who note that the interpretation of PCR as a measure of sentiment is problematic because when someone buys a put, you can’t tell whether they are expecting a downturn (bearish) or simply hedging a long position (bullish). They also note that if you use trading volume rather than number of options to calculate the PCR, you can’t tell if the volume is from newly minted positions, liquidations or just a heavy trading day, but using open interest instead can help with this. They also further note that PCR limits the bullish zone between 0 and 1, and the bearish zone between 1 and infinity, which makes it difficult to establish buy/sell signals.

But let’s take it for granted that you could somehow look at the PCR (tally-based or volume-based) of the VIX calculation options to gauge the amount of hedging. The chart below shows the respective proportions of puts and calls each month overlaid with the VIX close. Only about 20-30% of the options are calls with this figure rarely going above 40%. However, when this figure goes north of 40%, it coincides with a rise in VIX. That more puts than calls are used in the VIX calculation would imply that a rise in the VIX is due to hedging, but this isn't really the case. As you can see, the proportion of calls used in the calculation rises with the VIX. This happens because the VIX calculation is based on out-of-the-money options; when the market drops, puts go in-the-money, and more calls go out-of-the-money, so more calls than puts are used in the calculation at that time. So a rise in the VIX is not really a sign of hedging taking place.

How about the volume PCR? The chart below shows the average volume of puts relative to calls of S&P 500 options vs only the options used in the VIX calculation. There are some interesting things to note about this chart. First, both PCRs move together. You would expect the market PCR to be high when the VIX was high, even if the VIX calculation options’ PCR drops when the VIX rises, but both go down when the VIX rises. The PCRs were calculated using volume so what we are seeing is increased call volume, both in the overall market and in the VIX calculation options, when the VIX rises—the opposite of hedging. You would be excused for thinking this is a sign of speculation as the market goes down; afterall, greed is the fear of missing out; but you’ll see later that this isn’t the case.

Both charts so far show metrics that are agnostic to the total volume. That is why the S&P’s PCR remains around 2 despite an increase in total volume over time. If you really must see it, here is the total volume chart. Nothing special to note other than the gradual rise in total volume over time.

The VIX calculation does not consider volume. Earlier we saw from the VIX formula that an increase in the number of options used to calculate the VIX affects the figure. The reason this happens is that as more options are used, their bid-ask spreads and difference between consecutive options go into the calculation. Volume doesn’t come anywhere.

So, to recap. We've seen that a rise in the VIX coincides with a rise in the number and volume of calls compared to puts in the VIX calculation options, and this is because as the market drops, more calls than puts go out-of-the-money so more calls than puts are used in the calculation. Just how many calls and how many puts are shown in the chart below. It is different from the earlier chart because the latter one had the proportions, and this one has the total volume. Notice how the number of calls, and the VIX trend together.

The interesting this about this is that the increase in OTM calls is technical and not the result of speculation. It happens because of how the VIX is calculated, and in particular, because the market is going down. That is, the price of the S&P 500 affects the VIX figure. Later, you will see a chart of how much the VIX moves due to a 1% move in the VIX aka the VIX beta. So, the VIX does not measure the amount of hedging, but it indeed measures market sentiment and in fact measures the 'fear' factor by capturing how many puts previously available for hedging no longer are.

Therefore, all the VIX is good for, is a (flawed) measure of sentiment. If I were a conspiracy theorist, I would argue that the VIX was reinvented by Goldman Sachs to tell them when to expect a lot more market-making business. That's because as we shall see, there are two main reasons the VIX rises: either the market is(has) going (gone) down, or the bid-ask spread has widened—both of which are good for GS's business.

The reason I said flawed measure of sentiment is because the VIX is also affected by the quotes of the options, and these don’t always reflect investor behavior. As discussed before, the VIX can in theory be manipulated or just affected by any factors that would influence the quotes. Still, I think a better decomposition of the VIX into greed and fear would be to look directly at the prices of puts and calls used to calculate the VIX. The reason is because options quotes are more complicated; with other assets, you can either buy or sell, but with options, you can buy or sell calls or puts, and all these actions imply different sentiments. Buying calls and selling puts are both long positions but with different sentiments about the expected volatility, while selling calls and buying puts are also the short positions with different expectations for vol. You buy a call when you expect the asset to go up and volatility to increase, and you sell a put when you expect the asset to go up, but volatility to decrease.  

The VIX calculation uses the midpoint of the bid-ask spread, so the width of the spread doesn't matter if it has no skew. To check if there is a skew, and in which direction, you can check if the median bid-ask spread is higher than the corresponding mean. That is shown in the chart above. Normally, the put is positively skewed (more bid-ask spreads are higher than the average), while calls are negatively skewed (bid-ask spreads are lower than the average). But when the VIX spikes, either the put bid-ask spread widens (2015), and/or the call bid ask spread shrinks(2019, 2020, 2022), or both bid-ask spreads increase (Dec 2018).

The rises in the bid-ask spread coincide with VIX spikes. Call bid-ask spreads are typically higher than put bid-ask spreads which is because puts are generally more liquid than calls. When the VIX spikes, the call bid-ask spread rises much higher than the put bid-ask spread. A sign that calls are at that less liquid. This is important because we’ve seen that when the market drops, more calls than puts go into the VIX calculation, and deep OTM options get a higher weight in the final figure. Now we see that at the same time, calls have wider spreads so their net effect on the VIX calculation is quite high. To see this even better, I took the maximum bid-ask spread each month for the calls and the puts. You can see that most of the time they are quite close, but when the VIX spikes, the maximum call bid-ask spread is higher than it's put counterpart.

Considering this and the earlier observation that the number of calls used in the VIX calculation rises in a downturn, how it is a downturn that causes the VIX to rise and not the other way around. When it comes to the VIX and the S&P 500, the S&P 500s the VIX—the tail wags the dog. This result was documented in a 2014 study by (Ozair, 2014). To quote the author:

“First we perform the Granger Causality test. The test results, indicate that both time series the VIX minute returns and the SPX minute returns Granger Cause each other’s time series, although the F-statistics for testing the Causality of the SPX time series on the VIX time series is significantly higher than the F-statistics of the opposite causality relationship, implying (and also consistent with the VAR estimated model results discussed above) that the SPX causality on the VIX time series is of a greater impact and magnitude.”

What this means is that the VIX is best used as a real-time measure of sentiment than a volatility estimate. This explains why the VIX grossly overestimates volatility in normal times and underestimates it during a turmoil. It also explains why the VIX can overreact to a relatively small move in the S&P 500.

In a previous post I showed a chart of how much the S&P 500 moves for a 1% move in the VIX but given this new finding it is better to do it the other way around: how much the VIX moves for a 1% move in the S&P 500.  Here is a chart of that.

You can see that over time, the S&P 500’s influence on the VIX has diminished, even though the p-values remain statistically significant. Some research by Ambrus Group may help explain why it could be the result of an increase in 0DTE option trading.

We can also look at how much the VIX moves based on the median bid-ask spread for both calls and puts. Let’s first look at how wide those spreads typically are as shown in the table below.

You can see that call spreads are typically higher than the put spreads, and that the average spread and the range increase in the crisis years.

Here we see that even though the calls have higher spreads, puts are more often statistically significant than calls when it comes to influencing the VIX calculation, and move the VIX by a higher magnitude than do calls.

The discussion in the post explain why the recent VIX spike may have been an anomaly. I was limited by data because I couldn’t get some 2024 S&P 500 options, but this article and this post

explains why that may be the case.

PS: If you note some mistakes in my thinking or in the methodology, or want to reach out for any other reason, you can message me.

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Appendix & References

Ozair, M. (2014). WHAT DOES THE VIX ACTUALLY MEASURE? AN ANALYSIS OF THE CAUSATION OF SPX AND VIX. 3(2).

Serur, J. A., Dapena, J. P., & Siri, J. R. (2021). Decomposing the VIX Index into Greed and Fear (SSRN Scholarly Paper 3806521). https://doi.org/10.2139/ssrn.3806521

Son, L., & Robert, M. (2012). Predicting returns with the Put-Call Ratio. https://repository.up.ac.za/handle/2263/30616

Some extra charts and tables :)