A Journey to the Center of the VIX

How S&P 500 options trading activity affects the VIX figure.

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:

\(\sigma^2 = \frac{2}{T} \sum_{i=1}^n \frac{\Delta K_i}{K_i^2} e^{rT} Q(K_i) - \frac{1}{T} \bigg(\frac{F}{K_0} - 1 \bigg)^2\)

where:

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

\(F=Strike\;Price + e^{rT}.(Call\;Price - Put\;Price)\)

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:

\(\begin{align*} \sigma_{calls}^2 &= \frac{2}{T} \sum_{i=1}^n \frac{\Delta K_i}{K_i^2} e^{rT} Q(K_i) - \frac{1}{T} \bigg(\frac{F}{K_0} - 1 \bigg)^2 \\ \sigma_{putss}^2 &= \frac{2}{T} \sum_{i=1}^n \frac{\Delta K_i}{K_i^2} e^{rT} Q(K_i) - \frac{1}{T} \bigg(\frac{F}{K_0} - 1 \bigg)^2 \end{align*}\)

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.