To calculate the VIX are SPX calls and puts simply compared against the current price of the SPX?

Question

I understand the math behind the VIX calculation is relatively complex but I believe the concept is relatively simple. If I understand it correctly, the VIX is calculated in part by aggregating the prices of SPX calls and puts between 23 and 37 days into the future. To get the result (the VIX itself), is that aggregated price then compared to the current price of the SPX and the greater the difference (plus or minus, doesn't matter), the higher the VIX?

Answer 1

The VIX index is basically the square root of the theoretical fair variance swap strike. This logic was chosen because the metric allows to incorporate skewness and kurtosis (convexity) of the vol surface, which in turn adds additional information content to the number.

The intuition behind a Var swap is that a vanilla options trader, following a delta-hedging strategy, is essentially replicating the payoff of a weighted variance swap where the daily squared returns are weighted by the option’s dollar gamma, which is highest near the strike. Taking this argument one step further, a fair variance swap can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes. These weights are being inversely proportional to squared strikes, an application of the Black Scholes closed-form formula for gamma, which ensures results in constant dollar gamma as shown below.

Now, you do not have a continuum of strikes available, which is why the VIX is computed using discrete values. Specifically each strike that goes into the computation (there is some filtering), is computed as delta_K/k^2 * df * opt_val_mid.

• delta_k is half the difference between the adjacent strike prices on either side of 𝐾
• k^2 is strike squared
• df is the discount factor (computed with minutes to expiry accuracy from constant maturity treasury rate yield curve with cubic spline interpolation to get the yield on the expiration dates)
• opt_val_mid is the mid of the options bid ask at that k (only OTM calls and puts are used, except for ATM, which uses the average of call and put).

The sum of all such values for all strikes (meeting the filtering criteria) for the near term (below 30 days) and next term (above 30 days) is combined to a 30 day weighted average.

The white paper is very clear and even replicates the entire VIX calculation step by step (only minor details like getting the correct df are ignored).

However, we can make our life a lot easier by assuming we do have a continuum of strikes available. In the GIF below, I plot a fictional vol surface at the bottom, and compute the Fair variance swap strike as the integral of weighted prices of out-of-the-money options over all strikes. While this sounds incredible difficult, it is straightforward to follow graphically:

• The exact details, formula and most of the Julia code used here can be found in this answer from Quantitative Finance SE. For the purpose of this answer, just remember that the VIX is the area under the weighted OTM options (Puts red / Calls orange).
• A vol surface (according to Black Scholes) does not exists. IVOL is known and constant. This is the starting point of the GIF (IVOL is a flat horizontal line). In this case, IVOL and the VIX are identical.
• The vol surface exists mainly because there are fat tails, skewness, heteroscedasticity, jumps (crashes), and so forth. None of these real-world phenomena are featured in the Black Scholes formula. The market just developed ways to account for many of the shortcomings of Black Scholes. You can read plenty more details here.
• Adding skew (the line tilts to one side) or adding kurtosis (the curve bends upward), not only adjusts for the shortcoming in Black Scholes, but also adjusts the VIX (Fair Var Swap strike, shown in the chart title). Usually, VIX is higher, relative to ATM, the more pronounced skew and kurtosis are.

Since imgur does not allow larger files, I am also adding a screenshot of the GIF, so that it is easier to see the details.

Answer 2

Option prices encode expected volatility via a probability distribution of future SPX prices. The VIX is essentially a normalized standard deviation of that inferred distribution. Option prices don't give a single prediction for the future SPX price that could be "plus or minus, doesn't matter".

The closest thing to a single prediction is the expectation value of the distribution, which due to arbitrage is always tied to the current SPX price (adjusted for interest and dividends, similar to futures trading).

Answer 3

No, it's not that simple. The VIX is intended to be a measure of implied volatility in the market. Yes, the way you measure implied volatility is by comparing option prices (all else being equal, the higher the volatility, the higher the price of an option), so there is probably very high correlation between the VIX, and the price of options relative to the SPX (e.g. dividing), but it's not as simple as taking the total and subtracting off the SPX. The arithmetic difference is not as important as the relative difference.