VIX Formula Intuitive Breakdown

The VIX index is something I always thought was interesting. I looked at it pretty frequently. I was introduced to it during some classes at Penn, and learned more about it through a data science for finance course I took a while ago. Market fear is a decent way to put it, but I don't think it expresses what is under the hood for its mathematical expression. I decided it would be a good exercsie to derive this calculation to better understand the nature of the index, but also as a way to communicate how I interpret the components of the derivation to explain the intution behind its formula. The VIX (Volatility Index) formally measures expected market volatility over the next 30 days using S&P 500 option prices. Here, we break down the derivation into parts to better understand each component. This breakdown assumes some mathematical maturtiy and some finance knowledge, specifically put-call parity. I talked about put-call parity in my previous blog about Martingales.

VIX = 100 \times \sqrt{\frac{2 e^{r T}}{T} \sum_{i} (\frac{Δ K_{i}}{K_{i}^{2}} Q (K_{i})) - \frac{1}{T} (\frac{F}{K_{0}} - 1)^{2}}

VIX

= 100 \times \sqrt{} (

\frac{2 e^{r T}}{T}

\sum_{i} (\frac{Δ K_{i}}{K_{i}^{2}} Q (K_{i}))

- \frac{1}{T} (\frac{F}{K_{0}} - 1)^{2}

)

Variable Definitions

$r$ : The risk-free interest rate.

$T$ : Time to expiration of the options used in the VIX calculation.

$K_{i}$ : The strike price of an option contract.

$Δ K_{i}$ : The interval between adjacent strike prices.

$Q (K_{i})$ : Risk-neutral probability density for strike

K_{i}

$F$ : The forward price of the underlying S&P 500 index.

$K_{0}$ : The first strike price below the forward price

F

VIX Chart

Below is an interactive chart showing the relationship between the VIX (Volatility Index) and the S&P 500 over time. You can zoom in, hover over data points, and explore trends interactively.