The VIX index, often called the "fear index," is a measure of the implied volatility of S&P 500 index options. It is calculated and maintained by the CBOE (Chicago Board Options Exchange). The VIX represents the market’s expectation of stock market volatility over the next 30 days, and it can be used as a barometer of investor sentiment and market volatility.
The VIX index calculation is based on the prices of S&P 500 index options with near-term expiration dates. Specifically, it considers a weighted average of the prices of out-of-the-money call and put options across various strike prices. Here’s a step-by-step guide on how the VIX index is calculated:
1. Obtain option prices: Collect the mid-quote prices of call and put options on the S&P 500 Index for the nearest expiration month and the next nearest expiration month.
2. Calculate option deltas: Determine the price change of the option with respect to the price change in the underlying S&P 500 Index (e.g., using Black-Scholes model).
3. Select out-of-the-money options: Only consider options with deltas less than a specific threshold (e.g., 0.5) and discard other options. This filters out in-the-money options and focuses on out-of-the-money options, which are more sensitive to changes in volatility.
4. Calculate the option’s implied volatility: Compute the implied volatility for each out-of-the-money option using option pricing models, such as the Black-Scholes model.
5. Calculate the contribution of each option to the VIX index: The contribution of each option is calculated by taking the square of the option’s strike price, multiply it by the option’s price, and divide by the square of the strike price. This is then multiplied by a risk-free interest rate term structure.
Mathematically, the contribution of an option with strike price K and option price P at time t is:
$$\frac{P_t K^2 e^{RT}}{(K^2)^2}.$$
where R is the risk-free interest rate and T is the time to expiration.
6. Compute the VIX index: Finally, sum up the contributions of all out-of-the-money options, take the square root, and annualize the result by multiplying by 100.
The VIX index value can be expressed mathematically as:
$$VIX = 100 \times \sqrt{2\pi\sum_{i=1}^n\frac{P_i K_i^2 e^{R_i T_i}}{(K_i^2)^2}\times\frac{1}{T}},$$
where n represents the number of out-of-the-money options considered, and T is the time to expiration (in years).
It’s worth noting that the VIX index itself cannot be directly traded like stocks or other securities. Instead, traders and investors can use VIX futures, VIX options, and various Exchange Traded Products (ETPs), such as ETFs and ETNs, which are designed to track the VIX or take positions based on their market outlook and investment strategies.
Keep in mind that the VIX index is just one measure of market volatility and should be used in conjunction with other tools and analysis. It’s also important to be aware of the limitations and potential risks associated with trading VIX-related products, as they can exhibit complex behavior and may not provide the exact exposure to the VIX index as expected.