How the Vix is Calculated

VIX Variance Contribution Calculator

Calculation Result:

Weighted Option Price Contribution to Variance:

This value represents the contribution of a single representative option to the overall variance calculation used in VIX. The VIX itself is derived from summing many such contributions across a range of options and expirations.

The VIX, often referred to as the "fear index," is a real-time market index that represents the market's expectation of 30-day forward-looking volatility. It is derived from the prices of S&P 500 (SPX) index options and is published by the Chicago Board Options Exchange (CBOE). Unlike traditional volatility measures that look backward at historical price movements, the VIX is forward-looking, making it a crucial indicator for market sentiment and potential future market turbulence.

What Does the VIX Measure?

At its core, the VIX measures the implied volatility of a wide range of S&P 500 options. Implied volatility is the market's forecast of a likely movement in a security's price. When the VIX is high, it suggests investors expect significant price swings in the S&P 500 over the next 30 days, often associated with market uncertainty or fear. Conversely, a low VIX indicates expectations of calmer, less volatile market conditions.

The VIX Calculation Methodology: A Deep Dive

The VIX calculation is not based on a single option's price or a simple average. Instead, it's a sophisticated, model-free approach that aggregates the implied volatilities of a wide range of out-of-the-money (OTM) S&P 500 options across two near-term expiration dates. The methodology was significantly revised in 2003 to use a broader set of options, making it a more robust measure.

Key Components of the VIX Calculation:

1. S&P 500 (SPX) Options: The VIX uses both call and put options on the S&P 500 index.
2. Out-of-the-Money (OTM) Options: Only OTM options are included. For calls, these are options with strike prices above the current forward index level. For puts, these are options with strike prices below the current forward index level. The at-the-money (ATM) options are also included if they are the first OTM strike.
3. Two Near-Term Expirations: The calculation uses options from two consecutive expiration months. The first expiration must have at least 8 days but less than 30 days to expiration. The second expiration must have more than 30 days to expiration. This ensures the VIX always reflects a 30-day outlook.
4. Risk-Free Interest Rate (R): This is typically derived from U.S. Treasury bills or other short-term government securities, reflecting the cost of money over the options' life.
5. Forward Index Level (F): This is determined by finding the strike price where the absolute difference between the call and put prices is minimized for a given expiration. It represents the market's expectation of the S&P 500 index level at expiration.
6. Strike Interval (ΔK): For each strike price, a ΔK is determined. For the lowest strike, it's the difference between that strike and the next higher strike. For other strikes, it's half the difference between the adjacent strikes (the strike below and the strike above).
7. Option Midpoint Price (Q): For each option, the midpoint between its bid and ask price is used to ensure a fair representation of its market value.

The Core Formula: Calculating Variance

The VIX is essentially 100 times the square root of the expected variance of the S&P 500's return over the next 30 days. The variance for each expiration (σ²) is calculated using the following formula:

σ² = (2/T) * Σ (ΔK / K²) * e^(RT) * Q - (F/K₀ - 1)² / T

Where:

• T = Time to expiration (in years)
• K = Strike price of the option
• ΔK = Interval between strike prices
• e^(RT) = Discount factor, where 'e' is Euler's number, 'R' is the risk-free rate, and 'T' is time to expiration.
• Q = Midpoint of the bid-ask spread for the option
• F = Forward index level
• K₀ = The strike price immediately below the forward index level (F)

This formula sums the weighted contributions of all included OTM options for a given expiration. The term (ΔK / K²) * e^(RT) * Q represents the individual contribution of each option to the overall variance. The final term (F/K₀ - 1)² / T is a correction factor.

Combining the Two Expirations

Since the VIX aims for a 30-day volatility measure, the variances calculated for the two near-term expirations (one just under 30 days, one just over 30 days) are interpolated to create a single 30-day variance. This interpolated variance is then square-rooted and multiplied by 100 to arrive at the final VIX value.

Using the VIX Variance Contribution Calculator

The calculator above provides a simplified illustration of one crucial component of the VIX calculation: the Weighted Option Price Contribution to Variance. It allows you to input representative values for a single out-of-the-money option and see how these parameters contribute to the overall variance sum.

• Representative Strike Price (K): The strike price of an out-of-the-money option.
• Representative Option Midpoint Price (Q): The average of the bid and ask price for that option.
• Days to Expiration (T): The number of days until the option expires.
• Risk-Free Rate (R): The annual risk-free interest rate, expressed as a percentage.
• Typical Strike Interval (ΔK): The common spacing between strike prices for SPX options (e.g., 5 points).

By adjusting these inputs, you can observe how changes in option prices, time to expiration, or interest rates affect an individual option's contribution to the market's expected volatility. Remember, the actual VIX aggregates hundreds of such contributions across multiple strike prices and two expirations to produce its final value.