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Key Takeaways
- Implied correlation reconciles index and constituent implied variance through a weighted pairwise-correlation calculation.
- The full formula retains each component’s diagonal variance contribution.
- For the worked 50-stock basket, 18% index IV and 26.5% component IV give implied correlation 0.4504.
- Correlation times 100 converts 0.4504 to 45.04 index points.
- Compare consistent tenors, moneyness, baskets and volatility inputs before evaluating relative pricing.
What is implied correlation?
Implied correlation is the average pairwise correlation among an index’s constituents that option prices imply. It connects index option pricing with option pricing on the index’s constituents through the variance identity: individual-stock variance contributions plus weighted covariance contributions equal index variance. A high reading means options price the stocks to move together; a low reading means they price more independent moves.
Solve that identity using a consistent tenor, moneyness and basket. The result is one weighted pairwise-correlation measure. Each pair’s weight depends on both stock weights and both volatilities.
How is implied correlation calculated?
Let D be the sum of squared weights times stock variances. Let C be the squared weighted-average stock volatility minus D. Then implied correlation is index variance minus D, divided by C.
For 50 equal weights and 26.5% component IV, D is 0.001405 and C is 0.06882. At index IV of 18%, implied correlation is 0.4504. Multiplying by 100 gives 45.04 correlation-index points.
The shorter squared-volatility-ratio calculation gives 0.4614. It drops the diagonal term. Use the full identity when comparing concentrated baskets or small differences.
Formula
ρ_implied = (σ_index² − D) / C; D = Σ w_i² σ_i²; C = (Σ w_i σ_i)² − D.- Use consistent expirations and moneyness for index and constituent IV.
- D is the sum of each stock’s individual variance contribution.
- C is twice the sum of w_i × w_j × σ_i × σ_j over distinct pairs.
- With general pairwise correlations, the result is their weighted average using those pair contributions.
- The squared weighted-volatility-ratio shortcut drops D.
- Multiply a decimal correlation by 100 to express it in correlation-index points.
Worked Example
Use a stylized 50-stock index with equal 2% weights. All constituent options have 26.5% IV and the index option has 18% IV at the same 30-day tenor and moneyness. For the two realized scenarios, hold each constituent’s realized volatility at 26.5%.
- D = 50 × 0.02² × 0.265² = 0.001405
- C = 0.265² − 0.001405 = 0.06882
- Implied correlation = (0.18² − 0.001405) / 0.06882 = 0.4504
- Correlation-index scale = 0.4504 × 100 = 45.04
- At realized correlation 0.30: index volatility = √(0.001405 + 0.30 × 0.06882) = 14.85%
- At realized correlation 0.60: index volatility = √(0.001405 + 0.60 × 0.06882) = 20.66%
With component volatility fixed, the correlation change moves index volatility from 14.85% to 20.66%. The full calculation gives 0.4504 for the entry inputs. The ratio shortcut gives 0.4614 by dropping the diagonal contribution.
How correlation changes index volatility
Hold each stock’s realized volatility at 26.5%. At a common realized correlation of 0.30, the full identity gives index volatility of 14.85%. At correlation 0.60, it gives 20.66%.
Negative correlation is mathematically valid. An n-stock matrix with a common pairwise correlation has a lower bound of −1 / (n - 1). With 50 equal-volatility stocks and correlation −0.01, the same calculation gives 2.68% index volatility.
How traders use implied correlation
Cboe’s COR3M uses three-month constant-maturity at-the-money IV on SPX and its top 50 stocks. Compare matching maturities and strike conventions when reading another correlation measure.
Use this relationship to compare index options with single-stock options. For a dispersion trade, size the legs from their contract multipliers and dollar sensitivities. Use options flow to examine activity in the selected expirations and the options profit calculator to compare the individual structures. Sum every leg and hedge to evaluate the combined position.
Limitations of implied correlation
- Check the pairs your trade depends on. Implied correlation is a single weighted average, and the same reading covers very different pair and sector correlations.
- Judge the trade by realized correlation. Option premiums carry a risk premium and supply and demand, and a correlation trade settles on what the stocks actually do: with 26.5% stock volatility, 0.30 realized correlation delivers 14.85% index volatility against the 18% the index options priced, and 0.60 delivers 20.66%.
- Match the tenor, moneyness and basket before you compare readings. Each reading depends on all three, and COR3M uses SPX and its top 50 stocks.
- Use the full identity. The squared-volatility-ratio shortcut drops each stock’s own variance and overstates the result, as the 0.4614 versus 0.4504 comparison shows.
Bottom line
Implied correlation shows how closely options price an index’s stocks to move together, linking index and single-stock option prices in one number. Calculate it with the full variance identity on matched tenors, moneyness and baskets, and use it to compare relative pricing before sizing a dispersion trade.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Cboe: Cboe Announces Launch of New Cboe 3-Month Implied Correlation Index
- Driessen, Maenhout and Vilkov: Option-Implied Correlations and the Price of Correlation Risk
- Cboe: The Impact of Dispersion on Market Expectations and Volatility
- Cboe: Implied Correlation Indices
- Cboe Global Indices: COR3M daily price history
How entries are sourced, calculated and updated: editorial methodology.