Relative value: dispersion, skew trades and correlation

VOL · Chapter 413 min readAsked at Optiver, SIG, Citadel Securities, Akuna

Assumes Variance swaps, the log contract and what the VIX is.

After this lesson you should be able to

  • Derive implied correlation from index and single-name volatilities.
  • Explain what a dispersion trade is long and short.
  • Describe a skew trade and the risk it carries.

Once you can price volatility, the interesting trades are between volatilities rather than on one. Index against single names is a correlation trade; one strike against another is a skew trade; one expiry against another is a term-structure trade. Each isolates a parameter that is not directly quoted anywhere.

Equation 4.1

Index variance from its constituents

The index is a portfolio, so its variance carries a cross term — and that cross term is where correlation enters.

σI2=∑iwi2σi2+∑i≠jwiwjρijσiσj\sigma_I^2 = \sum_i w_i^2\sigma_i^2 + \sum_{i \ne j} w_iw_j\rho_{ij}\sigma_i\sigma_j
ρij\rho_{ij}
Pairwise correlations. Assuming a single ρ\rho gives the implied correlation.
wiw_i
Index weights.

Derivation 4.2

Implied correlation

Assume one common correlation and invert.

  1. σI2≈ρ(∑iwiσi)2+(1−ρ)∑iwi2σi2\sigma_I^2 \approx \rho\left(\sum_i w_i\sigma_i\right)^2 + (1-\rho)\sum_i w_i^2\sigma_i^2

    The two limits: ρ=1\rho = 1 gives the weighted average volatility, ρ=0\rho = 0 gives the diversified one.

  2. ρ≈σI2−∑iwi2σi2(∑iwiσi)2−∑iwi2σi2\rho \approx \frac{\sigma_I^2 - \sum_i w_i^2\sigma_i^2}{\left(\sum_i w_i\sigma_i\right)^2 - \sum_i w_i^2\sigma_i^2}
a number that is quoted nowhere but is implied by two things that are\text{a number that is quoted nowhere but is implied by two things that are}

Proposition 4.3

The dispersion trade

Sell index volatility and buy the constituents’ volatility, in variance-matched size. Since index variance sits below the weighted-average single-name variance by exactly the diversification the correlation provides, you are short correlation: you profit when the names move but move independently, and lose when they all move together.

Holds when

  • The trade is historically profitable because index implied correlation trades above realised — index put demand bids up index volatility specifically.
  • The losing scenario is a market-wide selloff, when correlation goes to one. It is therefore short the tail.
  • Weights must be variance-matched, not vega-matched, or the exposure drifts as volatilities move.

Example 4.4

An index of two equally weighted names implies 18%18\%. Each name implies 30%30\%. What is the implied correlation?

Show the worked solutionHide the worked solution

Worked solution

  1. Formula
    σI2=w2σ12+w2σ22+2w2ρσ1σ2\sigma_I^2 = w^2\sigma_1^2 + w^2\sigma_2^2 + 2w^2\rho\sigma_1\sigma_2
  2. Substitute
    w=0.5, σ1=σ2=0.30, σI=0.18w = 0.5, \ \sigma_1 = \sigma_2 = 0.30, \ \sigma_I = 0.18
  3. Solve
    0.0324=0.25(0.09)+0.25(0.09)+0.5ρ(0.09)0.0324 = 0.25(0.09) + 0.25(0.09) + 0.5\rho(0.09)
  4. 0.0324=0.045+0.045ρ0.0324 = 0.045 + 0.045\rho
  5. ρ=0.0324−0.0450.045\rho = \frac{0.0324 - 0.045}{0.045}
  6. Answer
    ρ=−0.28\rho = -0.28

Sanity check. Negative, which is a real answer here: two names at 30%30\% that were uncorrelated would give an index at 21.2%21.2\%, so an index implying 18%18\% requires them to offset. If this came from live quotes, the first thing to check is whether the index volatility is stale.

Why index volatility is persistently rich. The flow is one-directional. Institutions hedge portfolios by buying index puts, and there is no equivalent structural buyer of single-name puts across the whole market — so index implied volatility is bid relative to what its constituents imply, and the gap shows up as an implied correlation above what subsequently realises. The dispersion trade harvests that, and it has the shape of every flow-driven premium: a small, steady return with a large and correlated loss when the hedging demand is finally justified.

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