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Dispersion Trading: Index Variance, Constituent Variance, and Correlation

Evaluate dispersion with compatible variance inputs, implied-correlation checks, executable sizing, event and surface risk, settlement controls, and a complete P/L ledger.

Updated

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Dispersion trading takes opposite volatility or variance positions in an equity index and a basket of its constituents. A conventional long-dispersion book is long constituent volatility or variance and short index volatility or variance; reverse dispersion flips those signed exposures. The exact instruments, weights, notionals, maturities, surfaces, settlements, and hedge rules define the trade.

Dispersion is not a pure or riskless correlation claim. An at-the-money straddle quote, option-strip implied variance, and a variance-swap strike are different objects. Even a carefully normalized book retains volatility-level, skew, jump, Gamma, Delta-hedge, composition, execution, funding, settlement, and model risk.

A compatible variance and lifecycle ledger

For point-in-time constituent weights w_i with sum_i w_i = 1, compatible returns R_i, and index return approximation R_I = sum_i w_i R_i, the variance identity is sigma_I^2 = sum_i w_i^2 sigma_i^2 + 2 sum_{i<j} w_i w_j sigma_i sigma_j rho_ij. If one common scalar correlation is imposed, define A = sum_i w_i^2 sigma_i^2, B = 2 sum_{i<j} w_i w_j sigma_i sigma_j, and rho_bar = (sigma_I^2 - A) / B, provided B > 0.

  1. Lock the implementation: listed straddles, option strips, variance exposures, or another claim; long or short dispersion; index version; constituent universe; security lines; objective; loss budget; and whether a reduced basket creates tracking risk.
  2. Freeze point-in-time weights, return definition, price or total-return convention, currency, annualization, observation and maturity dates, forward, rates, dividends, and corporate-action treatment. Record every option or swap strike, surface, multiplier, notional, exercise style, settlement method, quote timestamp, and executable side.
  3. Put index and constituent inputs on one variance basis. Calculate A, B, index variance, and rho_bar; verify the weights, B > 0, numerical range, and positive-semidefinite feasibility. A scalar average compresses a full correlation matrix and is not a directly observed truth.
  4. Size signed index and constituent legs using variance notional, Vega, Gamma, multiplier, maturity, currency, and forward sensitivity rather than stock weights or contract counts alone. Record omitted names, integer rounding, gross Greeks, reduced-basket tracking, and residual exposure.
  5. Execute with synchronized bids and asks, package or portfolio controls, limits on partial fills and legging, and an explicit Delta-hedge plan. Record premium or swap value, collateral, margin, cash, fees, impact, borrow, funding, and rejected legs; opening credit and short-sale proceeds are not profit.
  6. Monitor weights, surfaces, term structure, skew, Greeks, earnings, mergers, litigation, halts, delistings, rebalances, corporate actions, borrow, margin, and settlement clocks. Recalculate sizing after prices, constituents, or sensitivities change.
  7. Attribute P/L separately to index variance, constituent variance, covariance or correlation residual, surface changes, Delta hedges, jumps, composition, funding, borrow, fees, and tax. Reconcile official cash-settled index values, physical stock-option exercise or assignment, final cash, positions, collateral, and broker or swap statements.

The identity requires aligned inputs; it does not turn quoted volatilities into a trade. Option strips can estimate a model-free-style variance measure under their methodology, while ATM straddles retain strike and surface exposure. A value of rho_bar outside the feasible region is a diagnostic for inconsistent inputs or an inadequate scalar model, not evidence of executable arbitrage.

Worked examples

  • Correlation changes index variance without changing stock volatility. Ten equal-weight stocks each have sigma_i = 30%. Under common correlation, sigma_I^2 = 0.1 x 0.30^2 + 0.9 x 0.30^2 x rho_bar. At rho_bar = 0.20, variance is 0.0252 and index volatility is sqrt(0.0252) = 15.874508%. At rho_bar = 0.80, variance is 0.0738 and volatility is 27.166155%. Constituent volatilities did not change; common movement did.
  • Non-equal weights and an implied-correlation inversion. Let weights be 0.50, 0.30, 0.20, constituent volatilities be 25%, 30%, 40%, and compatible index volatility be 24%. Then A = 0.030125, B = 0.056900, and rho_bar = (0.24^2 - 0.030125) / 0.056900 = 0.4828646749, or 48.286467%. This is a scalar-model output under synchronized inputs, not the full correlation matrix.
  • Vega sizing and neutrality drift. A constituent option basket has Vega +$40,000 per vol point; one index package has Vega +$16,000 per vol point. A continuous long-dispersion Vega target shorts 2.5 index packages. Shorting 2 leaves +$8,000 per vol point; shorting 3 leaves -$8,000 per vol point. If constituent Vega later falls to +$36,000 and index Vega to +$15,000 per package, the original -2.5 target has net Vega 36,000 - 2.5 x 15,000 = -$1,500 per vol point.
  • Variance P/L and implementation costs are separate. Give a constituent sleeve and an index sleeve each N_var = $100,000 per 1.00 decimal variance. The long constituent strike is 30% and realized volatility is 32%, so its payoff is 100,000 x (0.32^2 - 0.30^2) = +$1,240. The short index strike is 22%; at realized index volatility 20%, its payoff is 100,000 x (0.22^2 - 0.20^2) = +$840, for gross +$2,080. At index realized volatility 25%, that leg is -$1,410 and gross book P/L is -$170. If an option implementation crosses 20 stock contracts at $0.10/share x 100 and 5 index packages at $0.30 x 100 each way, one-way cost is $350 and round-trip cost is $700; the two net results become +$1,380 and -$870 before Delta hedging, funding, borrow, and tax.

Risks and controls

  • Wrong index, version, divisor, universe, security line, or weight vintage changes the calculation.
  • A reduced basket creates tracking and model risk rather than full replication.
  • Additions, deletions, splits, mergers, special dividends, halts, and delistings change identities and weights.
  • Missing squared weights, pair terms, or normalization corrupts the variance identity.
  • Scalar correlation can be infeasible, hide a non-PSD matrix, or compress materially different pair risks.
  • Volatility, decimal variance, variance points, and annualization units can differ by factors of 100 or 10,000.
  • Log versus simple returns, price versus total returns, calendars, windows, and day counts can mismatch.
  • Maturity, timestamp, forward, rates, dividends, borrow, currency, and FX can be inconsistent.
  • ATM quotes, skewed surfaces, option-strip variance, and variance-swap strikes are not interchangeable.
  • Variance notional, Vega, Gamma, multiplier, forward, and contract count can be sized inconsistently.
  • Integer sizing, omitted names, and moving Greeks cause neutrality and tracking drift.
  • Index downside skew and crash-protection premium can reprice independently of average correlation.
  • Correlation can jump toward one during a marketwide shock while index volatility rises sharply.
  • Earnings, mergers, litigation, and other idiosyncratic jumps can dominate individual legs.
  • Index and stock options can have different last trades, AM/PM clocks, official values, and settlements.
  • Physical stock-option exercise or assignment can create shares and strike funding while the index leg settles in cash.
  • Many-leg orders can partially fill, reject, leg, or leave an unintended directional book.
  • Bid-ask spreads, impact, liquidity withdrawal, adverse selection, and repeated Delta hedges can consume the edge.
  • Borrow, funding, margin, collateral, liquidation, counterparty, tax, and operational terms can change.
  • Data, model, P/L attribution, cash, positions, collateral, and broker records can fail to reconcile.

Common misconceptions

  • “Dispersion is pure correlation or arbitrage.” Volatility, skew, jumps, Greeks, execution, and lifecycle terms remain.
  • “Index weights determine option counts.” Variance notionals, Greeks, multipliers, maturities, forwards, and rounding also control sizing.
  • “Low implied correlation is a forecast that must mean-revert.” It is a model-dependent price output, not a realized-path guarantee.
  • “Initial Vega neutrality means risk neutrality and persists.” Gamma, surface, time, composition, and integer constraints move the exposure.
  • “Long dispersion is a guaranteed crash hedge, or profit proves the correlation forecast.” Crash correlation, index skew, stock events, hedges, and costs can drive the result.

Authoritative sources

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