Dispersion Trading: Index vs Single-Name Vol
Dispersion trading compares an index's implied variance with the weighted implied variances of its constituents. The usual position sells expensive index volatility and buys single-name volatility, seeking to monetize the correlation premium embedded in an index option. It is not a pure correlation swap: basket weights, skew, jump risk, corporate actions, and hedge conventions determine whether the realised result resembles the elegant identity.
The variance identity
For a fixed-weight basket with constituent returns ri and weights wi,
sigma_index^2 = sum_i w_i^2 sigma_i^2
+ sum_(i != j) w_i w_j rho_ij sigma_i sigma_j
Define average pairwise correlation as the second term divided by the cross-volatility weights. When index implied variance exceeds the single-name contribution at a common correlation assumption, the residual is implied correlation. The trade sells that residual by selling index variance and buying its components.
| Leg | Typical trade | Desired outcome | Dominant adverse move |
|---|---|---|---|
| Index | sell ATM straddle or variance | index realised vol below implied | market crash, correlation spike |
| Singles | buy weighted straddles or variance | constituent realised vol exceeds implied | idiosyncratic vols collapse |
| Hedge | delta hedge each option | isolate variance/correlation | gap and transaction-cost loss |
The relationship connects directly to volatility arbitrage dispersion, but it should be measured in variance units. Comparing a 20% index IV with a simple average of 30% stock IVs is wrong because both option payoff convexity and the basket identity operate on squared volatility.
Implied correlation estimate
With a diagonal approximation and a chosen constituent universe:
rho_impl = (sigma_I^2 - sum_i w_i^2 sigma_i^2) /
(sum_(i != j) w_i w_j sigma_i sigma_j)
This estimate is model-dependent. A cap-weighted index option reflects the official index methodology; a static equal-notional basket does not. The calculation must also use matched expiries, forwards, and option deltas. A 25-delta put basket against ATM index options is largely a skew trade, not a clean correlation comparison.
import numpy as np
def implied_correlation(index_vol, single_vols, weights):
v = np.asarray(single_vols, float)
w = np.asarray(weights, float)
diagonal = np.sum((w * v) ** 2)
cross_weight = (np.sum(w * v) ** 2) - diagonal
return (index_vol**2 - diagonal) / cross_weight
rho = implied_correlation(0.19, [0.31, 0.27, 0.35], [0.5, 0.3, 0.2])
Reject observations where cross_weight is near zero or the result lies far outside plausible bounds. Values above one or below zero can arise from stale quotes, inconsistent strikes, dividends, or genuine differences in skew rather than an executable arbitrage.
Constructing the book
The simplest implementation sells one index straddle and buys constituent straddles scaled to equalize vega. Vega matching limits initial parallel-vol exposure but does not match gamma or variance notional. For a variance perspective, choose single-name notionals such that their weighted variance contribution matches the index variance notional. Rebalance after large spot moves because delta, vega, and index weights drift.
| Scaling choice | What it neutralizes initially | What remains |
|---|---|---|
| Equal premium | cash outlay | almost all risk factors |
| Vega matched | parallel IV move | gamma and correlation mismatch |
| Variance matched | diagonal variance | skew, jumps, rebalancing |
| Beta/sector constrained | factor concentration | residual stock dispersion |
Use liquid names and a documented treatment for additions, deletions, mergers, special dividends, and earnings. An index may have hundreds of names while only a smaller liquid subset is tradable. The omitted-name residual should be estimated and booked as basis risk, not silently assumed away.
P&L mechanics
Over a short horizon, a delta-hedged option's P&L is approximately gamma times realised variance minus theta, with vega and smile terms added:
dPi ≈ 0.5 * Gamma * S^2 * (realized_variance - implied_variance) * dt
+ Vega * dIV + Vanna * dS * dIV + Volga * (dIV)^2
The dispersion book profits when the long singles realize sufficient idiosyncratic movement relative to their premiums and the short index remains diversified. It also benefits if implied correlation falls. Earnings can help a long single-name book, but they introduce discrete gaps that make delta hedging ineffective and can leave implied volatility overpriced after the event.
The difficult regime is a systemic selloff. Index skew steepens, index volatility rises sharply, and correlations approach one just when the short-index leg has negative convexity. Single names also rise in vol, but their long vegas may lag because their initially higher implieds fall less or because the index downside wing is repriced more violently.
Surface and execution risks
Surface marks must be coherent. Obtain forward and dividends first, fit or interpolate the implied volatility surface by delta and expiry, then price each trade using the same conventions. Avoid treating last trades as executable mids. Wide single-name spreads, stock borrow, exchange fees, and frequent delta rehedging can consume a modest theoretical correlation edge.
Stress at least three scenarios: a correlation shock with unchanged single-name vols, a broad 10% index gap with skew steepening, and a sector-specific shock that makes a few long names jump. Historical correlations understate crisis co-movement; combine history with option-implied scenarios and concentration limits.
Key takeaways
- Dispersion isolates the gap between index variance and constituent variance, often described as a correlation premium.
- Calculate and size in variance units with matched strikes, expiries, forwards, and weights.
- Vega matching is useful but does not make the book correlation-neutral.
- Systemic selloffs combine short-index convexity, skew repricing, and correlation spikes.
- Corporate actions, liquidity, surface conventions, and hedge costs can dominate a small theoretical edge.