---
description: "Dispersion trading goes long single-stock option straddles on index constituents and short the index option straddle, exploiting the empirical tendency for index implied volatility to exceed the theoretical volatility implied by constituent volatilities and correlations."
tags: [indexes, volatility, dispersion, options, straddle, correlation]
---

# Dispersion Trading in Equity Indexes

**Section**: 6.3 | **Asset Class**: Indexes | **Type**: Volatility / Arbitrage

## Overview
Dispersion trading exploits the empirical observation that the implied volatility of index options is typically higher than the theoretical index volatility implied by constituent implied volatilities and their correlations. The strategy goes long single-stock volatility (via straddles on each constituent) and short index volatility (via a short index straddle), profiting when realized correlations are lower than the correlation implied by index options pricing.

## Construction / Mechanics

**Theoretical index variance** from constituent volatilities and correlation matrix:
```
σ_I² = Σ_{i,j=1}^N w_i·w_j·σ_i·σ_j·ρ_ij                          (423)
```

where:
- w_i: index weights (assumed market-cap weighted)
- σ_i: implied volatility of stock i (from single-stock options)
- ρ_ij: sample (historical) correlation matrix (ρ_ii = 1)

Empirically, the implied index volatility ̃σ_I > σ_I, i.e., index options are priced at a premium to theoretical constituent volatility.

**Position construction** (near-ATM straddles, ~1 month to expiry):
- For each stock i in the index: **long n_i straddles** on stock i (near-ATM)
- **Short 1 index straddle** (near-ATM, index level P_I)

**Sizing** n_i (for market-cap-weighted index):
```
n_i = S_i · P_I / Σ_{k=1}^N S_k·P_k                                (424)
```

where S_i is shares outstanding for stock i. With this sizing, P_I = Σ_i n_i·P_i, so the index straddle payoff matches the sum of individual straddle payoffs as closely as possible.

All positions held until expiry (~1 month).

## Payoff / Return Profile
- Profits when constituent volatilities are high relative to index volatility (low realized correlation).
- Loses when correlation spikes (e.g., broad market selloff where all stocks fall together), causing the index to move as much as the constituents.
- The strategy can also be viewed as short the average pairwise correlation ρ_ij.

## Key Parameters / Signals
- ̃σ_I: implied index volatility (from index options)
- σ_I: theoretical index volatility from Eq. (423) using constituent implied vols + historical correlations
- Spread ̃σ_I - σ_I: the basis to be monetized
- Realized vs. implied correlation: the key P&L driver

## Variations

### 6.3.1 Dispersion Trading — Subset Portfolio
For large indexes (e.g., S&P 500), some constituent stocks may lack liquid single-stock options (typically smaller-cap stocks). These must be excluded from the bought portfolio. Additional reasons to use a subset:
- Reducing the number of single-stock positions lowers transaction costs.
- The sample correlation matrix ρ_ij is singular for typical lookback periods (e.g., 1 year of daily returns with N=500 stocks), making Eq. (423) unreliable.

**Stabilized correlation matrix** using a statistical risk model (PCA-based):
```
ψ_ij = ξ_i²·δ_ij + Σ_{A=1}^K λ^(A)·V_i^(A)·V_j^(A)               (425)
ξ_i² = 1 - Σ_{A=1}^K λ^(A)·[V_i^(A)]²                             (426)
```

where K < r (r = rank of ρ_ij) principal components explain systematic risk; ξ_i is the idiosyncratic (specific) risk of stock i. K is chosen via eRank.

Using ψ_ij instead of ρ_ij, the theoretical variance becomes:
```
σ_I² = Σ_{i,j=1}^N w_i·w_j·σ_i·σ_j·ψ_ij
      = Σ_i w_i²σ_i²ξ_i² + Σ_{A=1}^K λ^(A)·[Σ_i λ^(A)·V_i^(A)·w_i·σ_i]²   (427)
```

The subset long portfolio contains only the N_* stocks with the lowest w_i²σ_i²ξ_i² values (e.g., N_* = 100 for S&P 500), which are the stocks contributing least specific risk to the index.

## Notes
- This strategy sits at the boundary of index and volatility strategies; it can also be viewed as correlation trading.
- The strategy loses badly during market crises when correlations spike toward 1 (e.g., 2008), as the index moves almost as much as the individual stocks.
- The pairwise correlations ρ_ij are unstable out-of-sample; using a PCA-based risk model (subset variation) mitigates this.
- ATM options may not be available for all stocks; OTM options close to ATM can substitute.
- The strategy is typically implemented with options expiring in about 1 month, and all positions held to expiration.