Dispersion Trading: Index Variance vs Single-Name Variance

Introduction

Dispersion trading is a correlation trade that profits when realized correlations between index components diverge from what is implied by option markets. At its core it is taking opposite variance exposure in single stocks and the aggregate index, for example long single-name variance and short index variance, or the reverse.

Why does this matter to you as an advanced trader? Dispersion isolates correlation risk, which is a non-linear, path-dependent source of opportunity that often cannot be arbitraged away with simple delta hedging. If you understand how to size the trade and manage the key risks, dispersion can provide returns that are decorrelated from directional equity exposure.

In this article you will learn the mathematics behind dispersion, practical ways to build the trade with options or variance swaps, heuristics for sizing, concrete hedging and stop rules, plus worked examples using $SPY and single names such as $AAPL and $NVDA. What happens if realized correlation falls, and how do you size to survive volatility shocks? Those questions and more are covered below.

• Dispersion is fundamentally a correlation bet: long single-name variance and short index variance is long dispersion, profiting if realized correlation falls below implied.
• Index variance equals weighted sum of single-name variances plus covariance terms, so implied correlation can be backed out from market-implied variances.
• Common execution paths use variance swaps, or option-based replication using vega-weighted straddles across constituents and the index.
• Size using vega-neutral or variance-notional rules, monitor skew and liquidity, and keep dynamic hedges for delta and concentrated gamma risk.
• Use scenario analysis, stress tests, and stop-loss levels tied to correlation moves and realized variance to control tail risk.

What dispersion trading actually is

Dispersion trading expresses a view on correlation rather than direction. If you are long single-name variance and short index variance, you are long dispersion. You win if single stocks move more independently than the index implied, so realized correlation falls relative to the market's expectation.

Mathematically, for an index I made of n names with weights w_i, index variance is

Var(I) = sum_i w_i^2 Var_i + sum_{i != j} w_i w_j Cov_{ij}.

Rewriting covariance in terms of pairwise correlations rho_{ij} gives an implied average correlation. Traders use market-implied variances from options to compute an implied correlation that equates the index variance to the component variances. Dispersion is the trade on the difference between implied correlation and expected realized correlation.

Implied correlation formula

Assume Var_i are implied variances from single-name options, and Var_index is implied index variance. A useful estimator for the average implied correlation rho_hat is

rho_hat = (Var_index - sum_i w_i^2 Var_i) / sum_{i != j} w_i w_j sqrt(Var_i Var_j).

This reduces to a simpler form when you assume equal variances or equal weights, but the full formula is what you should use for real portfolios. If rho_hat is high, index variance is mostly coming from covariance, not idiosyncratic risk.

How to construct the trade

There are two standard execution routes. One is direct variance swap or variance-notional instruments on the index and on single names if those are available. The other is options-based replication, usually with at-the-money straddles or strangles to capture vega exposure.

Variance swap route

Variance swaps pay realized variance against a fixed variance strike. If you can buy variance swaps on a basket of single names and sell a variance swap on the index, you directly implement the correlation bet. Not all single names have liquid variance swaps though, and counterparty terms can vary.

Options replication route

When variance swaps are unavailable, you replicate variance exposure with a strip of options across strikes that approximate vega or variance notional. The typical approach is to buy ATM straddles on each constituent scaled to a target variance notional, and sell an ATM straddle on the index scaled to match vega or variance exposure.

Two common sizing rules are vega-neutrality and variance-notional parity. Vega-neutrality sizes positions so the net vega is near zero. Variance-notional parity aims to match the expected payoff profile of variance swaps, using options across strikes to replicate the same squared-vol payoff.

Sizing heuristics and risk allocation

Sizing dispersion trades is both art and science. You are taking correlation exposure, which is second-order relative to vol. That means small correlation moves can produce outsized P&L when vols are high. Size with capital preservation in mind.

1. Vega-weighted sizing: scale each single-name option position so that the sum of individual vegas equals the index vega sold. This reduces immediate vega imbalance, and gives you a clearer correlation-only exposure.
2. Variance-notional parity: compute notional variance exposures by using variance swap equivalents, then size single-name variance notionals to sum to the index variance notional.
3. Weight scaling: for replicating the index exactly, scale single-name exposures by the square of index weights, because index variance depends on w_i^2 terms. For example if a name has double the weight, its variance contribution scales with weight squared.
4. Concentration limits: cap exposure to the top 10 names, because large-cap constituents often dominate index variance, and you can end up concentrated on idiosyncratic moves.

As a rule of thumb, many desks start small, allocating 0.5% to 2% of risk capital to dispersion per idea, and scale up after live results and stress tests. You should back-test sizing across scenarios including single-name jumps, index crashes, and volatility regime shifts.

Risk controls and P&L drivers

Dispersion P&L decomposes into three main drivers: realized variance of constituents, realized index variance, and changes in implied vol/skew that affect option replication. You must monitor all three in real time.

Key risks

• Correlation risk, the primary exposure. If realized correlation rises above implied, a long dispersion position loses money.
• Skew and smile changes, especially single-name skew. Dispersion hedges that use ATM straddles are sensitive to skew steepness because realized returns are not symmetric.
• Jump and tail risk. Single-name gap moves generate realized variance spikes that can create large one-day P&L, especially if positions are concentrated.
• Liquidity and execution risk. Index options are usually deeper than single-name options for large names, so you may struggle to scale single-name positions without moving prices.
• Model and calibration risk. Implied correlation calculation depends on how you compute implied variances across strike and term. Errors create sizing mistakes.

Practical hedges and stop rules

Delta-hedge all option-based replication frequently. Use vega monitors to measure exposure drift. If you're long dispersion, cap losses to a predetermined fraction of capital, for example 25% of the initial allocation, and reduce sizing when realized correlation moves sharply against you.

Stress testing should include scenarios where a few names gap by 20 percent, and index falls 10 percent while vols spike. Decide on automatic de-risk triggers, such as a correlation move of 50 percent relative to implied, or daily losses beyond a set threshold.

Real-world examples

Worked examples make the concept concrete. Below are two scenarios, one simplified for clarity and one that mirrors a real index versus a small basket.

Example 1, simplified four-stock index

Assume a 4-name equal-weight index with weights w_i = 0.25 each. Market implied vols annualized are 30 percent for each single name, so Var_i = 0.30^2 = 0.09. The index implied vol is 18 percent, so Var_index = 0.18^2 = 0.0324.

Compute rho_hat. First sum w_i^2 Var_i = 4 * (0.25^2 * 0.09) = 4 * (0.0625 * 0.09) = 4 * 0.005625 = 0.0225. Next compute denominator sum_{i != j} w_i w_j sqrt(Var_i Var_j) = for equal weights this is (1 - sum w_i^2) * average volatility product. With equal weights the off-diagonal weight sum is 1 - 4*0.0625 = 0.75. The sqrt(Var_i Var_j) is 0.09. So denominator = 0.75 * 0.09 = 0.0675. Then rho_hat = (0.0324 - 0.0225) / 0.0675 = 0.0099 / 0.0675 = 0.1467, or about 14.7 percent implied correlation.

If you expect realized correlation to be 5 percent, long dispersion (long single-name variance, short index variance) should produce profit. If realized correlation turns out to be 25 percent you lose. This shows how low index vol relative to single-name vol implies low implied correlation, and vice versa.

Example 2, practical $SPY vs large-cap names

Consider using $SPY options as the index leg, and a basket of five large-cap names including $AAPL and $NVDA that together represent a substantial portion of $SPY variance. Suppose implied vol on $SPY is 16 percent annually and the weighted average implied vol of selected names is 28 percent. Using real weights and the full implied correlation formula gives you rho_hat. In practice you would compute Var_index from $SPY option-strip variance, and Var_i from each single-name variance strip.

Execution: buy variance-replicating strips on each single name scaled by their index weights squared and sell variance-replicating strip on $SPY so that net variance notional matches your target. Hedge away delta daily, and monitor skew shifts because single-name skews often steepen more in stress than index skew.

Common Mistakes to Avoid

• Ignoring skew and smile differences, which can make option-based replication deviate from true variance exposure. How to avoid: use full strip replication and monitor P&L attribution by strike.
• Mismatching notionals by weight and vega, causing unintended net vega or gamma. How to avoid: compute vega notional and variance-notional parity, then size positions accordingly.
• Underestimating concentration risk from top index names. How to avoid: cap single-name exposure, or include additional names to diversify idiosyncratic jump risk.
• Failing to stress-test for jumps and regime shifts. How to avoid: run scenario P&L with 10 to 30 percent single-name moves and correlated index crashes, then set stop-loss and de-risk triggers.
• Relying on backtests without transaction cost and liquidity modeling. How to avoid: include real-world slippage, borrowing costs, and bid-ask spreads in your simulation.

FAQ

Q: How does long dispersion make money if vols rise across the board?

A: If vols rise uniformly and correlation stays constant, long single-name variance and short index variance largely offset, producing little net profit. Long dispersion specifically benefits when individual variances increase relative to index variance because correlation falls, or when variance increases are concentrated in single names rather than the aggregate. You must monitor cross-asset vol moves and skew changes.

Q: Can I execute dispersion with ETFs instead of index options?

A: Yes, you can use ETF options such as $SPY to represent index variance, but you need liquid single-name options for replication. Using ETFs changes dividend and borrow characteristics, so adjust for those when computing variance replication. ETFs may provide deeper liquidity for the index leg, which helps execution.

Q: Is vega-neutral sizing always the best approach?

A: Vega-neutral sizing is a practical starting point because it reduces immediate exposure to uniform vol moves. However, if your view is driven by expected realized variance differences rather than implied vol shifts, variance-notional sizing may be preferable. Many traders use a hybrid approach and adjust dynamically.

Q: What monitoring metrics should I track intraday?

A: Track net vega, net gamma, daily realized variance flows, and an estimate of realized correlation. Also watch single-name skew changes, concentration metrics, and liquidity indicators like bid-ask width. Set automated alerts for correlation divergence beyond pre-set thresholds.

Bottom Line

Dispersion trading isolates correlation risk by taking opposite variance exposure between single names and an index. It can be attractive because it focuses on a second-order driver of returns that typical directional strategies do not capture. You should think of it as a correlation arbitrage where careful sizing, replication fidelity, and risk controls matter more than a simple directional thesis.

Start with small, vega-aware position sizes, replicate variance carefully across strikes to handle skew, and prepare for jump and liquidity scenarios with explicit stop rules. Back-test with realistic transaction costs and run scenario analyses so you know how the trade behaves when correlation moves sharply.

At the end of the day, dispersion is a nuanced, highly technical strategy. If you execute it with disciplined sizing and robust hedging, it can be a useful tool in an advanced trader's toolkit, offering a relatively pure play on correlation. Keep learning, monitor your assumptions, and adapt to changing market regimes.