# Dispersion and correlation

**ADV-27 · Advanced Options · Core**

**Lesson question:** Why can index volatility differ from component volatility?

Original fictional examples; educational writing, not live market data, orders or production approval.

## By the end

- Explain the mechanism in the lesson question.
- Reproduce the worked comparison and recognize its failure case.

**Prerequisites:** OPT-F01, OPT-F02, OPT-F03, OPT-F04

An index aggregates individual returns and their co-movement. Even volatile components can form a less volatile basket when they do not move together. Dispersion strategies compare index and component option exposures, but implementation is much more complex than this identity.

Correlation can rise during stress exactly when diversification is most needed.

## Understand the mechanism

For two assets, basket variance includes weighted individual variances plus twice the weighted covariance. Actual index weights, differing vols, rebalancing and more constituents matter.

An institutional dispersion trade is not simply 'buy stocks and short the index.' It concerns volatility exposures with hedge, weighting, surface and jump risks.

## Walk through the process

Understand the variance relationship before discussing positions. Use variance-consistent weights and compare implied versus realized quantities. Explain correlation and constituent-event risks.

A theoretical covariance calculation is not an executable strategy payoff or a promise of diversification through a gap.

## Model and implementation boundary

These are complete fictional reading examples, not live opportunities. Terminal identities assume the stated common settlement. Multi-expiry values require a surviving-leg model. Costs, assignment and financing can alter realized outcomes; a mathematical bound alone does not guarantee the same operational loss path.

## A worked example

Two fictional equal-weight assets each have 20% annual volatility. With correlation 0, basket variance is .5²×.2²+.5²×.2²=.02, giving 14.142% volatility. With correlation 1, covariance raises volatility to20%.

The graph varies correlation with fixed weights and constituent vols. It illustrates the relationship, not a forecast or a live dispersion opportunity.

## Calculation popup: Two-asset basket variance

```text
variance=w1²sigma1²+w2²sigma2²+2w1w2rho sigma1sigma2
```

Fixed half weights and20% constituent vols.

1. At rho0: variance=.25×.04+.25×.04=.02.

2. sqrt(.02)=.1414213562.

**Result:** About14.142% annualized volatility.

## Graph: Dispersion and correlation—worked comparison

**Horizontal:** Correlation assumption  
**Vertical:** Basket annualized volatility (%)

Fictional data from the written example. The chart organizes the comparison; its shape is not a forecast.

Complete numerical data are embedded in `scripts/advanced-options/ADV-27.js` and displayed as an accessible table in the HTML lesson.

## What would invalidate the reading?

A trader ignores changing weights or assumes low historical correlation persists. One constituent's gap can also dominate a component-option book.

## Check your understanding

### What is equal-weight basket volatility with two20%-vol assets at zero correlation?

**Answer:** Approximately14.142%.

**Why:** Square-root the variance; do not average volatilities directly.

### What is the critical failure to avoid?

**Answer:** A trader ignores changing weights or assumes low historical correlation persists. One constituent's gap can also dominate a component-option book.

**Why:** Explain this risk before treating the example as a decision rule. A correct calculation does not guarantee a favorable market outcome. **Critical risk check.**

## Takeaway

Return to the question: Why can index volatility differ from component volatility? State the observed facts, the assumptions and the failure condition separately.

## References and provenance

The cited references support definitions and conventions. The teaching prose, tickets and numerical scenarios are original; they are not attributed as historical market observations.

- [OIC Strategy Reference](https://www.optionseducation.org/strategies/all-strategies-en)

- [OCC Options Disclosure Document](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document)

- [OIC Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks)