Dispersion Trades Using Options
Dispersion Trades Using Options
Dispersion trading exploits the relationship between index option volatility and single-stock option volatility. When implied correlation is high relative to expected realized correlation, selling index volatility and buying single-stock volatility creates a position that profits from lower-than-expected correlation—stocks moving more independently than priced.
Definition and Key Concepts
Dispersion Trade Structure
Classic dispersion:
- Sell: Index option volatility (straddles or variance swaps)
- Buy: Single-stock option volatility (component straddles)
- Net position: Short implied correlation
Profit condition: Realized correlation < Implied correlation
Mathematical Relationship
Index variance formula: σ²_index = Σᵢ wᵢ² σᵢ² + 2 Σᵢ Σⱼ wᵢ wⱼ σᵢ σⱼ ρᵢⱼ
Simplified (equal weights, identical vols): σ²_index ≈ σ²_stock × [1/n + (1 - 1/n) × ρ]
Implied correlation: ρ_implied ≈ (σ²_index - weighted avg σ²_singles) / (cross terms)
Why Implied Correlation Is Often High
| Factor | Explanation |
|---|---|
| Structured product hedging | Dealers buy index vol to hedge worst-of |
| Correlation risk premium | Investors pay for crisis protection |
| Supply/demand imbalance | More index vol sellers than buyers |
| Mean reversion | Correlation reverts from crisis levels |
How It Works in Practice
Trade Setup
S&P 500 dispersion example:
- Index: S&P 500 at 5,000
- Components: Top 50 stocks by weight
- Tenor: 3 months
- Implied index vol: 16%
- Weighted avg singles vol: 28%
- Implied correlation: 0.35
Step 1: Sell index volatility Sell SPX straddle: $250 premium (25 vol points exposure)
Step 2: Buy component volatility Buy straddles on 50 stocks, vega-weighted to match index exposure
Vega matching: Total singles vega = Index vega × (1 / √implied correlation) If index vega = $10,000, singles vega = $10,000 / √0.35 = $16,900
Position Sizing
Per-stock vega allocation: Stock vega = (Stock weight² × Stock vol) / Σ(weights² × vols) × Total singles vega
Example for AAPL (7% weight, 25% vol): AAPL vega = (0.07² × 0.25) / (total) × $16,900 ≈ $600
Buy AAPL straddles with $600 vega exposure
P/L Mechanics
Daily P/L attribution:
| Source | Calculation |
|---|---|
| Index realized move | Short exposure × index move² |
| Singles realized moves | Long exposure × Σ(stock moves²) |
| Correlation effect | Net of above shows correlation |
| Theta | Time decay (often net positive) |
Worked Example
Full Dispersion Trade
Trade parameters:
- Notional: $1 million vega on singles
- Index vega sold: $600,000 (adjusted for correlation)
- Tenor: 60 days
- Entry implied correlation: 0.38
- Expected realized correlation: 0.30
Week 1 P/L breakdown:
| Day | SPX Move | Weighted Singles Move | Singles P/L | Index P/L | Net |
|---|---|---|---|---|---|
| Mon | -1.5% | -1.8% avg | +$3,200 | -$1,800 | +$1,400 |
| Tue | +0.8% | +0.6% to +1.2% | +$1,500 | -$800 | +$700 |
| Wed | -0.2% | -0.5% to +0.3% | +$600 | -$100 | +$500 |
| Thu | +1.0% | +0.5% to +1.8% | +$2,800 | -$1,200 | +$1,600 |
| Fri | -0.5% | -0.3% to -0.8% | +$900 | -$400 | +$500 |
| Week | +$9,000 | -$4,300 | +$4,700 |
Interpretation: Singles moved more than index (low correlation), generating profits.
Scenario Analysis
Trade outcomes by correlation:
| Realized Correlation | Index Realized Vol | Singles Avg Vol | Approx P/L |
|---|---|---|---|
| 0.20 | 13% | 28% | +$80,000 |
| 0.30 | 15% | 28% | +$40,000 |
| 0.38 | 17% | 28% | $0 |
| 0.50 | 20% | 28% | -$30,000 |
| 0.70 | 23% | 28% | -$70,000 |
| 0.90 (crisis) | 27% | 30% | -$150,000 |
Maximum loss: When correlation approaches 1, index moves as much as singles but you're short more gamma on index.
Greeks of Dispersion
| Greek | Singles Position | Index Position | Net |
|---|---|---|---|
| Delta | Approximately zero | Approximately zero | ~0 |
| Gamma | Long | Short | Long (net) |
| Vega | Long | Short | Depends on correlation |
| Theta | Negative | Positive | Usually positive |
Risks, Limitations, and Tradeoffs
Correlation Spike Risk
Crisis scenario (2008/2020 style):
- Normal correlation: 0.30
- Crisis correlation: 0.85
- Implied was: 0.38
Impact:
- Index realized vol jumps to match singles
- Short index position loses more than long singles gains
- Massive drawdown in short period
Structural Risks
| Risk | Description | Mitigation |
|---|---|---|
| Correlation spike | Sudden increase in correlation | Size conservatively |
| Execution cost | Bid-ask on many options | Trade liquid names |
| Rebalancing | Weights change as prices move | Regular adjustment |
| Single-stock gaps | Individual stock events | Diversify, monitor |
| Model risk | Wrong correlation estimate | Use robust methodology |
Operational Challenges
| Challenge | Description |
|---|---|
| Number of positions | 50-100 individual stock options |
| Roll management | Multiple expiration calendars |
| Delta hedging | Many positions to delta hedge |
| Margin requirements | Large gross notional |
| Reporting | Complex P/L attribution |
Common Pitfalls
| Pitfall | Description | Prevention |
|---|---|---|
| Wrong vega ratio | Mismatched singles to index | Careful calculation |
| Ignoring tail risk | Underweight crisis scenario | Stress test regularly |
| Over-leverage | Too large relative to capital | Conservative sizing |
| Stale weights | Index composition changes | Update quarterly |
| Illiquid names | Can't exit positions | Stick to liquid stocks |
Implementation Approaches
Variance Swap Dispersion
Alternative to options:
- Sell index variance swap
- Buy single-stock variance swaps
Advantages:
- Cleaner correlation exposure
- No delta hedging
- Linear in variance
Disadvantages:
- Less liquid
- Larger notionals
- Counterparty risk
ETF-Based Dispersion
Simplified approach:
- Sell sector ETF straddles
- Buy constituent ETF straddles
Example:
- Short XLK (tech sector) straddle
- Long AAPL, MSFT, NVDA, GOOGL straddles
Pro: Fewer positions Con: Less precise correlation exposure
Checklist and Next Steps
Pre-trade checklist:
- Calculate implied correlation from market prices
- Estimate expected realized correlation
- Verify premium exists (implied > expected)
- Size for worst-case correlation spike
- Select liquid single-stock options
- Plan execution sequence
Execution checklist:
- Execute index leg first (most liquid)
- Build singles portfolio over time
- Verify vega ratios match target
- Set up delta hedging
- Confirm margin requirements
Ongoing management:
- Monitor correlation daily
- Rebalance weights as needed
- Track P/L attribution
- Manage option rolls
- Report to risk management
Related articles:
- For correlation trading, see Correlation Trading and Basket Options
- For volatility surfaces, see Volatility Surface Construction Techniques