Delta Hedging Basics
Delta Hedging Basics
Delta hedging neutralizes the directional exposure of an options position by taking an offsetting position in the underlying asset. This technique is fundamental to options market-making and risk management, allowing traders to isolate volatility exposure from price direction.
Definition and Key Concepts
What Is Delta
Delta measures how much an option's price changes for a $1 move in the underlying asset:
| Option Type | Delta Range | Interpretation |
|---|---|---|
| Long call | 0 to +1 | Gains when underlying rises |
| Long put | -1 to 0 | Gains when underlying falls |
| Short call | -1 to 0 | Loses when underlying rises |
| Short put | 0 to +1 | Loses when underlying falls |
Example: A call option with delta = 0.50 gains approximately $0.50 when the stock rises $1.
The Hedge Ratio
The hedge ratio equals the delta of the options position:
Hedge formula: Shares to hedge = -Delta × Number of contracts × Contract multiplier
For a long call with delta 0.50 (100 contracts, 100 shares each): Shares to sell = -0.50 × 100 × 100 = -5,000 shares (short 5,000)
Delta-Neutral Position
A delta-neutral position has net delta of zero:
| Component | Position | Delta | Net Delta |
|---|---|---|---|
| Long 100 calls | +100 | +0.50 | +5,000 |
| Short stock | -5,000 | +1.00 | -5,000 |
| Portfolio | 0 |
The combined position neither gains nor loses from small underlying moves.
How It Works in Practice
Initial Hedge Setup
Step 1: Calculate position delta
- 100 call contracts × 100 multiplier × 0.50 delta = 5,000 share-equivalent exposure
Step 2: Determine hedge quantity
- Sell 5,000 shares to offset the long delta
Step 3: Execute hedge
- Short 5,000 shares at current market price
Rebalancing Requirements
Delta changes as the underlying moves (gamma effect):
| Stock Price | Call Delta | Position Delta | Shares Needed |
|---|---|---|---|
| $100 | 0.50 | +5,000 | -5,000 |
| $105 | 0.60 | +6,000 | -6,000 |
| $95 | 0.40 | +4,000 | -4,000 |
Rebalancing rule: When stock rises to $105, sell 1,000 more shares (adjust from -5,000 to -6,000).
Rebalancing Frequency
| Approach | Frequency | Transaction Costs | Hedge Accuracy |
|---|---|---|---|
| Continuous | Every tick | Very high | Perfect (theoretical) |
| Time-based | Hourly/daily | Moderate | Good |
| Threshold-based | When delta changes by X | Lower | Depends on threshold |
| Event-based | After significant moves | Lowest | Variable |
Worked Example
Trade details:
- Position: Long 50 ATM call contracts
- Underlying: Stock XYZ at $100
- Delta: 0.52
- Gamma: 0.04
- Contract size: 100 shares
- Bid-ask spread (stock): $0.02
Initial hedge: Position delta = 50 × 100 × 0.52 = 2,600 shares Hedge: Short 2,600 shares at $100
Day 1: Stock rises to $103 New delta: 0.52 + (0.04 × 3) = 0.64 New position delta: 50 × 100 × 0.64 = 3,200 shares Current hedge: -2,600 shares Action: Sell 600 more shares at $103
Day 2: Stock falls to $98 New delta: 0.64 - (0.04 × 5) = 0.44 New position delta: 50 × 100 × 0.44 = 2,200 shares Current hedge: -3,200 shares Action: Buy 1,000 shares at $98
P/L Analysis
| Component | P/L |
|---|---|
| Call options | +$4,500 (estimated) |
| Initial short (2,600 @ $100 → $98) | +$5,200 |
| Additional short (600 @ $103 → $98) | +$3,000 |
| Buyback cost (1,000 @ $98) | N/A (adjusts position) |
| Transaction costs (4,200 shares × $0.02) | -$84 |
| Net P/L | Approximately even |
The hedge neutralizes most directional P/L, leaving volatility exposure.
VaR of Hedged Position
Unhedged VaR (95%, 1-day): = Position delta × Stock volatility × Confidence factor = 2,600 × ($100 × 1.5% × 1.65) = $6,435
Hedged VaR (95%, 1-day): = Residual gamma/vega exposure only = ~$1,200 (primarily from discrete rebalancing gaps)
Delta hedging reduces VaR by approximately 80% in this example.
Risks, Limitations, and Tradeoffs
Transaction Costs
Each rebalance incurs costs:
| Cost Component | Impact |
|---|---|
| Bid-ask spread | $0.01-0.05 per share |
| Commission | $0.005-0.01 per share |
| Market impact | Variable with size |
Break-even analysis: If rebalancing costs $0.03/share and you rebalance 1,000 shares 20 times: Total cost = 20,000 × $0.03 = $600
This cost must be recovered from volatility trading profits.
Gamma Risk
Delta hedging does not eliminate gamma:
| Scenario | Effect |
|---|---|
| Large move (gap) | Delta changes before hedge adjusts; loss |
| Overnight gap | Cannot rebalance during close |
| Flash crash | Delta changes faster than execution |
Gamma risk is the risk of being "behind the curve" in rebalancing.
Model Risk
Delta calculations depend on model inputs:
| Input | Uncertainty |
|---|---|
| Implied volatility | Which strike/tenor to use? |
| Dividend estimate | Affects forward price |
| Interest rate | Minor impact |
| Skew adjustment | ATM vs. actual strike delta |
Using wrong delta leads to systematic hedge errors.
Common Pitfalls
| Pitfall | Description | Prevention |
|---|---|---|
| Stale delta | Using yesterday's delta | Recalculate before trading |
| Ignoring gamma | Underestimating rebalance needs | Monitor gamma exposure |
| Wrong multiplier | Contract size error | Verify contract specifications |
| Dividend surprise | Stock goes ex-div unexpectedly | Track dividend calendar |
Checklist and Next Steps
Pre-hedge checklist:
- Calculate current position delta
- Verify contract multiplier and lot size
- Check current stock price and bid-ask
- Determine hedge quantity
- Confirm borrow availability (if shorting)
- Calculate transaction cost estimate
- Set rebalancing thresholds
Ongoing hedge management:
- Monitor delta continuously
- Rebalance when threshold breached
- Track cumulative transaction costs
- Review hedge P/L attribution
- Adjust for corporate actions
- Document all rebalancing trades
Related articles: For advanced volatility trading, see Gamma Scalping and Volatility Trading.