How do delta and gamma interact in options hedging, and why is gamma risk dangerous?

Delta and gamma are the two most important Greeks for options risk management, and understanding their interaction is essential for FRM Part I. **Delta (First-Order Sensitivity)** - Measures the change in option price for a $1 change in the underlying - Call delta: ranges from 0 to +1 (ATM call ≈ 0.50) - Put delta: ranges from -1 to 0 (ATM put ≈ -0.50) - Delta-hedging means holding delta units of the underlying to offset option exposure **Gamma (Second-Order Sensitivity)** - Measures the rate of change of delta for a $1 change in the underlying - Gamma tells you how quickly your delta hedge becomes stale - Always positive for long options (both calls and puts) - Highest for at-the-money options near expiration **Why Gamma Is Dangerous:** Consider a market maker at Vanguard Derivatives who has sold 1,000 ATM call options on Crestfield Corp stock at $75, expiring in 3 days: | Metric | Value | |---|---| | Delta per option | 0.52 | | Gamma per option | 0.08 | | Position delta | -520 (short calls) | | Position gamma | -80 | To delta-hedge, the market maker buys 520 shares of Crestfield at $75. **Scenario: Stock jumps to $78 (+$3)** - New delta per option ≈ 0.52 + (0.08 × 3) = 0.76 - New position delta = -760 - But the hedge only has 520 shares → **under-hedged by 240 shares** - Loss from the gap: approximately 240 × $3 = **$720 in unhedged loss** The problem is that the short gamma position means delta moves against you faster than you can re-hedge. Near expiration, gamma for ATM options approaches infinity — delta can swing from 0.50 to 1.00 with tiny price moves, making hedging nearly impossible. **Delta-Gamma Hedging:** A delta-only hedge is a linear approximation. Adding gamma creates a quadratic approximation: **Change in option value ≈ Delta × ΔS + ½ × Gamma × (ΔS)²** To hedge both delta and gamma, you need two instruments: 1. **Another option** to neutralize gamma (you can't hedge gamma with the underlying alone — stock has zero gamma) 2. **The underlying** to neutralize the remaining delta **Example:** The market maker buys 400 ATM put options (gamma = +0.08 each, delta = -0.48 each): - Gamma from puts: +400 × 0.08 = +32 (partially offsets the -80) - Additional delta from puts: 400 × (-0.48) = -192 - Adjust stock position: buy 192 more shares to re-neutralize delta To fully gamma-neutralize, the maker would need 1,000 offsetting options with matching gamma. **Gamma and P&L:** - **Long gamma** = profit from large moves (bought options). You make money from re-hedging — buy low after drops, sell high after rises. - **Short gamma** = profit from small moves (sold options). You collect theta (time decay) but face catastrophic losses from large moves. **Exam Tip:** FRM Part I loves to test the relationship between gamma and time to expiration. Remember: gamma increases as expiration approaches for ATM options, making short gamma positions increasingly risky in the final days. Practice Greeks calculations in our FRM Part I question bank.