Gamma and Managing Convexity

Gamma measures how fast delta changes as the underlying moves. It represents the convexity of an option's payoff—the curvature that makes options different from linear instruments like stocks. Understanding gamma is essential for managing positions dynamically and anticipating how quickly hedges become stale.

Definition and Key Concepts

What Is Gamma?

Gamma is the rate of change of delta with respect to a $1 move in the underlying:

• High gamma: Delta changes rapidly
• Low gamma: Delta changes slowly

Mathematically: Gamma = Change in Delta / Change in Underlying Price

Long vs. Short Gamma

Long gamma benefits from large moves; your position becomes more favorable as the underlying trends.

Short gamma suffers from large moves; your position becomes more unfavorable as the underlying trends.

Gamma and Moneyness

Gamma is highest for ATM options near expiration:

Gamma and Time

As expiration approaches, ATM options see gamma explode. This "gamma risk" near expiration is why many traders close positions before the final week.

How It Works in Practice

Gamma's Effect on Delta

Example: You're long an ATM call with delta 0.50 and gamma 0.05.

As the stock rises, your delta increases—you become more long. As the stock falls, your delta decreases—you become less long. This asymmetric behavior is what makes long gamma positions profit from movement.

Long Gamma: Buying Convexity

When you own options, you're long gamma. Your P/L curve is convex:

• Small moves: Lose to theta (time decay)
• Large moves: Gain from delta accelerating in your favor

Profit Dynamics:

• Stock moves up: Calls gain delta, you become more long, gains accelerate
• Stock moves down: Calls lose delta, you become less long, losses decelerate

Short Gamma: Selling Convexity

When you sell options, you're short gamma. Your P/L curve is concave:

• Small moves: Gain from theta
• Large moves: Losses accelerate as delta moves against you

Short Iron Condor Example: The iron condor's short options create negative gamma. As the underlying approaches a short strike, delta moves against you faster. Near expiration, small moves can create large swings in position value.

Worked Example

Gamma Scalping Strategy

You believe XYZ ($100 stock) will be volatile but you're unsure of direction. You buy a straddle and hedge delta to profit from gamma.

Initial Position:

• Buy 10 XYZ $100 calls at $4.00 (delta: +0.52, gamma: 0.04 each)
• Buy 10 XYZ $100 puts at $3.75 (delta: -0.48, gamma: 0.04 each)
• Total premium: $7,750
• Position delta: (+520 - 480) = +40
• Position gamma: (0.04 + 0.04) × 10 × 100 = +80

Delta Hedge: Short 40 shares at $100

Hedged Position:

• Net delta: 0
• Net gamma: +80
• Net theta: -$35 per day

Scenario: Stock moves to $104

The position gained +$320 in delta-equivalent exposure as XYZ rose. To lock in this gamma gain:

Rebalance: Short 320 shares at $104

Now you've:

• Realized a $320 profit from the delta shift (gamma gain)
• Reset to delta-neutral
• Still long gamma for the next move

If XYZ falls back to $100:

• Delta drops back toward initial levels
• You buy back 320 shares at $100
• Profit: Sold at $104, bought at $100 = $400 per 100 shares × 3.2 lots = $1,280

This is gamma scalping: capturing profits from the underlying's oscillation while remaining delta-neutral.

Break-Even Volatility: Daily theta = $35. To break even, gamma gains must exceed $35/day. This requires sufficient realized volatility in the underlying.

Risks, Limitations, and Tradeoffs

Theta Bleed for Long Gamma

Long gamma positions pay for convexity through time decay. If the underlying doesn't move enough, theta losses exceed gamma gains. In the example above, losing $35/day requires regular volatile swings to overcome.

Short Gamma Blowup Risk

Short gamma positions face unlimited or substantial losses from large moves. Near expiration, gamma explodes, and even moderate moves can cause rapid losses. The 2018 "Volmageddon" event destroyed short volatility strategies that were short gamma.

Transaction Costs in Gamma Scalping

Frequent rebalancing incurs commissions and slippage. If you rebalance too often, costs consume gains. If you rebalance too rarely, you miss capturing gamma.

Gamma Peaks Near Expiration

ATM options experience maximum gamma in the final days before expiration. This creates "pin risk"—rapid P/L swings from small moves. Many traders avoid holding positions into expiration week to sidestep this risk.

Common Pitfalls

1. Underestimating gamma near expiration: A 0.50 delta option can become 0.90 delta or 0.10 delta with a small move on the final day.

2. Over-trading when gamma scalping: Transaction costs can exceed gamma gains if rebalancing is too frequent.

3. Ignoring theta for long gamma: Time decay is certain; realized volatility is not.

4. Holding short gamma through events: Earnings, Fed announcements, and other events can create large moves that devastate short gamma positions.

5. Assuming gamma is constant: Gamma itself changes with price and time.

Checklist for Managing Gamma

• Calculate net position gamma across all options
• Identify whether you're long or short gamma
• Assess proximity to expiration (gamma increases for ATM options)
• Compare expected theta loss to potential gamma gains
• Set rebalancing thresholds for delta hedging
• Plan for expiration week: close, roll, or accept gamma risk
• Avoid holding short gamma through known events
• Size positions so gamma-related losses are tolerable

Next Steps

Time decay is the cost of holding long gamma positions. See Theta Decay and Time-Based Trades for strategies that profit from theta and how to manage it.

For delta hedging techniques that address gamma's impact, review Using Delta as a Hedge Ratio.