Gamma Hedging: Definition, How It Works, and Vs. Delta Hedging

What is Gamma Hedging?

Gamma hedging is an options risk-management technique that keeps an options portfolio’s delta (the sensitivity of option value to small changes in the underlying) from changing materially as the underlying moves. In practice it means creating a portfolio whose rate of change of delta—its gamma—is close to zero (or to some targeted level). Gamma hedging is most often used to protect against large or rapid price moves and to stabilize hedges as expiration approaches.

Key takeaways

– Gamma is the rate of change of an option’s delta for a one‑point move in the underlying. Positive gamma means delta increases as the underlying rises and decreases as it falls; negative gamma is the opposite.
– A gamma-neutral portfolio has (near) zero aggregate gamma and therefore small sensitivity of delta to underlying moves.
– Delta hedging protects against small moves. Gamma hedging protects the delta hedge from changing when the underlying makes larger moves.
– Gamma hedging is implemented by trading additional options (or combinations of options) and then adjusting the underlying to control delta; it is dynamic and requires rebalancing.
– Costs and practical issues (transaction costs, discrete rebalancing, model risk, margin) often make gamma hedging complex and potentially expensive.

How gamma works (intuition and formula)

– Intuition: Delta tells you how much option price changes for a small move in the underlying. Gamma tells you how that delta itself will change if the underlying moves further. High gamma means an option’s delta is very sensitive to small changes in the underlying.
– Approximation: If ΔS is a small change in the underlying, the change in option delta ≈ Gamma × ΔS. The option price change for a finite move can be approximated by including gamma:
Option price change ≈ Delta × ΔS + 0.5 × Gamma × (ΔS)^2
– Black‑Scholes formula (for a European option) gives gamma as:
Gamma = φ(d1) / (S σ √T)
where φ(d1) is the standard normal pdf at d1, S = spot price, σ = volatility, T = time to expiration. (This shows gamma increases as time to expiry decreases and as volatility falls, other things equal.)

Gamma vs. Delta — the distinction

– Delta measures first‑order sensitivity: how much the option price moves per small move in the underlying.
– Gamma measures second‑order sensitivity: how much delta changes when the underlying moves.
– Delta hedging eliminates (or reduces) first‑order exposure; gamma hedging is about stabilizing that delta hedge against larger moves.

Why gamma matters in practice

– Near expiration, small underlying moves can cause large option price swings (gamma tends to increase), so delta hedges can break rapidly unless gamma is controlled.
– Traders long gamma (long options) benefit from frequent rebalancing in a volatile market (“gamma scalping”): buying low and selling high by trading the underlying to stay delta-neutral.
– Traders short gamma face the risk of large, adverse moves and continuous rebalancing costs; they are typically paid a premium (theta) for short gamma exposure.

Delta-gamma hedging (how the two are used together)

– Delta-gamma hedging means simultaneously managing both delta and gamma exposures:
1. Use options with offsetting gamma to reduce portfolio gamma toward target (often zero).
2. Use the underlying (or futures) to adjust delta to the desired level (often zero).
– Example flow: add or remove option contracts to adjust aggregate gamma; then buy/sell the underlying to neutralize the portfolio delta.

Practical steps to implement gamma hedging (step-by-step)

1. Define objectives and constraints
– Hedge horizon (intraday, multi-day, until expiration)
– Target gamma (e.g., gamma-neutral or a specific positive/negative gamma)
– Target delta (usually delta-neutral, but you might want a specific directional delta)
– Risk limits, capital, margin availability, transaction-cost limits

2. Calculate current Greeks for the portfolio

– For each option: get delta and gamma (and vega/theta as relevant).
– Compute portfolio totals: Total Gamma = Σ (gamma_i × contracts_i × contract_size)
Total Delta = Σ (delta_i × contracts_i × contract_size) + shares
– Note contract_size (e.g., 100 shares per option in U.S. equity options).

3. Decide instruments to use for gamma adjustment

– Common choices: other option strikes/maturities, option spreads, or futures (futures have zero gamma).
– To change gamma you must trade options (since underlying/futures have zero gamma).

4. Determine quantity of offsetting options

– Solve for the number of option contracts with opposite-signed gamma needed to reach the target gamma.
– Example: If portfolio gamma = +50 (in share-equivalents) and available option has gamma = −0.02 per contract (each contract = 100 shares), you need 50 / (100 × 0.02) = 25 contracts short of that option to neutralize gamma.

5. Rebalance delta with the underlying

– After adding option trades to reach the target gamma, recompute portfolio delta.
– Buy or sell shares (or futures) to bring delta to the target (e.g., zero).
– Remember: underlying trades change delta but not gamma.

6. Set rebalancing rules and triggers

– Decide when to rebalance (time-based e.g., daily, or trigger-based e.g., when delta moves by X%, or when spot moves by Y).
– Carefully weigh transaction costs against risk: more frequent rebalancing reduces risk but increases cost.

7. Monitor and manage secondary risks

– Vega: option trades used to change gamma will change vega (sensitivity to volatility). Be explicit about vega targets.
– Theta/time decay: long gamma positions often are long vega but lose time value; shorting options to reduce gamma may produce positive theta.
– Liquidity and slippage: large option trades can move prices. Use limit orders, spread trades, or execution algorithms as appropriate.

8. Backtest and paper-trade

– Before committing capital, backtest the rebalancing rules with historical data and simulate transaction costs.
– Paper-trade to ensure operational readiness.

Concrete numerical example

– Suppose you hold 10 long call contracts (contract_size = 100), each with:
delta = 0.40 and gamma = 0.05.
– Portfolio gamma (share-equivalents) = 10 × 100 × 0.05 = +50.
– Portfolio delta (share-equivalents) = 10 × 100 × 0.40 = +400 shares equivalent.
Goal: be gamma-neutral and delta-neutral.
– Find option to offset gamma: suppose you can short another option with gamma = 0.02 per contract.
– Number of contracts to short = desired gamma offset / (100 × 0.02) = 50 / 2 = 25 contracts short.
– After shorting 25 contracts (gamma now near zero), recompute delta:
– Delta from original longs = +400
– Delta from short 25 contracts (assume delta per contract = 0.20): −25 × 100 × 0.20 = −500
– Net delta = 400 − 500 = −100 (short 100 share-equivalents)
– To delta-neutralize, buy 100 shares of the underlying.
Notes: In practice you must use the actual deltas and gammas of the specific options you trade; the prior example is illustrative.

Gamma scalping (how long-gamma traders can profit)

– Traders long gamma and delta-neutral can make profits by actively rebalancing the underlying: when price falls, buy underlying (because delta becomes negative); when price rises, sell underlying (delta becomes positive).
– Profit comes from capturing the realized volatility (trading the swings) if realized volatility exceeds implied volatility paid for the options, minus transaction costs and time decay.
– Requires active trading and sufficient liquidity.

When gamma hedging is commonly used

– Around major news or earnings where rapid moves may occur.
– Near option expiration when gamma tends to be large.
– By market makers and professional options desks to keep inventories hedged.

Costs, limitations, and risks

– Transaction costs and bid–ask spreads: frequent rebalancing can be expensive.
– Discrete hedging: continuous adjustment is impossible; gaps and jumps can cause large P&L swings.
– Model risk: gamma and delta estimates come from models (e.g., Black‑Scholes); parameter error (volatility, interest rates) causes hedging errors.
– Secondary exposures: changing gamma alters vega and theta exposures—hedging one Greek may increase exposure to others.
– Margin and liquidity risk: shorting options to change gamma may incur high margin requirements.
– Short gamma risk: being short gamma is particularly dangerous in fast markets because losses can accelerate.

Gamma hedging vs. delta hedging — short comparison

– Delta hedging: trade underlying to neutralize first‑order price risk. Works well for small, incremental moves and is less costly to implement.
– Gamma hedging: trade options to reduce the rate at which the delta changes (second‑order risk). More expensive and complex because options (not the underlying) must be used to change gamma.
– Combined approach: most professional hedgers manage both delta and gamma (and other Greeks), balancing costs of hedging against risk tolerance.

Practical checklist for a trader

– Know your portfolio’s Greeks (delta, gamma, vega, theta).
– Define target gamma and delta, hedge horizon, and acceptable costs.
– Choose liquid hedging instruments (strikes/maturities with depth).
– Calculate the option quantities needed to reach target gamma.
– Neutralize delta with the underlying or futures.
– Set rebalancing triggers and monitor execution costs.
– Track realized vs. implied volatility to evaluate gamma scalping potential.
– Maintain records and stress-test for jump risk and model error.

If you want, I can:

– run a worked example using real option quotes you provide (strike, premium, implied volatility);
– produce a simple spreadsheet template for calculating the number of contracts needed to reach gamma neutrality and the underlying shares to delta-neutralize. Which would you prefer?