FRM Exam Part I · Option Sensitivity Measures: The "Greeks"

Gamma and Convexity of Options for FRM Part I

Updated 11 October 2026 · Fact-checked

Gamma is the rate of change of an option's delta when the underlying price moves by one unit. It equals the second derivative of option value with respect to price. It is highest for at-the-money options near expiry. To make a portfolio gamma neutral, trade options, not the underlying, because the underlying has zero gamma.

Understand Gamma and Convexity of Options

Delta tells you how much an option's value changes for a small move in the underlying. But delta is not constant. As the underlying moves, delta moves too. Gamma measures that change: gamma = change in delta ÷ change in underlying price.

Gamma is the second derivative of option value with respect to the underlying price. That is why it is also called the option's convexity. The option value curve bends. A long option has positive gamma, so its value curve is convex. A short option has negative gamma.

For a European call or put on a non-dividend stock, gamma is the same for both. This follows from put-call parity, since a call minus a put is a forward, which has zero gamma. Gamma is always positive for a long option and negative for a short option.

Gamma is highest when the option is at the money. Here delta shifts fastest, from near 0 to near 1 for a call. Deep in or out of the money, delta is almost flat, so gamma is small. As expiry nears, an at-the-money option's gamma grows very large, because a tiny price move can flip the option from worthless to in the money. Gamma for in-the-money and out-of-the-money options near expiry falls toward zero.

Gamma matters because delta hedging only works for small moves. A delta-neutral but long-gamma position gains from large moves in either direction. A short-gamma position loses. Gamma-neutral positions reduce the need to rebalance often. Since the underlying has zero gamma, you must add options to change gamma, and then re-adjust delta with the underlying.

Key formulas to remember

Gamma definition: Γ = ∂Δ/∂S = ∂²V/∂S²; Second derivative of option value V with respect to underlying price S.
Black-Scholes-Merton gamma (no dividends): Γ = N'(d1) ÷ (S × σ × √T); N'(d1) is the standard normal density at d1. Same for calls and puts with the same strike and expiry.
Gamma with a continuous yield q: Γ = e^(−qT) × N'(d1) ÷ (S × σ × √T); Use for index or currency options.
Delta change approximation: ΔΔ ≈ Γ × ΔS; New delta ≈ old delta + Γ × ΔS.
Taylor approximation of option value: ΔV ≈ Δ × ΔS + ½ × Γ × (ΔS)²; Ignores theta and vega. The gamma term is always positive for a long option.
Gamma-neutral hedge: w_T = −Γ_P ÷ Γ_T; Number of traded options w_T to add to a portfolio with gamma Γ_P, where Γ_T is the traded option's gamma.
Delta after gamma hedge: Δ_new = Δ_P + w_T × Δ_T; underlying position = −Δ_new; Re-neutralize delta with the underlying, which has zero gamma.

How to solve Gamma and Convexity of Options questions

Use this method for any gamma question, whether it asks for a delta change, a value change or a hedge.

1Identify the position: long or short, call or put, and units held. Sign matters for gamma.
2Write down the given Greeks: delta, gamma, and the price move ΔS. Check whether they are per option or per portfolio.
3For a delta change, compute Γ × ΔS and add it to the starting delta.
4For a value change, use ΔV ≈ Δ × ΔS + ½ × Γ × (ΔS)². Multiply by the number of options at the end.
5For gamma neutrality, compute w_T = −Γ_P ÷ Γ_T. A negative result means you sell the traded option.
6Recompute the portfolio delta including the new options, then trade the underlying to bring delta to zero.
7Check signs and size: a long option should have positive gamma, and the final portfolio gamma should be zero.

Quickest way: Gamma-neutral in three lines

When to use it: Use when the question gives portfolio gamma, traded option gamma and delta, and asks for the hedge trades.

Options needed = −Γ_P ÷ Γ_T. Keep the sign.
New delta = Δ_P + (options needed × Δ_T).
Underlying trade = −New delta. Positive means buy shares, negative means sell.

Common mistakes in Gamma and Convexity of Options

Trying to fix gamma by trading the underlying.
Students know the underlying hedges delta and assume it hedges everything.
Fix: The underlying has delta 1 and gamma 0. Only options change gamma.
Forgetting to rebalance delta after the gamma hedge.
Adding options changes portfolio delta, but the question seems finished after the gamma step.
Fix: Always do gamma first, then recompute delta and neutralize it with the underlying.
Dropping the ½ in the Taylor term.
The formula is confused with the delta-change formula ΔΔ = Γ × ΔS.
Fix: Value change uses ½ × Γ × (ΔS)². Delta change uses Γ × ΔS with no ½.
Thinking a put has negative gamma.
Puts have negative delta, so students assume everything is negative.
Fix: Long calls and long puts both have positive gamma. Only short positions have negative gamma.
Saying gamma is highest for deep in-the-money options.
Delta is highest there, and delta and gamma get mixed up.
Fix: Delta is largest in the money. Gamma is largest at the money, because that is where delta changes fastest.
Applying the sign of gamma to the wrong side of the hedge.
Portfolio gamma is negative, so students also take a negative traded-option position.
Fix: With Γ_P negative and Γ_T positive, w_T = −Γ_P ÷ Γ_T is positive, so you buy options.

Worked examples

Example 1

A portfolio of short options has delta 0 and gamma −8,000. A traded call has delta 0.50 and gamma 2.00. Build a gamma-neutral and delta-neutral position.

Show the solution

Options needed = −Γ_P ÷ Γ_T = −(−8,000) ÷ 2.00 = +4,000.
Buy 4,000 calls. Portfolio gamma = −8,000 + 4,000 × 2.00 = 0.
New delta = 0 + 4,000 × 0.50 = 2,000.
Underlying trade = −2,000, so sell 2,000 units of the underlying.

Answer: Buy 4,000 calls and sell 2,000 units of the underlying.

Example 2

A long position of 1 option has delta 0.40 and gamma 0.05. The underlying rises by 2. Estimate the new delta and the change in option value.

Show the solution

Delta change = Γ × ΔS = 0.05 × 2 = 0.10.
New delta = 0.40 + 0.10 = 0.50.
Value change ≈ Δ × ΔS + ½ × Γ × (ΔS)² = 0.40 × 2 + 0.5 × 0.05 × 4.
= 0.80 + 0.10 = 0.90.

Answer: New delta is 0.50 and the option value rises by about 0.90.

Exam tips

Know the signs cold: long options have positive gamma, short options negative, and the underlying has zero gamma.
Expect the question to ask for the hedge in two stages: options for gamma, then underlying for delta.
Remember that at-the-money gamma rises sharply as expiry approaches, while deep in or out of the money gamma falls toward zero.
Use the formula N'(d1) ÷ (S × σ × √T) to reason about direction: higher volatility lowers peak gamma, shorter time raises it.
If the answer options differ only by sign, recheck whether the question asks what you should buy or sell.

Practice questions from Option Sensitivity Measures: The "Greeks"

Gamma and Convexity of Options: frequently asked questions

What is the difference between gamma and delta?

Delta is the change in option value per unit move in the underlying. Gamma is the change in delta per unit move in the underlying. Delta is a first-order measure, gamma is second-order.

Why is gamma highest for at-the-money options?

Delta changes fastest around the strike, where the option moves from likely worthless to likely in the money. Far from the strike, delta is near 0 or 1 and barely moves, so gamma is small.

What happens to gamma of an at-the-money option near expiry?

It rises sharply. A small move in the underlying can swing delta from near 0 to near 1. This makes short at-the-money options near expiry hard to delta hedge.

How do you make a portfolio gamma neutral?

Trade an option so that the combined gamma is zero. The number of options is −Γ_P ÷ Γ_T. Then adjust the portfolio delta back to zero using the underlying.