Gamma - The Rate of Change of Delta

By EC Assets Research Team, Derivatives Strategy · Published 19 May 2026 · Updated 14 October 2026

Gamma: Gamma measures how an option's delta changes as the underlying price moves. It is the second derivative of option price with respect to underlying price, and the source of both convexity benefit and dynamic-hedging cost.

What Gamma Measures

Gamma is the rate at which an option's delta changes as the underlying moves. If a call has a delta of 0.53 and a gamma of 0.07, a one-point rise in the stock lifts its delta to about 0.60. Mathematically it is the second derivative of the option price with respect to the underlying.

Gamma is what makes options convex. A long option position has positive gamma: as the market moves in its favour its exposure grows, and as the market moves against it its exposure shrinks. A short option position has negative gamma and the opposite profile: limited gains, accelerating losses. Every option strategy is, at bottom, a decision about which side of that convexity to own and at what price.

How It Works

$$\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{e^{-qT},\varphi(d_1)}{S,\sigma\sqrt{T}}$$

Gamma depends on moneyness and time. For a stock at 100 with 20 percent volatility:

At-the-money gamma roughly doubles from 30 days to 7 days; five points away from the strike, a 7-day option has less gamma than a 30-day one.

Volatility cuts both ways. Doubling volatility to 40 percent halves the 30-day at-the-money gamma, from 0.0695 to 0.0347, because the same uncertainty is spread over a wider range of prices. For a strike 15 percent away it does the opposite, raising gamma from 0.0038 to 0.0177, because the option comes within reach of the money.

The economic meaning is clearest for a delta-hedged position. Over a short interval its profit and loss is approximately

$$\Delta P&L \approx \tfrac{1}{2},\Gamma,(\Delta S)^2 + \Theta,\Delta t$$

Gamma earns on the square of the move, whatever its direction; theta is the daily price of holding it.

Worked Example

Buy a 30-day at-the-money call on a stock at 100 with 20 percent implied volatility and zero interest, and hedge its delta. Gamma is 0.0695 and the option loses about 0.0381 of value per calendar day. The move that makes the two balance is √(2 × 0.0381 / 0.0695) = 1.05, which is exactly the daily move implied volatility predicts: 100 × 20% / √365 = 1.05.

A day with twice the expected move earns more than three days of theta; a quiet day loses most of one.

Over the life of the option the hedged position makes money if the stock's realised volatility exceeds the 20 percent implied in the price, and loses if it falls short. The volatility-to-move converter translates implied volatility into the expected daily move that sets this break-even. Collecting those gains by re-hedging is gamma scalping.

When It Applies (and Limitations)

It is a local measure. Gamma itself changes with spot, time and volatility. Over large moves or around expiry the quadratic approximation breaks down, and the third derivative, sometimes called speed, starts to matter.

Hedging is discrete. The gamma-theta balance assumes continuous re-hedging. Real desks hedge at intervals, so realised profit and loss depends on the path and the timing of hedges, not only on realised volatility.

Short gamma hides its risk in calm markets. A short-gamma book collects theta day after day while moves stay small, and gives back months of income in a single large move. The record looks smoothest just before it breaks.

Gamma concentrates at expiry. As expiry approaches, at-the-money gamma grows without limit while dollar gamma elsewhere vanishes. Large positions near big strikes on expiration day are hard to hedge and can move the underlying, which is the source of pin risk.

Why It Matters for Institutional Investors

It is the trade behind the volatility risk premium. Option sellers are short gamma and demand compensation for the convex losses they bear; on broad equity indices implied volatility has on average exceeded subsequently realised volatility, which is the volatility risk premium documented by Carr and Wu (2009). The premium is earned in quiet periods and paid back in crashes, as short-volatility products discovered in February 2018.

Dealer positioning moves markets. When dealers are net long gamma their hedging sells rallies and buys dips, damping volatility. When they are net short gamma their hedging chases the market and amplifies it. Estimates of aggregate gamma exposure are watched for this reason.

It is the right unit for buying protection. Out-of-the-money options cost less but carry less gamma near the current price; at-the-money options carry the most. Comparing gamma per unit of premium, not premium alone, shows what a hedge actually buys.

Risk limits need it. A book that is delta neutral but carries large negative gamma will be far from neutral after a big move. Gamma limits, and scenario tests across a range of spot moves, are standard for any desk that writes options. Limits are usually set in dollar gamma, the change in dollar delta for a 1 percent move in the underlying, so that books on different underlyings can be added together.

References

1. Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
2. Carr, P., & Wu, L. (2009). Variance Risk Premiums. Review of Financial Studies, 22(3), 1311-1341.
3. Ni, S. X., Pearson, N. D., & Poteshman, A. M. (2005). Stock Price Clustering on Option Expiration Dates. Journal of Financial Economics, 78(1), 49-87.
4. Taleb, N. N. (1997). Dynamic Hedging: Managing Vanilla and Exotic Options. Wiley.
5. Natenberg, S. (2015). Option Volatility and Pricing (2nd ed.). McGraw-Hill.