Gamma Scalping: Trading the Curvature of Delta

Delta hedging cancels out an option position's directional exposure — but only for an instant. The moment the stock moves, delta itself has already changed, and the hedge is stale. Gamma is the Greek that measures exactly how fast that happens, and gamma scalping is the trading strategy built entirely around exploiting it.

What gamma measures

Gamma is the rate of change of delta with respect to the stock price — ∂²C/∂S², the second derivative of the option's price, or equivalently, the first derivative of delta. For a call:

Γ = n(d1) / (S · σ · √T)

where n(d1) is again the standard normal density at d1. Gamma is the same for calls and puts with identical strike and expiration — unlike delta, which flips sign between the two, gamma doesn't care about direction, only about how fast delta is changing.

A gamma of 0.04 means: for a $1 move in the stock, delta itself changes by about 0.04. So an option with delta 0.50 and gamma 0.04, after the stock rises $1, has a new delta of roughly 0.54 — the hedge you set up ($50 of stock against 100 shares of option exposure) is now short about 4 shares of the correct hedge, purely from that one move.

Why gamma peaks at-the-money, near expiration

Gamma follows almost the same shape as vega — highest at-the-money, decaying for strikes far from the current price — but behaves oppositely with time: short-dated at-the-money options have the highest gamma, not the lowest. That's the opposite of vega's pattern, and it's worth sitting with why.

Near expiration, an at-the-money option is genuinely on a knife's edge — a small move in the stock is the difference between expiring worthless and expiring deep in the money, so delta swings from near 0 to near 1 over a very small range of stock prices. That's exactly what high gamma means: delta changing fast over a small move. With months left, the same option has more time to "figure itself out," so delta doesn't swing as violently for the same size of stock move — gamma is lower.

What gamma scalping actually is

A gamma scalper deliberately holds a long gamma position — typically long options, hedged delta-neutral with stock — and profits from continuously rebalancing that hedge as the stock moves, regardless of direction.

Walk through it: start delta-neutral. The stock rises. Because gamma is positive, delta rises too — the position is now net long delta (it'll gain if the stock keeps rising). To rehedge back to neutral, you sell some stock at this new, higher price. Now the stock falls back to where it started. Delta falls back too, back toward neutral in the other direction — so you buy back the stock, now at a lower price than you sold it. You sold high and bought low, purely from following the hedge, with no view on direction at all. That round trip is the "scalp," and it repeats every time the stock moves and gets rehedged.

The catch: this isn't free money. Being long gamma means being long the option, which means paying theta — time decay — every single day, whether the stock moves or not. Gamma scalping is a bet that the stock will move enough, often enough, that the accumulated scalping profit exceeds the theta paid to hold the position. It's a volatility trade dressed up as a hedging technique: profitable when realized volatility comes in higher than what you paid for (implied volatility, baked into the option's price), unprofitable when it comes in lower.

Computing gamma in Python

import math

def norm_pdf(x):
    return math.exp(-0.5 * x**2) / math.sqrt(2 * math.pi)

def gamma(S, K, T, r, sigma):
    d1 = (math.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * math.sqrt(T))
    return norm_pdf(d1) / (S * sigma * math.sqrt(T))

# At-the-money, 1 month left — high gamma
print(round(gamma(100, 100, 30/365, 0.05, 0.20), 5))

# At-the-money, 1 year left — noticeably lower gamma
print(round(gamma(100, 100, 1, 0.05, 0.20), 5))