Gamma

Formula

Γ = e^{−qT} φ(d1) / (S σ √T)

φ is the standard normal density.
d1 = [ln(S/K) + (r − q + σ²/2) T] / (σ √T)

Same number for the European call and the European put.
Units: change in delta per 1 point of spot. Not a percent.

Why the call and the put match

• Put-call parity says the call minus the put equals a discounted forward. Differentiating twice in S kills the forward and leaves Γ_call = Γ_put.
• Delta does not match. The call delta and the put delta differ by e^{−qT}. See delta.
• The Greeks calculator reports this gamma for both sides.

Pitfalls

• It blows up as expiry approaches for an option sitting on the strike, because of the √T in the denominator, and it goes to 0 for an option far from the strike. A gamma number with no spot, strike and expiry attached is not interpretable.
• P&L is not gamma times the move. The second-order Taylor term is ½ Γ (ΔS)², and it is a model term. Realised moves, hedges and vega P&L are all still there.
• Spot gamma. As with delta, this differentiates with respect to spot and includes e^{−qT}. Divide by that discount if you wanted a forward gamma and you meant to.

Python reference implementation

Requires scipy. Tested before publication; the generator re-executes it on every rebuild. MIT licensed. Download gamma.py · All snippets

from math import exp, log, sqrt
from scipy.stats import norm

def bs_gamma(S, K, T, r, sigma, q=0.0) -> float:
    """Black-Scholes-Merton gamma. Identical for the European call and put."""
    if S <= 0 or K <= 0 or T <= 0 or sigma <= 0:
        raise ValueError("S, K, T and sigma must be positive")
    d1 = (log(S / K) + (r - q + 0.5 * sigma * sigma) * T) / (sigma * sqrt(T))
    return float(exp(-q * T) * norm.pdf(d1) / (S * sigma * sqrt(T)))

FAQ

Why does gamma not depend on whether the option is a call or a put?

Call minus put is a forward under put-call parity, and a forward's second derivative with respect to spot is zero. Whatever curvature the call has, the put has the same curvature.

Is a high gamma good?

It means delta will move quickly if the spot moves, which is a property of that contract in this model. Long gamma and short gamma describe a position. The page does not score either as an advantage.