Gamma: Rate of Change of Delta

The Second Derivative

Delta tells you how much an option moves for a small stock price change. But delta itself changes as the stock moves — and gamma measures how fast. Formally:

Γ = ∂Δ / ∂S = ∂²V / ∂S²

Gamma is the second derivative of the option price with respect to the stock price. It measures the convexity of the option price curve. If a call has Δ = 0.50 and Γ = 0.06, then after a $1 rise in the stock, the new delta is approximately 0.56. After a $2 rise, it's approximately 0.62. The option gains delta as the stock rises — this is positive convexity, and it's what option buyers are paying for.

The Formula

Γ = n(d₁) / (S · σ · √T)

where n(d₁) is the standard normal PDF evaluated at d₁. Two crucial facts:

1. Gamma is always positive for long options — both calls and puts. If you're long options, you benefit from large moves in either direction (positive convexity).
2. The formula is identical for calls and puts. A call and put with the same strike and expiry have the same gamma. This follows from put-call parity: C − P = S − PV(K), so ∂²C/∂S² = ∂²P/∂S².

Where Gamma Lives: ATM Near Expiry

Gamma is not uniformly distributed across strikes and maturities. It is highest for at-the-money options close to expiration. Let's understand why intuitively:

Think about what delta does as expiration approaches for an ATM option. With 1 year left, a $1 move in the stock barely changes whether the option expires ITM or OTM — delta changes slowly. With 1 day left, a $1 move can flip the option from almost certainly OTM to almost certainly ITM — delta changes violently. This rapid delta change is high gamma.

Numerically: an ATM option with 1 year to expiration might have Γ = 0.02 (delta changes by 0.02 per $1 stock move). The same option with 1 week to expiration might have Γ = 0.15. The option is "pinned" near the strike — either side wins big, and small moves determine everything.

Gamma Risk: The Pin and the Explosion

High gamma near expiration creates two distinct risks that options professionals manage carefully:

Pin risk: When a heavily-traded option's strike coincides with the stock price near expiration, market makers who have sold those options must delta-hedge aggressively. As the stock moves above the strike, they buy shares (delta rising toward 1). As it moves below, they sell shares (delta falling toward 0). This creates a self-fulfilling "magnetism" — the stock can get pinned to the strike as hedgers chase delta on both sides. This phenomenon is called the "max pain" effect and is visible in large-cap stocks on monthly expiration Fridays (OpEx).

Gamma explosion: In the days before expiration, 0DTE (zero days to expiration) options have enormous gamma. A 5-point SPX move can change a 0DTE option's delta by 0.40 or more. Market makers short 0DTE options face potentially unlimited delta exposure from small moves. The rise of 0DTE trading — 0DTE options now account for over 40% of SPX options volume — has fundamentally changed how professionals think about intraday hedging.

Gamma Scalping: Profiting from Moves

A trader who is long gamma (long options, delta-hedged) can make money from volatility through a strategy called gamma scalping. The mechanics:

1. Buy ATM options and delta-hedge to neutrality.
2. When the stock rises, your delta becomes positive (gamma effect). Sell some stock to rebalance to delta-neutral. You've sold high.
3. When the stock then falls back, your delta becomes negative. Buy stock to rebalance. You've bought low.
4. Each round-trip rebalance locks in a small profit proportional to the stock move squared times gamma: P&L ≈ ½Γ(ΔS)².

The catch: you're paying theta every day for this privilege. Gamma scalping is profitable only if the realized volatility exceeds the implied volatility embedded in the options you bought. If IV = 20% but the stock only moves as if σ = 15%, you'll lose theta faster than you earn from rebalancing. This is the core question every options trader faces: is the option cheap or expensive relative to realized vol?

The Black-Scholes PDE Revisited

Recall the Black-Scholes PDE from the theta lesson:

Θ + ½σ²S²Γ + rSΔ − rV = 0

For a delta-hedged portfolio (net Δ = 0), this simplifies to:

Θ + ½σ²S²Γ = rV (roughly)

The term ½σ²S²Γ is the "gamma P&L" from stock moves, and Θ is the theta cost. In equilibrium, they balance. This equation is essentially saying: the expected gamma profits from continuous rebalancing exactly offset the theta cost, leaving a risk-free return. It's the mathematical statement that options are "fairly priced" under Black-Scholes.