The Greeks and Option Risk Management › Gamma: The Curvature of Delta

Gamma ($\Gamma$) measures the rate of change of delta with respect to changes in the underlying price. While delta tells you your current exposure to price movements, gamma tells you how that exposure will change as the price moves. This second-order sensitivity captures the essence of what makes options non-linear instruments, fundamentally different from stocks or forward contracts.

Mathematically, gamma is the second partial derivative of the option price with respect to the spot price:

$$\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}$$

where:

• $\Gamma$: option gamma
• $V$: option value
• $S$: underlying asset price
• $\Delta$: option delta

The equivalence between these two expressions follows directly from the definition: since delta is the first derivative of price with respect to spot, gamma is the derivative of delta, which is the second derivative of price. Gamma tells you how quickly your hedge (delta) will become stale as the underlying moves. A high gamma position requires frequent rebalancing to maintain delta neutrality.

Using the Black-Scholes framework, we derive gamma by differentiating delta with respect to the spot price $S_0$. Recall that for a call, $\Delta = N(d_1)$. To find gamma, we differentiate this expression with respect to the stock price:

$$\begin{aligned} \Gamma &= \frac{\partial \Delta}{\partial S_0} = \frac{\partial}{\partial S_0} N(d_1) \\ &= N'(d_1) \frac{\partial d_1}{\partial S_0} && \text{(chain rule)} \\ &= \frac{1}{\sqrt{2\pi}} e^{-d_1^2/2} \cdot \frac{1}{S_0 \sigma \sqrt{T}} && \text{(substitute derivatives)} \end{aligned}$$

The chain rule tells us to multiply the derivative of the outer function (the cumulative normal, whose derivative is the standard normal PDF) by the derivative of the inner function ($d_1$ with respect to $S_0$). The derivative $\frac{\partial d_1}{\partial S_0} = \frac{1}{S_0 \sigma \sqrt{T}}$ comes from recognizing that $d_1$ contains $\ln(S_0/K)$, and the derivative of the natural logarithm of $S_0$ is $1/S_0$.

This leads to the standard formula:

$$\Gamma = \frac{N'(d_1)}{S_0 \sigma \sqrt{T}} = \frac{e^{-d_1^2/2}}{S_0 \sigma \sqrt{2\pi T}}$$

where:

• $\Gamma$: option gamma
• $N'(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$: standard normal probability density function
• $S_0$: underlying price
• $K$: strike price
• $r$: risk-free interest rate
• $\sigma$: volatility
• $T$: time to expiration
• $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$: standardized moneyness factor

Calls and puts have identical gamma because of put-call parity. Since $C - P = S - Ke^{-rT}$ is linear in $S$, the second derivatives are equal. This makes intuitive sense: both calls and puts exhibit the same curvature in their payoff profiles; they simply face different directions.

Gamma captures the convexity or curvature in an option's value as a function of the underlying price. This curvature is what makes options fundamentally different from linear instruments like forwards. A forward contract has constant delta (equal to one) and zero gamma: its value moves one-for-one with the underlying at all prices. An option, by contrast, has variable delta that changes with the stock price, and this variability is measured by gamma.

Consider a delta-neutral portfolio. If gamma is positive, a large move in either direction benefits the position; the portfolio gains value whether the stock jumps up or crashes down. This is why being "long gamma" is attractive but costly: you pay time decay for the privilege of this favorable convexity. Conversely, being "short gamma" means you earn time decay but face unlimited downside if the market makes a large move. If you sell options, you are typically short gamma and must actively manage this exposure.

We implement the gamma calculation below. Note that the formula is the same for calls and puts, so we do not need an option type argument: