Straddle Gamma Across Price Movement

Worked Solution

How to Think About It: Before writing any formulas, think about what gamma means economically. Gamma measures how fast your delta changes as the stock moves. A straddle is a bet on realized volatility -- you want the stock to move a lot. Gamma tells you how much you profit from those moves (through delta-hedging or rebalancing). Intuitively, your gamma should be highest when the stock is near the strike (where both options are most sensitive to price changes) and should fall off as the stock moves far away (where the options become deep ITM or deep OTM and behave like linear instruments).

Quick Sanity Checks: - At $S = K$: Both options are ATM, maximum uncertainty about whether they expire ITM. Delta is changing rapidly. Gamma should be at its peak. - Deep ITM call / deep OTM put ($S \gg K$): Call delta is near 1, put delta is near 0. Neither is changing much. Gamma should be near zero. - Deep OTM call / deep ITM put ($S \ll K$): Call delta is near 0, put delta is near $-1$. Again, deltas are barely moving. Gamma near zero. - The gamma profile should be symmetric and bell-shaped around $S = K$.

Derivation:

A straddle consists of a long call and a long put at the same strike and expiry. Since put-call parity implies that calls and puts have the same gamma: $$\Gamma_{\text{call}} = \Gamma_{\text{put}} = \frac{N'(d_1)}{S \sigma \sqrt{T}}$$

where $N'(d_1) = \frac{1}{\sqrt{2\pi}} e^{-d_1^2 / 2}$ is the standard normal density evaluated at $d_1$.

The straddle gamma is: $$\Gamma_{\text{straddle}} = \Gamma_{\text{call}} + \Gamma_{\text{put}} = \frac{2 N'(d_1)}{S \sigma \sqrt{T}}$$

1. At-the-money ($S = K$):

When $S = K$ (and ignoring rates/dividends for simplicity), $d_1 \approx \frac{\sigma \sqrt{T}}{2}$, which is close to 0 for short-dated options. The normal density $N'(d_1)$ is near its maximum value of $1/\sqrt{2\pi} \approx 0.399$. So: $$\Gamma_{\text{straddle}}^{\text{ATM}} \approx \frac{2 \times 0.399}{K \sigma \sqrt{T}} = \frac{0.798}{K \sigma \sqrt{T}}$$

This is the peak gamma.

2. As $S$ moves away from $K$:

As $S$ moves away from $K$ in either direction, $|d_1|$ increases, and $N'(d_1) = \frac{1}{\sqrt{2\pi}} e^{-d_1^2/2}$ decreases (the Gaussian density has its peak at 0 and decays exponentially). Therefore, $\Gamma_{\text{straddle}}$ decreases.

The decay is roughly Gaussian in shape: gamma falls off like $e^{-d_1^2/2}$. Deep ITM or deep OTM, gamma approaches zero -- the options behave approximately like linear instruments (delta is pinned near $\pm 1$ or 0) and there is no curvature left.

The profile is symmetric around $S = K$ (approximately, ignoring the $1/S$ factor which creates slight asymmetry).

3. Effect of time to expiry:

• Far from expiry (large $T$): The gamma peak at ATM is lower (divided by $\sqrt{T}$) but broader. The options are sensitive over a wider range of spot prices.

• Near expiry (small $T$): The gamma peak becomes very tall (proportional to $1/\sqrt{T} \to \infty$) but very narrow. Gamma is concentrated in a tight band around the strike. This is the "gamma spike" that traders talk about near expiry -- if you are ATM, your delta is swinging wildly; if you are even slightly away from the strike, gamma is negligible.

• At expiry ($T \to 0$): Gamma approaches a Dirac delta function centered at $S = K$. The option payoff becomes a hockey stick with an infinite kink at the strike.

Practical Interpretation:

A straddle holder profits from realized moves through gamma: each day, the gamma P&L is approximately $\frac{1}{2} \Gamma (\Delta S)^2$. This profit is maximized when gamma is highest, i.e., when the stock is near the strike. If the stock drifts far from the strike, the straddle becomes a directional position with low gamma and low theta -- it is no longer a good volatility bet.

Traders managing straddle books near expiry are acutely aware of the gamma spike. If you are ATM with hours to go, a 1% move creates a huge delta change that must be hedged immediately. This is why gamma risk near expiry dominates the risk management conversation.

Answer: Straddle gamma is $\frac{2 N'(d_1)}{S \sigma \sqrt{T}}$, which is bell-shaped and centered at $S = K$. It peaks ATM (where $N'(d_1)$ is maximized) and decays toward zero as $S$ moves deep ITM or deep OTM. Near expiry, the peak becomes taller and narrower; far from expiry, it is shorter and broader.