The Near-Dated Option Has More Gamma, Until You Move Nine Percent Away

At the money the shorter maturity always wins, and a single negative derivative settles it for every volatility and every maturity: curvature runs 0.06907 against 0.02814 for one month against six. Ten percent out of the money the order reverses, 0.01854 against 0.02352, and the two curves cross 8.845 percent above the strike. What forces a crossover to exist is a conservation law, since every option in the family carries exactly the same total curvature and can only choose how to spread it.

Two calls sit on the same share with the same strike. One expires in a month, the other in six. Which one bends more, meaning which has the larger second derivative of price against share price? Ask the question in a room and the answer comes back instantly, and it comes back wrong, because the answer is a conditional and the room has heard only half of it.

The half everyone has heard is true. At the money the near-dated option wins, and it wins by a lot: with a 20 percent annual swing and the share exactly on the strike, the curvature runs 0.06907 for one month against 0.02814 for six, a factor of 2.4546. Move the strike ten percent away and the order flips. Same share, same volatility, same pair of maturities, and now the six-month option carries 0.02352 against the one-month's 0.01854.

Why the near date feels like the answer

The reflex is built on a real picture. A call's payoff has a kink at the strike, and as expiry approaches the price curve has to converge to that kink. A kink is infinite curvature, so something must blow up, and what blows up is exactly the quantity being compared. Near the strike, near expiry, the price graph turns almost as sharply as the payoff itself.

That is why the answer “the shorter one” survives so much scrutiny. It is not a guess. It is a correct observation about the one place the question is most often asked, and the word that makes it false is the silent always attached to it.

Maturity and volatility are the same variable

Everything below is cleaner in two combinations rather than in four separate parameters. Rates are zero throughout, there are no dividends, and the option is on one share.

Total volatility and log-moneyness

Write $u = \sigma\sqrt{T-t}$ for the total volatility left in the option and $m = \ln(S/X)$ for the log-moneyness. Curvature is $\Gamma = \varphi(d_1)/(S u)$ with $d_1 = m/u + u/2$ , so it depends on maturity and volatility only through their product $u$ . Halving the volatility and quadrupling the time left gives the identical number.

Expanding the square in the exponent splits it into three terms that never mix, and this one line carries the whole article:

\frac{d_1^{\,2}}{2} \;=\; \frac{m^2}{2u^2} \;+\; \frac{m}{2} \;+\; \frac{u^2}{8}

(1)

The middle term does not involve $u$ , so it cancels out of any comparison between two maturities at the same moneyness. What is left is a competition between $m^2/(2u^2)$ , which punishes small $u$ when the share is away from the strike, and $u^2/8$ together with the $1/u$ prefactor, which punish large $u$ . Written out:

\Gamma \;=\; \frac{1}{\sqrt{2\pi}\,S\,u}\,\exp\!\left(-\frac{m^2}{2u^2} - \frac{m}{2} - \frac{u^2}{8}\right)

(2)

Every option carries the same total curvature

Before comparing two numbers it helps to know that the comparison cannot be about size. Sweep the curvature across every possible share price and add it up. Since curvature is the derivative of the hedge ratio, the sweep telescopes:

\int_0^\infty \Gamma(S)\,\mathrm{d}S \;=\; \Delta(\infty) - \Delta(0) \;=\; 1 - 0 \;=\; 1

(3)

The integral is over the share price, and it comes out to one for every maturity and every volatility. So the same total curvature is issued to every option in the family, and the only freedom is how to spread it. A taller spike is a narrower spike. That single sentence forces a crossover to exist, because two curves of equal area cannot have one sitting above the other everywhere.

Fig. 1 — Both curves enclose exactly the same area. The near-dated option piles its allowance onto the strike and the far-dated one spreads it thin, so they have to cross.

At the money the short date wins, with no exceptions

Set $m = 0$ . Equation (2) collapses to a function of $u$ alone, and its derivative is not a borderline case:

\frac{\mathrm{d}}{\mathrm{d}u}\!\left(\frac{e^{-u^2/8}}{u}\right) \;=\; -\,e^{-u^2/8}\left(\frac{1}{u^2} + \frac{1}{4}\right) \;<\; 0

(4)

Both terms in the bracket are positive and the exponential is positive, so the expression is strictly negative for every $u > 0$ . There is no region of the parameter space where an at-the-money option gains curvature by having more time left. Ratios follow directly from equation (2): one month against six at a 20 percent swing gives 2.4546, and the same pair at a 50 percent swing gives 2.4816. The prefactor $u_6/u_1 = \sqrt{6} \approx 2.4495$ does almost all of the work, and the exponential contributes the rest of the third decimal.

Where the order reverses

Away from the strike the picture changes shape rather than degree. Differentiating equation (2) in $u$ at fixed $m$ and clearing denominators gives a quadratic in $u^2$ whose positive root is the location of the peak:

\frac{u^4}{4} + u^2 - m^2 = 0 \quad\Longleftrightarrow\quad u_\star^2 = 2\left(\sqrt{1+m^2}-1\right)

(5)

Curvature at fixed moneyness rises to that peak and falls afterwards. At $m = 0$ the peak sits at $u_\star = 0$ , which is why the at-the-money case looks monotone: it is the boundary case, with the rising half of the hump squeezed out of existence. At $m = 0.2$ the peak has moved out to $u_\star = 0.199$ , comfortably beyond the 0.0577 of a one-month option, so on that side of the peak more time still buys more curvature.

Setting the two gammas equal and taking logarithms leaves a linear equation in $m^2$ , so the crossover is exact rather than numerical:

m^2 \;=\; \frac{\ln(u_2/u_1) + \bigl(u_2^2 - u_1^2\bigr)/8}{\dfrac{1}{2u_1^2} - \dfrac{1}{2u_2^2}}

(6)

For one month against six at a 20 percent swing, $u_1 = 0.057735$ and $u_2 = 0.141421$ , and (6) returns $|m| = 0.084757$ . That is 8.845 percent above the strike in price, or 8.126 percent below it. The asymmetry in price is only the asymmetry of the logarithm; in log-moneyness the two crossings are exactly symmetric. A crossover exists at every one of 180 volatility and maturity-pair settings tested, so “it depends” is not an artefact of this example.

Fig. 2 — Above the horizontal line the near-dated option has more curvature. Below it, the six-month option does, and by the edges of the frame it has sixty times as much.

With the share at 100 and the strike at 110 the reversal is not a technicality. Six-month curvature is 0.023517 against the one-month's 0.018544, a factor of 1.2681 in favour of the longer option. Anyone who answered “the shorter one” has the sign of a hedge wrong, not a decimal.

Where the comparison stops meaning anything

Let the near maturity run to zero at fixed moneyness and the two branches separate violently. At the money, curvature grows like $1/\sqrt{T-t}$ , and it does so without bound: 1.995, then 199.5, then 19947 as the time left falls through $10^{-4}$ , $10^{-8}$ and $10^{-12}$ of a year. Away from the money it collapses to zero at the same maturities, since the $m^2/(2u^2)$ term in equation (2) beats every prefactor. In the limit the whole unit of curvature has been squeezed into a single point, which is the kink itself, and neither number is something a desk can hedge.

Two smaller cautions. Equation (3) is a statement about integrating over the share price, and it says nothing about totals over time; the phrase “total curvature is one” is only true with that qualifier attached. And the 2.4546 belongs to one month against six at a 20 percent swing. The direction of the at-the-money comparison is universal, by equation (4), while its size is not.

Sources and further reading

Wikipedia: Greeks (finance) for the standard sensitivities, and Black–Scholes model for the price whose second derivative is being taken.
Wikipedia: Log-normal distribution, which is the terminal law that turns equation (2) into a density in disguise.
Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy81 (1973), 637–654.

Every figure quoted here was recomputed as a second difference of the call price itself, with no curvature formula anywhere in that code path, agreeing to seven decimal places at 315 grid points. The conservation law in equation (3) was checked by quadrature in log space at six maturity and volatility pairs, landing on 1.00000 at each, and the crossover was located by bisection on the real gammas across five maturity pairs, matching the closed form to nine decimal places.