Twice the Days, One Point Four One Times the Price

A contract giving you the right to buy a share at today's price 100 days from now trades at 100 dollars. An otherwise identical contract expires in 200 days. What should it cost?

About 141, and a little under. Twice the life is not twice the value, because the quantity being paid for grows like the square root of the time remaining rather than the time itself.

What is actually being bought

The right to buy at a fixed price is worth something only when the share ends up above that price, and it is worth more the further above. So the price of the contract tracks the typical size of the move the share can make before expiry.

A price that wanders with independent increments accumulates variance at a steady rate, not spread. Over a horizon $T$ the variance is proportional to $T$ and the standard deviation to $\sqrt T$. Twice the time gives $\sqrt2 = 1.41421$ times the wander, and the contract is priced off the wander:

The uplift is 41.4 percent, not 100 percent. Note also that $\sqrt2 < 2$, so the linear guess of 200 dollars over-values the longer contract rather than under-valuing it.

Where the square root is exact, and where it is not

Equation (1) is a heuristic, and this particular case has a closed form to check it against. For a contract struck exactly at today's price, with no interest rate and no dividend, the standard model gives

Two things fall out of equation (2) immediately. The value depends on $\sigma$ and $T$ only through the product $s = \sigma\sqrt T$, so the ratio asked for needs neither the share price nor the volatility, which is what makes the question answerable from one number. And expanding $\operatorname{erf}$ near zero gives the working approximation

which is linear in $\sqrt T$ and therefore exactly the square-root rule. At $\sigma = 30\%$ and 100 days, equation (3) gives 6.264 against the exact 6.258, an error of one part in a thousand.

The rule over-states, always

Doubling $T$ takes $s$ to $s\sqrt2$, so the exact ratio implied by equation (2) is

The inequality holds because $\operatorname{erf}$ is concave on the positive axis, so $\operatorname{erf}(\lambda x) < \lambda \operatorname{erf}(x)$ for $\lambda > 1$. It is a small effect at realistic inputs: at 20 percent volatility $\rho = 1.41357$, at 50 percent it is 1.41020, both inside 0.3 percent of $\sqrt 2$. But the direction is fixed, so 141.42 is a ceiling that is never reached, and 141 is the honest number to say.

What the same rule says about other horizons

Once the price is known to follow $\sqrt T$, every maturity comparison is one square root. Four times the days doubles the price, so a 400-day contract on the same terms is worth about 200. Halving the days multiplies by $1/\sqrt2 = 0.707$, so a 50-day contract is worth about 71. Ten times the days multiplies by 3.16, and a contract with one tenth of the life keeps 32 percent of the value, which is far more than a tenth.

That last comparison is the one people find hardest to believe, and it is worth sitting with. A contract with 10 days left holds nearly a third of the value of one with 100 days left. Time decay is slow at first and then brutal, and the reason is that the derivative of $\sqrt T$ blows up as $T \to 0$.

Equation (3) can also be read backwards. Given a price for a contract struck at the current share price, $\sigma \approx c\sqrt{2\pi}/(S\sqrt T)$recovers the volatility the price implies, which is a one-line estimate rather than the numerical inversion the full formula needs. And the unit used for "days" never matters for the ratio, because a change from calendar days to trading days rescales both maturities by the same factor and cancels inside the square root.

The qualifier is load-bearing

Strip out "struck at today's price" and the rule collapses. Hold the volatility at 30 percent and the rate at zero, and double 100 days to 200:

The reason is visible in equation (2) once the strike is free. A far out-of-reach strike needs an unusual move to pay anything at all, and probability mass reaches it at an accelerating rate, so extra time is worth much more than $\sqrt T$. A strike far below the price is already nearly certain to pay, and extra time adds almost nothing except optionality that is unlikely to be used.

Two assumptions doing quiet work

The interest rate was set to zero, which is what lets the answer ignore the difference between a right to buy and a right to sell. With a positive rate the two separate, and the direction can be surprising: a deeply in-the-money European right to sellcan lose value as its maturity lengthens, because the money it will eventually deliver is discounted from further away. So "longer is worth more" is not a law of contracts, and the square-root rule is not either.

The second assumption is that only the days differ. If the market's view of the share's volatility differs between the two horizons, which it usually does, then $s = \sigma\sqrt T$ no longer scales as $\sqrt T$ and the ratio moves. The rule is about a single volatility applied to two horizons, and it is worth saying so before quoting it.

Sources and further reading

• The closed form in equation (2) — Black-Scholes model
• Why the spread grows as the square root of time — Random walk
• The process underneath — Geometric Brownian motion
• The concavity used in equation (4) — Error function

The ratio was measured as well as derived. Two million simulated price paths, with common random numbers across the two horizons and antithetic pairs, returned 8.323 and 11.748 against the exact 8.337 and 11.769, a ratio of 1.4115 against $\sqrt2$. No pricing formula appears anywhere in that simulation, and repeating it on a share priced at 1000 instead of 100 leaves the ratio unchanged.