The 1900 Option Formula: Why an At-the-Money Call Is Worth Two Fifths of a Swing

A share sits at $\$100$. Over the coming year its price will move with a standard deviation of $\$20$, quoted in dollars rather than in percent. Interest rates are zero. What is a fair price for a call struck at $\$100$?

The tempting answer arrives in two steps and both of them sound safe. The option pays only when the share finishes above 100, which happens half the time. When it does pay, it pays roughly one swing, so roughly 20. Half of 20 is 10.

The second step is where the money goes. The correct price is $\$7.98$, and the gap is not a rounding artefact: 10 sits more than 25 percent above the truth. The formula that fixes it was written down in 1900, in a doctoral thesis titled Théorie de la spéculation, and it is one line long.

A swing quoted in dollars

The dynamics here are arithmetic rather than geometric. The share price is driven by

where $\sigma_A$ carries units of dollars per square root of time. That is the whole difference from the model everyone learns first, where the noise term is $\sigma S\,dW$ and the volatility is a percentage. Here a swing of twenty dollars means twenty dollars whether the share trades at 100 or at 20.

Notice what has already vanished. The drift $\mu$ never appears again, because setting $r = 0$ pins the risk-neutral drift at zero regardless of what the real-world drift happens to be. Nothing in the pricing depends on whether the company is thriving.

The whole price is one integral

At the money the strike equals $S(t)$, so the payoff is

and the price is that swing multiplied by $\mathbb{E}[\max(Z,0)]$. One integral, and it is the kind you can do by hand, because $x\varphi(x)$ is minus the derivative of $\varphi(x)$:

The expected positive part of a standard normal is the height of its own density at the peak. Substituting gives the 1900 formula in the form it is usually quoted:

With $\sigma_A = 20$ and one year to run, that lands between 7.97 and 7.98. As a mental rule it is even cheaper to carry: the constant sits between 0.3989 and 0.3990, so two fifths of the swing is accurate to about a quarter of a percent. Two fifths of 20 gives 8.00 against the exact 7.9788, which is high by 0.265 percent.

Why half a swing is the wrong half

The reflex was right about the probability and wrong about the size. Split the expectation into the two things it is made of, a chance of paying and an average payment conditional on paying:

The upper half of a bell curve averages 0.798 of one standard deviation, not a full one. So the option pays half the time, and when it pays it hands over four fifths of a swing rather than a whole swing. Half of four fifths is two fifths exactly, which is why (2) and (4) are the same statement written twice.

The mistake is worth naming precisely, because it recurs. A candidate who says “half the swing” has implicitly replaced a random payoff by the single number that a payoff would be if it always arrived at full size. That substitution is the reflex, and the bell curve refuses it: most of the paying outcomes are small ones.

Away from the money

Nothing above needed the strike to sit at the current price. Write the standardised moneyness

and the price becomes $\sigma_A\sqrt{T-t}$ times an integral over the region where the payoff is alive, meaning $Z > -d$. That integral splits into a piece you already did and a piece that is a tail probability:

using $\varphi(-d) = \varphi(d)$ and $1 - \Phi(-d) = \Phi(d)$. The general 1900 price is therefore

which collapses to (3) at $d = 0$, since the second term dies and the first is $\varphi(0)$. The two limits are the sanity check worth running. Deep in the money $\varphi(d) \to 0$ and $\Phi(d) \to 1$, so the price tends to $\sigma_A\sqrt{T-t}\,d = S - X$, the intrinsic value. Numerically the convergence is brisk: at ten swings in the money the price agrees with an intrinsic value of 200 to eight decimal places, and ten swings out of the money it has fallen to $1.5 \times 10^{-23}$.

Equation (5) also hands over put-call parity for free. A put is worth $\sigma_A\sqrt{T-t}\,[\varphi(d) + (-d)\Phi(-d)]$, by the same integral run with $d$ replaced by $-d$, and subtracting gives $d\,[\Phi(d) + \Phi(-d)] = d$. In dollars that is $S - X$, exactly the parity relation at a zero rate.

The reason it fell out of use, and why it came back

A normal terminal distribution puts positive probability on a negative price, which is nonsense for a share and is the honest reason the model was displaced. How much nonsense depends entirely on the ratio of the swing to the price. For a 100-dollar share with a 20-dollar annual swing the probability of finishing below zero is $2.9 \times 10^{-7}$, small enough to ignore. Shrink the share to 20 dollars and keep the same swing and that probability is 15.9 percent, which no amount of goodwill will excuse.

The second failure is in the tails, and it points the other way from what people expect. The density in (5) decays like $e^{-d^2/2}$ in the price itself, faster than the lognormal tail, so far-out strikes come out cheap relative to a market that prices crash risk. The $1.5 \times 10^{-23}$ above is not a small number, it is an absurd one.

Between the two models there is no contest at the money, because there is no disagreement. Take the at-the-money price under (3) and divide it by the lognormal price at a matched volatility, and the ratio runs 1.00667, 1.00167, 1.00042, 1.00010 and 1.00000 as the swing falls through 40, 20, 10, 5 and 1 percent. The 1900 formula is the small-volatility limit of the one that replaced it, which is why the two fifths rule survives on desks that have never heard of arithmetic Brownian motion.

And the flaw is sometimes the point. A spread between two assets can be negative, a calendar spread can be negative, and short rates in several currencies went below zero and stayed there. For those underlyings a model that forbids negative values is the wrong one, and (5) is what gets quoted instead. The formula that lost on shares came back on the products where its defect is a feature.

Sources and further reading

• Louis Bachelier, Théorie de la spéculation (1900), the doctoral thesis examined by Henri Poincaré, where Brownian motion is used to price options five years before Einstein wrote about it in physics.
• Mark Davis and Alison Etheridge, Louis Bachelier's Theory of Speculation: The Origins of Modern Finance (2006), a full English translation with commentary.
• Wikipedia: Bachelier model, Louis Bachelier and Half-normal distribution for the mean in equation (4).

Every constant here was checked three ways before publication: as an exact integral, by deterministic quadrature with no special function in the code path, and by simulation. The general formula (5) was compared against quadrature at eleven values of the moneyness rather than only at the money.