A Coin Flip Between Two Volatilities Is Enough to Make a Smile

Let a fair coin decide at the start of the year whether the share runs at 15 or 35 percent, price calls in that world, then read the volatilities back out with the constant-volatility formula: 28.43, 25.91, 24.97, 25.68 and 27.16 percent across five strikes. The floor sits at the money and below the 25 percent average of the two regimes, which one second derivative settles without any numerics. The usual explanation for the wings is refuted here, because the coin-flip world is less likely to clear 130 than a flat 25 percent and its option is still worth 27 percent more.

Take one share on one day and a row of listed calls at different strikes. Invert the constant-volatility formula on each price to recover the volatility it implies. If the formula described the world, every inversion would return the same number, because there is only one share and it has only one volatility. Instead the numbers trace a curve with a floor near the money and higher wings on both sides.

The usual next sentence is that traders use a more elaborate model, which is true and explains nothing. What follows builds the crudest possible departure from constant volatility, prices options inside it, and reads the volatilities back out. A smile appears, its floor sits below the average of the two regimes, and the standard explanation for the wings turns out to be false in the very place it is usually offered.

What a flat line would require

Implied volatility

Given a market price $C^{\text{mkt}}$ for a call at strike $X$ , the implied volatility is the unique $\sigma$ solving $C^{\mathrm{BS}}(\sigma; X) = C^{\text{mkt}}$ . It is unique because the price is strictly increasing in $\sigma$ , so the map can be inverted by bisection with no derivative anywhere.

The definition is a change of units and nothing more. It quotes a price in volatility instead of dollars, using the constant-volatility formula as the dictionary. Nothing about that requires the formula to be true, which is exactly why the practice survives the formula being false.

A flat line across strikes would require the terminal distribution of the share to be the lognormal that a single constant volatility produces. Anything else, and the dictionary will return different numbers at different strikes, because different strikes read different parts of the distribution.

The crudest non-constant volatility there is

Here is the whole model. The share starts at 100, rates are zero, and the horizon is one year. At time zero a fair coin decides whether the year runs at 15 percent or at 35 percent, and then the year runs at that volatility from start to finish. Nothing is stochastic after the flip.

Both branches are martingales with the same starting value, so the mixture is a martingale too, and there is no arbitrage to argue about. A call in this world is priced by taking the expectation over both branches, which with a fair coin is an average of two ordinary prices:

C_{\text{mix}}(X) \;=\; \tfrac{1}{2}\,C^{\mathrm{BS}}(X;\,15\%) \;+\; \tfrac{1}{2}\,C^{\mathrm{BS}}(X;\,35\%)

(1)

Mixing two variances is already enough to leave the lognormal family. For a fair mixture of two zero-mean normals whose variances are in the ratio $v$ , the kurtosis is exact:

\kappa \;=\; \frac{6\,(1+v^2)}{(1+v)^2} \;=\; 3 \;+\; \underbrace{\frac{3\,(1-v)^2}{(1+v)^2}}_{\text{excess}}

(2)

The excess term vanishes only at $v = 1$ , so any non-constant volatility gives a log return with fatter tails than the single normal carrying the same total variance. Here $v = (15/35)^2 = 9/49$ , which gives $\kappa = 3723/841 = 4.4269$ , an excess of 1.4269 over the normal. The mechanism is not exotic and it is not a modelling choice. It is what averaging two variances does.

The implied volatilities, strike by strike

Price equation (1) at five strikes and invert each price with the constant-volatility formula. The prices are 31.183, 18.836, 9.9353, 5.0579 and 2.7342 at strikes 70, 85, 100, 115 and 130, and the volatilities they imply are:

\begin{array}{c|ccccc} X & 70 & 85 & 100 & 115 & 130 \\ \hline \sigma_{\text{imp}} & 28.43\% & 25.91\% & 24.97\% & 25.68\% & 27.16\% \end{array}

(3)

A curve, with its lowest point at the money. Sweeping 201 strikes from 60 to 160 confirms the minimum is exactly the at-the-money one and not an artefact of the five strikes chosen.

Fig. 1 — The dictionary returns a different number at every strike. The floor is at the money, and it is below the 25 percent average of the two volatilities in play.

The cost of ignoring the curve is a dollar figure rather than a decimal. Fit a single volatility to all five prices by least squares and the best available is 25.82 percent, which still misprices the 130 strike by 13.45 percent. Calibrate instead to the at-the-money price alone, giving 24.97 percent, and the 130 strike comes out at 2.1405 against its true 2.7342, so 21.71 percent too cheap. One flat number cannot price a row that was not generated by a flat number, and the residual does not shrink with care.

Why the floor sits below the average

That the at-the-money implied volatility is 24.97 rather than 25.00 looks like a rounding curiosity and is actually forced. At the money with a zero rate the call price has a closed form with a single error function:

C^{\mathrm{BS}}(S;\,\sigma) \;=\; S\,\mathrm{erf}\!\left(\frac{\sigma\sqrt{T}}{2\sqrt{2}}\right), \qquad \frac{\partial^2 C}{\partial \sigma^2} \;=\; -\,\frac{\sigma\, e^{-\sigma^2 T/8}}{4}\sqrt{\frac{T^3}{2\pi}}\,S

(4)

The second derivative is strictly negative for every positive volatility, so the price is concave in volatility. Jensen's inequality then does the rest: the average of the prices at 15 and 35 percent is strictly below the price at their average of 25 percent. Numerically the mixture is 9.9353 against 9.9476, about 1.2 cents lower, and inverting that lower price necessarily returns a volatility below 25 percent.

No numerics were needed for the direction, only the sign of one second derivative. It also says the floor of the smile is a general phenomenon in mixtures rather than a feature of this particular pair.

The explanation that does not survive measurement

The sentence usually offered for the wings is that fat tails make the share more likely to finish out there, so the far strikes cost more. It is wrong, and this toy world refutes it with a measurement rather than an argument.

The chance of clearing 130 is lower in the coin-flip world than under a flat 25 percent: 10.583 percent against 12.011 percent. And its call is worth 2.7342 against 2.1486, which is 27.2 percent more. Less likely to pay, and worth more. The two survival curves do not even cross until about 141.5, and the densities themselves stay below the flat one until 166.

Fig. 2 — On a logarithmic axis the tails are readable. The coin-flip world is thinner than the flat model between about 116 and 166, which includes the 130 strike, and only becomes fatter further out.

What the extra money buys is the size of the payoff rather than the chance of collecting it. In the coin-flip world, the paths that finish above 130 are overwhelmingly the ones that drew the wild year, and those finish a long way above. Dividing the option value by the probability of finishing in the money gives the average payoff conditional on collecting:

\frac{2.7342}{0.10583} = 25.84 \qquad\text{against}\qquad \frac{2.1486}{0.12011} = 17.89

(5)

Forty-four percent more per winning path. That is the entire source of the wing, and it is why the honest phrasing is that when the share moves it moves further, not that it is more likely to get there. The distinction matters the moment someone tries to trade the reasoning: a digital option, which pays on the chance and not on the size, is worth less in the coin-flip world at that strike.

What this toy world cannot produce

The mixture is deliberately crude and its limits are worth stating, since a reader who takes it for a real model will draw two wrong conclusions.

It cannot produce a skew. A coin decided once at time zero is symmetric in the log price, so the smile it makes is close to symmetric in log-moneyness, and the asymmetry visible in Fig. 1 is the asymmetry of plotting against strike rather than against the logarithm. Real equity surfaces slope, and the sign of the slope follows the correlation between volatility and price, which this model does not contain because volatility here never moves after the flip.

It also has nothing to say about maturity. One coin governs the whole year, so every horizon inherits the same mixture and the term structure is flat by construction. A model where volatility itself diffuses produces a smile that flattens with maturity, which is a separate observed fact and needs separate machinery.

What survives is the part that matters. The price row from equation (1) is arbitrage-free: every price sits between its intrinsic value and the share price, and the row is decreasing and convex in the strike at every strike checked. So the smile is not a pricing error waiting to be corrected. It is what a correct price row looks like when it is read through a dictionary written for the wrong distribution.

Sources and further reading

Wikipedia: Volatility smile and Implied volatility, for the observation and the definition respectively.
Wikipedia: Kurtosis and Jensen's inequality, which supply equations (2) and (4).
Bruno Dupire, “Pricing with a Smile”, Risk7 (1994), 18–20, for the step from a smile to a volatility that depends on price and time.

The inversion was shown to be well posed before it was used, by checking that the price is strictly increasing in volatility at 1,400 volatilities per strike. A 300,000-path simulation of the same two-regime world reproduces every point of equation (3) to within half a volatility point, and a single-regime control returns a flat 25.000000 percent, so the inverter does not manufacture smiles of its own.