Breeden-Litzenberger Formula

A single option price tells you one thing about the future. But an expiry does not have one option, it has fifty, spread across strikes from far below the market to far above it. Taken together those fifty prices are not fifty separate opinions. They are one object: the market's complete probability distribution for where the underlying will finish. The Breeden-Litzenberger result is the recipe for getting that distribution back out, and it needs nothing more than subtraction.

The bookmaker's ladder

A bookmaker takes bets on tomorrow's temperature, but only in one form: "will it end above X?" He quotes a price for every threshold. Above 20 degrees costs 40 cents on the dollar; above 25 degrees costs 28 cents.

You already know what he thinks about the band between them. Buy the first bet and sell the second for a net 12 cents, and you receive a dollar exactly when the temperature lands between 20 and 25. Twelve cents for a dollar means the market puts 12 percent on that band. Do it for every neighbouring pair and the whole distribution falls out, band by band, from nothing but a price list. Option strikes are that ladder, and Breeden-Litzenberger is that subtraction done properly.

First difference: the odds of finishing above a strike

Let $C(K)$ be today's price of a call struck at $K$, all with the same expiry $T$. Let $r$ be the risk-free rate, and let $Q$ denote the risk-neutral probabilities from Risk-Neutral Pricing. Then

$$\frac{\partial C}{\partial K} \;=\; -\,e^{-rT}\, Q\!\left(S_T > K\right).$$

In plain English: the slope of the call price as you walk up the strikes is, apart from a minus sign and a discount factor, the probability of finishing above that strike. The minus sign is there because calls get cheaper as strikes rise; the discount factor is there because the probability pays out at expiry, not today.

That slope is tradable. A tight call spread, long one strike and short the one just above, is essentially a bet paying $1 if you finish above $K$, and its price is the discounted probability (Digital (Binary) Options).

Second difference: the density itself

Differentiate once more and the probability of a single point appears:

$$\frac{\partial^2 C}{\partial K^2} \;=\; e^{-rT}\, q(K),$$

where $q(K)$ is the risk-neutral probability density at level $K$, that is, how much probability sits per index point right around $K$. In plain English: the curvature of the call-price-versus-strike curve, grossed up for discounting, is the market's probability distribution.

Nobody differentiates a screen, so use the discrete version, which is the one desks actually run. Pick a spacing $h$ between strikes:

$$q(K) \;\approx\; e^{rT}\,\frac{C(K - h) - 2\,C(K) + C(K + h)}{h^{2}}.$$

In plain English: buy one call below, sell two at the strike you care about, buy one above, and divide what it cost by the strike spacing squared. That combination is a butterfly, and you have just measured a probability with it.

Here is why the $h^2$ appears, with no calculus at all. A butterfly's payoff is a tent: zero at $K - h$, rising to a peak of $h$ at $K$, back to zero at $K + h$. Its area is base times height over two, which is $2h \times h / 2 = h^{2}$. So the butterfly is worth roughly "the chance of landing near $K$" multiplied by that area, discounted. Rearrange and you have the formula.

Worked example 1: the odds of finishing above 4025

An index expires in six months, rates are 4 percent, so $e^{rT} = e^{0.02} = 1.0202$. Two quoted calls:

• $C(4000) = 210.00$
• $C(4050) = 182.00$

Take the slope: $(182.00 - 210.00)/50 = -0.56$. In plain English, each extra index point of strike knocks 56 cents off the call.

Now apply the formula. The probability of finishing above the midpoint, 4025, is $0.56 \times 1.0202 = 0.571$. The market puts about a 57 percent risk-neutral chance on the index closing above 4025.

The same number priced as an instrument: a digital call paying $1 above 4025 should trade near $0.56, which is the 0.571 chance discounted back. If you see it at $0.48, either you have found an arbitrage against the call spread or, far more likely, one of the three prices is stale.

Worked example 2: how much probability sits around 4000

Add one more strike to the ladder:

• $C(3950) = 240.00$
• $C(4000) = 210.00$
• $C(4050) = 182.00$

Step 1, the butterfly. $240.00 - 2(210.00) + 182.00 = 2.00$. Buying that structure costs $2.00.

Step 2, the spacing. $h = 50$, so $h^{2} = 2500$.

Step 3, the density. $q(4000) \approx 1.0202 \times 2.00 / 2500 = 0.000816$ per index point.

Step 4, turn it into something readable. Probability of landing within roughly 25 points either side of 4000, a 50-point window, is $0.000816 \times 50 = 0.041$, about 4.1 percent.

Check it the quick way. The butterfly cost $2.00 and its best possible payoff is $50, at exactly 4000, so $2/50 = 4\%$ before discounting. Same answer, done in your head.

Pick "butterfly" below and drag the width slider. Narrow the tent and you are asking about a tighter window, so it gets cheaper and the probability it measures shrinks. Widen it and you buy a broader slice of the distribution.

The arbitrage check comes free

Suppose instead the middle call were quoted at 213.00. Then $240 - 426 + 182 = -4.00$: the butterfly costs a negative amount, so it pays you to own a structure that can never lose. The formula returns a negative probability density, which is impossible, so the quotes are impossible too. This is the same statement as the convexity condition in No-Arbitrage Bounds On Option Prices, arrived at from the other direction: a surface that produces a negative density is not slightly wrong, it is unusable.

Where the shape comes from

The recovered density is almost never a neat bell curve. It leans left with a fat left tail, because index puts are persistently expensive; that is the volatility skew in a different coordinate system. Below, drag the skew slider: raising it lifts implied vol on the low strikes, which raises those call prices and, through the second difference, thickens the left tail of the recovered density.

What this means in practice

Desks pull an implied density before every scheduled event. Ahead of a central bank meeting or an earnings print the extracted distribution often turns visibly bimodal, two humps rather than one, which no single volatility number could ever tell you.

It is also a pricing tool. Once you have $q$, any European payoff at that expiry is an integral against it: $V = e^{-rT}\int f(K)\,q(K)\,dK$. That is how odd structured payoffs stay consistent with the vanilla market, and it is the machinery behind the variance swap replication that produces the VIX. Market-implied tail probabilities, the chance of a 20 percent drawdown by year end, come from the same calculation.

Key terms

• Risk-neutral density — the probability distribution implied by option prices, not by history.
• Digital call — pays a fixed amount above a strike; priced by the first derivative in strike.
• Butterfly — long one call each side, short two in the middle; priced by the second difference.
• Strike spacing $h$ — the gap between adjacent strikes; the density divides by its square.
• Negative density — an impossible output, and a reliable sign of a bad quote or a bad fit.