Breeden–Litzenberger: Prices to Probabilities

The market does not just quote a price. If you know how to read it, the market quotes an entire probability distribution.

That claim sounds too strong the first time you hear it. A single stock price is one number — how could one number contain a distribution? The answer is that a stock price does not. But an option chain does. Across all the strikes trading at one expiry, the market is making a separate bet at every possible future price level, and the prices of those bets, taken together, trace out exactly how likely the market thinks each outcome is. The Breeden–Litzenberger result is the precise statement of how to pull that distribution back out. It is one of the most beautiful facts in derivatives, and it is what powers the RND Mountain you see in SVQ.

Let us build the intuition before we touch the calculus, because the calculus is almost an afterthought once you see the picture.

Think about a call option struck at one hundred. Its value comes from the chance the underlying finishes above one hundred, and by how much. Now think about a call struck one dollar higher, at one hundred and one. It is worth a little less, because it pays off in slightly fewer scenarios. The difference in price between those two adjacent calls tells you something specific: it isolates the value of the outcomes that fall in the one-dollar slot between them. Slide along the strikes, take these differences, and you are measuring how much the market pays for each thin slice of future price. That is a probability in disguise — slices the market pays a lot for are slices it thinks are likely.

So the first difference across strikes — how fast the call price falls as you raise the strike — encodes the probability of finishing above a given level. And if the first difference encodes a cumulative probability, then the difference of the difference — the second difference, the curvature — encodes the probability density itself, the chance of landing right at a particular price. That is the whole idea. The risk-neutral density is the curvature of the call-price curve in the strike direction.

Now the formalism, which simply makes that precise.

A European call expiring at time capital-T, struck at capital-K, has a value today that we write as capital-C of capital-K. Under the risk-neutral measure, that value is the discounted expected payoff: the discount factor e-to-the-minus-r-T times the average, over all terminal prices, of how far the underlying ends up above the strike. Write that expectation as an integral against the risk-neutral density, the function we will call little-f. Differentiate the call price once with respect to the strike, and the payoff's slope hands you the discounted probability that the underlying finishes above the strike — that is the discounted complementary cumulative distribution. Differentiate a second time, and you land on the density itself. Cleaning up the discount factor gives the Breeden–Litzenberger formula, and it is exactly the anchor for this lesson:

little-f of capital-K equals e-to-the-r-T times the second derivative of capital-C with respect to capital-K.

Read it slowly. The risk-neutral density at any price level is the second strike-derivative of the call price at that level, scaled up by the growth factor e-to-the-r-T to undo the discounting. The curvature of the price curve is the probability. No model of volatility, no assumption of normality, no Black–Scholes required — the result holds for whatever distribution the market is actually pricing. Breeden and Litzenberger proved it in nineteen seventy-eight, and it has anchored model-free density extraction ever since.

A word on what kind of probability this is, because it is the single most important caveat in the whole subject. The density you recover is the risk-neutral density, not the real-world one. It is the distribution the market prices as if it believed, after baking in risk aversion. Because investors pay up for protection against crashes, the risk-neutral density is fatter in the left tail than the true physical density — it over-weights disaster relative to what will actually happen on average. So when you read an RND, you are reading a blend of genuine forecast and the price of fear. Holding those two apart is the heart of this entire course, and it is the difference between a number that is real evidence an asset will move and a number that is mostly the market paying for insurance.