Implied Distributions: What Option Prices Know

3 min read

An option chain is not just a menu of trades — it's a photograph of the market's probability distribution for the underlying at expiry. Extracting that photograph is the Breeden–Litzenberger result, and it ties together the butterflies, smiles, and no-arbitrage bounds from earlier lessons into one picture.

The result

Let $C(K)$ be the price of a call at strike $K$ (same expiry, rates set aside for clarity). Then:

$\frac{\partial C}{\partial K} = -\,P(S_T > K), \qquad \frac{\partial^2 C}{\partial K^2} = f(K)$

— the first derivative in strike reads off the (risk-neutral) tail probability, and the second derivative is the probability density itself. The market's entire implied distribution, recovered from prices.

Why: butterflies are probability

No calculus needed — the finite-difference version is a trade. The butterfly $C(K - h) - 2C(K) + C(K+h)$ , scaled by $1/h^2$ , has a payoff that's a narrowing tent around $K$ : in the limit, a spike paying \ $1 exactly at$ S_T = K $. Its price must therefore be the probability (density) of landing there. The option-strategies lesson said "a butterfly buys probability mass at a price" — this is that sentence made exact. Similarly, a tight call spread$ \frac{C(K) - C(K+h)}{h} $approximates a **digital** paying \$ 1 if $S_T > K$ : its price is the tail probability. Every no-arbitrage shape constraint from the first lesson now has a probabilistic reading: calls decreasing in strike ⇔ probabilities are non-negative; butterflies non-negative ⇔ the density is non-negative; convexity of $C(K)$ ⇔ the distribution is a distribution.

Reading the smile as a distribution

Run the extraction on real equity-index options and the implied density is visibly not lognormal: a fatter left tail (crash mass) and a thinner right tail than Black–Scholes assumes. That is the volatility smile, translated from vol-space to probability-space — two languages for one disagreement with the model. Concrete uses:

Event pricing: before earnings or an election, the implied density often goes bimodal; the market literally displays its two scenarios and their weights. "What does the market think this drug trial is worth?" is answerable from the chain.
Crash odds: the price of a far-OTM put spread bounds the (risk-neutral) probability of a move beyond it — the standard way to quote "what are the odds of a 20% drawdown by year-end, per the market?"
The caveat that must accompany every such reading: these are risk-neutral probabilities — real-world odds times the market's pricing of pain in those states (the binomial lesson's "prices, not forecasts"). Crash states carry a risk premium, so implied crash probabilities systematically exceed physical ones. The gap isn't error; it's the price of insurance — and naming it is what separates a careful answer from a naive one.

The interview version

"Estimate the market-implied probability that the stock finishes above 110." — Price the tight 110/111 call spread; if it costs \ $0.30 per \$ 1 of width, the implied probability is ~30%. "Your 100-strike butterfly quotes at a negative price — thoughts?" — Free money and a broken market: buy it, and tell the desk the quotes violate static no-arbitrage (a negative density). "Are these real probabilities?" — Risk-neutral, so no: reweighted by marginal utility; the left tail is inflated by exactly the premium people pay for crash insurance. Three answers, one theorem — this is among the highest-leverage results in the whole options course.