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SVQ · Option-implied distributions

Risk-neutral density (Breeden–Litzenberger)

Also called: RND, implied probability density, Breeden–Litzenberger density, option-implied distribution

The risk-neutral density is the probability distribution of the future underlying price implied by option prices, recovered from their curvature in strike.

Definition

The risk-neutral density (RND) is the probability distribution for the underlying's price at an expiry that, when used to discount expected payoffs at the risk-free rate, reproduces the observed option prices. Breeden and Litzenberger (1978) showed it can be read off the second derivative of call prices with respect to strike.

It is "risk-neutral" because it includes the market's risk premia: it is the distribution under which prices are fair, not a forecast of real-world frequencies.

Intuition

A very tight butterfly spread pays one unit only if the underlying finishes near its middle strike. Its price, scaled by the strike spacing, is therefore the discounted probability of finishing there. Repeating this across strikes traces out the whole distribution.

The RND makes the smile tangible. Downside skew shows up as a fat left tail; high curvature shows up as fat tails on both sides; an event can even produce two humps.

How it's calculated

q(K) = erT · ∂²C(K, T) / ∂K²

Because raw prices are discrete and noisy, practitioners first fit an arbitrage-free implied volatility smile (for example SVI or SSVI), convert it back to call prices on a fine strike grid, and differentiate numerically. The result must integrate to one and stay non-negative; a negative region signals butterfly arbitrage in the fit.

From the density you can compute implied moments (mean, variance, skewness, kurtosis) and probabilities such as P(ST < K) as the area under the curve, which also equals erT ∂C/∂K + 1.

Example: with strikes 5 apart, calls at 95, 100 and 105 priced 7.10, 4.20 and 2.30 give a butterfly cost of 7.10 − 8.40 + 2.30 = 1.00. Dividing by the squared strike spacing (25) gives an undiscounted density of about 0.04 per unit of price near 100, or roughly a 20% probability of finishing within 2.5 points either side.

How to read it

Use the RND to see how much probability the market prices beyond particular levels and how that changes across expiries. Stacking densities by expiry shows how uncertainty spreads with time.

Remember that the tails are mostly extrapolation from the fitted wings, and that risk-neutral probabilities overweight bad states relative to real-world ones because investors pay to insure against them.

Common mistakes

  • Treating risk-neutral probabilities as real-world forecasts; they embed risk premia, especially in the left tail.
  • Differentiating raw quotes twice, which amplifies noise into spurious humps.
  • Trusting the tails beyond the last liquid strike.
  • Ignoring dividends and the forward when centering the distribution.

In Saqrid

SVQ computes a risk-neutral probability density for each expiry and stacks them into a 3D probability mountain across expiries.

Its Risk Lab also tracks implied-moment time series.

Open SVQ.

  • Butterfly arbitrageButterfly arbitrage exists when option prices are not convex in strike, so a butterfly spread would cost less than zero to own.
  • Durrleman conditionThe Durrleman condition tests whether an implied variance smile implies a non-negative probability density, the smile form of no butterfly arbitrage.
  • Expected moveThe expected move is the size of price change that options imply over a horizon, usually derived from implied volatility or the at-the-money straddle.
  • Volatility smile and skewThe volatility smile and skew describe how implied volatility varies with strike for a single expiry, revealing how the market prices tail moves.
  • Probability coneA probability cone projects a range of future prices that widens with the square root of time, scaled by volatility and a chosen confidence level.

Go deeper

Saqrid Institute lesson: Breeden–Litzenberger: Prices to Probabilities. Lesson text is open to read without an account.

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