# Breeden-Litzenberger Formula: Extracting Risk-Neutral Density from Options

Derive risk-neutral tail probabilities and density from European call prices, apply finite differences, and manage arbitrage and data-noise risks.

Canonical: https://wiki.fcontext.com/options/breeden-litzenberger-formula/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

The **Breeden-Litzenberger relationship** recovers state-price information from European call prices across strikes for one expiration. If `C(K,T)` is a sufficiently smooth no-arbitrage call-price function, with constant continuously compounded rate `r`, then:

`−∂C(K,T)/∂K = e^(−rT) Q(S_T > K)`

`∂²C(K,T)/∂K² = e^(−rT) f_Q(K)`

Therefore `f_Q(K) = e^(rT) ∂²C/∂K²`, where `f_Q` is the risk-neutral terminal-price density. The formula extracts prices of future states, not a direct real-world forecast.

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## Why strike curvature contains a density

A call pays `max(S_T − K, 0)`. Raising `K` removes a thin slice of payoff whenever the option finishes above the strike, so the negative first strike derivative corresponds to a discounted digital-call price or tail probability. Differentiating that tail probability again reveals local density.

No-arbitrage call prices must be nonincreasing and convex in strike. Convexity means `∂²C/∂K² ≥ 0`; a negative second derivative implies a negative state price and signals inconsistent inputs or a butterfly-arbitrage opportunity under ideal execution assumptions.

Markets list discrete strikes, so practitioners fit an arbitrage-aware curve or surface and use finite differences. For equally spaced strikes with interval `ΔK`:

`∂²C/∂K² ≈ [C(K−ΔK) − 2C(K) + C(K+ΔK)] / (ΔK)²`

The numerator is the price of a narrow call butterfly. Its payoff is concentrated around the middle strike, which explains the density interpretation.

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## Three-strike finite difference

Suppose synchronized European calls with one expiration have these prices:

| Strike | Call price |
| ---: | ---: |
| `$95` | `$8.20` |
| `$100` | `$5.00` |
| `$105` | `$2.80` |

With `ΔK = $5`, curvature near `$100` is:

`[$8.20 − 2 × $5.00 + $2.80] / $5² = $1.00 / 25 = 0.04 per dollar`

If discounting is negligible, the local risk-neutral density estimate is about `0.04 per dollar`. With nonzero rates, multiply by `e^(rT)`. This point estimate is not a probability of finishing exactly at `$100`; probability requires integrating density across a price interval.

If the middle quote were `$5.60`, the numerator would become negative. That would not prove investors expect a “negative probability”; it would first indicate stale quotes, bid-ask inconsistency, American exercise effects, or an arbitrage-violating fit.

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## Extraction checklist

- Fix one expiration and use synchronized, executable European-style option quotes.
- Estimate discount factors, forwards, dividends, settlement, and exercise terms consistently.
- Prefer liquid out-of-the-money quotes and use put-call parity carefully when converting puts to calls.
- Remove obvious errors and fit prices subject to monotonicity and convexity constraints.
- Differentiate the fitted price curve; raw second differences magnify small quote errors.
- Repeat extraction at bid, mid, and ask and across plausible smoothing choices.
- Confirm the density is nonnegative, integrates near one, and has a mean consistent with the forward.
- Treat sparse-strike tails as model-dependent extrapolation, not observed information.

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## Common misconceptions

- “Risk-neutral probability is the market's objective forecast.” It includes state pricing and risk preferences.
- “A density of 0.04 means a 4% chance of exactly $100.” Density must be integrated over an interval.
- “Differentiating raw midpoints is enough.” Second derivatives amplify noise and stale quotes.
- “A smooth chart proves accurate data.” The smoothing method can manufacture tail shape.
- “Negative density should simply be clipped to zero.” Clipping can hide arbitrage and destroy price consistency.
- “The formula applies unchanged to American equity options.” Early exercise and discrete dividends break the simple European relationship.

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## Related topics

- [Implied Volatility](/options/implied-volatility/)
- [Black-Scholes Model](/options/black-scholes-model/)
- [Options Chain](/options/options-chain/)

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## Authoritative sources

- [Prices of State-Contingent Claims Implicit in Option Prices](https://doi.org/10.1086/296025) - Douglas T. Breeden and Robert H. Litzenberger
- [Characteristics and Risks of Standardized Options](https://www.theocc.com/company-information/documents-and-archives/options-disclosure-document) - Options Clearing Corporation
- [SPX Options Product Specifications](https://www.cboe.com/tradable_products/sp_500/spx_options/specifications/) - Cboe