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

For educational purposes only; not investment advice.

Direct answer

Section titled “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.

Why strike curvature contains a density

Section titled “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.

Three-strike finite difference

Section titled “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.

Extraction checklist

Section titled “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.

Common misconceptions

Section titled “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.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”