Breeden–Litzenberger
Definition · Level 10 · Derivatives pricing
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Result that the second derivative of call prices with respect to strike, grossed up by e^(rT), is the risk-neutral density: a tight butterfly prices the chance of landing near its body.
Example
Convex call prices ⇔ non-negative implied probabilities.
Where Tradecraft teaches it
Level 10 · Derivatives pricing, in the lesson “No-arbitrage bounds & trick questions”: Price limits, limit cases, early exercise, the full sign table and strike convexity — the interview favourites.
Related terms
- Arbitrage boundsPrice limits any option must respect or someone locks in riskless profit: call ≤ S, European put ≤ K·e^(−rT), call ≥ max(0, S − K·e^(−rT)) without…
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- Forward-start optionOption whose strike is set at a later date (e.g. ATM in 6 months, expiring in 12).
- Put-call parity with dividendsC − P = S·e^(−qT) − K·e^(−rT) = e^(−rT)·(F − K).
- Zero-strike callEuropean call you always exercise, so it is worth the share itself: spot minus the PV of dividends paid before expiry (a prepaid forward).
- 0.4 ruleAt-the-money-forward call or put ≈ 0.4 × S × σ × √T (small dividends, modest σ√T), because 1/√(2π) ≈ 0.4.