Breeden–Litzenberger: Strike Derivatives and Butterfly Claim Values

Summary

The document asks how the Breeden–Litzenberger relationship connects call option prices across strikes to prices of state-contingent claims. It presents a finite-difference expression built from calls at a central strike and nearby strikes, then asks why the resulting quantity is divided by the strike spacing. The question also seeks an interpretation of the second derivative of a call price with respect to strike as the value of a butterfly position.

The author’s concern is dimensional and economic: the derivative appears to represent value per unit of payoff, while an idealized butterfly over an infinitesimal strike interval seems to require a very large position. No answer, derivation, or empirical illustration is supplied. The document therefore identifies the interpretation problem but does not explain the limiting argument, payoff scaling, discounting, or how the resulting quantity relates to a risk-neutral density.

Key ideas

• The document presents a finite-difference formula using call prices at neighboring strikes.
• It asks why the finite-difference expression is normalized by the strike interval.
• It questions how the second strike derivative of a call price maps to a butterfly claim’s value.
• The source contains no resolution or derivation of the scaling interpretation.

Tags

# Breeden and Litzenberger formula for pricing state-contingent claims


# Breeden and Litzenberger formula for pricing state-contingent claims












I am reading these two papers Prices of State-Contingent Claims Implicit in Option Prices and Implied Risk-Neutral Distribution: A Comparison of Estimation Methods. I understand how we get the formula $$P(M, T, \Delta M) = \frac{\left(c(M+\Delta M)-c(M)\right)-\left(c(M)-c(M-\Delta M)\right)}{\Delta M},$$ which looks like Equation (1) in Breeden and Litzenberger's paper, but why do we divide $P$ with $\Delta M$? And how the second order derivative of the call with respect to $M$ is the value of a butterfly strategy? What I see is that $\frac{\partial^2c}{\partial M^2}$ is the value per one dollar of the payoff of a infinity large share of the butterfly.

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