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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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  • $\begingroup$ The trick is the central difference, $$\frac{\partial f}{\partial x^2}\approx \frac{f(x-\Delta x)-2f(x)+f(x+\Delta x)}{(\Delta x)^2}.$$ A butterfly option trading strategy simply longs calls with high and low strike and shorts two calls with intermediate strike, i.e. $C(M+\Delta M)-2C(M)+C(M-\Delta M)$. $\endgroup$ – Kevin Apr 4 at 21:34
  • $\begingroup$ Hi. Thnx for this. I am aware of the central; difference and the butterfly strategy. Although, I am a bit week with maths and I do not understand why the value of the butterfly (V) is $\frac{\partial^2c}{\partial M^2}$ and simply V = ${\left(c(M+\Delta M)-c(M)\right)-\left(c(M)-c(M-\Delta M)\right)}$. I mean how dividing both sides with ${Δx^2}$ it's the same. $\endgroup$ – F.G Apr 4 at 21:48
  • $\begingroup$ The butterfly itself is worth $c(M-\Delta M)-2C(M)+c(M+\Delta M)$. It's just a linear combination of option prices with same maturity. This portfolio pays $\Delta M$ (if $S_T=M$) or $0$ otherwise if $\Delta M$ is small enough (in the limit $\Delta M\to0$). Traditional Arrow-Debreu prices pay 1 dollar in one particular state of nature, so B&L rescale that butterfly strategy and consider the portfolio $\Pi=\frac{c(M-\Delta M)-2C(M)+c(M+\Delta M)}{\Delta M}$ which now pays 1 if $S_T=M$. B&L then observe that dividing this portfolio by $\Delta M$ gives a second-order central difference. $\endgroup$ – Kevin Apr 4 at 22:26
  • $\begingroup$ All these are very clear to me. What I miss is how V = V/ΔΜ = $\frac{\partial^2c}{\partial M^2}$. "B&L then observe that dividing this portfolio by ΔM gives a second-order central difference." So $\frac{\partial^2c}{\partial M^2}$ is the value of Π/ΔΜ not the butterfly. How then are they related? $\endgroup$ – F.G Apr 4 at 22:44
  • $\begingroup$ Of course, $V\neq\frac{V}{\Delta M}$, unless $\Delta M=1$. Thus, $\frac{\partial^2 c}{\partial M^2}$ is not the value of a butterfly strategy, but of a scaled/adjusted butterfly which divides the position by $\Delta M^2$. As it turns out (due to the central difference), $\frac{\partial^2 c}{\partial M^2}$ is then the value of an Arrow-Debreu security. $\endgroup$ – Kevin Apr 4 at 22:47

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