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For a given maturity, given three equally spaced option strikes $K_1,K_2,K_3$ a "butterfly" combination consists of shorting 2 of the middle strike calls and buying one each of the "wing" or lateral calls. This position has a positive cost i.e. $c_1+c_3-2 c_2 >=0$ (why? because it has a positive payoff for $S_T\approx K_2$ and zero payoff elsewhere). In ...

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I'll assume rates to be 0, so any $\text{Call}=\text{Put}+S-K$ so in the end you need to have $p_1+p_3-2p_2$, theoretically

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Note that \begin{align*} \max(S-K, 0) = S-K + \max(K-S, 0). \end{align*} Then, \begin{align*} &\ \max(S-K_1, 0)+ \max(S-K_3, 0) - 2 \max(S-K_2, 0)\\ =&\ S-K_1 + \max(K_1-S, 0) + S-K_3 + \max(K_3-S, 0)\\ &\qquad -2(S-K_2) - 2\max(K_2-S, 0)\\ =&\ 2K_2 - (K_1+K_3) + \max(K_1-S, 0)+\max(K_3-S, 0)- 2\max(K_2-S, 0)\\ =&\ \max(K_1-S, 0)+\max(K_3-...

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Into the first equation we can substitute $C$ and $P$ as given by the other two equations, we get: $(S-K)^+ -(K-S)^+ +TV_C - TV_P = S-PV(K)$ $S-K+TV_C-TV_P=S-PV(K)$ $TV_C-TV_P=K-PV(K)$ If interest rates are zero then $PV(K)=K$ and then we indeed have $TV_C=TV_P$ Note: as suggested in the comments above a slightly different definition of Intrinsic Value ...

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See my post: Implied interest rate using put-call parity. Maybe it helps. Liquidity is an issue for OTM and results should be more consistent using most liquid points (ATM).

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