If a butterfly in the limit represents a probability (by the Breeden-Litzenberger result), what can be said about the relative likelihood of a random variable $S_0$ from the price of a vanilla-option constructed butterfly with practical strikes (i.e. not infinitesimally close)?
For example, with the price of the butterfly $$B = C(S,K-\delta,t-T,\sigma, r) - 2*C(S,K,t-T,\sigma, r) + C(S,K+\delta,t-T,\sigma, r) $$
where $\delta = \text{{width of Butterfly}}/2$, can we say via some relationship of $B$ and $\delta$-or-$K$ the likelihood of being within a range at expiration?
I'm interested in what we can say about the likelihood with only the data from the butterfly spread. An approximation, like the one for the Black-Scholes eq'n: $C(S,t)\approx 0.4Se^{-r(T-t)}\sigma\sqrt{T-t}$.
Here's a first attempt:
Let's take $S=10$, $K=10$, $\delta=1$, $t-T=0.25$, $\sigma=0.25$, and $r=0.01$ with no dividends or cashflows until expiration.
$$B = $1.15 - 2\cdot $0.51 + $0.174 = $0.304$$
So we have $B=\$0.34$ and $\delta=\$1$. We assume the bet is fair - losses are equally as likely as profits. So the market thinks this "trade" has equal chances of making money as losing money. The butterfly has a breakeven profit in the range $[K-\delta +B, K+\delta-B]=[\$9.34,\$10.66]$. If we had a uniform distribution function, we can say the market has a 50-50 chance of being within or outside of this range. We have already assumed a normal distribution using the BSM prices (can this be relaxed?), so this is just an approximation. Can we do better?
What assumptions do we need? For example, I would assume we cannot ignore skew when $\delta$ is not approaching zero.
Edit - Note, the probability of expiring within a range and the probability of the trade being profitable are different.