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I agree with Gordon's deduction if the stock price is distributed that way under risk neutral measure. With sufficiently large K it should be monotone, but for other cases, there could be different cases. I think it can be helpful if you create a list of options with different strikes, time to maturity and spot prices to observe the multivariate relationship ...

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The implied Black-Scholes skew will be downward sloping in the limit on both the left and the right. (I believe @Gordon's derivation claiming upward slope may have a sign error somewhere). Left Side For the left side it is sufficient to note that the lognormal model has no density below zero while the normal model has strictly positive density in that ...

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Since $S_T = S_0 + \sigma W_T$, \begin{align*} C &:= E\left((S_T-K)^+ \right)\\ &= E\left((S_0+\sigma W_T-K)^+ \right)\\ &=\int_{\frac{K-S_0}{\sigma \sqrt{T}}}^{\infty}(S_0+\sigma\sqrt{T} x-K) \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx\\ &=(S_0-K)\Phi\left(\frac{S_0-K}{\sigma \sqrt{T}}\right)+\frac{\sigma\sqrt{T}}{\sqrt{2\pi}}e^{-\frac{(S_0-K)...

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In that case, the problem becomes a non-trivial stopping time problem. Consider a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with the natural filtration of a standard Brownian motion $W_t^\mathbb{P}$. Assuming a geometric Brownian motion for the underlying asset, one gets  S_t = S_0 \exp\left((\mu-\frac{1}{2}\sigma^2)t + \...

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