Roger Lee (2004): the right-wing slope of total variance in log-moneyness is at most $2$, and is tied analytically to the largest finite moment of the underlying.
If the implied total variance $w(k, T) = \sigma_{BS}^2(k, T) T$ grows linearly in log-moneyness $k$ as $k \to \infty$, the slope is bounded above by $2$ (Roger Lee 2004). The bound is sharp: slope $= 2$ corresponds to a distribution with no positive moments, slope $= 0$ corresponds to all moments finite (lognormal-like). The full statement gives an exact analytic formula for the largest moment $p^*$ of $S_T$ given the slope $\beta_R$, making the smile shape directly readable as a tail-heaviness number. Calibration loops use this as a sanity check: a slope that pushes against $\beta_R = 2$ implies a tail with no variance, usually a flag that the fit is over-extrapolating.
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Worked example
A fitted SVI slice has right-wing total-variance slope $\beta_R = 1.0$ in log-moneyness (i.e. as $k \to \infty$, $w(k)/k \to 1$). Use Lee's formula to find the supremum $p^*$ such that $E[S_T^{1 + p^*}] < \infty$, i.e. the largest moment of $S_T$ that is finite.
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Reflection
Lee's formula is symmetric: there's an analogous $\beta_L$ for the left wing tied to *negative* moments. Why does this asymmetry matter for equity surfaces specifically (where left wing tends to be steeper than right) and what does the relative size $\beta_L > \beta_R$ say about the underlying's downside risk?