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Let $x=\log(S/K)$ be the log-moneyness.  The Lee 2004 moment formulae provide the relation between the highest exponent $p$ such that $ES^{1+p}_T <\infty$ and implied volatility $I(x)$.  More precisely $\beta_R := \limsup \frac{I^2(x)}{|x|/T}$ lies in [0,2] and the exponent has the formula $1/2\beta_R + \beta_R/8 - 1/2$.