Another Look at the Log Contract

Abstract

Motivated by the recent Log Moment formula of \cite{raval_jacquier23}, we look at the power of absolute Log Contract
$|\log(S_T/F_T)|^q$, $q\geq 1$, and derive its corresponding replicating strip of Vanilla options. Unlike the standard Log Contract where the Delta of the replicating portfolio synthesizes a pure short position in realized asset variance, here
the position changes between short and long depending on the asset level. We also derive a generalization of the standard Log Contract replication, where the payoff $K\mapsto (\ln(K))^q I_{K\geq 1}-(-\ln(K))^q I_{K< 1}$ is parameterized by $q>0$.

Keywords: Log Contract, Moment Formula

JEL Classification: G13