Modeling the Bid and Ask Prices of Options

Comonotone additivity for two price economy bid and ask prices motivates combining bid prices for call options with the ask prices for puts and the converse to construct two densities (termed lower and upper) reflected by these prices. Bilateral gamma models are fit to estimate these the lower and upper densities for 183 underlying equity assets on 26 days in 2020 for four maturities. Distances between the densities are reported along with measures for the upper density being higher in the convex order. Cases are presented when the upper density reflects an upper random variable that is an independent multiplicative shift over the lower one and when there is a multiplicative mean preserving spread between them that is not independent. It is observed that the imposition of an independent multiplicative shift may be too restrictive in general. Convex orders across maturities are then reported on. Finally martingale models mixing independent and identically distributed shocks with components based on self similarity are fit to the option surface across all strikes and maturities