Modeling the bid and ask prices of options

Dilip B. Madan, Wim Schoutens and King Wang

Abstract

Comonotone additivity for a two-price economy (ie, bid and ask prices) motivates the combination of bid prices for call options with ask prices for puts and the combination of ask prices for call options with bid prices for puts in order to construct two densities (termed lower and upper densities) that are reflected by these prices. In this paper, the two densities, scaled to a unit mean, are related, with the upper generally being higher than the lower in the convex order. Bilateral gamma models are fitted to estimate the lower and upper densities for 183 underlying equity assets on 26 days in 2020 for four maturities. The distances between the densities are reported, along with measures for the upper density being higher in the convex order. The paper presents the cases 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, leading to martingale models mixing independent and identically distributed shocks with self-similarity. Novel theoretical and empirical problems for the joint modeling of the two prices prevailing in two-price markets are studied and partially solved.