Avellaneda-Stoikov and Cartea-Jaimungal as One Framework: A Forced Uniqueness Theorem for Inventory Market Making

Abstract

In inventory market making, the running-penalty coefficient $\varphi$ of the Cartea-Jaimungal framework and the risk-aversion parameter $\gamma$ of the Avellaneda-Stoikov framework are typically treated as independent free parameters, calibrated separately. We show that they are in fact not independent. A small set of axioms on the market maker's dynamic preference functional, namely cash-additivity, normalization, concavity, strong dynamic consistency, and law-invariance, forces the preference functional to be the entropic certainty-equivalent on liquidation-adjusted terminal wealth, parametrized by a single positive scalar $\gamma$. The Avellaneda-Stoikov framework is the unique representative of this axiom class. The Cartea-Jaimungal framework is its second-order Taylor expansion in inventory magnitude, with the running coefficient forced to $\varphi =\gamma {\sigma }^2/2$ and (under a mild regularity condition on the liquidation cost) the terminal coefficient forced to $\alpha =\frac{1}{2}{L}^{\prime \prime }(0)$. The two frameworks, typically presented as competing alternatives with the choice between them driven by tractability, are different manifestations of a single underlying object. The forced relation is invertible, $\gamma =2\varphi /{\sigma }^2$, giving a consistency cross-check on independently calibrated desk parameters.

arXiv comment63 pages, 1 figure (two panels). Submitted to Mathematical Finance