Large liquidity expansion of super-hedging costs

TL;DR

This paper analyzes how super-hedging costs in a financial market increase with liquidity costs, providing a Taylor expansion and explicit calculations for European options.

Contribution

It introduces a Taylor expansion approach for super-hedging costs considering liquidity effects, extending previous PDE characterizations.

Findings

Explicit first-order term for European Call options

Bounds for expansion order in European Digital Options

Quantitative insights into liquidity impact on super-hedging costs

Abstract

We consider a financial market with liquidity cost as in \c{C}etin, Jarrow and Protter [2004], where the supply function $S^{\epsilon}(s,\nu)$ depends on a parameter $\epsilon\geq 0$ with $S^0(s,\nu)=s$ corresponding to the perfect liquid situation. Using the PDE characterization of \c{C}etin, Soner and Touzi [2010] of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of $\epsilon$. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option.