Large liquidity expansion of super-hedging costs

Abstract

We consider a financial market with liquidity cost as in [Çetin, Jarrow and Protter, Finance and Stochastics 8 (2004), 311–341], where the supply function S^ε(s,ν) depends on a parameter ε≥0 with S⁰(s,ν)=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [Finance and Stochastics 14(3) (2010), 317–341], 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 ε. 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.