We consider a financial market with liquidity cost as in Çetin, Jarrow and Protter [2004], where the supply function S{\epsilon}(s,{\nu}) depends on a parameter {\epsilon}\geq0 with S0(s,{\nu})=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, 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.