Large liquidity expansion of super-hedging costs

Abstract

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 $\epsilon \ge 0$ with $S^0(s,\nu )=s$ 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.