Pricing European Options in a Discrete Time Model for the Limit Order Book

In this paper we build a discrete time model for the structure of the limit order book, so that the price per share depends on the size of the transaction. We deduce the value of a portfolio when the investor trades using market orders and a bank account with different interest rates for lending and borrowing. We also deduce conditions to rule out arbitrage and solve the problem of pricing and hedging an European call option with physical delivery. It is shown that contrary to the perfectly liquid setting, the price of a European call is not given by an expectation, but can be expressed as an optimization problem on a set of equivalent probability measures.