Liquidity in a binomial market

Official URL: http://dx.doi.org/10.1111/j.1467-9965.2010.00462.x

We study the binomial version of the illiquid market model introduced by Cetin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we characterize the liquidity premium that results from the model. In Cetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L2-approximating sense. Cetin, Soner, and Touzi investigated the super-replication problem using the same supply curve model but under some restrictions on the trading strategies. They showed that the super-replicating cost differs from the Black-Scholes value of the claim, thus proving the existence of liquidity premium. In this paper, we study the super-replication problem in discrete time but with no assumptions on the portfolio process. We recover the same liquidity premium as in the continuous-time limit. This is an independent justification of the restrictions introduced in Cetin, Soner, and Touzi. Moreover, we also propose an algorithm to calculate the options price for a binomial market.