Arbitrage-Free Interval and Dynamic Hedging in an Illiquid Market

Abstract

This paper provides two modified pricing PDEs for a general European option under liquidity risk, by which two modified hedges are derived. It is shown that the hedge errors of the two modified hedges approach zero as the trading time interval converges to zero inclusive of liquidity costs. An arbitrage-free interval is identified and in contrast to transaction costs, the liquidity cost is proved to be finite even if trading is continuous. Numerical results are presented on option pricing and the moments of hedge errors with both Black-Scholes hedge and one of the modified hedge. The results indicate that under liquidity risk, the modified option hedge developed in this paper is much superior to the Black-Scholes hedge. The bigger the liquidity risk, the more significant the advantages. In fact, the modified hedge leads to not only a much less hedge error but also to a lower replication costs than the classical Black-Scholes hedge.

Keywords: Liquidity Modelling, Liquidity Costs, Arbitrage-Free Interval, Modified Hedge

JEL Classification: G11, G13