Option hedging for small investors under liquidity costs

Following the framework of Çetin et al. (Finance Stoch. 8:311-341, 2004), we study the problem of super-replication in the presence of liquidity costs under additional restrictions on the gamma of the hedging strategies in a generalized Black-Scholes economy. We find that the minimal super-replication price is different from the one suggested by the Black-Scholes formula and is the unique viscosity solution of the associated dynamic programming equation. This is in contrast with the results of Çetin et al. (Finance Stoch. 8:311-341, 2004), who find that the arbitrage-free price of a contingent claim coincides with the Black-Scholes price. However, in Çetin et al. (Finance Stoch. 8:311-341, 2004) a larger class of admissible portfolio processes is used, and the replication is achieved in the L² approximating sense.