Option replication with transaction cost under Knightian uncertainty

To price options using replication in imperfect markets, both Knightian uncertainty and transaction cost have to be taken into account. In this paper, we put an uncertainty factor into volatility, assume investors minimize the root mean square error of replication when they choose hedging ratio, and derive European option price by a recursive procedure. To avoid high transaction cost caused by continuous hedging, we establish a discrete and binomial replication model considering both uncertainty and transaction cost. Numerical examples imply that option price contains both risk premium and uncertainty premium, and it is an approximately linearly increasing function of transaction cost but a nonlinearly increasing function of uncertainty. Additionally, both uncertainty and transaction cost have effects on the price of the at-the-money option, but they almost have no impact on the price of deeply in-the-money or out-of-the-money options. Empirical analysis of the Shanghai 50ETF options market indicates that the Black–Scholes model tended to underestimate the market price, whereas our model better estimates market prices.