Explicit formulas for the minimal variance hedging strategy in a martingale case

We explicitly compute closed formulas for the minimal variance hedging strategy in discrete time of a European option and for the variance of the corresponding hedging error under the hypothesis that the underlying asset is a martingale following a Geometric Brownian motion. The formulas are easy to implement, hence the optimal hedge ratio can be employed as a valid substitute of the standard Black-Scholes delta and the knowledge of the variance of the total error can be a useful tool for measuring and managing the hedging risk.