In the present paper we provide a closed form expression for the distribution of the extrema of a Brownian motion observed at discrete times. We reduce the evaluation problem to a Wiener-Hopf equation that we solve analytically. We discuss the relationship of our result with the well known Spitzer identity. Finally we apply the result to the analytical evaluation of discrete lookback options in the Black-Scholes setting. The results from our analytical formulae are compared with those from other numerical methods available in the literature.