On the correlation of the supremum and the infimum and of maximum gain and maximum loss of Brownian motion with drift

Investors are naturally interested in the supremum and the infimum of stock prices, also in the maximum gain and the maximum loss over a time period. To shed light on these relatively complicated aspects of sample paths, we consider Brownian motion with and without drift. We provide explicit calculations of the correlation between the supremum and the infimum of Brownian motion with drift. We establish a number of results concerning the distributions of maximum gain and maximum loss. We present simulation studies of maximum gain and of maximum loss of Brownian motion with a range of values for the drift. We conjecture that the correlation between maximum gain and maximum loss has a minimum value of −0.5 at drift 2.