The maximum loss and the supremum of fractional brownian motion

In financial markets, the supremum of the price of an asset and the maximum possible loss as a measure of risk are important variables for investors. In literature, there are some bounds on the expected value of supremum and of maximum possible loss of fBm. For now, there are no exact solutions because with today's techniques, exact values are not possible to obtain. In this thesis, we provide a new theoretical bound on the expected value of maximum loss using Sudakov-Fernique inequality. Using discretization method and considering order statistics of multivariate normal variables, we obtain some numerical bounds on the expected value of maximum loss and of supremum of fBm. We also obtain some simulation results on the expected value of maximum loss and of supremum of fBm using Cholesky method. Finally we provide comparison of the established bounds.Keywords: Fractional Brownian Motion, Maximum Loss, Sudakov-Fernique Inequality, Markov Inequality, The Cholesky Method.