  We show that the distribution of the square of the supremum of reflected
fractional Brownian motion up to time a, with Hurst parameter-H greater than
1/2, is related to the distribution of its hitting time to level $1,$ using the
self similarity property of fractional Brownian motion. It is also proven that
second moment of supremum of reflected fractional Brownian motion up to time
$a$ is bounded above by $a^{2H}.$ Similar relations are obtained for the
supremum of fractional Brownian motion with Hurst parameter greater than 1/2,
and its hitting time to level $1.$ What is more, we obtain an upper bound on
the complementary probability distribution of the supremum of fractional
Brownian motion and reflected fractional Brownian motion up to time a, using
Jensen's and Markov's inequalities. A sharper bound is observed on the
distribution of the supremum of fractional Brownian motion by the properties of
Gamma distribution. Finally, applications of the given results to the financial
markets are investigated and partial results are provided.
