On Supremum, Infimum, Maximum Gain and Maximum Loss of Brownian Motion with drift and of Fractional Brownian Motion
نویسندگان
چکیده
In finance one of the primary issues is managing risk. Related to this issue and maybe for hedging, investors are naturally interested in the expected values of supremum, infimum, maximum gain and maximum loss of risky assets and the relations between them. Price of a risky asset, stock, can be modeled using Brownian motion and fractional Brownian motion. In this study, we first present the marginal and joint distributions of supremum, infimum, maximum gain and maximum loss of Brownian motion with drift, 0. As an extension of this work, we provide calculations of the expectations and correlation between them for Brownian motion with drift. We give results related to these distributions over various time horizons. We also present numerical studies of Brownian motion with drift and we collect some conjectures on the relation between maximum gain and maximum loss of stock prices. We introduce some bounds on the expected values and distributions of supremum and of maximum loss of fractional Brownian motion. We present large deviation results on maximum loss of fractional Brownian motion, which is also an alternative model of risky asset to Brownian motion.
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