Hi Guys,

Just wondering if you could suggest applications of distribution of the supremum of a fractional Brownian motion process with a drift ?

Also if you could possibly recommend how to approach this problem, that would be much appreciated.


  • $\begingroup$ Where does this question come from? If you know what a fractional BM with drift is, then why not try to study the distribution of its supremum? (I assume you mean the process $Y_t = \sup_{o\leq s\leq t} X_s$.) I really don't understand what you mean by "application" and "project" $\endgroup$ – Yemon Choi Feb 8 '12 at 0:39
  • $\begingroup$ Firstly thanks for your reply. My question was mearly to find out where such a distribution could be used in real world. I am an honours student and i am considering this as a thesis project, i was told that this is an old unsolved problem but was never given any motivation behind this project. I am just trying to fill in some blanks here. If you share your suggestions or knowldege it would be much appreciated. $\endgroup$ – Comic Book Guy Feb 8 '12 at 2:14

An application that springs to mind immediately is option evaluation. Suppose I offer you to buy a contract to me: after three months I pay you the maximum value of the price of an asset. How much are you willing to pay this contract ? If you model the price of the assert by a drifted brownian motion, then you'll probably want to estimate the distribution of this maximum and take the expected value as a first guess of this maximum price. This will also be my first guess at the minimum price at which I will be willing to sell the contract. This is a non arbitrage price.

Note to purists : of course one will object that each of the parties can hedge himself, and that the distribution has to be corrected by a risk neutral argument (which will probably discard the initial drift and replace it by a zero risk rate drift instead).

N.b. : this is a real world application, exotic option traders buy and sell like contracts everyday.

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