SERVING THE QUANTITATIVE FINANCE COMMUNITY

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

### Maximum of Brownian Motion

Hi!I would like to present a solution for how to calculate where and I'd really appreciate if you guys would be so kind to pointed out where my reasoning has flaws (if it has any).First of all we need to find the probability density function of . To do this we use the Reflection Principle which tells us that and in other words . Which tells us that the density function is And now the expectation can be calculated:Is this reasoning valid?Thank you in advance!

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

### Maximum of Brownian Motion

Perhaps one can find the distribution of easier by reasoning like this:where

ThinkDifferent
Posts: 659
Joined: March 14th, 2007, 1:09 pm

### Maximum of Brownian Motion

QuoteOriginally posted by: DoubleTroublePerhaps one can find the distribution of easier by reasoning like this:where the approach in your first post is a standard one ( i didn't check for exact calculations). what you are doing in the second post doesn't seem to lead anywhere. in fact, starting from the second equality sign it becomes nonsense.... The confusion comes from the fact that W_t is not equal to sqrt(t) N. Both have the same distribution though.
Last edited by ThinkDifferent on August 1st, 2011, 10:00 pm, edited 1 time in total.

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

### Maximum of Brownian Motion

QuoteOriginally posted by: ThinkDifferentwhat you are doing in the second post doesn't seem to lead anywhere. in fact, starting from the second equality sign it becomes nonsense.... Yeah, I see why now! Thanks for your input!
Last edited by DoubleTrouble on August 1st, 2011, 10:00 pm, edited 1 time in total.

mirthgrief
Posts: 1
Joined: August 1st, 2011, 8:29 am

### Maximum of Brownian Motion

Hey I wonder if you could name 5 to 6 physics-related models other than the brownian motion models that is used in quantitative finance?Also, do these models work well in extreme, non-random situations such as the 2008 subsidary mortgage crisis or in other financial turmoils?

DoubleTrouble
Topic Author
Posts: 83
Joined: November 9th, 2010, 9:46 pm

### Maximum of Brownian Motion

I realized today that I made a slight error in my first post. That density formula holds only for so the integral should range from 0 to infinity which gives the previous result divided by 2.Right?