SERVING THE QUANTITATIVE FINANCE COMMUNITY

 
User avatar
BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

Distribution of max BM - BM

March 16th, 2015, 10:49 am

Hi,Can anybody help me with deriving the distribution of [$]max_{0 \leq s \leq t} W_s - W_t[$]?Thanks,Bernd
 
User avatar
LocalVolatility
Posts: 128
Joined: May 27th, 2009, 10:07 am
Location: Amsterdam
Contact:

Distribution of max BM - BM

March 16th, 2015, 1:02 pm

Let [$]M_t = \max_{0 \leq u \leq t} W_u[$]. You know that [$]M_t[$] and [$]W_t[$] have the joint density[$]f_{M_t, W_t}(m, w) = \frac{2 (2m - w)}{t \sqrt{2 \pi t}} \exp \left\{ -\frac{(2m - w)^2}{2 t} \right\}[$]. Then I'd just take it from there..[$]\mathbb{P} \left\{ M_t - W_t \geq x \right\} = \int_0^\infty \int_{-\infty}^{m - x} f_{M_t, W_t}(m, w) \mathrm{d}w \mathrm{d}m[$]
 
User avatar
BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

Distribution of max BM - BM

March 16th, 2015, 3:01 pm

Thanks. I had hoped for some elegant trick though :-)
 
User avatar
BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

Distribution of max BM - BM

March 17th, 2015, 11:54 am

QuotePr( max(ws) - wt > a) = Pr( max(ws) > a) How do you justify that?
Last edited by BerndSchmitz on March 16th, 2015, 11:00 pm, edited 1 time in total.
 
User avatar
Alan
Posts: 10206
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Distribution of max BM - BM

March 17th, 2015, 6:12 pm

QuoteOriginally posted by: BerndSchmitzThanks. I had hoped for some elegant trick though :-)One elegant trick would note that it is well-known that [$]M_t - W_t \stackrel{d}{=} |W_t|[$], where M is the BM maximum process and |W| is reflected BM. The pdf for the latter is [$]p(t,x) = \frac{2 \, e^{-x^2/(2 t)}}{\sqrt{2 \pi t}}[$] on [$](0,\infty)[$] and so is also the density of the distribution you seek.
Last edited by Alan on March 16th, 2015, 11:00 pm, edited 1 time in total.
 
User avatar
BerndSchmitz
Topic Author
Posts: 242
Joined: August 16th, 2011, 9:48 am

Distribution of max BM - BM

March 23rd, 2015, 7:53 am

Thanks a lot Alan. This was the kind of answer I was looking for.Is it intuitive why [$]M_t-W_t=^d|W_t|[$]? Or can you recommend any good (online available?) ressources for that?
 
User avatar
Alan
Posts: 10206
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Distribution of max BM - BM

March 23rd, 2015, 2:12 pm

You're welcome. To me, it is certainly not "obvious", merely plausible. Certainly both sides have non-negative support andhit zero a lot. You'll have to google -- I looked it up in Bertoin's book on Levy processes.
 
User avatar
savr
Posts: 48
Joined: January 21st, 2013, 3:28 pm

Distribution of max BM - BM

March 24th, 2015, 4:23 pm

The equality in law P(Mt) = P(Mt-Wt) (is this what you mean?) is what is true.The equality almost surely Mt - Wt = |Wt| could not be true, for instance with >0 prob Wt can be <0 and Mt >0.
 
User avatar
savr
Posts: 48
Joined: January 21st, 2013, 3:28 pm

Distribution of max BM - BM

March 25th, 2015, 5:15 pm

Terminology is the same whether r.v. s or processes. To sum up, in continuous time:for processes: Mt - Wt ~ |Wt| , and for fixed t ~ Mtin discrete time (binomial tree)for fixed t: Mt - Wt ~ Mt !~ |Wt|In a trinomial tree at least |Wt| and Mt have the same support, so there is some hope of redefining things to make them have the same law.
Last edited by savr on March 24th, 2015, 11:00 pm, edited 1 time in total.
ABOUT WILMOTT

PW by JB

Wilmott.com has been "Serving the Quantitative Finance Community" since 2001. Continued...


Twitter LinkedIn Instagram

JOBS BOARD

JOBS BOARD

Looking for a quant job, risk, algo trading,...? Browse jobs here...


GZIP: On