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 FORUMS > Technical Forum < refresh >
 Topic Title: Variance of integrated Wiener process Created On Tue Jun 20, 06 09:14 AM Topic View: Branch View Threaded (All Messages) Threaded (Single Messages) Linear

PiotrW
Member

Posts: 70
Joined: Mar 2004

Tue Jun 20, 06 09:14 AM

Hello,

Mr Zhang in his paper “Theory of Continuously Sampled Asian Option Pricing” wrote during derivation of a formula for backward-starting geometric Asian option (page 6):

Perhaps someone knows why it is so.

Regards,
PiotrW

http://www.eaber.org/intranet/documents/23/228/CUHK_Zhang_00.pdf

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Without application, knowledge is pointless. /R. Gervais
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DavidF
Member

Posts: 154
Joined: May 2006

Tue Jun 20, 06 09:32 AM

I'm sorry to reply you in words but I don't know anything about LaTex...

You have two ways to see that:
first you can just compute the expectation of the square of this integral, making it a double integral with Fubini's theorem (and use the property of the brownian motion that
E[WsWt]=min(s,t)
second you can apply Ito's lemma to the fonction (T-t-tau)*Wtau and it will tranform your integral to a stochastic integral

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PiotrW
Member

Posts: 70
Joined: Mar 2004

Fri Jun 23, 06 06:27 PM

But do you know how to calculate it (there are two tau's as integration vatiables, and once you integrated internal integration there is no tau inside \int symbol for the second integration.

(basing on Fubini's theorem)

Regards,
PiotrW

-------------------------
Without application, knowledge is pointless. /R. Gervais
QUANTitude - market risk Training, IT & Consulting
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GoGoFa
Member

Posts: 47
Joined: Mar 2005

Mon Jun 26, 06 10:58 AM

Of course you have two different variables of integration:

E[(\int W_t d t)^2] = E[\int W_s ds \int W_t dt] = E[\int\int W_s W_t dt ds] = \int\int E[W_sW_t] dt ds by Fubini
Now use E[W_sW_t]=min(s,t) and you're through.

PiotrW
Member

Posts: 70
Joined: Mar 2004

Thu Jun 29, 06 01:35 PM

GoGoFa,

all right but what are the integration boundaries? Are they changed?

Regards,
PiotrW

PS. DavidF - oh, you have changed your 'face'

-------------------------
Without application, knowledge is pointless. /R. Gervais
QUANTitude - market risk Training, IT & Consulting
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ppauper
Senior Member

Posts: 58390
Joined: Nov 2001

Fri Jun 30, 06 03:27 PM

hehehehe.... he said wiener

horacioaliaga
Senior Member

Posts: 325
Joined: Aug 2005

Mon Jul 03, 06 05:57 PM

This is what GoGoFa meant:

Edited: Mon Jul 03, 06 at 06:27 PM by horacioaliaga

PiotrW
Member

Posts: 70
Joined: Mar 2004

Tue Jul 04, 06 08:02 AM

Thank you horacioaliaga.

Have a good day everybody.

PiotrW

-------------------------
Without application, knowledge is pointless. /R. Gervais
QUANTitude - market risk Training, IT & Consulting
quantitude.pl