CQF - Information Sessions & Free Sample Lectures

Forum Navigation:

magazine

FORUMS > Technical Forum < refresh >
Topic Title: Variance of integrated Wiener process
Created On Tue Jun 20, 06 09:14 AM
Topic View:

View thread in raw text format


PiotrW
Member

Posts: 70
Joined: Mar 2004

Tue Jun 20, 06 09:14 AM
User is offline View users profile

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

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



DavidF
Member

Posts: 154
Joined: May 2006

Tue Jun 20, 06 09:32 AM
User is offline

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

-------------------------
Lebanon will survive !
 
Reply
   
Quote
   
Top
   
Bottom
     



PiotrW
Member

Posts: 70
Joined: Mar 2004

Fri Jun 23, 06 06:27 PM
User is offline View users profile

Thanks DavidF for an answer.

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
quantitude.pl
 
Reply
   
Quote
   
Top
   
Bottom
     



GoGoFa
Member

Posts: 47
Joined: Mar 2005

Mon Jun 26, 06 10:58 AM
User is offline

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.
 
Reply
   
Quote
   
Top
   
Bottom
     



PiotrW
Member

Posts: 70
Joined: Mar 2004

Thu Jun 29, 06 01:35 PM
User is offline View users profile

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
quantitude.pl
 
Reply
   
Quote
   
Top
   
Bottom
     



ppauper
Senior Member

Posts: 47989
Joined: Nov 2001

Fri Jun 30, 06 03:27 PM
User is online View users profile

hehehehe.... he said wiener
 
Reply
   
Quote
   
Top
   
Bottom
     



horacioaliaga
Senior Member

Posts: 315
Joined: Aug 2005

Mon Jul 03, 06 05:57 PM
User is offline View users profile

This is what GoGoFa meant:


Edited: Mon Jul 03, 06 at 06:27 PM by horacioaliaga
 
Reply
   
Quote
   
Top
   
Bottom
     



PiotrW
Member

Posts: 70
Joined: Mar 2004

Tue Jul 04, 06 08:02 AM
User is offline View users profile

Thank you horacioaliaga.

Have a good day everybody.

PiotrW


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

View thread in raw text format
FORUMS > Technical Forum < refresh >

Forum Navigation:

© All material, including contents and design, copyright Wilmott Electronic Media Limited - FuseTalk 4.01 © 1999-2014 FuseTalk Inc. Terms & Conditions