Actuarial Outpost Ito's Lemma general question
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 Investment / Financial Markets Old Exam MFE Forum

#1
09-18-2008, 12:39 AM
 langstafftigerpizza Member Join Date: Jan 2007 Studying for nothing right now Favorite beer: heineken Posts: 420
Ito's Lemma general question

I wonder how is (dS)^2, which is the middle term, calculated.

For example, ASM page 279 14G, 2nd last line, how does (dZ)^2 become dt?

#2
09-18-2008, 12:50 AM
 Actiger Member SOA CCA AAA Join Date: May 2007 Location: NYC Studying for Married Life Posts: 1,373

This is the multiplication rule of the SDE.

dZ^2 = dt
dt^2 = 0
dt*dZ = 0
#3
09-18-2008, 01:13 AM
 langstafftigerpizza Member Join Date: Jan 2007 Studying for nothing right now Favorite beer: heineken Posts: 420

Quote:
 Originally Posted by Actiger This is the multiplication rule of the SDE. dZ^2 = dt dt^2 = 0 dt*dZ = 0
Still confused. Can you explain a little further?
Also for Quiz 14-6, the solution shows (dY)^2 = 0.4^2dt?

Thanks
#4
09-18-2008, 05:36 AM
 jraven Member Join Date: Aug 2007 Location: New Hampshire Studying for nothing! College: Penn State Posts: 1,311

Quote:
 Originally Posted by langstafftigerpizza Still confused. Can you explain a little further? Also for Quiz 14-6, the solution shows (dY)^2 = 0.4^2dt? Please help. Thanks
The idea is that if, say, $dY = 0.1 \,dt + 0.4 \,dZ$ (I don't have a copy of the manual on-hand to see what it really uses), then

$(dY)^2 = (0.1 \,dt + 0.4 \,dZ)^2 = (0.1)^2 \,(dt)^2 + 2 (0.1) (0.4) \,dt\,dZ + (0.4)^2 \,(dZ)^2$

Then you use the multiplication table that Actiger gave to change that to

$(dY)^2 = (0.1)^2 (0) + 2 (0.1) (0.4) (0) + (0.4)^2 \,dt = (0.4)^2 \,dt$

As for why the multiplication table is what it is... that's a little (or a lot) complicated, and of no use whatsoever in understanding the material. You just need to know the multiplication table that Actiger provided.
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#5
09-18-2008, 08:25 PM
 langstafftigerpizza Member Join Date: Jan 2007 Studying for nothing right now Favorite beer: heineken Posts: 420

thanks a lot jraven, I guess I will just memorize the multiplication table, and it is good to go.
#6
09-24-2008, 10:50 PM
 Fermat83 Member Join Date: May 2008 Posts: 295

Quote:
 Originally Posted by langstafftigerpizza I wonder how is (dS)^2, which is the middle term, calculated. For example, ASM page 279 14G, 2nd last line, how does (dZ)^2 become dt? Thanks in advance!
Just memorize the multiplication table and everything will be fine. If the multiplication table seems weird its because it is. You are dealing with stochastic
random variables, and to manipulate them with calculus and see what there doing instantaneously, some bright boys had to come up with some new axioms to deal with these strange objects. I actually read a very theoretical book on the subject and it was kind of interesting but did nothing to help with the exam.
#7
09-25-2008, 11:14 AM
 volva yet Note Contributor Join Date: Feb 2006 Location: Nomadic Studying for GHC/DMAC College: PSU '07 Favorite beer: Oskar Blues Old Chub Scotch Ale Posts: 4,948

Quote:
 Originally Posted by jraven The idea is that if, say, $dY = 0.1 \,dt + 0.4 \,dZ$ (I don't have a copy of the manual on-hand to see what it really uses), then $(dY)^2 = (0.1 \,dt + 0.4 \,dZ)^2 = (0.1)^2 \,(dt)^2 + 2 (0.1) (0.4) \,dt\,dZ + (0.4)^2 \,(dZ)^2$ Then you use the multiplication table that Actiger gave to change that to $(dY)^2 = (0.1)^2 (0) + 2 (0.1) (0.4) (0) + (0.4)^2 \,dt = (0.4)^2 \,dt$ As for why the multiplication table is what it is... that's a little (or a lot) complicated, and of no use whatsoever in understanding the material. You just need to know the multiplication table that Actiger provided.
I am extremely interested in why, and there is no source that will tell me why. I am just forced to know that these rules are the law and I must abide. I'm not a fan of this as it does not help me truly understand the bridge between stochastic stock price modeling and the financial derivatives based on stock prices that is Ito's Lemma.
#8
09-25-2008, 11:55 AM
 raidersfan Member SOA Join Date: May 2007 Studying for MLC & C Favorite beer: NewCastle Posts: 40

Quote:
 Originally Posted by colby2152 I am extremely interested in why, and there is no source that will tell me why. I am just forced to know that these rules are the law and I must abide. I'm not a fan of this as it does not help me truly understand the bridge between stochastic stock price modeling and the financial derivatives based on stock prices that is Ito's Lemma.
Quote:
 Originally Posted by colby2152 I am extremely interested in why, .
No you're not.

Quote:
 Originally Posted by colby2152 and there is no source that will tell me why. .
Yes there is.

Quote:
 Originally Posted by colby2152 I am just forced to know that these rules are the law and I must abide. .
This should be the least of your worries on this exam.

Quote:
 Originally Posted by colby2152 it does not help me truly understand the bridge between stochastic stock price modeling and the financial derivatives based on stock prices that is Ito's Lemma..
No such bridge exists.
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#9
09-25-2008, 06:34 PM
 Fermat83 Member Join Date: May 2008 Posts: 295

Quote:
 Originally Posted by colby2152 I am extremely interested in why, and there is no source that will tell me why. I am just forced to know that these rules are the law and I must abide. I'm not a fan of this as it does not help me truly understand the bridge between stochastic stock price modeling and the financial derivatives based on stock prices that is Ito's Lemma.

There are plenty of books on theory of this stuff. I was also annoyed that I had to take these multiplication rules and itos Lemma as is with no explanation of why. I spent a few days digging in a very theoretical book and saw why stochastic random variables model stock behavior well and as for stochastic calculus it was increadibly abstract and strange. Overall it was a waste of study time, and I'm guessing thats why they don't go into this stuff. I've realized you can't be an applied mathematician and also fully understand the theoretical aspects behind everything without getting 3 hours of sleep and having 0 social life. Theoretical mathematicians develoup the tools and Applied mathematicians use them to solve complicated problems in the real world.
#10
09-25-2008, 11:53 PM
 Nonpareil Note Contributor Join Date: Nov 2006 Location: Rocket City Studying for Exam C Posts: 833

Quote:
 Originally Posted by colby2152 I am extremely interested in why, and there is no source that will tell me why. I am just forced to know that these rules are the law and I must abide. I'm not a fan of this as it does not help me truly understand the bridge between stochastic stock price modeling and the financial derivatives based on stock prices that is Ito's Lemma.
Look at page 658 of DM, which has a decent heuristic explanation.
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