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junior24
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Joined: March 1st, 2012, 1:03 pm

Wilmott's Book Quantitative Finance 2nd Ed

July 19th, 2018, 1:12 pm

Hi everyone,
Was reading Part One ( # 7.2 DERIVATION OF THE FORMULAE FOR CALLS, PUTS AND SIMPLE DIGITALS) from : Paul Wilmott on Quantitative Finance 2nd Ed. 

I can't get how it's done to transform the equation from dU/dTo  = 0.5sigma²S² d²U/dS² ... to : 
dU/dTo = 0.5sigma²d²U/dEta² + (r-0.5sigma²) dU/dEta

Actually it's the secondary derivation that I can't get : d²/dS² = exp(-2Eta) d²/dEta² - exp(-2Eta)d/dEta
How does it come one has exp(-2Eta) !!

Any idea ?
Much appreciate !
  
Just because you've shorted your hedge, yours legs won't get longer.
 
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ppauper
Posts: 11729
Joined: November 15th, 2001, 1:29 pm

Re: Wilmott's Book Quantitative Finance 2nd Ed

July 19th, 2018, 1:49 pm

I don't have the book at hand, but assume there's a transformation along the lines of [$]S=X e^{\eta}[$] and [$]\eta=\log(S/X)[$]
and then chain rule
[$]\frac{\partial V}{\partial S}=\frac{\partial\eta}{\partial S}\,\frac{\partial V}{\partial\eta}=S^{-1}\,\frac{\partial V}{\partial\eta}[$]
then
[$]\frac{\partial^{2} V}{\partial S^{2}}=\frac{\partial\;}{\partial S}\left[S^{-1}\,\frac{\partial V}{\partial\eta}\right]=-S^{-2}\,\frac{\partial V}{\partial\eta}+S^{-2}\,\frac{\partial^{2} V}{\partial\eta^{2}}[$]
and use [$]S^{-2}=X^{-2}e^{-2\eta}[$]
 
junior24
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Posts: 4
Joined: March 1st, 2012, 1:03 pm

Re: Wilmott's Book Quantitative Finance 2nd Ed

July 20th, 2018, 7:30 am

Thanks that was the trick : dV/dS = dSmthing /dS * dV/dSmthing !!
How ever I found it useless to solve the Black Scholes equation knowing we can solve it numerically speaking with M-Carlo 

So why is it so popular ?
Just because you've shorted your hedge, yours legs won't get longer.
 
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bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: Wilmott's Book Quantitative Finance 2nd Ed

July 20th, 2018, 10:16 am

Thanks that was the trick : dV/dS = dSmthing /dS * dV/dSmthing !!
How ever I found it useless to solve the Black Scholes equation knowing we can solve it numerically speaking with M-Carlo 

So why is it so popular ?
Probably for the same reason many people travel by airplane when they could just walk.
 
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Paul
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Joined: July 20th, 2001, 3:28 pm

Re: Wilmott's Book Quantitative Finance 2nd Ed

July 20th, 2018, 12:24 pm

LOL!
 
junior24
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Posts: 4
Joined: March 1st, 2012, 1:03 pm

Re: Wilmott's Book Quantitative Finance 2nd Ed

July 20th, 2018, 12:37 pm

Another question regarding the derivation of calls/puts and digitals (#7.2 page 110) :
The initial equation is said to be :

dV/dt + r.S.dV/dS + 1/2.sigma^2.S^2.d"V/dS"  = r.V
and then by using : V = exp(-r.(T-t)).U the right elements just disappear.. Do not understand.

Any help ?
Just because you've shorted your hedge, yours legs won't get longer.
 
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ppauper
Posts: 11729
Joined: November 15th, 2001, 1:29 pm

Re: Wilmott's Book Quantitative Finance 2nd Ed

July 20th, 2018, 12:53 pm

The last one was the chain rule.
This one is another trick, the derivative of a product
[$]\frac{\partial V}{\partial t}=\frac{\partial\;}{\partial t}\left[e^{-r(T-r)}U\right]=re^{-r(T-r)}U+e^{-r(T-r)}\frac{\partial U}{\partial t}[$]

the first term [$]re^{-r(T-r)}U=rV[$]  will cancel the term on the right