Initial value PDE: the Black-Scholes model (part b)

kreedman

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Sep 19, 2007
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I solved part a of the question. I need a hint on how to start solving part b:

(b) Show further that transforming:

. . . . .\(\displaystyle \L v\, =\, e^{\kappa y\, +\, \beta s} w(s,\, y)\)

...where:

. . . . .\(\displaystyle \L \gamma \,=\, \frac{1\, -\, \kappa}{2}\, ,\,\,\mbox{ } \beta\, =\, -\frac{(\kappa \,+\, 1)^2}{4}\)

...yields the PDE problem:

. . . . .\(\displaystyle \L w_s \,=\, w_{yy},\,\,\mbox{ } -\infty \,< y\, < \, \infty,\,\, s\, \geq \, 0\). . . . .
(1.30)

. . . . .\(\displaystyle \L w(0,\,y)\, =\, \mbox{max}\left(e^{\frac{1}{2}(\kappa\, +\, 1)y}\, -\, e^{\frac{1}{2}(\kappa\, +\, 1)y}\, ,\,0\right)\)

Thanks,

Jim

FreeMathHelp thread for part a

screen-shot of entire page
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Edited by stapel -- Reason for edit: Converting fuzzy graphic into clear text.
 
Re: Deff Help

kreedman said:
http://picasaweb.google.ca/ThefHomework/20070922Hw3/photo#5113109915853094082

OK, I solved part a of the question. I need a hint on how to start solving part b.

Thanks,

Jim

Just like before, find w_s & w_yy and then show 1.29 and 1.30 are equivalent.

Lot of messy algebra - but like my teacher used say - "it will build your character..."
 
kreedman said:
So:

dv/dx = d/x [w * e^(vy+Bs)] ??

Remember these are partial derivatives - so the expressions become quite long.
 
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