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May 27th, 2016, 05:23 PM   #1
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American Put Options - Journal Article Help

In the article "The American Put Option and Its Critical Stock Price" Bunch and Johnson (2000),
the author posed this expression as a representation of an American put option:

$\displaystyle
P= p+ \int_{0}^{T} rXe^{-rt}N(-d_{2}(S,Sc,t))dt

$

where

$\displaystyle

d_{2}(S,Sc,t) = \frac{\log (S/Sc) +(r-\frac{1}{2}\sigma^2 )}{\sigma \sqrt{t}}
$


Now in the article, they say that this can be shown to yield the following:

$\displaystyle \frac{Sc}{X}=e^{(r+(1/2)\sigma ^2)\tau -g\sigma \sqrt{\tau }}$

where

$\displaystyle \tau \equiv T-t$

$\displaystyle
g=\pm \sqrt{2\log\frac{\sigma ^2}{\frac{2r}{\sqrt{a}}x\, \log\, xe^{-a(r+(1/2)\sigma ^2)^2\tau /(2\sigma ^2)}}}

$

and

$\displaystyle \gamma \equiv \frac{2r}{\sigma ^2}$

$\displaystyle a=1-\frac{A}{1+\frac{(1+\gamma )^2}{4}\sigma ^2\tau } $

$\displaystyle A=\frac{1}{2}\left ( \frac{\gamma }{1+\gamma } \right )^2 $

$\displaystyle x= \frac{X}{Sc}$

Question is how did they do that?

Note: I have solved this once, but I threw the piece of paper , don't ask me why...

Last edited by skipjack; May 27th, 2016 at 09:26 PM.
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