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May 27th, 2016, 06:23 PM  #1 
Member Joined: Sep 2013 Posts: 83 Thanks: 0  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 $\displaystyle \tau \equiv Tt$ $\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 10:26 PM. 

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