# American Option price formula assuming a logLaplace distribution?

What are $d_1$ and $d_2$ for Laplace? may be running before walking.

When I tried to use the equations provided, the pricing became extremely lopsided, with the calls being routinely double puts. This is extremely unrealistic.

My guess was correct that a distribution closer to the ideal (whatever it is) would remove the volatility smile, proven here with European options assuming logLaplace (formula 1 page up): http://books.google.com/books?id=cb8B07hwULUC&pg=PA297&lpg=PA296#v=onepage&q&f=false

However, it looks like the fundamental assumptions even going back to risk-neutral have to be changed because all BS depends upon lognormality at some point whatever the derivation.

I've searched and searched, but I can't find anything that's worked out American option prices assuming a logLaplace distribution and not lognormality.

What is the American option price formula for a call assuming a logLaplace Distribution and not lognormality?

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Just few basic "sanity checks": What do you mean "calls being routinely double puts"? If you're looking at ITM calls, they certainly can be double the price of puts, or even more. Or are you comparing ATMF? Did you accidentally add "drift" into your model? Try with r=0. – onlyvix.blogspot.com Nov 7 '13 at 18:14

Have you looked at using Laplace in a Monte Carlo simulation? Here is how you price American style options within a MC framework:

http://www2.math.uu.se/research/pub/Jia1.pdf

and the Longstaff, Schwartz paper: http://escholarship.org/uc/item/43n1k4jb#page-1

Regarding the discretization of a process that draws its random variables from a Laplace distribution I can only suggest ideas as I have not myself worked with those in regards to a MC discretization:

You can draw RV from a uniform distribution and generate a laplace distributed RV X = μ - b * sgn(U) * ln(1-2 * |U|). This only concerns the RV generation itself. You need to make other transformations as well but I am not aware this distribution has been much applied to pricing financial derivatives. The following may help, though they use a mixture of Normal-Laplace distributions:

http://asianfa2012.mcu.edu.tw/fullpaper/10305.pdf

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