American Option price formula assuming a logLaplace distribution?

Question

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?

Answer 1

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