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?