Monte Carlo Pricing of American Options with Laplace Returns
Summary
The document asks whether American options can be priced under a log-Laplace distribution rather than the lognormal assumption used in Black–Scholes. It reports that an attempted formula produced an implausible imbalance between call and put prices, and notes that a European-option formula under a log-Laplace assumption had been found elsewhere. No American-option closed-form solution or pricing results are supplied.
The reply suggests Monte Carlo simulation for American-style pricing and points to the Longstaff–Schwartz approach, which estimates continuation values to handle early exercise. It also sketches inverse-transform sampling to generate Laplace-distributed random variables from uniform draws. This describes a possible simulation building block, not a complete pricing procedure: the reply says further process transformations and discretization choices are needed and notes limited familiarity with applying the distribution to derivative pricing. The proposed route therefore needs careful model specification and validation.
Key ideas
- The author asks how to price American options when returns follow a log-Laplace distribution.
- The reply suggests Monte Carlo simulation and the Longstaff–Schwartz approach for handling early exercise.
- Uniform random draws can be transformed to generate Laplace-distributed variables.
- The document gives no complete discretization scheme, closed-form formula, or pricing evidence.
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Full text
# American Option price formula assuming a logLaplace distribution? # 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? ## Answer by Matt Wolf (score 4) https://quant.stackexchange.com/a/7294 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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