black scholes - What are $d_1$ and $d_2$ for Laplace?

What are the formulae for d1 & d2 using a Laplace distribution?

Answer

Your question is interesting because I thought that the only chance with Lévy-processes is to use Fourier-transform approaches (see e.g. Cont,Tankov).

But in the paper Option Pricing for Log-Symmetric Distributions of Returns by Fima C. Klebaner· Zinoviy Landsman they consider models, where the log of the price has a symmetric distribution. In Corollary 3.2 they propose an approximate formula if the log of the price follows the Laplace distribution where

$$d_{1,2} = \frac{\ln(S_0/K) + (r \pm log(1-\sigma^2/2))T}{\sigma \sqrt{T}}.$$ They write $\log$ but I guess this is just an $\ln$ as before. But please try yourself and/or read the first pages of the paper to be sure.