I'm working through Laplace distributions within a stock model. The stock price is modelled with a geometric Laplace motion along the lines of
$(1-\frac{\sigma^2}{2}\cdot dt)*exp(r*dt-\sigma*\sqrt{dt}*f_L)$
def laplacestock(r, sigma, mu, dt = 1):
return (1 - 0.5 * pow(sigma, 2) * dt) * math.exp(r * dt + sigma * sqrt(dt) * laplace(mu, sigma)) # Generates a laplacian increment at each call
with $\sigma$ scale factor, $dt$ time step, $r$ interest rate and $f_L$ a RNG with Laplace distribution (from numpy.random in python). In code this gives me what I expect:
(100 steps over 1000 code iterations. I know the image is not exactly correct bc Laplace distributions are not stable but it is good enough here.)
But when I want to plot the function with the Mac built-in Grapher Application, I don't get the same graph. Here I just replace the Laplace RNG with its distribution, $\frac{1}{2\sigma}\exp(-\frac{|x|}{\sigma})$. That then gives me
Where do I go wrong in my thinking? How do I express a geometric Laplace distribution in a formula?
