Laplace Returns, Wiener Processes, and Lévy Models in Finance

# Why do we usually use normal distribution and not Laplace distribution to generate stochastic process?


# Why do we usually use normal distribution and not Laplace distribution to generate stochastic process?












When working with a stochastic process based on brownian motion, the increments have normal (gaussian) distribution.

However, it seems that a Laplace distribution, with density:

$$f(t) = \frac{\lambda}{2} e^{-\lambda |t|} \qquad (t \in \mathbb R)$$

would fit much more returns of EUR/USD for example than a normal distribution. (Especially, it has fatter tails than normal distribution, as required).

Here in blue is the density of returns, based on 10 years of historical data of 5-minutes chart of EUR/USD. In green, the density of a Laplace distribution.

## Question:

Are there some financial models, in which the stochastic process used is:

$$d \, X_t = ... + c \, d \, W_t$$

where $d\, W_t$ has a Laplace distribution instead of a normal distribution?

## Answer by Neeraj (score 4)

https://quant.stackexchange.com/a/23295

It is very natural to think that why assumption of Normal distribution is made for stochastic process $W_t$ when other more appropriate and valid distribution is available specially for modelling stock price. Before writing answer to your question explicitly, first look at definition of Wiener process:

Wiener Process: A Wiener process $W_t$, relative to family of information set {$\mathscr{F}_t$}, is a stochastic process such that:

- $W_t$ is square integrable martingale with $W_0=0$ and $$\mathbb{E}\big[(W_t-W_s)^2 \big]=t-s, \quad s\leq t$$

- The path of $W_t$ is continuous over $t$.

The following definition lead to following characteristics of Wiener process:

- $W_t$ has independent increments because it is martingale.

- $W_t$ has zero mean and and mean of every increment equals zero.

- $W_t$ has variance $t$

- $W_t$ has continuous paths.

Note that, in the above definition nothing is said about the distribution of increments. Normal distribution follows from the assumption stated in the definition. This is known as famous Levy Theorem. If assumptions under Wiener process are satisfied then Levy theorem prove that Wiener increments, $W_t - W_s$, are normally distributed with mean zero and variance $|t-s|$.

In short, for a stochastic process having continuous path and independent increments (natural properties of stock price), normality is not an assumption but gets derived from the basics assumption of Wiener process. And this is the reason no other assumption about the distribution of $dW_t$is made in literature because normality is not an assumption at all.

## Answer by Raskolnikov (score 4)

https://quant.stackexchange.com/a/38592

If you're willing to drop the requirement to have continuous paths, or rather, if you're willing to relax it, it is possible to have a bigger class of stochastic processes called Lévy processes. The requirement for it to work is that the probability distribution of your variable is infinitely divisible. The easiest way to formulate this is in terms of the characteristic function

$$\phi_X(u)=\mathbb{E}[\exp(iuX)]$$

If for any positive integer $n$ the characteristic function $\phi_X(u)$ is the $n$th power of a characteristic function, then we have the infinite divisibility property. For such distributions, it is possible to construct Lévy processes, i.e. a process for which $X_0=0$ and the increments $X_{s+t}-X_s$ have $\phi_X(u)^t$ as their characteristic function.

The Laplace distribution possesses the infinite divisibility property, its characteristic function is

$$\phi(u)=(1+\lambda^{-2}u^2)^{-1}$$

and there are indeed random variables with as characteristic functions powers of this. Those variables possess a variance gamma or generalized Laplace distribution and the Laplace distribution is of course a special case of those distributions.

The associated process is known as a variance gamma process and it was introduced in financial mathematics by Madan, Seneta, Carr and Chang in the 90's.

There is also a jump-diffusion model called the Kou model which has jump sizes which are Laplace distributed.