Building Fat-Tailed Terminal Distributions with a Binomial Tree

Summary

This note asks whether a binary price model can produce a non-normal, fat-tailed terminal distribution. The response describes constructing a model from a stochastic differential equation chosen to have a fat-tailed terminal distribution, then translating it into a partial differential equation and an explicit finite-difference scheme. With a sufficiently wide price grid relative to the time grid, the central node is said to be unaffected by boundary conditions.

The finite-difference scheme is interpreted as a trinomial tree, and each trinomial step is replaced by two binomial half-steps. The proposed construction yields a binomial tree whose terminal distribution matches that of the selected SDE. This is a conceptual recipe rather than a worked numerical example: it provides no calibration, convergence demonstration, or specific fat-tailed model. The stated grid condition and boundary claim are part of the answer’s construction and are not independently supported in the text.

Key ideas

• A binary tree need not be limited to the distribution produced by independent, identically distributed moves.
• Start with an SDE whose terminal distribution has the desired tail behavior and convert it to a PDE.
• An explicit finite-difference scheme can be represented as a trinomial tree.
• Replacing each trinomial step with two binomial half-steps is proposed as a way to construct a matching terminal distribution.
• The document gives no numerical demonstration or validation of the construction.

Tags

# Can binary model lead to non-normal distribution?


# Can binary model lead to non-normal distribution?












If we suppose an instrument goes up or down 1 tick per $\Delta t$ (binary model), its long term distribution will be normal, per the Central Limit Theorem.

However, suppose we model as follows:

- The first tick is up or down with 50% probability.



The Central Limit Theorem doesn't apply here, because these are no longer independent random variables.

However, as this post shows us, the resulting distribution is still normal.

My question: can I construct a binary model that yields a non-normal distribution, ideally a "fat tailed" distribution?

## Answer by Brian B (score 2)

https://quant.stackexchange.com/a/2329

Yes you can.

Begin by choosing your favorite stochastic differential equation with a fat-tailed terminal distribution, for example a local volatility model. Use the usual techniques to convert to a partial differential equation (PDE).

Construct an explicit finite difference scheme for solving the PDE, and make your number $M$ of grid points in time $\tau$ sufficiently large compared to the number of grid points $N$ in asset price. Specifically, you need $$ N\geq 2M. $$ Your choice of boundary conditions has no influence on the central node since the grid is so wide.

This explicit scheme is therefore trivially equivalent to a trinomial tree. Now, noting that one step of a trinomial tree can be constructed from two steps of a binomial tree, insert those binomial half-steps to obtain a $2M$ step binomial tree whose terminal distibution matches that of your SDE.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.