Bulls vs Bears: a Trinomial Model of a Financial Asset

TL;DR

This paper introduces a trinomial model for financial assets that incorporates short-selling bounds, providing formulas for moments of returns and methods to fit model parameters to observed data.

Contribution

It formalizes a new trinomial model with short-selling constraints and derives formulas for moments, including cases with many investors, enhancing asset price modeling.

Findings

Derived formulas for moments of logarithmic returns.

Provided methods to fit model parameters to observed moments.

Extended the model to include multiple investors and generalized moments.

Abstract

We present a variation of the well-known binomial model of asset prices. This variation incorporates a bound to short-selling, inspired by a model from Gunduz Caginalp[2]. We formalize this model and prove a formula for all the moments of the logarithmic returns. We also derive a formula for the case with infinitely many investors. As an application of the model, we show how to compute parameters in order to approximate given moments, enabling the modeling of skewness and excess kurtosis. Finally, we generalize the model and give the corresponding formula for the moments of the logarithmic returns, and the algorithm for fitting given moments.