Bulls vs Bears: a Trinomial Model of a Financial Asset

Paper: arXiv 2505.18723

Authors: Nahuel I. Arca

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.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation of log-return moments for a novel agent-inspired trinomial model incorporating short-selling bounds. However, it lacks empirical validation, backtesting, or real-market data application, relying entirely on theoretical proofs and algorithmic formulations.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Binomial Asset Model with Short-Selling Bounds] --> B[Incorporate Caginalp Bound into Trinomial Framework]
    B --> C[Derive Formulas for Log Return Moments & Infinite Investor Case]
    C --> D[Develop Moment Matching Algorithm]
    D --> E[Inputs: Given Asset Moments Skewness & Excess Kurtosis]
    E --> F[Generalize Model & Fit Parameters to Empirical Data]
    F --> G[Outcome: Flexible Model Capturing Non-Normal Return Features]