Paper: arXiv 2503.18165

Authors: Dario Crisci, Sebastian E. Ferrando, Konrad Gajewski

Abstract

An agent-based modelling methodology for the joint price evolution of two stocks is put forward. The method models future multidimensional price trajectories reflecting how a class of agents rebalance their portfolios in an operational way by reacting to how stocks’ charts unfold. Prices are expressed in units of a third stock that acts as numeraire. The methodology is robust, in particular, it does not depend on any prior probability or analytical assumptions and it is based on constructing scenarios/trajectories. A main ingredient is a superhedging interpretation that provides relative superhedging prices between the two modelled stocks. The operational nature of the methodology gives objective conditions for the validity of the model and so implies realistic risk-rewards profiles for the agent’s operations. Superhedging computations are performed with a dynamic programming algorithm deployed on a graph data structure. Null subsets of the trajectory space are directly related to arbitrage opportunities (i.e. there is no need for probabilistic considerations) that may emerge during the trajectory set construction. It follows that the superhedging algorithm handles null sets in a rigorous and intuitive way. Superhedging and underhedging bounds are kept relevant to the investor by means of a worst case pruning method and, as an alternative, a theory supported pruning that relies on a new notion of small arbitrage.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents an innovative non-probabilistic, agent-based trajectory framework for relative asset pricing and superhedging without relying on standard stochastic assumptions. It combines advanced measure-free mathematical concepts with dynamic programming algorithms and practical backtests, offering a strong theoretical foundation paired with empirical risk-reward analyses.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Joint 2-Stock Price Trajectories & Superhedging Without Probabilistic Assumptions] --> B[Inputs: Relative Stock Prices via Numeraire & Agent Rebalancing Rules]
    B --> C[Construct Trajectory Space Scenarios & Identify Arbitrage/Null Sets]
    C --> D[Prune Trajectories via Worst-Case & Small Arbitrage Methods]
    D --> E[Compute Relative Superhedging/Underhedging Bounds using Dynamic Programming on Graphs]
    E --> F[Outcomes: Robust Model Validation, Objective Risk-Reward Profiles & Rigorous Option Bounds]