Paper: arXiv 2410.04748

Authors: Jagdish Gnawali, W. Brent Lindquist, Svetlozar T. Rachev

Abstract

We introduce a fairly general, recombining trinomial tree model in the natural world. Market-completeness is ensured by considering a market consisting of two risky assets, a riskless asset, and a European option. The two risky assets consist of a stock and a perpetual derivative of that stock. The option has the stock and its derivative as its underlying. Using a replicating portfolio, we develop prices for European options and generate the unique relationships between the risk-neutral and real-world parameters of the model. We discuss calibration of the model to empirical data in the cases in which the risky asset returns are treated as either arithmetic or logarithmic. From historical price and call option data for select large cap stocks, we develop implied parameter surfaces for the real-world parameters in the model.

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper resolves a fundamental issue in existing trinomial option pricing literature by restoring market completeness via a stock’s perpetual derivative, linking real-world and risk-neutral parameters. It features rigorous discrete-time mathematical formulations alongside empirical calibration to historical stock and call option data.

Research Flowchart

  flowchart TD
    A[Research Goal: Option Pricing & Market Completeness via Perpetual Derivatives] --> B[Model Setup: Recombining Trinomial Tree with Stock, Perpetual Derivative, & Riskless Asset]
    B --> C[Methodology: Replicating Portfolio & Risk-Neutral Measure Derivation]
    D[Empirical Inputs: Historical Stock Prices & Call Option Data] --> E
    C --> E[Calibration: Arithmetic & Logarithmic Asset Return Models]
    E --> F[Outcomes: Unique Parameter Mappings & Implied Parameter Surfaces]

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1 of 42 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.