Paper: arXiv 2411.13937

Authors: Kittisak Chumpong, Khamron Mekchay, Fukiat Nualsri, Phiraphat Sutthimat

Abstract

This paper introduces an analytical formula for the fractional-order conditional moments of nonlinear drift constant elasticity of variance (NLD-CEV) processes under regime switching, governed by continuous-time finite-state irreducible Markov chains. By employing a hybrid system approach, we derive exact closed-form expressions for these moments across arbitrary fractional orders and regime states, thereby enhancing the analytical tractability of NLD-CEV models under stochastic regimes. Our methodology hinges on formulating and solving a complex system of interconnected partial differential equations derived from the Feynman-Kac formula for switching diffusions. To illustrate the practical relevance of our approach, Monte Carlo simulations for process with Markovian switching are applied to validate the accuracy and computational efficiency of the analytical formulas. Furthermore, we apply our findings for the valuation of financial derivatives within a dynamic nonlinear mean-reverting regime-switching framework, which demonstrates significant improvements over traditional methods. This work offers substantial contributions to financial modeling and derivative pricing by providing a robust tool for practitioners and researchers who are dealing with complex stochastic environments.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a dense mathematical framework deriving analytical closed-form solutions for fractional conditional moments using coupled PDE systems and Feynman-Kac equations under regime switching. While it includes Monte Carlo simulations to validate the math and applies the results to derivative pricing (e.g., VIX futures), it lacks empirical backtesting on historical market data.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br/>Derive analytical formula for fractional conditional moments of NLD-CEV process with regime switching"] --> B["<b>System Formulation</b><br/>Incorporate continuous-time finite-state Markov chain & nonlinear drift"]
    
    B --> C["<b>Theoretical Framework</b><br/>Apply Feynman-Kac formula for switching diffusions"]
    
    C --> D["<b>Mathematical Derivation</b><br/>Formulate & solve interconnected system of PDEs"]
    
    D --> E["<b>Validation & Computation</b><br/>Run Monte Carlo simulations for regime-switching process"]
    
    E --> F["<b>Key Outcomes</b><br/>Exact closed-form expressions & improved derivative valuation"]