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Analytical Formula for Fractional-Order Conditional Moments of Nonlinear Drift CEV Process with Regime Switching: Hybrid Approach with Applications

Kittisak Chumpong, Khamron Mekchay, Fukiat Nualsri, Phiraphat Sutthimat

TL;DR

The paper develops an analytical framework for fractional-order conditional moments of the $m$-state regime-switching nonlinear drift CEV (NLD-CEV) process by formulating a hybrid system of PDEs via the Feynman-Kac approach. It derives exact closed-form expressions for these moments across arbitrary fractional orders and regime states, including backward recursive schemes and specializations to two-state switches, along with unconditional moment limits as time to maturity grows. The methodology is validated against Monte Carlo simulations and applied to VIX futures and options pricing using a Laguerre expansion informed by the conditional moments, demonstrating substantial computational efficiency gains over simulation-based methods. The results provide a robust, tractable tool for derivative pricing and risk management in environments with nonlinear variance elasticity and regime shifts, and they offer a foundation for extending regime-switching analyses to broader financial contexts.

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.

Analytical Formula for Fractional-Order Conditional Moments of Nonlinear Drift CEV Process with Regime Switching: Hybrid Approach with Applications

TL;DR

The paper develops an analytical framework for fractional-order conditional moments of the -state regime-switching nonlinear drift CEV (NLD-CEV) process by formulating a hybrid system of PDEs via the Feynman-Kac approach. It derives exact closed-form expressions for these moments across arbitrary fractional orders and regime states, including backward recursive schemes and specializations to two-state switches, along with unconditional moment limits as time to maturity grows. The methodology is validated against Monte Carlo simulations and applied to VIX futures and options pricing using a Laguerre expansion informed by the conditional moments, demonstrating substantial computational efficiency gains over simulation-based methods. The results provide a robust, tractable tool for derivative pricing and risk management in environments with nonlinear variance elasticity and regime shifts, and they offer a foundation for extending regime-switching analyses to broader financial contexts.

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.

Paper Structure

This paper contains 17 sections, 12 theorems, 106 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The $m$-state regime-switching NLD-CEV process
  3. The nonlinear drift constant elasticity of variance process
  4. The NLD-CEV process with regime switching
  5. Conditional moments: A hybrid system of PDEs
  6. Hybrid system of PDEs when $\beta\in[0,2)$
  7. Hybrid system of PDEs when $\beta\in(2,\infty)$
  8. The two-state regime-switching NLD-CEV process when $\beta\in[0,2)$
  9. The two-state regime-switching NLD-CEV process when $\beta\in(2,\infty)$
  10. Unconditional moments
  11. Euler--Maruyama method for SDEs with regime switching
  12. Simulation algorithm
  13. Accuracy and efficiency
  14. Financial applications on the VIX
  15. Futures pricing strategy: Offsetting positions
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose that $R_t$ follows the system of SDEs model3 on $[0,T]$ and Assumptions Assumption4 and Assumption5 hold. The function $U_{\alpha,i}^{\langle n\rangle} := U_{\alpha,i}^{\langle n\rangle}(\tau,R)$ defined in aim, for $i \in \mathcal{M}_m$ and $\tau = T - t \geq 0$, are the solutions of the fo for $(\tau,R) \in [0,T] \times (0,\infty)$ subject to the initial condition $U_{\alpha,i}^{\langle

Figures (3)

  • Figure 1: Sample paths of the two-state regime-switching NLD-CEV process using the EM method for SDEs.
  • Figure 2: Conditional moments of the regime-switching NLD-CEV process.
  • Figure 3: Future prices for the CIR process with the two-state regime-switching.

Theorems & Definitions (24)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 14 more