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Asymptotic properties and drift parameter estimations of the ergodic double Heston model based on continuous-time observations

Mohamed Ben Alaya, Houssem Dahbi, Hamdi Fathallah

TL;DR

This work analyzes a three-factor double Heston diffusion in an affine diffusion framework to address both probabilistic and statistical properties. It first classifies the long-run behavior of the model into subcritical, critical, and supercritical regimes, establishing a unique stationary distribution and exponential ergodicity under suitable drift conditions. It then develops continuous-time MLE and CLSE methods for the drift parameters, proving strong consistency and asymptotic normality in the ergodic (subcritical) setting, with explicit limit covariance structures derived from stationary behavior. Numerical simulations corroborate the theoretical results, showing faster convergence for MLE relative to CLSE and illustrating the practical applicability of the inference procedures for calibrated stochastic-volatility models.

Abstract

The double Heston model is one of the most popular option pricing models in financial theory. It is applied to several issues such that risk management and volatility surface calibration. This paper deals with the problem of global parameter estimations in this model. Our main stochastic results are about the stationarity and the ergodicity of the double Heston process. The statistical part of this paper is about the maximum likelihood and the conditional least squares estimations based on continuous-time observations; then for each estimation method, we study the asymptotic properties of the resulted estimators in the ergodic case.

Asymptotic properties and drift parameter estimations of the ergodic double Heston model based on continuous-time observations

TL;DR

This work analyzes a three-factor double Heston diffusion in an affine diffusion framework to address both probabilistic and statistical properties. It first classifies the long-run behavior of the model into subcritical, critical, and supercritical regimes, establishing a unique stationary distribution and exponential ergodicity under suitable drift conditions. It then develops continuous-time MLE and CLSE methods for the drift parameters, proving strong consistency and asymptotic normality in the ergodic (subcritical) setting, with explicit limit covariance structures derived from stationary behavior. Numerical simulations corroborate the theoretical results, showing faster convergence for MLE relative to CLSE and illustrating the practical applicability of the inference procedures for calibrated stochastic-volatility models.

Abstract

The double Heston model is one of the most popular option pricing models in financial theory. It is applied to several issues such that risk management and volatility surface calibration. This paper deals with the problem of global parameter estimations in this model. Our main stochastic results are about the stationarity and the ergodicity of the double Heston process. The statistical part of this paper is about the maximum likelihood and the conditional least squares estimations based on continuous-time observations; then for each estimation method, we study the asymptotic properties of the resulted estimators in the ergodic case.

Paper Structure

This paper contains 12 sections, 12 theorems, 119 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Classification result
  4. Stationarity
  5. Exponential ergodicity
  6. Maximum likelihood estimation
  7. Consistency of the MLE
  8. Asmptotic behavior of the MLE
  9. Conditional least squares estimation
  10. Consistency of the CLSE
  11. Asymptotic behavior of the CLSE
  12. Numerical results

Key Result

Proposition 3.1

Let $(Z_t)_{t\in\mathbb{R}_{+}}$ be the unique strong solution of model01 satisfying $\mathbb{P}(Y_0\in\mathbb{R}_{++}^2)=1$ and $\mathbb{E}\left(\left\Vert Z_0\right\Vert_1\right)<\infty$. Then, for all $t\in\mathbb{R}_+$, $Z_t$ is integrable and $\mathbb{E}(Z_t)$ converges if $b_{11},b_{22},\theta

Figures (5)

  • Figure 1: Asymptotic behavior of $E(Z_T)$ in the subcritical case: $(b_{11}, b_{22}, \theta) = (1, 3, 2)$.
  • Figure 2: Asymptotic behavior of $E(Z_T)$ in the critical case for different parameter sets.
  • Figure 3: Asymptotic behavior of $E(Z_T)$ in the supercritical case.
  • Figure 4: MLE: Distribution of the scaled error: $\sqrt{T}(\hat{\tau}_i-\tau_i)$, with $T=10^3$.
  • Figure 5: CLSE: Distribution of the scaled error: $\sqrt{T}(\check{\tau}_i-\tau_i)$, with $T=3\times10^5$.

Theorems & Definitions (23)

  • Proposition 3.1
  • proof
  • Definition 3.1
  • Proposition 4.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • proof
  • Theorem 6.2
  • ...and 13 more