Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian

TL;DR

This work introduces the iVi scheme, an integrated Volterra implicit method for simulating integrated Volterra square-root processes and Volterra Heston models using Inverse Gaussian increments. It handles $L^1$ kernels with singularities by relying solely on integrated kernel quantities and preserves the non-decreasing property of the integrated process, $U$. The authors prove weak convergence by recasting the scheme as a stochastic Volterra equation with a measure kernel, establishing tightness and a stability framework; weak convergence is guaranteed under weak uniqueness, e.g., for completely monotone kernels. Numerically, the iVi scheme achieves high accuracy with few time steps, and convergence improves as the Hurst index $H$ decreases toward $-1/2$, which aligns with the limiting Inverse Gaussian regime. The results are demonstrated for both the integrated variance process and the Volterra Heston model, including option pricing via Fourier techniques and contact with hyper-rough dynamics, offering a robust, efficient tool for non-Markovian volatility modeling in finance.

Abstract

We introduce a novel simulation scheme, iVi (integrated Volterra implicit), for integrated Volterra square-root processes and Volterra Heston models based on the Inverse Gaussian distribution. The scheme is designed to handle $L^1$ kernels with singularities by relying solely on integrated kernel quantities, and it preserves the non-decreasing property of the integrated process. We establish weak convergence of the iVi scheme by reformulating it as a stochastic Volterra equation with a measure kernel and proving a stability result for this class of equations. Numerical results demonstrate that convergence is achieved with very few time steps. Remarkably, for the rough fractional kernel, unlike existing schemes, convergence seems to improve as the Hurst index $H$ decreases and approaches $-1/2$.

Figures (4)

• Figure 1: Implied volatility slice for call options of the hyper-rough Heston model \ref{['eq:HestonS']}, with maturity $T=1$. Parameters as in Case 2 of Table \ref{['tab:parameter_cases']}, with varying Hurst index between $0.1$ and -$0.49$. iVi scheme in shades of orange with the number of corresponding time steps with 1 million sample paths.
• Figure 2: The parameters are $a = 0.1$, $b = -0.3$, $c = 0.2$, $V_0 = 0.04$ with a varying Hurst index $H \in \{0.1,-0.1, -0.4\}$ for each row. $T = 0.2$ and $200$ time steps.
• Figure 3: Errors on Laplace transform of $\hat{U}_T$ in terms of the number of time steps for the four cases with $T = 1$ and 1 million sample paths. Plain line for the iVi scheme, blue dotted line for the benchmark.
• Figure 4: ATM Call options on $S$: error in prices in terms of number of time steps for the four cases with 1 million sample paths. Plain line for the iVi scheme, blue dotted line for the benchmark.

Theorems & Definitions (43)

• Proposition 1.1
• proof
• Definition 1.2
• Theorem 1.3
• proof
• Proposition 1.4
• proof
• Example 1.5
• Proposition 1.6
• proof