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VIX Simulation in the Quadratic Rough Heston Model

Article Quant Q&A · Author: FRT
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Summary

The document describes a problem simulating the VIX in a quadratic rough Heston framework intended for joint calibration to S&P 500 and VIX smiles. It specifies spot dynamics driven by stochastic variance, with variance defined as a quadratic function of a latent process. That process follows a Volterra equation with a power-law kernel, deterministic input, and volatility-dependent noise.

The author cites an infinite-dimensional Markovian representation that rewrites the latent process after a chosen time using a shifted deterministic input, and notes a representation involving a Mittag-Leffler density. They ask how to use these forms to express VIX as a function of the current state and input. The post supplies model equations and references methods from prior work, but gives no simulation procedure, calibration results, or resolution. It is useful as a model specification and statement of the computational challenge, while leaving implementation and numerical accuracy open.

Key ideas

  • The model defines variance as a quadratic function of a latent Volterra process.
  • A power-law kernel and volatility-dependent noise govern the latent process.
  • The cited Markovian representation introduces a time-shifted deterministic input.
  • The post asks how to derive and simulate VIX from the state but offers no solution or numerical evidence.

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VolatilityDerivatives pricingStatistics
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# Simulation of VIX index (Joint calibration VIX/SPX)


# Simulation of VIX index (Joint calibration VIX/SPX)












I work on the paper "The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem", deal with the Quadratic Rough Heston, and I have an issue to simulate VIX_T. The model is define as follows : $$dS_t = S_t \sqrt{V_t}dW_t,$$ $$V_t = a(Z_t-b)^2 + c,$$ $$Z_t = \int_0^t K(t-s)\lambda(\theta_0(s)-Z_s)\, ds + \int_0^t K(t-s)\lambda\eta\sqrt{V_s}\, dW_s,$$ where $\theta_0(\cdot)$ is a deterministic function, and $K(t) = t^{\alpha-1}/\Gamma(\alpha)$. In subsection 3.3 "Infinite dimensional Markovian representation" on this paper, it is """shown""" that $$Z_{t_0+t} = \int_0^t \dfrac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\lambda(\theta_{t_0}(s) - Z_{t_0+s})\, ds + \int_0^t \dfrac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\lambda\eta\sqrt{V_{t_0+s}}dW_{t_0+s},$$ where $\theta_{t_0}(u) = \theta_0(t_0+u) + \frac{\alpha}{\lambda \Gamma(1-\alpha)}\int_0^{t_0} (t_0+v-u)^{-1-\alpha}(Z_v-Z_{t_0})\, dv.$ After that, it is said that thanks to this forms we could be able to express the VIX at time t as a function of $S_t$ and $\theta_t$. I known several methods to do this computation quite easily, based on the article of S. Romer. Aditionally, I known that $$Z_t = \int_0^t f^{\alpha,\lambda}(t-s)\theta_0(s)\, ds + \int_0^t f^{\alpha,\lambda}(t-s)\eta\sqrt{V_s}\, dW_s,$$ where $f^{\alpha,\lambda}$ is the density of Mittag-Leffler function, but I don't know if it is very helpful. Could someone tell me how to proceed ? Thanks a lot.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.