VIXRHeston 0.1.1

VIX term structure in the rough Heston model via Markovian approximation

VIXRHeston

VIXRHeston is a lightweight Python package for computing the VIX term structure in a rough Heston–type setting via a Markovian (lifted Heston) approximation. Leveraging the key theoretical result that, under the lifted model, the squared VIX can be expressed analytically (in particular, as a linear function of the Markovian variance components), the package evaluates the quantity

VIX^2_{t,τ} = (1/τ) ∫_{t}^{t+τ} E_t[V_s] ds

efficiently across maturities, without requiring nested Monte Carlo simulation. This enables fast generation of the full VIX term structure and is suitable for calibration workflows where repeated VIX evaluations are needed.

Rough Heston dynamics (risk-neutral)

Under the risk-neutral measure ( \mathbb{Q} ), the rough Heston model specifies the index and variance dynamics as [ \frac{dS_t}{S_t} = r,dt + \sqrt{V_t},dB_t, ] [ V_t = g_0(t) + \int_0^t K(t-s)\Big(\lambda(\theta - V_s),ds + \nu\sqrt{V_s},dW_s\Big), ] where (B_t) and (W_t) are Brownian motions with instantaneous correlation (dB_t,dW_t=\rho,dt).
Here (r) is the risk-free rate, (\lambda) is the mean-reversion speed, (\theta) is the long-run variance level, (\nu) is the vol-of-vol, and (K(\cdot)) is a fractional (rough) kernel (typically of the form (K(u)\propto u^{H-\frac12}) with Hurst index (H\in(0,\tfrac12))). The deterministic function (g_0(t)) encodes the forward variance curve / initial variance condition.

Installation

From PyPI

pip install VIXRHeston

Quick start

import numpy as np
from VIXRHeston import vec_c, vec_x
from VIXRHeston import squared_VIX

H,n = 0.1,2
alpha, rn= H+0.5, 1+10*(1/(n)**0.9)
c, x = vec_c(n,rn,alpha), vec_x(n,rn,alpha)
t, tau = 0, 1/12
rho, lamb,theta,nu,V0 = -0.7,0.1,0.03,0.3, 0.01
VIX2 = squared_VIX(t, c, x, V0,lamb,theta,nu,rho, tau)
VIX  = np.sqrt(VIX2)
print(VIX)

License

MIT License. See LICENSE.

References

Ye, Y., Fan, Z., & Kwok, Y. K. (2026). VIX term structure in the rough Heston model via Markovian approximation. Journal of Futures Markets, forthcoming.