Approximating Rough Heston with a Lifted Multi-Factor Model
Summary
The document presents the lifted Heston approximation as a practical route to simulating rough Heston dynamics. Rough Heston replaces the usual variance evolution with a fractional, Volterra-style square-root process, whose memory kernel makes direct simulation mathematically demanding. The lifted construction approximates that kernel with a finite combination of factors, yielding a coupled system of conventional stochastic differential equations for asset price and variance-related state variables.
The description gives the factor dynamics and parameter relationships used to form the approximation, including a correlation between the asset’s Brownian motion and the variance driver. A response suggests standard schemes such as Milstein as an initial way to simulate the resulting finite-dimensional system, and mentions that libraries may support Heston simulation schemes. However, it provides no worked implementation, convergence study, or comparison of simulation accuracy and speed. The factor approximation and numerical scheme therefore require validation for a particular use, especially when fidelity to rough-volatility behavior matters.
Key ideas
- Rough Heston uses a fractional square-root variance process with memory.
- The lifted approximation represents the variance dynamics with a finite set of factors.
- Asset and variance shocks can be correlated through a shared Brownian driver.
- Classical SDE schemes such as Milstein can serve as starting points for simulating the lifted system.
- The exchange gives no implementation or evidence about approximation accuracy and computational cost.
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# Simulating the Rough Heston
# Simulating the Rough Heston
I found this paper here https://arxiv.org/abs/1810.04868, "The Lifted Heston", but since I'm not an expert in stochastic volterra processes , nor in fractional ricatti equations, the math is beyond me. If anyone could explain to me step-by-step the process to simulate paths described in the paper (I know it's an approximation), or better yet, share a repo, I would greatly appreciate it.
## Answer by crow (score 6, accepted)
https://quant.stackexchange.com/a/59107
The rough Heston process is a Heston process that has had its variance process process replaced with a fractional square-root diffusion \begin{equation} d S_t = S_t \sqrt{V_t} d B_t \end{equation} \begin{equation} V_t = V_0 + \frac{\int_0^t (t - s)^{H - \frac{1}{2}} \lambda (\theta - V_s) d s + \int_0^t (t - s)^{H - \frac{1}{2}} v \sqrt{V_s} d W_{} }{\Gamma \left( H + \frac{1}{2} \right)} \end{equation}
The lifted Heston model is a finite linear combination of conventional stochastic volatility models with $n$ factors driving the variance process given by the system of stochastic differential equations \begin{equation} d S_t^n = S_t^n \sqrt{V_t^n} d B_t \end{equation} \begin{equation} V_t^n = g_0^n (t) + \sum_{i = 1}^n c_i^n U_t^{n, i} \end{equation} \begin{equation} d U_t^{n, i} = (- x_i^n U_t^{n, i} - \lambda V_t^n) d t + v \sqrt{V_t^n} d W_t \end{equation} with \begin{equation} S_0^n > 0 \end{equation} \begin{equation} U_0^{n, j} = 0 \forall 1 \ldots n \end{equation} and \begin{equation} B = \rho W + \sqrt{1 - \rho^2} W^{\perp} \end{equation} such that \begin{equation} (W, W^{\perp}) \end{equation} is a standard 2-dimensional Wiener process on a fixed filtered probability space with correlation $\rho \in [- 1, + 1]$ and parameters $g_0^n, \lambda, \nu \in \mathbb{R}_+, c_i^n, x_i^n \geqslant 0$. The weights $x_i^n$ and $c_i^n$ are functions of $\alpha = H + \frac{1}{2} \forall i \in 1 \ldots n$ given by \begin{equation} x_i^n = \left( \frac{1 - \alpha}{2 - \alpha} \right) \left( \frac{r_n^{2 - \alpha} - 1}{r_n^{1 - \alpha} - 1} \right) r_n^{i - 1 - \frac{n}{2}} \end{equation} \begin{equation} c_i^n = \frac{(r_n^{1 - \alpha} - 1) r_n^{(\alpha - 1) \left( 1 + \frac{n}{2} \right)}}{\Gamma (\alpha) \Gamma (2 - \alpha)} \end{equation} where \begin{equation} r_n = 1 + \frac{10}{n^{0.9}} \forall n \geqslant 1 \end{equation}
## Answer by JohnDoe (score 4)
https://quant.stackexchange.com/a/57277
Isn't this model just a bunch of classical Heston volatility processes, driven by the same Brownian motion? In this case, you can use some common schemes like Milstein. At least as a starter to toy with the model. If speed/accuracy is an issue, there probably exist some clever solutions as well.
## Answer by rvignolo (score 1)
https://quant.stackexchange.com/a/59144
> ... or better yet, share a repo ...
Yes, of course. This library is probably the best library in the world for SDEs.
In less than 10 lines you can simulate a Heston model using various simulation schemes.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.