Lifted Heston Model

• The Lifted Heston Model is an advanced stochastic volatility framework that approximates rough volatility using multiple square-root processes in a Markovian structure.
• Its novel integrated variance implicit (IVI) scheme with Constrained Linear Projection efficiently simulates complex dynamics, ensuring stability with larger time steps.
• The model is pivotal in pricing volatility derivatives and exotic options, enabling robust risk management and precise model calibration in quantitative finance.

The Lifted Heston Model is an advanced stochastic volatility model derived as a Markovian approximation of the rough Heston model, designed to better capture the path dependency of volatility using a set of state processes. This model is particularly useful in the financial industry for pricing exotic options and volatility derivatives.

Model Description

The Lifted Heston Model extends the classical Heston model by incorporating multiple square-root processes that drive volatility with shared stochastic factors. This transforms what would typically be a non-Markovian system into a Markovian framework by modeling the instantaneous variance as a sum of state processes:

$V_t = g_0(t) + \sum_{n=1}^{N} \omega_n U_t^n,$

where $g_0(t)$ is deterministic, $U_t^n$ are square-root processes, and the asset price $S_t$ and variance $V_t$ follow:

• $$d(\ln S_t) = (r - \frac{1}{2}V_t)dt + \sqrt{V_t} dW^1_t$$,
• $$dU_t^n = -x_n U_t^n dt - \lambda V_t dt + \nu \sqrt{V_t} dW^2_t$$.

The correlation between the Brownian motions $W^1_t$ and $W^2_t$ is $\rho$. The parameters $\omega_n$ and $x_n$ are chosen to create a Markovian approximation that retains the rough characteristics of the original model as $N \to \infty$.

Simulation Challenges

Simulating the Lifted Heston Model is computationally intensive due to the need to handle the multidimensional system of state processes with fine discretization steps. Traditional Euler–Maruyama schemes struggle with instability and computational cost, making it difficult to achieve accurate results with large time steps in complex, high-dimensional systems.

Novel Simulation Scheme

The paper introduces an "integrated variance implicit" (IVI) scheme using Constrained Linear Projection (C-LP) to efficiently simulate the model:

• Instead of directly simulating $V_t$, the scheme focuses on the integrated variance $X_{s,t} = \int_s^t V_u du$ and its quadratic variation $Z_{s,t} = \int_s^t \sqrt{V_u} dW^2_u$.
• By establishing the linear relation $X_{s,t} = \alpha + \beta Z_{s,t}$ involving the expected conditional moments, the method uses inverse Gaussian distributions known to model first passage times of Brownian motion for sampling.

The constrained projection ensures the positivity of $V_t$, optimizing the simulation in the $L^2$ sense, allowing accurate result with larger time steps.

Efficiency and Accuracy

The C-LP scheme shows near-exact accuracy for the integrated variance. Unlike conventional methods, it handles significant time step sizes while maintaining stability and robustness. Numerical experiments indicated that the method converges efficiently, making it suitable for pricing volatility derivatives like VIX options.

Applications and Implications

The Lifted Heston Model's efficient simulation scheme finds practical applications in the pricing of volatility products and exotic derivatives. It enables robust and efficient exploration of parameter spaces in risk management and model calibration, making it a valuable tool in quantitative finance for those dealing with complex volatility models.

The novel approach outperforms traditional simulation and achieves computational efficiency by simulating the integrated variance directly using the designed distribution, offering a practical and reliable solution for real-world financial modeling challenges.