[Paper Review] Dual control Monte Carlo method for tight bounds of value function under Heston stochastic volatility model

Jingtang Ma, Wenyuan Li+1 more|arXiv (Cornell University)|Oct 28, 2017

Stochastic processes and financial applications11 references3 citations

TL;DR

This paper proposes a dual control Monte Carlo method to compute tight lower and upper bounds for the value function in utility maximization under the Heston stochastic volatility model with general utility functions. By leveraging a specific dual control form, γₜ = c(t)√vₜ, the method enables closed-form upper bounds for power, non-HARA, and Yarri utilities, achieving high accuracy and efficiency with small duality gaps even for non-affine utilities.

ABSTRACT

The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution of the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yarri utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme.

Motivation & Objective

• To address the lack of efficient numerical methods for solving the high-dimensional, nonlinear HJB equation in utility maximization under the Heston stochastic volatility model with general utility functions.
• To develop a computationally efficient approach for computing reliable lower and upper bounds on the primal value function when closed-form solutions are unavailable.
• To identify a specific dual control structure that enables analytical tractability of the upper bound for a broad class of utility functions, including non-HARA and Yarri utilities.
• To demonstrate the robustness, accuracy, and efficiency of the proposed method through extensive numerical experiments across multiple utility types and parameter sets.

Proposed method

• The method employs weak duality to derive upper bounds via the dual control problem and lower bounds via feasible primal controls, forming tight bounds on the primal value function.
• A specific dual control form, γₜ = c(t)√vₜ with piecewise constant c(t), is introduced to enable closed-form upper bounds for power, non-HARA, and Yarri utilities.
• The upper bound is computed using Monte Carlo simulation with path number 10,000 and time steps 100, while the lower bound uses 100,000 paths and 100 time steps.
• For Yarri utility, the Fourier-cosine method is applied to accelerate the computation of the upper bound, leveraging the affine structure of the Heston model.
• The method uses a sampling strategy for the dual control parameter c, uniformly distributed in [−0.5, 0.5], to test robustness and optimize performance.
• Numerical validation involves comparing bounds across multiple examples with varying parameters, including wealth, variance, and volatility-of-volatility, to assess accuracy and convergence.

Experimental results

Research questions

• RQ1 Can a dual control Monte Carlo method be designed to efficiently compute tight lower and upper bounds for the value function in the Heston model with general utility functions?
• RQ2 Does the choice of dual control γₜ = c(t)√vₜ lead to analytically tractable upper bounds for non-HARA and Yarri utilities under the Heston model?
• RQ3 How does the gap between the lower and upper bounds scale with the number of sampling paths and time steps, and can it be minimized efficiently?
• RQ4 Is the proposed method robust across diverse parameter sets, including varying interest rates, volatility mean reversion, and correlation parameters?

Key findings

• For Yarri utility, the method achieves a relative difference of only 0.112% between lower and upper bounds when using γₜ = c√vₜ with 20 sampling times, indicating high tightness.
• With 20 sampling times, the mean relative difference between bounds is reduced to 0.434%, and the mean absolute difference is 5.8369×10⁻³, demonstrating consistent accuracy.
• The upper bound computation using the Fourier-cosine method is significantly faster than pure Monte Carlo, with UB computation time dropping to 0.248 seconds for γₜ = c√vₜ.
• The lower bound computation with 100,000 paths takes approximately 2.37×10³ seconds, while the upper bound with 10,000 paths takes only 6.08 seconds, showing computational efficiency.
• The method remains robust across 10 randomly sampled parameter sets, with mean relative difference below 0.5% even when parameters like κ, θ, ρ, and ξ are varied within realistic ranges.
• The 3D visualization of optimal strategies and terminal wealth distribution confirms the method's ability to capture complex dynamic behavior under stochastic volatility.