Single- and Multilevel Quadrature with Error Control for Fourier Pricing under the Rough Heston Model

Abstract

Unlike the classical Heston model, Fourier pricing under the rough Heston model requires solving a fractional Riccati equation at every quadrature point. Since the required resolution varies with model parameters and quadrature point, a single uniform time discretization can be inefficient. We develop single- and multilevel Gauss-Laguerre quadrature methods that balance the time discretization and Fourier quadrature errors. Both methods scale the laguerre weight to the estimated Fourier integrand decay. The single-level method allocates a prescribed tolerance between the two errors. The multilevel method splits the integrand into a level-zero term and level differences, selecting quadrature points separately at each level. Suppose that the Fourier integrand discretization error is O(Δt^p), that evaluating the characteristic function once costs O(Δt^−β), and that the algebraic Gauss-Laguerre quadrature error is O(N^−s_SL/2), where s_SL is the smoothness index. Under this estimate and assumptions on the regularity and decay of level differences, we prove that the proposed single-level method requires O(ε^{−(β/p+2/s_SL)}) computational work to achieve accuracy ε, whereas the proposed multilevel method requires O(ε^−β/p) computational work. We also study root-exponential Gauss-Laguerre error models for practical multilevel quadrature allocation. Numerical experiments support the observed fractional Riccati and Fourier integrand convergence rates and root-exponential quadrature behavior, and show substantial reductions in quadrature cost from the proposed scaling. The multilevel method provides clear computational savings over the single-level method. We further benchmark the multilevel fractional Riccati method against the BL2 Markovian approximation and report lower total CPU time in the tested configurations.

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BibTeX RIS

BibTeX

@misc{hammouda2026single,
  title = {{Single- and Multilevel Quadrature with Error Control for Fourier Pricing under the Rough Heston Model}},
  author = {Chiheb Ben Hammouda and Abderrahmene Ben Romdhane and Michael Samet and Raul F. Tempone},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00438},
  eprint = {2609.00438},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00438},
}

RIS

TY  - UNPB
TI  - Single- and Multilevel Quadrature with Error Control for Fourier Pricing under the Rough Heston Model
AU  - Chiheb Ben Hammouda
AU  - Abderrahmene Ben Romdhane
AU  - Michael Samet
AU  - Raul F. Tempone
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00438
UR  - https://arxiv.org/abs/2609.00438
AB  - Unlike the classical Heston model, Fourier pricing under the rough Heston model requires solving a fractional Riccati equation at every quadrature point. Since the required resolution varies with model parameters and quadrature point, a single uniform time discretization can be inefficient. We develop single- and multilevel Gauss-Laguerre quadrature methods that balance the time discretization and Fourier quadrature errors. Both methods scale the laguerre weight to the estimated Fourier integrand decay. The single-level method allocates a prescribed tolerance between the two errors. The multilevel method splits the integrand into a level-zero term and level differences, selecting quadrature points separately at each level. Suppose that the Fourier integrand discretization error is O(Δt^p), that evaluating the characteristic function once costs O(Δt^−β), and that the algebraic Gauss-Laguerre quadrature error is O(N^−s_SL/2), where s_SL is the smoothness index. Under this estimate and assumptions on the regularity and decay of level differences, we prove that the proposed single-level method requires O(ε^−(β/p+2/s_SL)) computational work to achieve accuracy ε, whereas the proposed multilevel method requires O(ε^−β/p) computational work. We also study root-exponential Gauss-Laguerre error models for practical multilevel quadrature allocation. Numerical experiments support the observed fractional Riccati and Fourier integrand convergence rates and root-exponential quadrature behavior, and show substantial reductions in quadrature cost from the proposed scaling. The multilevel method provides clear computational savings over the single-level method. We further benchmark the multilevel fractional Riccati method against the BL2 Markovian approximation and report lower total CPU time in the tested configurations.
ER  -

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