Approximation and regularity results for the Heston model and related processes

Paper: arXiv 2504.21658

Authors: Edoardo Lombardo

Abstract

This Ph.D. thesis explores approximations and regularity for the Heston stochastic volatility model through three interconnected works. The first work focuses on developing high-order weak approximations for the Cox-Ingersoll-Ross (CIR) process, essential for financial modelling but challenging due to the square root diffusion term preventing standard methods. By employing the random grid technique (Alfonsi & Bally, 2021) built upon Alfonsi’s (2010) second-order scheme, the work proves that weak approximations of any order can be achieved for smooth test functions. This holds under a condition that is less restrictive than the famous Feller’s one. Numerical results confirm convergence for both CIR and Heston models and show significant computational time improvements. The second work extends the random grid technique to the log-Heston process. Two second-order schemes are introduced (one using exact volatility simulation, another using Ninomiya-Victoir splitting under a the same restriction used above). Convergence to any desired order is rigorously proven. Numerical experiments validate the schemes’ effectiveness for pricing European and Asian options and suggest potential applicability to multifactor/rough Heston models. The third work investigates the partial differential equation (PDE) associated with the log-Heston model. It extends classical solution results and establishes the existence and uniqueness of viscosity solutions without relying on the Feller condition. Uniqueness is proven even for certain discontinuous initial data, relevant for pricing instruments like digital options. Furthermore, the convergence of a hybrid numerical scheme to the viscosity solution is shown under relaxed regularity (continuity) for the initial data. An appendix includes supplementary results for the CIR process.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The thesis employs advanced stochastic calculus and PDE theory, including high-order weak approximations and viscosity solutions, indicating high mathematical complexity. While numerical experiments are presented for validation, the work is theoretical in nature, focusing on proofs and convergence analysis without integrating real-world data or providing full backtesting frameworks, resulting in moderate empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Develop efficient approximations &<br>regularity results for Heston/CIR models"] --> B["Work 1: High-Order Weak Approx<br>for CIR Process"]
  A --> C["Work 2: Random Grid Extension<br>to Log-Heston Process"]
  A --> D["Work 3: PDE Analysis &<br>Hybrid Scheme for Log-Heston"]

  B --> E{"Methodology"}
  E --> E1["Random Grid Technique<br>Alfonsi & Bally (2021)"]
  E --> E2["Alfonsi (2010) Second-Order Scheme"]

  C --> F{"Methodology"}
  F --> F1["Exact Volatility Simulation"]
  F --> F2["Ninomiya-Victoir Splitting"]

  D --> G{"Methodology"}
  G --> G1["Viscosity Solution Theory"]
  G --> G2["Hybrid Numerical Scheme"]

  E1 & E2 --> H["Computational Process<br>Monte Carlo Simulations"]
  F1 & F2 --> H
  G2 --> I["Computational Process<br>PDE Discretization"]

  H --> J["Findings: High-Order Convergence<br>Validated for CIR/Heston<br>Feller Condition Relaxed"]
  I --> K["Findings: Existence/Uniqueness<br>Viscosity Sol. for Discontinuous Data<br>Hybrid Scheme Convergence"]

  J & K --> L["Outcomes: Efficient Pricing<br>European/Asian Options<br>Extensions to Multifactor/Rough Heston"]

Cited References (23)

1. Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation (2023) — link
2. Multifactor approximation of rough volatility models (2019)
3. High order approximations of the Cox–Ingersoll–Ross process semigroup using random grids (2023) — link
4. High order approximations of the log-Heston process semigroup (2024) — link
5. Approximation of Stochastic Volterra Equations with kernels of completely monotone type (2024)
6. Weak markovian approximations of rough heston — → 2309.07023
7. Numerical stability of a hybrid method for pricing options (2019) — link
8. A hybrid tree/finite-difference approach for Heston–Hull–White-type models (2017)

1 of 23 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.