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Approximation and regularity results for the Heston model and related processes

Edoardo Lombardo

TL;DR

This work develops high-order weak approximation methods for Heston-type models by leveraging random-grid boosting of simple, stable schemes. It first provides high-order weak approximations for the CIR volatility process under a relaxed condition and then extends the approach to the log-Heston semigroup, using splitting methods and random grids to achieve arbitrary even orders of convergence. The PDE analysis for the log-Heston model, including classical and viscosity solutions without Feller-type restrictions, ties the probabilistic schemes to robust numerical solvers. Collectively, the results yield faster, more accurate option-pricing tools (European, Asian) and offer promising avenues for multifactor and rough-Heston extensions, with numerical experiments illustrating substantial computational gains and applicability beyond conservative parameter regimes.

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.

Approximation and regularity results for the Heston model and related processes

TL;DR

This work develops high-order weak approximation methods for Heston-type models by leveraging random-grid boosting of simple, stable schemes. It first provides high-order weak approximations for the CIR volatility process under a relaxed condition and then extends the approach to the log-Heston semigroup, using splitting methods and random grids to achieve arbitrary even orders of convergence. The PDE analysis for the log-Heston model, including classical and viscosity solutions without Feller-type restrictions, ties the probabilistic schemes to robust numerical solvers. Collectively, the results yield faster, more accurate option-pricing tools (European, Asian) and offer promising avenues for multifactor and rough-Heston extensions, with numerical experiments illustrating substantial computational gains and applicability beyond conservative parameter regimes.

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.
Paper Structure (74 sections, 85 theorems, 561 equations, 14 figures, 8 tables)

This paper contains 74 sections, 85 theorems, 561 equations, 14 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Weak approximation of solutions of SDEs
  3. Weak approximations schemes
  4. The Euler-Maruyama scheme
  5. Weak error analysis and splitting operator techniques
  6. The Ninomiya-Victoir scheme
  7. Boosting techniques
  8. Richardson-Romberg extrapolation
  9. Random grids techniques
  10. Thesis contribution
  11. Resume of Chapter \ref{['Chapter_CIR']}
  12. On the adapted version of \ref{['H2_bar_intro']}
  13. On the adapted version of \ref{['H1_bar_intro']}
  14. Numerical experiments
  15. Resume of Chapter \ref{['Chapter_Heston']}
  16. ...and 59 more sections

Key Result

Theorem 1.1.1

Let $b,\sigma$ be four times continuously differentiable on ${\mathbb R}^d$ with bounded partial derivatives. Assume $f:{\mathbb R}^d\rightarrow {\mathbb R}$ is four times differentiable such that $f$ and its derivatives have polynomial growth. Then there exists $C>0$ such that for every $x\in{\math

Figures (14)

  • Figure 1: Parameters: $x=0.0$, $a=0.2$, $k=0.5$, $\sigma=0.65$, $f(z)=\exp(-10 z)$ and $T=1$ ($\frac{\sigma^2}{2a}\approx 1.06$). Graphic ( a) shows the values of $\hat{\mathcal{P}}^{1,n}f$, $\hat{\mathcal{P}}^{2,n}f$, $\hat{\mathcal{P}}^{3,n}f$ as a function of the time step $1/n$ and the exact value. Graphic ( b) draws $\log(|\hat{P}^{i,n}f-P_Tf|)$ in function of $\log(1/n)$: the regressed slopes are 1.86, 3.93 and 5.87 for the second, fourth and sixth order respectively.
  • Figure 2: Parameters: $x=0.3$, $a=0.4$, $k=1$, $\sigma=0.4$, $f(z)=\exp(-8 z)$ and $T=1$ ($\frac{\sigma^2}{2a}= 0.2$). Graphic ( a) shows the values of $\hat{\mathcal{P}}^{1,n}f$, $\hat{\mathcal{P}}^{2,n}f$, $\hat{\mathcal{P}}^{3,n}f$ as a function of the time step $1/n$ and the exact value. Graphic ( b) draws $\log(|\hat{P}^{i,n}f-P_Tf|)$ in function of $\log(1/n)$: the regressed slopes are 1.90, 3.93 and 5.77 for the second, fourth and sixth order respectively.
  • Figure 3: Parameters: $x=10$, $a=10$, $k=1$, $\sigma=0.23$, $f(z)=\exp(- z)$ and $T=1$ ($\frac{\sigma^2}{2a}\approx 0.0026$). Graphic ( a) shows the values of $\hat{\mathcal{P}}^{1,n}f$, $\hat{\mathcal{P}}^{2,n}f$, $\hat{\mathcal{P}}^{3,n}f$ as a function of the time step $1/n$ and the exact value. Graphic ( b) draws $\log(|\hat{P}^{i,n}f-P_Tf|)$ in function of $\log(1/n)$: the regressed slopes are 1.96, 4.00 and 6.02 for the second, fourth and sixth order respectively.
  • Figure 4: Test function: $f(x,s)=(K-s)^+$. Parameters: $S_0=100$, $r=0$, $x=0.25$, $a=0.25$, $k=1$, $\sigma=0.65$, $\rho=-0.3$, $T=1$, $K=100$ ($\frac{\sigma^2}{2a}= 0.845$). Graphic ( a) shows the values of $\hat{\mathcal{P}}^{1,n}f$, $\hat{\mathcal{P}}^{2,n}f$, $\hat{\mathcal{P}}^{3,n}f$ as a function of the time step $1/n$ and the exact value. Graphic ( b) draws $\log(|\hat{P}^{i,n}f-P_Tf|)$ in function of $\log(1/n)$: the regressed slopes are 1.34, 4.00 and 6.02 for the second, fourth and sixth order respectively.
  • Figure 5: $L^2$-square error in function of the execution time in seconds. Test function: $f(x,s)=(K-s)^+$. Parameters in graphic ( a) : $S_0=100$, $r=0$, $x=0.4$, $a=0.4$, $k=1$, $\sigma=0.1$, $\rho=-0.3$, $T=1$, $K=100$. Parameters in graphic ( b) : $S_0=100$, $r=0$, $x=0.1$, $a=0.1$, $k=1$, $\sigma=0.63$, $\rho=-0.3$, $T=1$, $K=100$.
  • ...and 9 more figures

Theorems & Definitions (158)

  • Theorem 1.1.1
  • Definition 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4
  • Proposition 1.1.5
  • Remark 1.1.6
  • Proposition 1.1.7
  • Corollary 1.1.8
  • Theorem 1.1.9
  • Example 1.1.10
  • ...and 148 more