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High order approximations and simulation schemes for the log-Heston process

Aurélien Alfonsi, Edoardo Lombardo

TL;DR

This work develops high-order weak approximation schemes for the log-Heston model by integrating Alfonsi and Bally's random-grid boosting with two robust base schemes: an exact Strang-splitting scheme and a Ninomiya–Victoir-based scheme. The authors prove that, for any\nu\ge 1, boosted estimators $\hat{\mathcal{P}}^{\nu,n}$ achieve weak convergence at rate $O(n^{-2\nu})$, under suitable regularity and moment-growth conditions, and provide a detailed implementation and numerical validation for European and Asian options. The framework accommodates both standard and rough/multi-factor extensions by using a discretized kernel representation and Strang splitting to maintain positivity and stability. Practical results show substantial variance reductions via clever coupling of coarse/fine grids, and the approach extends to a suite of financial models, offering a versatile path toward high-accuracy simulations in complex volatility dynamics.

Abstract

We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this method with either the Ninomiya-Victoir scheme (2008) or a second-order scheme that samples exactly the volatility component, and we show rigorously that we can achieve then any order of convergence. We give numerical illustrations on financial examples that validate the theoretical order of convergence. We also present promising numerical results for the multifactor/rough Heston model and hint at applications to other models, including the Bates model and the double Heston model.

High order approximations and simulation schemes for the log-Heston process

TL;DR

This work develops high-order weak approximation schemes for the log-Heston model by integrating Alfonsi and Bally's random-grid boosting with two robust base schemes: an exact Strang-splitting scheme and a Ninomiya–Victoir-based scheme. The authors prove that, for any\nu\ge 1, boosted estimators achieve weak convergence at rate , under suitable regularity and moment-growth conditions, and provide a detailed implementation and numerical validation for European and Asian options. The framework accommodates both standard and rough/multi-factor extensions by using a discretized kernel representation and Strang splitting to maintain positivity and stability. Practical results show substantial variance reductions via clever coupling of coarse/fine grids, and the approach extends to a suite of financial models, offering a versatile path toward high-accuracy simulations in complex volatility dynamics.

Abstract

We present weak approximations schemes of any order for the Heston model that are obtained by using the method developed by Alfonsi and Bally (2021). This method consists in combining approximation schemes calculated on different random grids to increase the order of convergence. We apply this method with either the Ninomiya-Victoir scheme (2008) or a second-order scheme that samples exactly the volatility component, and we show rigorously that we can achieve then any order of convergence. We give numerical illustrations on financial examples that validate the theoretical order of convergence. We also present promising numerical results for the multifactor/rough Heston model and hint at applications to other models, including the Bates model and the double Heston model.
Paper Structure (15 sections, 8 theorems, 105 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 8 theorems, 105 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results
  3. Second order schemes for the log-Heston process
  4. From the second order scheme to higher orders by using random grids
  5. A second-order scheme for the log-Heston model
  6. Proof of the main result
  7. Preliminary results on the norm
  8. The Cauchy problem of the Log-Heston SDE
  9. Proof of \ref{['H1_bar']} and \ref{['H2_bar']}
  10. Numerical results
  11. Implementation
  12. Pricing of European and Asian options
  13. Estimators variance and schemes coupling
  14. Towards higher order approximations of Rough Heston process
  15. Further applications to financial models

Key Result

Theorem 2.1

Let $\hat{P}_t$ be either $\hat{P}_t^{Ex}$ defined by def_PEx or $\hat{P}_t^{NV}$ by def_PNV. Let $T>0$, $n\in {\mathbb N}^*$ and $h_l=T/n^l$. Let $\hat{\mathcal{P}}^{1,n}=\hat{P}_{h_1}^{[n]}$, $\hat{\mathcal{P}}^{2,n}$ be defined by def_boost2 and $\hat{\mathcal{P}}^{\nu,n}$ the further approximati

Figures (6)

  • Figure 1: Test function: $f(x,y)=(K-e^x)^+$. Parameters: $S_0=e^{x}=100$, $r=0$, $y=0.2$, $a=0.2$, $b=1$, $\sigma=0.5$, $\rho=-0.7$, $T=1$, $K=105$. Statistical precision $\varepsilon=5$e-4 (too small to be visible on the plots). Graphic ( a) shows the Monte Carlo estimated values of $\hat{\mathcal{P}}^{NV,1,n}f$, $\hat{\mathcal{P}}^{NV,2,n}f$ as a function of the time step $1/n$ and the exact value. Graphic ( b) draws $|\hat{\mathcal{P}}^{NV,\nu,n}f-P_Tf|$ in function of $1/n$ (in log-log scale): the regressed slopes are 1.89 and 4.27 for the second and fourth order respectively.
  • Figure 2: Test function: $f(x,y)=(K-e^x)^+$. Parameters: $S_0=e^x=100$, $r=0$, $y=0.1$, $a=0.1$, $b=1$, $\sigma=1.0$, $\rho=-0.9$, $T=1$, $K=105$. Statistical precision $\varepsilon=5$e-4 (too small to be visible on the plots). Graphic ( a) shows the Monte Carlo estimated values of $\hat{\mathcal{P}}^{Ex,1,n}f$, $\hat{\mathcal{P}}^{Ex,2,n}f$ as a function of the time step $1/n$ and the exact value. Graphic ( b) draws $|\hat{P}^{Ex,\nu,n}f-P_Tf|$ in function of $1/n$ (in log-log scale): the regressed slopes are 1.89 and 4.26 for the second and fourth order respectively.
  • Figure 3: Test function: $f(x,y,i)=(K-i/T)^+$. Parameters: $e^{x}=100$, $r=0$, $y=0.2$, $a=0.2$, $b=2$, $\sigma=0.5$, $\rho=-0.7$, $T=1$, $K=100$. Statistical precision $\varepsilon=5$e-4. Graphic ( a) shows the Monte Carlo estimated values of $\hat{\mathcal{P}}^{NV,1,n}f$, $\hat{\mathcal{P}}^{NV,2,n}f$ as a function of the time step $1/n$. Graphic ( b) draws $|\hat{\mathcal{P}}^{NV,\nu,2n}f-\hat{\mathcal{P}}^{NV,\nu,n}f|$ in function of $1/n$ (in log-log scale): the regressed slopes are 1.85 and 4.30 for the second and fourth order respectively.
  • Figure 4: Test function: $f(x,y,i)=(K-i/T)^+$. Parameters: $e^x=100$, $r=0$, $y=0.1$, $a=0.1$, $b=1$, $\sigma=1.0$, $\rho=-0.9$, $T=1$, $K=100$. Statistical precision $\varepsilon=5$e-4. Graphic ( a) shows the Monte Carlo estimated values of $\hat{\mathcal{P}}^{Ex,1,n}f$, $\hat{\mathcal{P}}^{Ex,2,n}f$ as a function of the time step $1/n$. Graphic ( b) draws $|\hat{\mathcal{P}}^{Ex,\nu,2n}f-\hat{\mathcal{P}}^{Ex,\nu,n}f|$ in function of $1/n$ (in log-log scale): the regressed slopes are 1.72 and 3.98 for the second and fourth order respectively.
  • Figure 5: Test function: $f(x,y)=(K-e^x)^+$. Statistical precision $\varepsilon=1$e-3. Graphic ( a) shows the Monte Carlo estimated values of $\hat{\mathcal{P}}^{NV,1,n^2}f$, $\hat{\mathcal{P}}^{NV,2,n}f$ as a function of the execution time. Parameters: $S_0=e^{x}=100$, $r=0$, $y=0.2$, $a=0.2$, $b=1$, $\sigma=0.5$, $\rho=-0.7$, $T=1$, $K=105$. Graphic ( b) shows the Monte Carlo estimated values of $\hat{\mathcal{P}}^{Ex,1,n^2}f$, $\hat{\mathcal{P}}^{Ex,2,n}f$ as a function of the execution time. Parameters: $S_0=e^{x}=100$, $r=0$, $y=0.1$, $a=0.1$, $b=1$, $\sigma=1.0$, $\rho=-0.9$, $T=1$, $K=105$.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 7 more