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Reduced-Order Modeling for Heston Stochastic Volatility Model

Sinem Kozpınar, Murat Uzunca, Bülent Karasözen

TL;DR

This work targets efficient option pricing under the Heston stochastic volatility model by comparing two reduced-order modeling approaches built on snapshots of a full-order model. The full-order model uses discontinuous Galerkin discretization in space and backward Euler in time to solve a 2D diffusion-convection-reaction PDE derived from the Heston framework, including a log-transform to $(v,x)$ coordinates and domain localization. POD-Galerkin provides an intrusive, energy-ordered reduced basis, while DMD delivers a non-intrusive, equation-free representation via Koopman modes and eigenvalues; the study assesses accuracy and speed-ups on European, butterfly, and digital options. Across the tested cases, POD generally yields higher accuracy for a given number of modes, but DMD offers much larger computational speed-ups, highlighting a trade-off between accuracy and efficiency when selecting MOR techniques for financial pricing tasks.

Abstract

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

Reduced-Order Modeling for Heston Stochastic Volatility Model

TL;DR

This work targets efficient option pricing under the Heston stochastic volatility model by comparing two reduced-order modeling approaches built on snapshots of a full-order model. The full-order model uses discontinuous Galerkin discretization in space and backward Euler in time to solve a 2D diffusion-convection-reaction PDE derived from the Heston framework, including a log-transform to coordinates and domain localization. POD-Galerkin provides an intrusive, energy-ordered reduced basis, while DMD delivers a non-intrusive, equation-free representation via Koopman modes and eigenvalues; the study assesses accuracy and speed-ups on European, butterfly, and digital options. Across the tested cases, POD generally yields higher accuracy for a given number of modes, but DMD offers much larger computational speed-ups, highlighting a trade-off between accuracy and efficiency when selecting MOR techniques for financial pricing tasks.

Abstract

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

Paper Structure

This paper contains 13 sections, 47 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Full order model
  3. Heston model as diffusion-convection-reaction equation
  4. Variational form of Heston's model
  5. Discontinuous Galerkin discretization in space
  6. Reduced-order modeling
  7. Proper orthogonal decomposition
  8. Dynamic mode decomposition
  9. Numerical results
  10. European call option
  11. Butterfly spread
  12. Digital option
  13. Conclusions

Figures (7)

  • Figure 1: Relative price error for $v_0=0.1683$ and $S_0=1$ (left), relative Frobenious error (right)
  • Figure 2: ROM-FOM errors: 8 POD modes (left), 12 DMD (Chen) modes (middle), 18 DMD (Tu) modes function (right).
  • Figure 3: Speed-up factors
  • Figure 4: Relative price error for $v_0=0.1683$ and $S_0=1$ (left), relative Frobenious error (right).
  • Figure 5: FOM-ROM errors: 14 POD modes (left), 23 DMD (Chen) modes (middle), 8 DMD (Tu) modes (right).
  • ...and 2 more figures

Theorems & Definitions (1)

  • Remark 1