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A time-stepping deep gradient flow method for option pricing in (rough) diffusion models

Antonis Papapantoleon, Jasper Rou

TL;DR

The paper addresses high-dimensional European option pricing under Markovian approximations of rough volatility by recasting the pricing PDE as an energy minimization and solving it with a time-stepping neural network framework. The TDGF method discretizes time, derives a variational energy at each step, and trains per-step neural networks to approximate the solution, leveraging warm starts from previous steps and enforcing asymptotic price behavior. Across Black–Scholes, Heston, and lifted Heston models, the approach achieves around $10^{-3}$ accuracy and offers faster training than the DGM approach, with competitive online evaluation and scalability up to moderate numbers of variance factors. The work demonstrates practical viability for high-dimensional option pricing in diffusion and Markovian lift models, while pointing to future work on regimes with many variance factors and nontrivial correlations.

Abstract

We develop a novel deep learning approach for pricing European options in diffusion models, that can efficiently handle high-dimensional problems resulting from Markovian approximations of rough volatility models. The option pricing partial differential equation is reformulated as an energy minimization problem, which is approximated in a time-stepping fashion by deep artificial neural networks. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness, and adheres to a priori known bounds for option prices. The accuracy and efficiency of the proposed method is assessed in a series of numerical examples, with particular focus in the lifted Heston model.

A time-stepping deep gradient flow method for option pricing in (rough) diffusion models

TL;DR

The paper addresses high-dimensional European option pricing under Markovian approximations of rough volatility by recasting the pricing PDE as an energy minimization and solving it with a time-stepping neural network framework. The TDGF method discretizes time, derives a variational energy at each step, and trains per-step neural networks to approximate the solution, leveraging warm starts from previous steps and enforcing asymptotic price behavior. Across Black–Scholes, Heston, and lifted Heston models, the approach achieves around accuracy and offers faster training than the DGM approach, with competitive online evaluation and scalability up to moderate numbers of variance factors. The work demonstrates practical viability for high-dimensional option pricing in diffusion and Markovian lift models, while pointing to future work on regimes with many variance factors and nontrivial correlations.

Abstract

We develop a novel deep learning approach for pricing European options in diffusion models, that can efficiently handle high-dimensional problems resulting from Markovian approximations of rough volatility models. The option pricing partial differential equation is reformulated as an energy minimization problem, which is approximated in a time-stepping fashion by deep artificial neural networks. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness, and adheres to a priori known bounds for option prices. The accuracy and efficiency of the proposed method is assessed in a series of numerical examples, with particular focus in the lifted Heston model.
Paper Structure (15 sections, 1 theorem, 38 equations, 6 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 1 theorem, 38 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Energy minimization and a deep gradient flow method
  4. Time discretization
  5. Variational formulation
  6. Neural network approximation
  7. Examples
  8. Black--Scholes model
  9. Heston model
  10. Lifted Heston model
  11. Implementation details
  12. Numerical results
  13. Accuracy
  14. Training and computational times
  15. Conclusion

Key Result

Lemma 3.2

Assume that $A$ is symmetric and positive semi-definite. Then, $U^k$ solves eq:discretization2 if and only if $U^k$ minimizes

Figures (6)

  • Figure 6.1: Errors of the two methods in the Black--Scholes model against time, with interest rate $r=0.05$ and volatility $\sigma = 0.25$.
  • Figure 6.2: Error of the two methods in the Heston model against time, with $r=0.0$ and $\eta = 0.1$, $\rho=0.0$, $\kappa=0.01$, $V_0 = 0.03$ and $\lambda =2.0$.
  • Figure 6.3: Errors of the two methods in the lifted Heston model with $n=1$ variance process against time, with $r=0.0$, $\eta=0.1$, $\rho=0.0$, $\kappa=0.01$, $V_0=0.01$ and $\lambda=2.0$.
  • Figure 6.4: Errors of the two methods in the lifted Heston model with $n=20$ variance process against time, with $r=0.0$, $\eta=0.1$, $\rho=0.0$, $\kappa=0.01$, $V_0=0.01$ and $\lambda=2.0$.
  • Figure 6.5: Errors of the two methods in the lifted Heston model with $n=1$ variance process against time, with $r=0.0$, $\eta=0.3$, $\rho=-0.7$, $\kappa=0.02$, $V_0=0.02$ and $\lambda=0.3$.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Remark 3.1
  • Lemma 3.2
  • proof