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Deep Neural Operator Learning for Probabilistic Models

Erhan Bayraktar, Qi Feng, Zecheng Zhang, Zhaoyu Zhang

TL;DR

The work develops a rigorous deep neural-operator framework to approximate pricing operators that map input payoff functions to pricing functions in broad probabilistic models, including forward–backward SDEs and PDEs with free boundaries. It proves universal approximation with explicit network-size bounds under global Lipschitz and integrability/tail assumptions, and extends the theory to both European and American option pricing, establishing Lipschitz properties for the pricing operators. By combining branch and trunk networks (and a neural product extension), the approach provides provable approximations of operators with explicit complexity, enabling retraining-free generation of prices and optimal stopping boundaries for new payoffs. The numerical demonstrations on baskets of American options show that the learned operator yields accurate stopping boundaries for unseen strikes, highlighting potential for fast, scalable valuation and boundary estimation in stochastic control and financial engineering.

Abstract

We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining.

Deep Neural Operator Learning for Probabilistic Models

TL;DR

The work develops a rigorous deep neural-operator framework to approximate pricing operators that map input payoff functions to pricing functions in broad probabilistic models, including forward–backward SDEs and PDEs with free boundaries. It proves universal approximation with explicit network-size bounds under global Lipschitz and integrability/tail assumptions, and extends the theory to both European and American option pricing, establishing Lipschitz properties for the pricing operators. By combining branch and trunk networks (and a neural product extension), the approach provides provable approximations of operators with explicit complexity, enabling retraining-free generation of prices and optimal stopping boundaries for new payoffs. The numerical demonstrations on baskets of American options show that the learned operator yields accurate stopping boundaries for unseen strikes, highlighting potential for fast, scalable valuation and boundary estimation in stochastic control and financial engineering.

Abstract

We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining.

Paper Structure

This paper contains 12 sections, 11 theorems, 141 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Assumptions
  3. Deep Operator Learning Architecture
  4. Neural scaling of operator learning
  5. Function Approximation
  6. Functional Approximations
  7. Operator Approximation
  8. European Option Pricing Operator
  9. European option pricing
  10. Deep neural operator for American option pricing and PDE with free boundary
  11. Operator approximation for American option pricing operator
  12. Algorithm

Key Result

Lemma 2.7

Let $\{\Omega_k\}_{k=1}^M$ be an open cover of a compact smooth manifold $\mathcal{M}$ . There exists a $C^{\infty}$ partition of unity $\{\omega_k\}_{k=1}^M$ that subordinates to $\{\Omega_k\}_{k=1}^M$ such that $support(\omega_k)\subset \Omega_k$ for any $k$.

Figures (2)

  • Figure 1: Exercise boundaries for American put options.
  • Figure :

Theorems & Definitions (27)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Input space $\mathcal{G}$
  • Definition 2.4: Output space $\mathcal{U}$
  • Remark 2.5
  • Definition 2.6: Lipschitz functional
  • Lemma 2.7: Theorem 13.7(ii) of tu2011manifolds
  • Lemma 3.1
  • Theorem 4.1
  • proof
  • ...and 17 more