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Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks

Ivan Guo, Nicolas Langrené, Jiahao Wu

TL;DR

This paper tackles American option pricing by jointly solving the primal stopping problem and its dual via neural networks, eliminating nested simulations and enabling high-dimensional Bermudan pricing. It introduces two approaches: Method I, which uses a suite of networks to simultaneously approximate continuation values and martingale increments at each exercise date, and Method II, which uses a single global network with time as a state variable and alternates training with stopping updates. The methods yield tight lower and upper bounds and provide directly usable hedging strategies through the dual martingale, improving variance reduction and offering risk-management tools beyond pricing. Numerical results across Black–Scholes and Heston-type settings, including high-dimensional max-call and geometric-put options, demonstrate accurate bounds, efficient computation, and robust hedging, underscoring the practical value for pricing in complex, multi-factor markets.

Abstract

In this paper, we introduce two novel methods to solve the American-style option pricing problem and its dual form at the same time using neural networks. Without applying nested Monte Carlo, the first method uses a series of neural networks to simultaneously compute both the lower and upper bounds of the option price, and the second one accomplishes the same goal with one global network. The avoidance of extra simulations and the use of neural networks significantly reduce the computational complexity and allow us to price Bermudan options with frequent exercise opportunities in high dimensions, as illustrated by the provided numerical experiments. As a by-product, these methods also derive a hedging strategy for the option, which can also be used as a control variate for variance reduction.

Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks

TL;DR

This paper tackles American option pricing by jointly solving the primal stopping problem and its dual via neural networks, eliminating nested simulations and enabling high-dimensional Bermudan pricing. It introduces two approaches: Method I, which uses a suite of networks to simultaneously approximate continuation values and martingale increments at each exercise date, and Method II, which uses a single global network with time as a state variable and alternates training with stopping updates. The methods yield tight lower and upper bounds and provide directly usable hedging strategies through the dual martingale, improving variance reduction and offering risk-management tools beyond pricing. Numerical results across Black–Scholes and Heston-type settings, including high-dimensional max-call and geometric-put options, demonstrate accurate bounds, efficient computation, and robust hedging, underscoring the practical value for pricing in complex, multi-factor markets.

Abstract

In this paper, we introduce two novel methods to solve the American-style option pricing problem and its dual form at the same time using neural networks. Without applying nested Monte Carlo, the first method uses a series of neural networks to simultaneously compute both the lower and upper bounds of the option price, and the second one accomplishes the same goal with one global network. The avoidance of extra simulations and the use of neural networks significantly reduce the computational complexity and allow us to price Bermudan options with frequent exercise opportunities in high dimensions, as illustrated by the provided numerical experiments. As a by-product, these methods also derive a hedging strategy for the option, which can also be used as a control variate for variance reduction.
Paper Structure (25 sections, 1 theorem, 41 equations, 8 figures, 11 tables, 2 algorithms)

This paper contains 25 sections, 1 theorem, 41 equations, 8 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The lower bound of the option price
  4. The upper bound of the option price
  5. The hedging strategy
  6. Valuing an American option numerically
  7. Method I: multiple neural networks
  8. The regression rule
  9. The stopping strategy
  10. The update rule
  11. The whole process
  12. Discussion on the convergence
  13. Method II: one global neural network
  14. The whole process
  15. Discussion on the convergence
  16. ...and 10 more sections

Key Result

Proposition A.1

Given the option has not been exercised at $t\in[0, T)$. Let $\tau^{*} \in \mathcal{T}_{t}$ be the optimal stopping time. The martingale increment $\int_{t}^{\tau^{*}} H(S_{u}) \ \mathrm{d}W_{u}$ can be used as control variate to reduce variance.

Figures (8)

  • Figure 1: Estimates of continuation functions, martingale increment functions and hedging ratio of a 1D American-style put option with $50$ exercise dates (same parameters as the one in Section \ref{['sec: numerical_results']}). Each line represents a function at a step. Left: the continuation function; Middle: the martingale increment function; Right: the hedging ratio. The colorbar represents the step: 0 is the initial time and 50 is the maturity.
  • Figure 2: Price bounds (Left) and corresponding running times (Right) of a 5D max-call Bermudan option with different numbers of substeps using both method I and II.
  • Figure 3: Changes in the estimated bounds and training time with different numbers of timesteps used in training. The option priced is a 1D American-style put option.
  • Figure 4: Left: changes in results from different numbers of batches used among updates of the stopping strategy when we generate fresh data for training. Right: changes of the results using different methods: blue line: original method II; red line: method II with variation 5 and 25 batches were used among updates; brown line: method I with variation 1. The top and bottom two plots correspond to the 1D put option and the 5D max-call option, respectively.
  • Figure 5: Hedging errors for the 1D American-style put option along 100,000 paths by directly using the model we trained via method I.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Remark 3.1
  • Remark 3.2
  • Proposition A.1
  • proof