Paper: arXiv 2309.14784

Authors: Francesca Biagini, Lukas Gonon, Niklas Walter

Abstract

We derive quantitative error bounds for deep neural networks (DNNs) approximating option prices on a $d$-dimensional risky asset as functions of the underlying model parameters, payoff parameters and initial conditions. We cover a general class of stochastic volatility models of Markovian nature as well as the rough Bergomi model. In particular, under suitable assumptions we show that option prices can be learned by DNNs up to an arbitrary small error $\varepsilon \in (0,1/2)$ while the network size grows only sub-polynomially in the asset vector dimension $d$ and the reciprocal $\varepsilon^{-1}$ of the accuracy. Hence, the approximation does not suffer from the curse of dimensionality. As quantitative approximation results for DNNs applicable in our setting are formulated for functions on compact domains, we first consider the case of the asset price restricted to a compact set, then we extend these results to the general case by using convergence arguments for the option prices.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous theoretical proofs proving that deep neural networks can overcome the curse of dimensionality when approximating option prices under rough stochastic volatility models. However, it is purely mathematical and theoretical, lacking empirical validation, real market data backtests, or calibration implementations.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive DNN Approximation Bounds for Rough/SV Option Pricing without Curse of Dimensionality] --> B[Inputs: d-dim Asset Vector, Model/Payoff Parameters, Accuracy ε]
    B --> C[Step 1: Restrict Asset Prices to a Compact Domain for DNN Bounds]
    C --> D[Step 2: Extend Bounds to Unbounded Domain via Price Convergence Arguments]
    D --> E[Compute Network Size Growth vs Dimension d and Accuracy ε]
    E --> F[Outcomes: Network Size Grows Sub-Polynomially in d and 1/ε for Markovian & Rough Bergomi Models]