A rigorous quantitative finance and deep learning research benchmark exploring how Physics-Informed Neural Networks (PINNs) break the Curse of Dimensionality ($O(N^d)$ memory/compute bottleneck) and enforce financial no-arbitrage bounds during Out-of-Distribution (OOD) tail-risk extrapolation.
The Mathematical Problem: Multi-Asset Option Pricing
In multi-asset derivative markets, the pricing manifold of a European basket option $V(\mathbf{S}, t)$ over $d$ underlying assets $\mathbf{S} = (S_1, \dots, S_d)$ is governed by the multi-dimensional Black-Scholes Partial Differential Equation (PDE):
$$ \frac{\partial V}{\partial t} + \frac{1}{2} \sum_{i=1}^d \sum_{j=1}^d \rho_{ij} \sigma_i \sigma_j S_i S_j \frac{\partial^2 V}{\partial S_i \partial S_j} + r \sum_{i=1}^d S_i \frac{\partial V}{\partial S_i} - r V = 0 $$subject to terminal payoff condition:
$$ V(\mathbf{S}, T) = \max\left( \sum_{i=1}^d w_i S_i - K, \; 0 \right) $$The Dual Computational Challenge
- Curse of Dimensionality ($O(N^d)$): Classical mesh solvers (Finite Difference Methods) explode exponentially beyond $d \ge 3$. High-accuracy Monte Carlo ($60,000$ paths) handles high dimensions but requires $\sim 21.5\text{ ms}$ per contract, creating an overnight batch bottleneck for large portfolios.
- AI Failure in Tail Risk: Purely data-driven neural networks (
VanillaNN) only memorize near-the-money (NTM) quotes. During market crashes, asset prices diverge into deep out-of-the-money regions where pure networks hallucinate wildly ($>15\%$ relative error) and violate no-arbitrage laws.
Architectural Comparison & Benchmarks
We benchmark three deep learning architectures against high-precision Monte Carlo ground truth:
graph TD
A[Asset Price Domain S, t] --> B[Neural Network Surrogate]
B --> C[Loss Computation]
C --> D[Data Loss: Market Quotes]
C --> E[PDE Loss: Black-Scholes Residual via AutoDiff]
C --> F[IC Loss: Exact Hard-Constrained Payoff ICPINN]
| Model Architecture | Parameterization | Inference Latency | NTM Error (Relative) | OOD Tail-Risk Error |
|---|---|---|---|---|
| High-Precision Monte Carlo | 60,000 paths (Exact Baseline) | $\sim 21.5\text{ ms}$ | $0.00\%$ | $0.00\%$ |
| Vanilla Deep Neural Network | Pure data-driven MLP | $0.006\text{ ms}$ | $1.12\%$ | $16.48\%$ (Arbitrage Violation) |
| Standard PINN | Soft PDE Residual Regularization | $0.006\text{ ms}$ | $0.48\%$ | $1.85\%$ |
| Initial Condition PINN (ICPINN) | Hard-Constrained Payoff Boundary | $0.006\text{ ms}$ | $0.21\%$ | $0.74\%$ |
Key Highlights
- $\sim 3,500\times$ Speedup: Compresses pricing latency from $21.5\text{ ms}$ to $0.006\text{ ms}$ per contract, enabling real-time risk management and instant Greeks computation ($\Delta, \Gamma$) via PyTorch automatic differentiation.
- Physics Regularization as Noise Filter: Unlabelled collocation points evaluate the PDE residual across the asset price continuum without needing ground-truth labels, binding network predictions to physical conservation laws.