Add Black-Scholes example with documentation

[litchki]

Description

This PR adds a new example demonstrating Physics-Informed Neural Networks (PINNs) for pricing European call options under the Black-Scholes model. This complements the existing physics/engineering examples with a fundamental application in computational finance.

Problem Overview

The example solves the Black-Scholes PDE in time-to-maturity form ($\tau = T - t$):

$$ \frac{\partial V}{\partial \tau} = \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S} - r V $$

with appropriate boundary and initial conditions for a European call option.

Changes

1. Source Code (examples/pinn_forward/black_scholes_call.py):

   • Implements the Black-Scholes PDE.
   • Uses Sobol sequence sampling with powers of 2 ($N=2048$) to improve domain coverage and avoid sampling warnings.
   • Implements a two-stage training strategy (Adam for 20k iters + L-BFGS).
   • Includes validation against the analytical solution.

2. Documentation (docs/demos/pinn_forward/black_scholes.rst):

   • Added a comprehensive documentation page with mathematical formulation.
   • Explains implementation details (Geometry, Network Architecture, Hyperparameters).

3. Index Update (docs/demos/pinn_forward.rst):

   • Updated the table of contents to include the new example.
   • Removed the old raw link to keep the index clean.

Performance & Validation

• L2 relative error: ~8.5e-4 (0.00085)
• Mean PDE residual: ~0.069
• Training time: ~7 minutes (GPU) / ~15 minutes (CPU)

Testing Checklist

• Code runs without errors on PyTorch/TensorFlow backends.
• Output format matches existing examples (e.g., heat.py).
• Validated against analytical solution (scipy.stats.norm).
• Docstrings and comments added (PEP 8 style).
• No Sobol warnings (uses powers of 2: 2048, 64, 128).
• Documentation builds correctly.

Reference

Tanios, R. (2021). "Physics Informed Neural Networks in Computational Finance: High Dimensional Forward & Inverse Option Pricing". ETH Zurich Master's Thesis. https://doi.org/10.3929/ethz-b-000491555

[echen5503]

In this area, it is probably helpful to denote that x[:, 0] is stock prices, x[:, 1] are taus, y is stock prices.

Furthermore, it is likely best to write dV instead of dy.

[litchki]

Thank you for the suggestion! I have updated the notation for clarity and consistency.

To clarify: in this example, y represents the option price V(S, tau),
not the stock price — I believe that was a small typo in the review comment.

The variables are now annotated as follows:

• x[:, 0]: asset price S
• x[:, 1]: time to maturity tau
• y: option price V(S, tau) (network output)

I also renamed dy_* to dV_* in both the Python example and the
corresponding .rst documentation.

[echen5503]

If you are on board with the comment on R67 (same place in docs), change for consistency

[litchki]

Done, updated consistently with the changes in the .rst documentation.

[echen5503]

@litchki still working on this?

[litchki]

@litchki still working on this?
Yes! Sorry for the delay — I've just pushed the updates addressing your earlier review comments. Thank you for the follow-up!

[echen5503]

The user might not know what sobol sequence is and hence have no idea what you're talking about when you say powers of two for sobol sequence warnings. Linking to a source or directly explaining would help.

[litchki]

Thanks for the suggestion — I've updated the Sampling Strategy section to briefly
explain what Sobol sequences are (low-discrepancy quasi-random sequences providing
more uniform domain coverage than pseudorandom sampling), and added references to
both the original Sobol paper and the SciPy documentation for the power-of-two
requirement.

Please let me know if you'd prefer a more explicit explanation for first-time readers.

[echen5503]

Checked documentation on docs build, ran code. All good.