pde-pricer

Physics-informed neural networks for pricing PDEs, with classical reference pricers (closed form, Monte Carlo, finite difference) and a documented negative-result variational-quantum experiment.

Status

The supported slice is a European Black–Scholes PINN: versioned checkpoint, installed CLI (pde-pricer train|price|validate), and a committed artifact under artifacts/bs_european_call/. Basket, Heston, Merton, American, barrier, and Dupire PINNs are research-stage. Their reference engines have independent cross-checks; trained-PINN held-out errors are reported as measured, not as market-grade accuracy.

There is no live market feed, web API, or paid data dependency.

Demo

git clone https://code.rylanmalarchick.com/rylanmalarchick/pde-pricer.git
cd pde-pricer
python -m venv .venv
source .venv/bin/activate
pip install -e ".[classical,dev]" -c constraints.txt

make check

pde-pricer validate --checkpoint artifacts/bs_european_call/checkpoint.pt \
  --S-min 70 --S-max 130 --max-mae 0.25 --max-vega-mae 8
pde-pricer price --checkpoint artifacts/bs_european_call/checkpoint.pt \
  --S 100 --t 0 --sigma 0.2 --greeks

python scripts/summarize_artifacts.py

make check is format, lint, typecheck, the default (fast) pytest suite, and a wheel smoke test that the CUDA kernel source is packaged. validate should print a passing grid report matching artifacts/bs_european_call/validation.json. summarize_artifacts.py reprints the tables below from those JSON files.

Results

Tables are generated by python scripts/summarize_artifacts.py. Do not edit them by hand.

European Black–Scholes PINN (supported slice)

Artifact	Grid	Price MAE	Rel MAE	Vega MAE	Pass
artifacts/bs_european_call/	S∈[70,130] (31 pts)	0.072	1.54%	4.36	yes

Reproduce training: see artifacts/bs_european_call/README.md.

Reference engines

Engine	Check	Max abs error
Heston	quadrature vs COS (4 regimes)	4.98e-06
Merton	adaptive Poisson tail vs 200-term sum	1.66e-11
Basket	pathwise delta vs CRN bump	8.20e-05
Dupire	inverse local-vol MAE (valid cells)	0.0146
Dupire	reprice MAE	7.33e-04

Regenerate: python scripts/validate_advanced_references.py and python scripts/calibrate_local_vol.py.

Research-stage PINNs (held-out, 3 seeds × 3000 epochs)

PINN	Price rel MAE %	Delta MAE	Residual RMS
heston	5.27 ± 1.11	0.0335	143.76
merton	3.96 ± 1.44	0.0207	0.96
american	45.16 ± 0.00	0.1278	n/a
basket	7.85 ± 2.94	0.0059	0.71

Heston price error is single-digit while the PDE residual stays large. American did not leave the payoff-only initialization (identical metrics across seeds; residual omitted). See artifacts/p4_heldout_reports/.

Quantum ablation (parameter-matched basket)

Arm	Params	Mean test MAE
quantum hybrid	219	19.83
classical (matched)	219	13.36
paired t p-value		0.44
quantum advantage		false

Three seeds, equal budget. The hybrid is retained as a negative result, not a headline. python scripts/ablate_quantum_basket.py (needs the quantum extra).

Models in the tree

Model	PDE type	Dimension	Notes
Black-Scholes	PDE	2D	Closed form + FD + supported PINN
Basket	PDE	6D (5 assets + time)	Monte Carlo reference
Merton	PIDE	2D	Poisson jumps, tail-controlled sum
Heston	PDE	3D	Stochastic variance; COS cross-check
American	Free boundary	2D	CRR/PSOR references; PINN research-stage
Barrier	PDE	2D	Down-and-out and up-and-out closed forms

Install extras

classical (Torch + plotting), quantum (PennyLane), notebook, gpu, dev, all. Core install is NumPy/SciPy only; CPU analytical / MC / FD import without Torch.

# Optional: retrain research-stage PINNs (long; GPU recommended)
python scripts/evaluate_heldout.py --seeds 0 1 2 --epochs 3000 --device cuda

# Optional: other training scripts
python scripts/train_basket.py --epochs 5000 --n_interior 15000 --eval
python scripts/train_merton.py --epochs 3000 --eval
python scripts/train_heston.py --epochs 3000 --eval
python scripts/train_american.py --epochs 3000 --eval
python scripts/train_calibration.py --epochs 2000
pip install -e ".[quantum]" -c constraints.txt
python scripts/train_hybrid.py --epochs 300 --n-qubits 4 --n-layers 2

Project Structure

pde-pricer/
├── src/pde_pricer/
│   ├── cli.py               # pde-pricer train|price|validate
│   ├── checkpoint.py        # versioned European BS checkpoint schema
│   ├── pde/                 # PDE definitions
│   ├── classical/           # PINN architectures
│   ├── quantum/             # VQC integration (experimental)
│   ├── pricing/             # MC, FD, analytical engines
│   │   ├── mc_gpu.py        # CUDA Monte Carlo, pathwise Greeks
│   │   └── csrc/mc_gpu.cu   # fused European and basket kernels
│   ├── data/
│   └── validation/          # Greeks, grids, held-out reports, ablation
├── artifacts/              # committed checkpoints and JSON reports
├── tests/
├── scripts/
├── notebooks/
├── Makefile                 # make check / test-fast / smoke
├── constraints.txt
└── docs/
    ├── theory.md
    └── architecture.md

Mathematical Background

Forward Problem: Multi-Asset Black-Scholes

\frac{\partial V}{\partial t} + \sum_i rS_i\frac{\partial V}{\partial S_i} + \frac{1}{2}\sum_{i,j} \rho_{ij}\sigma_i\sigma_j S_i S_j \frac{\partial^2 V}{\partial S_i \partial S_j} - rV = 0

Inverse Problem: Dupire Calibration

Given market prices C(K,T), recover local volatility \sigma(K,T):

\sigma^2(K,T) = \frac{2\left(\frac{\partial C}{\partial T} + rK\frac{\partial C}{\partial K}\right)}{K^2 \frac{\partial^2 C}{\partial K^2}}

Jump-Diffusion (Merton)

\frac{\partial V}{\partial t} + (r-\lambda\kappa)S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV + \lambda\int_0^\infty [V(SJ,t) - V(S,t)]g(J)dJ = 0

Full derivations: docs/theory.md.

Testing

make check          # format + lint + typecheck + fast tests + wheel smoke
make test-fast      # default suite (not slow/gpu/quantum)
make test-slow      # longer numerical harness
make test-all       # everything except gpu

References

• Raissi et al. "Physics-informed neural networks" (2019). arXiv:1711.10561
• Stamatopoulos et al. "Option Pricing using Quantum Computers" (2020). arXiv:1905.02666
• Dupire. "Pricing with a Smile" (Risk, 1994)
• Schuld & Petruccione. "Machine Learning with Quantum Computers" (Springer, 2021)

License

MIT License. See LICENSE.