Status: Complete
GitHub: github.com/rylanmalarchick/quantum-derivatives-trader
Overview
This project uses physics-informed neural networks (PINNs) to solve partial differential equations in derivatives pricing. PINNs embed the governing PDE directly into the neural network loss function, enabling mesh-free solutions that scale to high dimensions where traditional finite difference methods fail.
Key insight: For a 5-asset basket option, finite difference requires ~10^12 grid points (infeasible). PINNs solve the same 6D PDE with 15,000 collocation points.
Key Results
| Problem | Dimension | Method | Result |
|---|---|---|---|
| Basket option | 6D (5 assets + time) | PINN | 3.9% error vs MC, exact Greeks via autodiff |
| Volatility calibration | Inverse problem | PINN | 4.5% vol surface recovery under no-arbitrage constraints |
| Merton jump-diffusion | 2D PIDE | PINN | Gauss-Hermite quadrature for jump integral |
| Heston stochastic vol | 3D PDE | PINN | Correlation, mean reversion, vol-of-vol |
| American options | Free boundary | PINN | Penalty method for early exercise |
| Barrier options | Path-dependent | PINN | Down-and-out calls with analytical validation |
High-Dimensional Pricing (5-Asset Basket)
The multi-asset Black-Scholes PDE solved in 6 dimensions (5 assets + time):
| Aspect | Finite Difference | PINN |
|---|---|---|
| Grid points (5 assets) | ~10^12 | 15,000 |
| Memory | Infeasible | ~3 GB |
| Greeks | Numerical noise | Exact (autodiff) |
| Cross-gammas | Expensive | Free |
Result: Final training loss 1.40, 3.9% error vs Monte Carlo at S0.
Volatility Surface Calibration (Inverse Problem)
Given market prices C(K,T), recover local volatility via Dupire's equation:
| Metric | Value |
|---|---|
| Price fit | 2.22% mean relative error |
| Vol surface recovery | 4.51% error |
| Constraints | Smoothness + no-arbitrage (convexity, calendar) |
| Training | ~2 min (3000 epochs) |
Advanced Models
Merton Jump-Diffusion (PIDE): Partial integro-differential equation with Gauss-Hermite quadrature for the jump integral (log-normal jump distribution).
Heston Stochastic Volatility (3D PDE): 3D coupled PDE with correlation, mean reversion, and vol-of-vol. Feller condition enforced to ensure variance stays positive.
American Options (Free Boundary): Early exercise via penalty method. The PINN learns both the option value and the optimal exercise boundary.
Barrier Options (Path-Dependent): Down-and-out call options with analytical formula validation. Most recent addition with comprehensive test coverage.
Architecture
| Directory | Description |
|---|---|
src/pde/ |
PDE definitions (7 models) |
src/pde/black_scholes.py |
1D European |
src/pde/basket.py |
N-asset basket (6D) |
src/pde/dupire.py |
Local vol calibration |
src/pde/merton.py |
Jump-diffusion (PIDE) |
src/pde/heston.py |
Stochastic vol (3D) |
src/pde/american.py |
Free boundary |
src/pde/barrier.py |
Path-dependent (NEW) |
src/classical/ |
PINN architectures |
src/classical/pinn.py |
Standard MLP PINN |
src/classical/pinn_basket.py |
High-dimensional with LHS |
src/classical/pinn_calibration.py |
Inverse problem |
src/quantum/ |
VQC integration (experimental) |
src/quantum/variational.py |
Hardware-efficient ansatz |
src/quantum/hybrid_pinn.py |
Quantum-classical hybrid |
src/quantum/hybrid_basket.py |
High-dim hybrid |
src/pricing/ |
Reference implementations |
src/pricing/analytical.py |
Black-Scholes closed-form |
src/pricing/monte_carlo.py |
MC with antithetic/control variates |
src/pricing/finite_difference.py |
Crank-Nicolson |
src/validation/ |
Greeks validation |
tests/ |
306 comprehensive tests |
docs/theory.md |
944 lines of mathematical derivations |
notebooks/ |
7 analysis notebooks |
Testing Philosophy
"Tests are the specification. Write them first, write them thoroughly."
pytest tests/ -v # 306 tests passing
| Category | Tests | Coverage |
|---|---|---|
| PDE residuals | 45 | Physics constraints, boundary conditions |
| PINN models | 68 | Forward pass, gradients, training |
| Quantum circuits | 32 | Output ranges, parameter updates |
| Greeks validation | 33 | Analytical vs autodiff |
| Basket/calibration | 89 | MC validation, vol recovery |
| Advanced models | 55 | Merton, Heston, American, Barrier |
Quantum-Hybrid (Experimental)
The project includes experimental integration of variational quantum circuits (VQCs) as function approximators within the PINN architecture:
| Component | Status |
|---|---|
| VQC trains and produces gradients | Working |
| Hardware-efficient ansatz | Implemented |
| Data re-uploading circuit | Implemented |
| Practical pricing accuracy (<1% error) | Not yet achieved |
Honest assessment: The quantum component shows promise on simple problems but doesn't yet outperform well-tuned classical PINNs on production pricing tasks. Current results: ~22% relative error (market tolerance is <1%). This is research exploring when/why quantum expressivity might help.
Technology Stack
- Deep Learning: PyTorch 2.10, PennyLane 0.44
- HPC: JAX (JIT, vmap), GPU acceleration
- Quantum: PennyLane (adjoint differentiation), hardware-efficient ansatz
- Scientific: NumPy, SciPy (Gauss-Hermite, numerical integration)
- Sampling: Latin Hypercube Sampling for high-dimensional collocation
- Testing: pytest, 306 tests
Why This Matters
For Quantitative Finance
- High-dimensional pricing: Basket options, rainbow options require methods that scale beyond 3D
- Model calibration: Every trading desk calibrates daily; PINNs give smooth, arbitrage-free surfaces
- Greeks computation: Autodiff gives exact Δ, Γ, Θ, ν, cross-gammas—no numerical noise
For Research
- Honest quantum exploration: Testing VQC expressivity on real problems, not toy examples
- Reproducible benchmarks: 306 tests, documented theory (944 lines), readable code
- Production patterns: Clean architecture, comprehensive testing, mathematical documentation
Links
"The goal is not to prove quantum advantage, but to rigorously investigate when it might exist—while solving real problems with classical methods that work today."