elworthy 0.1.3

Bismut-Elworthy-Li Malliavin weights for Monte Carlo Greeks: Rust core, Python API

elworthy

Why elworthy over Numba

Numba @njit is excellent for general-purpose Python loop acceleration. Where elworthy is unique:

Property	Numba pathwise	elworthy BEL
Smooth payoff delta (call)	correct, fast	correct, ~equal speed
Digital / barrier delta	silently returns 0 (Dirac in f')	unbiased, hits analytic within stderr
Weight derivation	user writes the integrator	symbolic synthesis from the SDE
Output type	Numba arrays (no autograd)	plain NumPy → Torch / JAX backprop free

Demo: python python/benchmarks/demo_digital_payoff_correctness.py shows Numba pathwise digital delta = 0.000 (bias -0.0197) vs elworthy BEL 0.0198 (bias +0.0001) against analytic 0.0197.

A Rust JIT compiler that specialises Bismut-Elworthy-Li formulas into SIMD kernels for unbiased Monte Carlo Greeks on non-stationary SDEs. Python bindings ship on PyPI: pip install elworthy. The Python API returns per-path Malliavin weights as NumPy arrays so users compose payoffs in NumPy / PyTorch / JAX and stay inside their autodiff framework of choice.

Named after K. David Elworthy, co-author of the Bismut-Elworthy-Li integration-by-parts formula:

$$ \partial_x \mathbb{E}[f(X_T) \mid X_0 = x] = \mathbb{E}\left[ f(X_T) \cdot \frac{1}{T} \int_0^T \sigma^{-1}(X_s)^\top \partial_x X_s \mathrm{d}W_s \right]. $$

elworthy takes a symbolic SDE, symbolically differentiates its coefficients, synthesises the Malliavin weight for a requested Greek, and lowers the entire inner-loop body (state update + weight accumulation + payoff) to a single kernel via Cranelift. Each SIMD lane carries one independent Monte Carlo path.

Status

v0.1. Ships:

• Full scalar Cranelift JIT with exp, log, sin, cos, sqrt via libm.
• 2-lane SIMD VectorKernel (Cranelift F64X2) covering every Fun variant per lane; F64X4 scaffolded behind the simd_avx2 feature flag with runtime CPU detection.
• In-memory structural-hash kernel cache + disk-persisted AST cache at $XDG_CACHE_HOME/elworthy/.
• Euler-Maruyama and Milstein discretisations for scalar SDEs; multi-dimensional Euler driver with full-truncation clamping for CIR/variance-type components.
• Greek drivers:
  • Delta, constant-flow Bismut-Elworthy-Li (GBM / ABM).
  • Delta, general tangent-flow Bismut-Elworthy-Li (any scalar SDE).
  • Delta, multi-dim pathwise tangent flow (Heston and friends).
  • Rho, vega, and arbitrary parameter Greeks, pathwise (smooth payoffs, any scalar SDE).
  • Rho and vega for GBM via the likelihood-ratio Malliavin weight, valid for non-smooth payoffs (digitals, barriers); derivation machine-checked with SymPy.

Architecture

elworthy/
├── elworthy-expr/      symbolic AST, canonicalisation, CSE
├── elworthy-diff/      automatic symbolic differentiation
├── elworthy-weight/    Bismut-Elworthy-Li weight synthesis
├── elworthy-codegen/   Expr -> Cranelift IR lowering + scalar interpreter
├── elworthy-rt/        kernel cache, SIMD RNG, Monte Carlo driver
└── elworthy/           CLI + examples

Each subcrate has its own README.

Performance

On a development laptop (x86_64, single core, release profile):

Scenario	throughput (M path-steps/s)	speedup vs interpreter
GBM price, tree-walking interpreter	6.8	1.0x
GBM price, scalar Cranelift JIT	150	22x
GBM price, 2-lane SIMD JIT	187	27x
GBM price + Bismut-Elworthy-Li delta	152	22x
Heston price (2-D)	55	8x
Heston price + pathwise delta (2-D, full Jacobian)	16	2.3x

Kernel cache hit: 64 ns per retrieval vs ~100 us for a cold Cranelift compile, a 1500x speedup on calibration inner loops.

Cross-validation. The JIT+BEL stack is cross-validated against Black-Scholes closed form (inline, libm::erf) and the independent blackscholes crate (v0.24). On a European call at S_0 = K = 100, r = 0.05, sigma = 0.2, T = 1, 200 000 Milstein paths x 512 steps, elworthy's Monte Carlo price and Bismut-Elworthy-Li delta agree with both analytic references within four Monte Carlo standard errors. Full table in BENCHMARK.md.

Reproduce with:

cargo test --release -p elworthy-rt --test benchmark -- --nocapture --ignored

See BENCHMARK.md for full methodology, caveats, and reproducibility notes.

Install

cargo add elworthy-rt elworthy-expr

Or the CLI:

cargo install elworthy

Quick start (CLI)

elworthy gbm       --backend jit --paths 10000
elworthy gbm-delta --paths 40000

Expected output for gbm-delta at default parameters:

price   ~ 105.12xx (stderr 0.14xx) | closed form 105.1271
delta   ~   1.05xx (stderr 0.02xx) | closed form 1.0513   [Bismut-Elworthy-Li]

SELinux note (Fedora / RHEL)

The JIT backend requires execmem permission so Cranelift can map newly generated code as executable. Under SELinux enforcing this is denied by default for user binaries and you will see

Error: cranelift module error: Backend error: unable to make memory readable+executable

Workarounds:

• Run via cargo test (the test harness domain already grants execmem).
• Temporarily relax policy with sudo setsebool -P selinuxuser_execheap 1.
• Use --backend interp for a JIT-free run.

Ubuntu, Debian, macOS, and most CI runners do not hit this restriction.

Next milestones

• VectorKernel4 (AVX2 F64X4) behind the simd_avx2 feature.
• General-SDE Malliavin parameter weight (without relying on a closed-form transition density).
• Gamma via second-order Bismut-Elworthy-Li.
• QuantLib reference benchmarks for Heston delta.

Licence

Apache License 2.0. See LICENSE.