Benchmarks

5D Black-Scholes Performance

All benchmarks use 5D Black-Scholes: \(V(S, K, T, \sigma, r)\) with 11 Chebyshev nodes per dimension (\(11^5 = 161{,}051\) grid points).

Accuracy

Method	Price Error	Greek Error (max)
Chebyshev Barycentric	0.000%	1.980%
MoCaX Standard (C++)	0.000%	1.980%
FDM (Crank-Nicolson)	0.803%	2.234%

Both Chebyshev methods achieve machine-precision price accuracy and identical Greek errors, as they compute the same unique interpolating polynomial.

Timing

Method	Build Time	Price Query	Price + 5 Greeks
Chebyshev Barycentric	0.35s	0.065 ms	0.29 ms
MoCaX Standard (C++)	1.04s	0.47 ms	2.85 ms
Analytical (direct)	N/A	0.01 ms	0.06 ms

Evaluation Method Comparison

Within PyChebyshev, multiple evaluation paths exist:

Method	Price Only	Price + 5 Greeks	Notes
eval()	~45 ms	~270 ms	Python loops, full validation
fast_eval()	~10 ms	~95 ms	Deprecated — JIT scalar loops
vectorized_eval()	0.065 ms	0.39 ms	Recommended — BLAS GEMV
vectorized_eval_multi()	—	0.29 ms	Shared weights across derivatives

vectorized_eval() is the recommended default. Use vectorized_eval_multi() when computing multiple derivatives at the same point.

Why BLAS beats JIT

fast_eval() uses Numba JIT to compile scalar barycentric interpolation loops. But vectorized_eval() restructures the algorithm into matrix-vector products (BLAS GEMV), replacing 16,105 Python loop iterations with 5 BLAS calls for a 5D problem. Optimized BLAS (OpenBLAS/MKL) running a single GEMV is fundamentally faster than JIT-compiled scalar loops — the data access pattern is more cache-friendly and leverages SIMD vectorization at the hardware level. fast_eval() is deprecated and will be removed in a future version.

Tensor Train (TT) vs MoCaX Extend

ChebyshevTT and MoCaX MocaxExtend both build Chebyshev interpolants in TT format for the same 5D Black-Scholes problem. PyChebyshev uses TT-Cross (maxvol pivoting); MoCaX uses rank-adaptive ALS on a random subgrid. Both use ~7,400--8,000 function evaluations.

Build

Metric	PyChebyshev TT	MoCaX TT
Build time	0.35s	5.73s
Function evaluations	7,419	8,000
TT ranks	[1, 11, 11, 11, 7, 1]	(not exposed)
Compression ratio	43.4x	N/A

Price Accuracy (50 random test points)

Metric	PyChebyshev TT	MoCaX TT
Mean error	0.002%	0.093%
Max error	0.014%	0.712%
Median error	0.001%	0.045%

Evaluation Speed (1000 random points)

Method	PyChebyshev TT	MoCaX TT
Single eval	0.065 ms	--
Batch eval	0.004 ms	0.246 ms

Greeks Accuracy (10 scenarios, FD vs analytical)

Greek	PyChebyshev avg error	MoCaX avg error
Delta	0.029%	0.379%
Gamma	0.019%	1.604%

To reproduce: uv run --with tqdm --with blackscholes python compare_tensor_train.py (requires MoCaX C++ library; PyChebyshev results are shown regardless).