Computing Greeks

PyChebyshev computes option Greeks (and any partial derivatives) analytically using spectral differentiation matrices — no finite differences needed.

Derivative Specification

Derivatives are specified as a list of integers, one per dimension. Each integer is the derivative order with respect to that dimension.

For a 5D function \(V(S, K, T, \sigma, r)\):

Greek	derivative_order	Mathematical
Price	[0, 0, 0, 0, 0]	\(V\)
Delta	[1, 0, 0, 0, 0]	\(\partial V / \partial S\)
Gamma	[2, 0, 0, 0, 0]	\(\partial^2 V / \partial S^2\)
Vega	[0, 0, 0, 1, 0]	\(\partial V / \partial \sigma\)
Rho	[0, 0, 0, 0, 1]	\(\partial V / \partial r\)

Example: Black-Scholes Greeks

from pychebyshev import ChebyshevApproximation

def black_scholes_call(x, _):
    S, K, T, sigma, r = x
    # ... your pricing function here
    return price

cheb = ChebyshevApproximation(
    black_scholes_call, 5,
    domain=[[80, 120], [90, 110], [0.25, 1.0], [0.15, 0.35], [0.01, 0.08]],
    n_nodes=[11, 11, 11, 11, 11],
)
cheb.build()

point = [100, 100, 1.0, 0.25, 0.05]

# All Greeks at once (most efficient)
price, delta, gamma, vega, rho = cheb.vectorized_eval_multi(point, [
    [0, 0, 0, 0, 0],
    [1, 0, 0, 0, 0],
    [2, 0, 0, 0, 0],
    [0, 0, 0, 1, 0],
    [0, 0, 0, 0, 1],
])

How It Works

1. At build time: Pre-compute differentiation matrix \(D\) from node positions
2. At query time: Apply \(D\) to the function value tensor before barycentric interpolation
3. For second derivatives: apply \(D\) twice (\(D^2 \mathbf{f}\))
4. Interpolate the resulting derivative values to the query point

This provides exact derivatives of the interpolating polynomial. Because the differentiation matrix computes derivatives of the degree-\(n\) polynomial \(p(x)\) exactly (within machine precision), and \(p(x)\) converges spectrally to \(f(x)\), the derivative \(p'(x)\) also converges spectrally to \(f'(x)\) (Trefethen 2013, Ch. 11).

See Berrut & Trefethen (2004) for the derivation of the differentiation matrix formulas. For the full theory of spectral differentiation, see Trefethen (2013), Approximation Theory and Approximation Practice, SIAM, Chapter 11.

Tensor Train derivatives

ChebyshevTT uses finite differences instead of analytical derivatives, because the spectral differentiation matrix requires the full tensor (which TT avoids storing). FD derivatives are still accurate to within a few hundredths of a percent for first derivatives. See Tensor Train: Derivatives.

Accuracy

With 11 nodes per dimension on a 5D Black-Scholes test:

Greek	Max Error
Delta	< 0.01%
Gamma	< 0.01%
Vega	~1.98%
Rho	< 0.01%

Vega has slightly higher error because volatility sensitivity involves a product of multiple terms, but remains well within practical tolerance.

References

• Berrut, J.-P. & Trefethen, L. N. (2004). "Barycentric Lagrange Interpolation." SIAM Review 46(3):501--517.
• Trefethen, L. N. (2013). Approximation Theory and Approximation Practice. SIAM. Chapter 11.