finite-difference

Finite-difference option pricing for the SABR stochastic-volatility model via the Douglas-scheme ADI method of von Sydow et al. (2010).

von Sydow et al. (2010). ADI finite difference schemes for option pricing in the Heston model with correlation. International Journal of Numerical Analysis and Modeling, 7(2), 303–320.

What it solves

European call and put options under the SABR model — a four-parameter (α, β, ρ, ν) stochastic-volatility framework — whose pricing PDE is two-dimensional and has no closed form. The package solves that PDE on a Cartesian (S, v, t) grid using the Douglas Alternating Direction Implicit scheme: implicit sweeps in S and v are alternated each step, with the cross-derivative term handled explicitly. One-dimensional Black–Scholes solvers (explicit, implicit, Crank–Nicolson, American) ship as baselines and as the validation reference for the SABR limit.

Installation

pip install mafn-finite-difference

Version 0.2.0

This release adds SABR benchmark validation, automatic SABR grid-boundary selection, and an Extended Heston ADI solver with package-level tests.

Quick start

from finite_difference import price_sabr_option

# 1-year ATM SABR call: alpha=0.2, beta=1, rho=-0.3, nu=0.4
price, V, S, v = price_sabr_option(
    S0=100.0, K=100.0, T=1.0, r=0.05,
    alpha=0.2, beta=1.0, rho=-0.3, nu=0.4,
    option_type="call",
)
print(f"SABR call price: {price:.5f}")

API reference

price_sabr_option(S0, K, T, r, alpha, beta, rho, nu, ...) -> (price, V, S, v)

High-level pricer. Builds the (S, v, t) grid, runs the Douglas-ADI sweeps, and bilinearly interpolates the value at (S0, alpha).

Returns:

ADISolver(sabr, grid, K, r, T, option_type)

Lower-level access. Call solve(theta=0.5, verbose=False) to get the full (V, S, v) surface — useful when you need the surface, not just one interpolated price.

1-D Black–Scholes solvers

In finite_difference/solvers/. Each takes (S_max, K, r, sigma, T, M, N, option_type) and returns (S, V) at t = 0.

bs_price(S, K, T, r, sigma, option_type) -> float

Closed-form Black–Scholes price; the validation reference for the SABR solver in its β = 1, ν → 0 limit.

License

MIT.

Demo notebook

Full SABR/ADI tutorial — derivation, solver, surface plots, convergence study, parity check, parameter sensitivity, and the implied-volatility smile — is in notebooks/07_sabr_adi.ipynb.