mafn-finite-difference 0.1.0

Finite-difference option pricing in 1-D (Black-Scholes) and 2-D (SABR) with a Douglas-scheme ADI solver.

finite-difference

Finite-difference option pricing for the SABR stochastic-volatility model via the Douglas-scheme ADI method of in 't Hout & Foulon (2010).

in 't Hout, K., & Foulon, S. (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

(mafn-finite-difference is a temporary placeholder for the published PyPI name.)

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).

Parameter	Type	Description
S0	float	Initial asset price
K	float	Strike
T	float	Time to maturity (years)
r	float	Risk-free rate
alpha	float	Initial SABR vol level (>0)
beta	float	CEV exponent in [0, 1]
rho	float	Correlation in [-1, 1]
nu	float	Vol-of-vol (>0)
M, L, N	int	Grid points in S, v, t (default 80 / 40 / 80)
S_max, v_max	float, optional	Domain bounds (sensible defaults)
option_type	str	"call" or "put"
theta	float	ADI weight; 0.5 is second-order accurate
verbose	bool	Print per-step progress

Returns:

Field	Type	Description
price	float	Bilinearly-interpolated value at (S0, alpha)
V	(M+1, L+1) array	Full price surface at t = 0
S	(M+1,) array	Asset-price grid
v	(L+1,) array	Volatility grid

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.

Function	Module	Notes
solve_explicit	explicit	CFL-limited; fastest per step
solve_implicit	implicit	Unconditionally stable, O(Δt)
solve_crank_nicolson	crank_nicolson	Unconditionally stable, O(Δt²)
solve_american	american	American put via projected CN

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.