PDE Solvers¶

VALAX solves the Black-Scholes PDE using Crank-Nicolson finite differences in log-spot space, with tridiagonal systems solved via lineax.

The Black-Scholes PDE¶

In log-spot space \(x = \ln S\):

\[\frac{\partial V}{\partial t} + \left(r - q - \frac{\sigma^2}{2}\right) \frac{\partial V}{\partial x} + \frac{\sigma^2}{2} \frac{\partial^2 V}{\partial x^2} - rV = 0\]

The PDE is solved backward in time from the terminal payoff using Crank-Nicolson (\(\theta = 0.5\)), which is unconditionally stable and second-order accurate in both time and space.

Usage¶

import jax.numpy as jnp
from valax.instruments import EuropeanOption
from valax.pricing.pde import pde_price, PDEConfig

option = EuropeanOption(strike=jnp.array(100.0), expiry=jnp.array(1.0), is_call=True)

price = pde_price(
    option,
    spot=jnp.array(100.0),
    vol=jnp.array(0.20),
    rate=jnp.array(0.05),
    dividend=jnp.array(0.02),
    config=PDEConfig(n_spot=400, n_time=400, spot_range=4.0),
)

Configuration¶

Parameter	Default	Description
`n_spot`	200	Number of spatial grid points (interior)
`n_time`	200	Number of time steps
`spot_range`	4.0	Grid extends `spot_range * vol * sqrt(T)` standard deviations from `ln(spot)`

Finer grids give more accurate prices at the cost of computation time.

Greeks via Autodiff¶

The entire PDE solve — grid construction, Crank-Nicolson time-stepping, tridiagonal solves, and interpolation — is differentiable:

import jax

delta = jax.grad(lambda s: pde_price(option, s, vol, rate, div, config))(spot)
vega = jax.grad(lambda v: pde_price(option, spot, v, rate, div, config))(vol)

Implementation Details¶

Log-spot grid: Uniform spacing in \(x = \ln S\) ensures equal resolution across all spot levels.
Tridiagonal solver: Each Crank-Nicolson time step requires solving a tridiagonal linear system, handled by lineax.Tridiagonal().
jax.lax.scan: The backward time-stepping loop uses lax.scan for JIT compilation — no Python-level loop overhead.
Boundary conditions: Derived from BS asymptotics for very small/large spot.