Models

Stochastic process definitions used by the Monte Carlo engine. Each model is an equinox.Module containing the process parameters.

BlackScholesModel

Geometric Brownian Motion: \(dS = (r - q) S\, dt + \sigma S\, dW\)

from valax.models import BlackScholesModel

model = BlackScholesModel(
    vol: Float[Array, ""],       # annualized volatility
    rate: Float[Array, ""],      # risk-free rate
    dividend: Float[Array, ""],  # continuous dividend yield
)

HestonModel

Stochastic volatility with mean-reverting variance:

\[dS = (r - q) S\, dt + \sqrt{V} S\, dW_1$$ $$dV = \kappa(\theta - V)\, dt + \xi \sqrt{V}\, dW_2, \quad \text{Corr}(dW_1, dW_2) = \rho\]

from valax.models import HestonModel

model = HestonModel(
    v0: Float[Array, ""],       # initial variance
    kappa: Float[Array, ""],    # mean reversion speed
    theta: Float[Array, ""],    # long-run variance
    xi: Float[Array, ""],       # vol of vol
    rho: Float[Array, ""],      # spot-vol correlation (typically negative)
    rate: Float[Array, ""],     # risk-free rate
    dividend: Float[Array, ""], # continuous dividend yield
)

Notes:

• Path generation works in log-spot space for numerical stability.
• Variance is floored at zero (jnp.maximum(v, 0)) to prevent square root of negative numbers — this is the absorption scheme.
• All parameters are differentiable, enabling gradient-based calibration.

SABRModel

Stochastic Alpha Beta Rho model for volatility smiles:

\[dF = \alpha F^\beta\, dW_1, \quad d\alpha_t = \nu \alpha_t\, dW_2, \quad \text{Corr}(dW_1, dW_2) = \rho\]

from valax.models import SABRModel

model = SABRModel(
    alpha: Float[Array, ""],  # initial volatility
    beta: Float[Array, ""],   # CEV exponent (0 = normal, 1 = lognormal)
    rho: Float[Array, ""],    # forward-vol correlation (typically negative)
    nu: Float[Array, ""],     # vol of vol
)

Notes:

• beta controls the backbone: \(\beta = 0\) gives normal dynamics, \(\beta = 1\) gives lognormal. Equity typically uses \(\beta = 0.5\); rates often use \(\beta = 0\).
• Pricing uses Hagan's asymptotic implied vol expansion fed into Black-76 (sabr_price).
• Monte Carlo paths available via generate_sabr_paths using diffrax.
• All parameters are differentiable — Greeks via jax.grad work through the full SABR → Black-76 chain.
• Calibration via calibrate_sabr in valax.calibration.