Model Calibration

VALAX calibrates model parameters to market data using gradient-based optimization. Because all pricing functions are differentiable via JAX, the optimizer gets exact Jacobians for free — no finite-difference bumping.

SABR Calibration

Fit SABR parameters (\(\alpha\), \(\rho\), \(\nu\)) to an observed volatility smile. Beta is typically fixed.

import jax.numpy as jnp
from valax.calibration import calibrate_sabr

# Market data: strikes and observed implied vols
strikes = jnp.array([80., 85., 90., 95., 100., 105., 110., 115., 120.])
market_vols = jnp.array([0.28, 0.26, 0.24, 0.22, 0.21, 0.205, 0.21, 0.22, 0.235])
forward = jnp.array(100.0)
expiry = jnp.array(1.0)

# Calibrate with beta fixed at 0.5
fitted, sol = calibrate_sabr(
    strikes, market_vols, forward, expiry,
    fixed_beta=jnp.array(0.5),
)

print(f"alpha={float(fitted.alpha):.4f}")
print(f"rho={float(fitted.rho):.4f}")
print(f"nu={float(fitted.nu):.4f}")

Solvers

Three backends are available:

Solver	Method	Best for
"levenberg_marquardt"	Least-squares (optimistix)	Default — fast, exploits Jacobian
"bfgs"	Quasi-Newton (optimistix)	Fallback when LM struggles
"optax_adam"	Gradient descent (optax)	Research / experimentation

# Use BFGS instead
fitted, sol = calibrate_sabr(
    strikes, market_vols, forward, expiry,
    fixed_beta=jnp.array(0.5),
    solver="bfgs",
)

Weighted Calibration

Emphasize ATM strikes by passing per-strike weights:

weights = jnp.exp(-0.5 * ((strikes - forward) / 10.0) ** 2)
fitted, sol = calibrate_sabr(
    strikes, market_vols, forward, expiry,
    fixed_beta=jnp.array(0.5),
    weights=weights,
)

Heston Calibration

Fit Heston parameters to option prices. Requires a pricing function to be injected — this allows using any Heston pricer (semi-analytic, Monte Carlo, or a neural surrogate).

from valax.calibration import calibrate_heston

fitted, sol = calibrate_heston(
    strikes, market_prices, spot, rate, dividend, expiry,
    pricing_fn=my_heston_pricer,
)

How It Works

Parameter Transforms

Model parameters have natural constraints (e.g., \(\alpha > 0\), \(-1 < \rho < 1\)). Rather than using constrained optimization, VALAX reparametrizes to unconstrained space:

Constraint	Transform	Inverse
\(x > 0\)	softplus	inverse softplus
\(a < x < b\)	scaled sigmoid	logit
\(-1 < x < 1\)	tanh	arctanh

The optimizer works in unconstrained \(\mathbb{R}^n\); results are mapped back to valid parameters automatically.

from valax.calibration import positive, correlation, bounded

# Custom transforms
pos = positive()                    # x > 0
corr = correlation()                # -1 < x < 1
frac = bounded(0.0, 1.0)           # 0 < x < 1

Loss Functions

Calibration minimizes weighted residuals in implied-vol space (default) or price space:

\[\min_\theta \sum_i w_i \left( \sigma^\text{model}(K_i; \theta) - \sigma^\text{market}(K_i) \right)^2\]

Vol-space calibration is more stable than price-space for SABR because it removes the option's moneyness dependence from the residuals.