Volatility Surface Calibration and Option Pricing Under Regime, Repair, and Numerical Constraints

01 / System boundary

The calibration is one part of the model

The project began as an implied-volatility fitting exercise. It became a study of the full chain of conditional decisions that determine a quoted surface and a theoretical price: carry, contract family, inversion, objective geometry, repair, exercise style, numerical resolution, and benchmark design.

02 / Data, carry, and routing

Economic coordinates before surface parameters

European index carry is estimated expiry by expiry from put-call parity. American equity and ETF carry uses the zero curve and a projected cash-dividend schedule. The release bundle excludes licensed quote chains and credentials; synthetic workflows and sanitized aggregate evidence remain reproducible.

Contract families are classified before calibration on settlement, root class, and expiry archetype. This prevents a dense or well-behaved family from masking a tail in a different convention.

03 / Implied volatility and raw SVI

Bracketed inversion and price-aligned weights

Implied volatility is solved as a monotone price root with explicit economic bounds and failure states. The initial bracket is [10-6, 3], with guarded expansion to 5 and 8, Brent tolerance 10-8, and an 80-iteration limit. Lattice inversion searches for the lowest valid model point when very small volatility makes the risk-neutral probability invalid.

# src/volsurf/core/implied_vol.py
root, details = brentq(
    objective,
    effective_low,
    high,
    xtol=tolerance,
    rtol=max(4.0 * math.ulp(1.0), tolerance),
    maxiter=max_iterations,
    full_output=True,
    disp=False,
)

Raw SVI is fitted by bounded, deterministic multi-start L-BFGS-B in total variance. The Jacobian converts quote-price half-spread into local total-variance uncertainty. It aligns the objective with price sensitivity, but does not guarantee a globally arbitrage-free surface or remove model misspecification.

04 / Static-arbitrage diagnostics

Calibration and repair are separate operations

The fitted slices are evaluated on a regular (T, k) grid. Calendar repair applies least-squares isotonic projection down each maturity column. Strike rows are then adjusted by constrained SLSQP until the discrete butterfly diagnostic satisfies its configured tolerance. Raw and repaired outputs are both retained.

# src/volsurf/surfaces/repair.py
for column in range(grid.shape[1]):
    original = grid[:, column]
    projected = isotonic_increasing(original)
    changed += int(np.sum(
        np.abs(projected - original) > 1.0e-14
    ))
    grid[:, column] = projected

result = minimize(
    objective,
    np.maximum(row, variance_floor),
    method="SLSQP",
    bounds=[(variance_floor, None)] * row.size,
    constraints=constraints,
)

05 / Benchmark governance

Paired contexts, exact support, and tail behavior

Average quote-level RMSE was not a sufficient promotion rule. A dense chain could dominate an aggregate, a deployment basket could reuse calibration observations, and a favorable family could offset a severe adverse date. The benchmark was rebuilt around family-date contexts and predeclared policies.

These audits reduce specific endogeneity concerns on the saved panels. They do not prove exogeneity on unobserved history, and they do not turn a small, irregular archive into a broad regime sample.

06 / Localized tail finding

A weak refinement, limited to nonstandard weeklies

The persistent European failure was concentrated in short-dated weekly_pm_nonstandard tails rather than average fit across all families. A weak price-local SVI refinement was allowed to change at most one eligible slice and was accepted only when price error improved without violating variance, parameter, butterfly, calendar-neighbor, or support guards.

07 / Alternatives and negative results

Useful research without global promotion

Several alternatives remained informative even when they did not become canonical. They identify where the baseline is vulnerable and where more data would be needed before changing policy.

08 / American option pricing

Approximate normalization, explicit exercise downstream

American equity and ETF surfaces retain Black implied-volatility normalization because American-aware alternatives did not generalize on the available panels. That approximation is confined to calibration coordinates. Downstream theoretical prices preserve American exercise.

# src/volsurf/pricing/early_exercise.py
american = price_crr(
    **common, exercise_style=ExerciseStyle.AMERICAN
)
european = price_crr(
    **common, exercise_style=ExerciseStyle.EUROPEAN
)
premium = american.require_price() \
    - european.require_price()

A positive model premium is not a trading signal. Borrow, dividend uncertainty, exercise operations, transaction costs, and model error can dominate the amount.

09 / Numerical validation

Tree and finite-difference convergence

Two apparent CRR-versus-PDE failures were traced to the comparison engine. The 400-step CRR price was within 0.00138 of its 3,200-step family reference. The original 400 × 400 CN-PSOR grid at a five-spot domain was materially under-resolved.

// cpp/src/pricing.cpp
const double unconstrained =
    residual / diagonal[index];
const double relaxed = solution[index]
    + settings.psor_omega
    * (unconstrained - solution[index]);
const double projected =
    std::max(obstacle[index], relaxed);

maximum_change = std::max(
    maximum_change,
    std::abs(projected - solution[index])
);
solution[index] = projected;

if (maximum_change
    < settings.psor_tolerance) {
    return {solution, iteration};
}

10 / Closing cross-market panel

Valid prices, strict quality failure retained

The closing accounting run covers INTC, NDX, NVDA, RUT, SPX, and SPY over eight August 2025 dates. Each asset contributes 400 prices. European indices use Black from the canonical repaired surface; American equities and ETF contracts use CRR with American exercise.

The pooled RMSE is 3.5122 and MAE is 1.4620, but neither is scale free. They are system-accounting statistics, not a universal accuracy claim. The release gate remains failed because the policy permits zero quote-quality failures; the threshold was not relaxed for presentation.

11 / Complete technical note

The report is the primary project record

The 55-page note contains the full mathematical and empirical path, including the parts that do not fit naturally on a portfolio page.

Carry and forward coordinatesContract-family taxonomyIV inversion and SVI calibrationJacobian-consistent weightingStatic-arbitrage repairBenchmark endogeneityBoundary and support analysisAlternative surfaces and repoAmerican normalizationEarly-exercise premiumCRR, LR, and CN-PSORClosing evidence and limits

Read complete PDFEquation indexClaim evidence →Source registry →

12 / Limitations and final state

Finished within a narrow evidence boundary