Volatility markets and model calibration

Option markets quote implied volatility across expiries, strikes and, for swaptions, underlying swap tenors. QuantSupport constructs surfaces and cubes from these observations. Pricers can query them directly or use them as targets when calibrating a stochastic model.

Calibration converts the selected volatility quotes into option prices, then fits model parameters to those prices. The resulting parameters describe the model’s dynamics and use units appropriate to that model.

Coordinates and interpolation

VolatilitySurfaceConfiguration selects expiry/smile quotes. VolatilityCubeConfiguration adds underlying tenor, as needed for swaptions. The volatility type is Black or Normal; the smile coordinate can be strike, delta or log-moneyness. The generic Strike::resolve maps Absolute, Atm and Relative to \(K\), \(F\) and \(F+\text{spread}\). Consumers do not all use that helper: the Black and Hull–White caplet/cap pricers interpret Relative(p) multiplicatively as \(F(1+p)\). Prefer an absolute strike when specifying an unambiguous contractual rate.

Configured identifiers supply the actual coordinates. Equivalent tenors such as 1Y and 12M cannot define duplicate nodes. Builders reject empty sets, nonpositive expiry/tenor, nonfinite keys and negative or nonfinite volatilities. The selected bid/mid/ask level supplies the quoted values.

Interpolated surfaces first interpolate within each smile, then across expiry. Cubes interpolate smile, tenor and finally expiry. Rectangular grids reproduce bilinear/trilinear interpolation, but sparse grids can have different strikes per slice and different tenor sets per expiry. Exact nodes reproduce quoted values without needing a neighboring slice.

ExtrapolationPolicy::Reject is the default. A query must lie within each selected slice; a singleton coordinate must match exactly. Flat clamps to boundaries and treats singleton axes as constant. Both reject nonfinite queries. Interpolation is in the implemented volatility coordinates, not an automatic guarantee of arbitrage-free total variance.

An FX market can store one pair orientation. OrientedFxVolSurface resolves a reciprocal request and maps inverse strike with \(K\mapsto1/K\). Smile-type metadata elsewhere does not guarantee universal conversion between strike, delta and moneyness. In particular, the current equity option pricer passes a resolved strike directly; configure a compatible surface rather than assume a delta inversion is being solved.

Source instruments and risk pillars

Constructed volatility objects retain calibration-instrument identities. These encode option expiry, tenor and strike and allow model calibration to select a meaningful basket. Risk pillars identify differentiable inputs whose labels appear in sensitivity results. An interpolated quote grid commonly uses the same identifiers for both purposes, but the roles are conceptually different.

CalibrationBasket selects expiries, tenors and a strike rule from the configured source. Omitted axes include all available values. Every explicitly requested value must exist and participate after the other filters are applied. A caplet surface’s tenor is the index observation tenor; a cube’s tenor is the underlying swap tenor. A three-month caplet source cannot satisfy a six-month basket by silently reinterpreting those contracts.

Parameter sources

ParameterSource::Fixed supplies a complete model parameter set. ParameterSource::Calibrated names a surface or cube and a semantic basket. Hull–White and LGM use Gaussian sigma parameters; lognormal Brownian motion uses a volatility parameter. These are distinct units and parameterizations, even when they are ultimately fitted to the same option premium.

Gaussian calibration converts each selected caplet or swaption’s Black or Normal volatility into an option price. It fits a constant sigma to each instrument independently, then assembles the fitted values into a piecewise schedule. Repricing under that combined schedule can differ from the targets because the individual fits do not jointly solve its interval volatilities.

Hull–White quality reports retain instrument identity, target volatility/price, fitted price, sigma, forward and strike. Their fitted prices describe the individual constant-sigma solves, not repricing under the assembled schedule.

For lognormal Brownian motion, term-volatility stripping instead enforces

\[ \int_0^{T_i}\sigma(u)^2\,du=\sigma_{\rm term}(T_i)^2T_i. \]

Negative forward variance is rejected. The stripped interval volatilities reproduce the accumulated variance implied by the selected term quotes.

The Gaussian calibration Jacobian uses diagonal forward differences with absolute volatility bump \(h=10^{-6}\):

\[ \frac{\partial\sigma_i}{\partial\nu_j} \approx\delta_{ij} \frac{\sigma_i(\nu_i+h)-\sigma_i(\nu_i)}{h}. \]

This numerical response holds curve inputs and calibration strikes fixed. It is not a full joint schedule-calibration Jacobian. Direct pricing may still have curve risk. Cube calibration selects at most one underlying tenor per expiry and must respect the source domain. Materialized numerical simulations introduce further derivative limits, explained in Stochastic models.