ABCD math and covariance

These additive Rust-only helpers cover Medium inventory rows 36 + 35. Existing models, defaults and bindings are unchanged. No calibration, LMM wiring, complete market model or release is included.

Shape and distinct defaults

f(t) = (a + b*t)*exp(-c*t) + d, with f(t)=0 for finite t<0.

Type	Default (a,b,c,d)
math::abcdmathfunction::AbcdMathFunction	(0.002,0.001,0.16,0.0005)
termstructures::volatility::AbcdFunction	(-0.06,0.17,0.54,0.17)

Both provide new, with_defaults, Default, coefficient getters and checked value. Math additionally exposes the free coefficient validate function. c must be strictly positive; zero is not a supported constructor shortcut. d and a+d must be non-negative. Negative a or b can be valid, but a negative stationary minimum cannot. Finite coefficients are mandatory. The negative-slope minimum is validated in the log domain to avoid overflow. With b<0,d=0, a mathematically negative eventual tail is rejected even if native exponential underflow would conceal it.

Rate covariance

For fixing times T,S and observation time u:

• instantaneous_covariance(u,T,S) returns f(T-u)*f(S-u).
• covariance(t1,t2,T,S) integrates that product to min(t2,T,S).
• variance(t1,t2,T) is the integrated variance, not an average variance. Duration normalization is not supplied by these helpers.
• AbcdSquared::new(a,b,c,d,T,S) stores an immutable integrand. Despite its name, T and S need not be equal.

Finite signed times are allowed. Negative observation times are not silently clipped at zero. At a fixing time the instantaneous value includes f(0); strictly after either fixing covariance is zero. Integrated zero-width and already-cut-off intervals return zero. Reversed bounds return an error.

use libitofin::termstructures::volatility::AbcdFunction;

let shape = AbcdFunction::with_defaults()?;
let covariance = shape.covariance(0.0, 1.0, 2.0, 3.0)?;
let variance = shape.variance(0.0, 1.0, 2.0)?;
assert!(covariance >= 0.0 && variance >= 0.0);
# Ok::<(), libitofin::errors::QlError>(())

Numerical boundaries and oracle

Integration uses analytic decaying exponential moments in local interval coordinates. For c*duration < 0.5, it integrates the product of local 33-term Taylor expansions of the volatility values, preserving near-zero initial values with expm1. At this threshold the omitted exponential term is below 0.5^33/33!; this is a fixed analytic expansion, not adaptive quadrature. It does not subtract large native primitives or replace positive small c with exactly zero. This avoids the original formula's long-maturity exponential overflow and small-decay cancellation in ordinary representable arithmetic.

All numerical queries return QlResult. Non-finite inputs/results, negative computed values, unrepresentable lags, durations, moments or intermediate expressions return errors, even when exact real arithmetic could yield a finite answer. There is no arbitrary precision, negative-variance clamp or universal relative-accuracy guarantee near zero; ordinary floating-point underflow can return zero. Cutoff shortcuts validate every input first.

Original source pin: 9863b578af0caa4cecabf697196533e84a8308b6. The directly exported math/value/covariance operations are checked against the compiled QuantLib 1.43 wheel, a separate revision. AbcdSquared delegates to the native covariance operation but is not exported by that wheel. High-precision/quadrature checks cover small decay and short intervals. See tests/fixtures/medium2_abcd/README.md for reproduction.

This is the indexed fork's bounded subset, not the complete native ABCD API. Primitive, derivative, maximum, rolling-window coefficients, normalized volatility and calibration APIs are intentionally deferred. Native primitive normalization/support quirks and native extrema conventions are consequently not exposed or claimed as covered.