Struct Slice 

struct Slice {
    expiry: f64,
    forward: f64,
    strikes: Vec<f64>,
    quotes: Vec<f64>,
    slopes: Vec<f64>,
    weights: Vec<f64>,
    atm: Atm,
}

One slice’s market data in the formula’s convention.

Fields

expiry: f64

forward: f64

strikes: Vec<f64>

quotes: Vec<f64>

slopes: Vec<f64>

d(restated quote)/d(grid quote): one where the conventions agree.

weights: Vec<f64>

atm: Atm

Implementations

impl Slice

fn read( grid: &RateVolSurface, node: usize, expiry: f64, tenor: f64, forward: f64, held: Option<f64>, spec: &SabrSpec, ) -> Result<Self, SabrCalibrationError>

The slice at grid node node: its strikes, its quotes restated in the spec’s convention, the ATM quote where the grid has one, and the normalised vega weights.

fn vega<R: Scalar>(&self, i: usize, vol: R, spec: &SabrSpec) -> R

The undiscounted vega of quote i read at vol vol — the weight a vol error carries in price terms — in mode R, so the fit’s quote Jacobian can see the weights move with the quotes.

fn held(&self) -> Option<f64>

σ_ATM where it is held rather than fitted.

fn vols<R: Scalar>( &self, spec: &SabrSpec, theta: [R; 3], ) -> Result<Vec<R>, SabrError>

The SABR vols at the slice’s strikes for natural parameters (σ_ATM, ρ, ν) in mode R.

fn natural_derivatives( &self, spec: &SabrSpec, theta: [f64; 3], ) -> Option<(Vec<[f64; 3]>, Vec<[[f64; 3]; 3]>)>

∂σᵢ/∂θₐ and ∂²σᵢ/∂θₐ∂θ_b at natural parameters theta: a Jet2 pass along each axis and along each pair’s sum, the mixed partials by polarisation.

fn natural(&self, u: &[f64], atm: f64) -> [f64; 3]

Natural parameters from the unconstrained vector over the free set.

fn unconstrained(&self, theta: [f64; 3]) -> Vec<f64>

The unconstrained vector over the free set at natural theta.

fn evaluate( &self, spec: &SabrSpec, u: &[f64], atm: f64, ) -> (Vec<f64>, Vec<Vec<f64>>)

Weighted residuals and their Jacobian in the unconstrained parameters.

fn errors(&self, spec: &SabrSpec, theta: [f64; 3]) -> Option<(f64, f64)>

The weighted RMS error and the largest absolute error at theta.

fn guess(&self, spec: &SabrSpec) -> [f64; 3]

The three-point explicit guess off Hagan’s small-moneyness expansion of the Black vol, σ(K) ≈ σ₀(1 + s₁x + s₂x²) at x = ln(K/F) with s₁ = −½(1 − β − ρλ) and s₂ = [(1 − β)² + (2 − 3ρ²)λ²]/12: a quadratic through the three strikes nearest the forward, solved for ρλ and λ², and ν = λσ₀. A slice the expansion cannot read (a non-positive shifted forward, fewer than three strikes) starts flat.

fn butterfly_ok(&self, spec: &SabrSpec, theta: [f64; 3]) -> bool

The butterfly check: undiscounted calls on 41 strikes over ±4 ATM standard deviations, their second strike differences non-negative. The strikes are spaced in the formula’s own domain — arithmetically where it reads any strike (β = 0), geometrically in the shifted strike otherwise, at the ATM vol’s Black equivalent — so none falls where the formula is undefined.

fn fit(&self, spec: &SabrSpec) -> Option<([f64; 3], usize, f64, f64, bool)>

The fit: natural parameters, restarts spent, RMS and largest error, and whether the acceptance bound was met. None when no start converged to a point the expansion reads.

fn free(&self) -> &'static [usize]

The free parameters’ indices into (σ_ATM, ρ, ν).

fn polish(&self, spec: &SabrSpec, theta: [f64; 3]) -> [f64; 3]

Newton on the first-order condition g = 0 from a converged Levenberg–Marquardt point: LM stops on a gradient test, a few exact steps finish what it leaves, so the implicit-function Jacobian reads a point that is stationary to the last digit. A step that leaves the parameters’ domain or does not shrink the gradient is not taken.

fn condition( &self, spec: &SabrSpec, theta: [f64; 3], ) -> Option<(Vec<f64>, Vec<Vec<f64>>)>

The first-order condition over the free parameters at theta: gₐ = Σᵢ wᵢ(σᵢ − qᵢ)∂ₐσᵢ and its exact slope Aₐ_b = Σᵢ wᵢ(∂ₐσᵢ∂_bσᵢ + (σᵢ − qᵢ)∂ₐ∂_bσᵢ).

fn quote_jacobian( &self, spec: &SabrSpec, theta: [f64; 3], ) -> (Vec<[f64; 3]>, [f64; 3])

d(σ_ATM, ρ, ν)/d(quote) on the converged slice, by the implicit function theorem on the first-order condition over the free parameters, the weights held.

With gₐ = Σᵢ wᵢ(σᵢ − qᵢ)∂ₐσᵢ = 0 over the free a, and Aₐ_b = Σᵢ wᵢ(∂ₐσᵢ∂_bσᵢ + (σᵢ − qᵢ)∂ₐ∂_bσᵢ) its slope in every θ_b: a quote moves g by −w_q∂ₐσ_q directly and, where it is the held ATM vol, by Aₐ₀ through σ_ATM; the free parameters answer with −A_ff⁻¹ of that. A quote also moves the weights, its own vega being read at it, which moves g by Σᵢ ∂wᵢ/∂q·(σᵢ − qᵢ)∂ₐσᵢ. With that and the residual-curvature term kept the result is exact off a perfect fit — it is what bumping a quote and refitting converges to.

Each row is against the grid’s own quote: the restatement’s slope is applied last. Beside the rows comes the response to a σ_ATM held from an ATM matrix, d(σ_ATM, ρ, ν)/dσ_ATM, zero where σ_ATM is fitted.