regit-svi

Arbitrage-free SVI volatility surfaces. Zero-dependency, pure Rust.

What it does

regit-svi parametrises the implied volatility smile and surface with the Stochastic Volatility Inspired (SVI) family, calibrates it to market quotes, and certifies the result free of static arbitrage.

It covers three parametrisations — Raw SVI (Gatheral 2004), SVI Jump-Wings, and the Surface SVI / SSVI of Gatheral & Jacquier (2014) — two calibration engines, and explicit butterfly and calendar-spread arbitrage checks.

Every formula is hand-rolled from primary paper sources with no external dependencies. A regulator, quant auditor, or new engineer can open any source file and trace every number to a citable derivation in MATH.md.

Why this crate exists

A volatility surface is the input to every option pricer and every risk engine. Markets quote implied volatilities at a sparse, discrete grid of strikes and maturities — but pricing and risk need a continuous surface.

The naive fix is to interpolate. Interpolation silently injects static arbitrage. Spline a smile through five quotes and the implied risk-neutral density goes negative between them — a butterfly spread with negative cost. Interpolate across maturities independently and the total-variance curves cross — a calendar spread that pays to hold. A surface with either defect produces mispriced exotics, unstable Greeks, and risk numbers that cannot be trusted, and the defect is invisible unless you test for it.

SVI solves this at the parametrisation level. A raw SVI slice is a five-parameter curve in total variance whose convexity and wing behaviour are controlled by construction. SSVI goes further: it parametrises the whole surface so that absence of butterfly and calendar-spread arbitrage reduces to closed-form inequalities on three parameters — the surface is arbitrage-free by construction, not by repair.

regit-svi implements that machinery, and — crucially — ships the verification alongside it: the butterfly function g(k) and the calendar-monotonicity test are exposed as first-class checks, so any surface, however it was built, can be certified or flagged.

This sits within Regit OS: regit-svi is the surface layer. It consumes implied volatilities recovered by regit-blackscholes and produces a clean, arbitrage-checked surface for pricing and risk downstream — quant-proven math (Gatheral, Gatheral-Jacquier, primary-source derivations) made auditable.

Quick start

[dependencies]
regit-svi = "1.0"

use regit_svi::{Quote, calibration::quasi_explicit};

// Market quotes for one maturity: (log-moneyness, total variance, weight).
let quotes = [
    Quote::new(-0.20, 0.0512, 1.0).unwrap(),
    Quote::new(-0.10, 0.0432, 1.0).unwrap(),
    Quote::new( 0.00, 0.0400, 1.0).unwrap(),
    Quote::new( 0.10, 0.0420, 1.0).unwrap(),
    Quote::new( 0.20, 0.0480, 1.0).unwrap(),
];

// Calibrate a raw SVI slice (quasi-explicit, de Marco-Martini).
let fit = quasi_explicit::calibrate(&quotes).unwrap();
println!("RMSE: {:.2e}", fit.rmse);

// Evaluate total variance and implied volatility anywhere on the slice.
let w   = fit.slice.total_variance(0.05);
let vol = fit.slice.implied_vol(0.05, 1.0).unwrap(); // time to expiry = 1y

// The calibration result carries a butterfly-arbitrage certificate.
assert!(fit.butterfly_free);

See examples/quickstart.rs for a complete working example covering conversions, the SSVI surface fit, and the arbitrage checks.

Parametrisations

All conversions that exist between them are closed-form: Raw ↔ Jump-Wings is bijective; every SSVI slice maps to a raw slice. See MATH.md §4–6.

Calibration

Two engines, by design complementary:

The default pipeline calibrates with the quasi-explicit method, then optionally refines with Levenberg–Marquardt. SSVI surfaces are calibrated jointly across maturities with the Theorem 4.1/4.2 no-arbitrage inequalities enforced throughout. See MATH.md §10–12.

Arbitrage checks

The risk-neutral density p(k) implied by a slice is exposed directly for inspection. See MATH.md §7–9.

Architecture

src/
  lib.rs                       # Module declarations + re-exports
  types.rs                     # Log-moneyness, total variance, market quotes
  errors.rs                    # Typed errors — parametrisation, conversion, calibration
  math.rs                      # Nelder-Mead, Levenberg-Marquardt, linear LS, Brent

  raw.rs                       # Raw SVI: w(k), derivatives, ATM quantities
  jw.rs                        # SVI Jump-Wings parametrisation
  ssvi.rs                      # Surface SVI: w(k, θ), φ functions, no-arb conditions
  convert.rs                   # Raw <-> JW, SSVI -> Raw conversions

  arbitrage.rs                 # Butterfly g(k) + calendar-spread checks
  density.rs                   # Risk-neutral density from a slice

  calibration/
    mod.rs                     # Default pipeline + CalibrationResult
    quasi_explicit.rs          # de Marco-Martini / Zeliade quasi-explicit
    least_squares.rs           # Direct Levenberg-Marquardt
    ssvi.rs                    # Joint SSVI surface calibration

  surface.rs                   # Multi-slice surface assembly + interpolation

One file, one domain. Each function is pure, deterministic, and composable.

Testing

cargo test                      # 258 tests
cargo run --example quickstart  # End-to-end slice + surface workflow
cargo bench                     # Criterion benchmarks

145 unit tests — golden values, finite-difference derivative checks, parametrisation round-trips, the optimiser primitives (Nelder-Mead on Rosenbrock, Brent on known roots), and calibration parameter recovery.

32 integration tests across seven suites: hand-computed golden anchors; Raw ↔ Jump-Wings and SSVI → Raw round-trip identities; an arbitrage oracle including the Axel Vogt slice (a documented butterfly-arbitrage example); risk-neutral density mass and positivity; synthetic-data calibration recovery for both engines and the SSVI surface fit; multi-slice surface assembly; and proptest invariants (w ≥ 0, w'' > 0, Raw→JW→Raw identity, SSVI Theorem 4.2 slices passing the butterfly scan).

81 doc-tests — every public item carries a runnable example.

Code quality

• #![forbid(unsafe_code)] crate-wide
• clippy::pedantic with zero warnings
• Every public function documented with its mathematical reference
• No unwrap() or panic!() in library code — all failure paths typed
• Deterministic: same input produces bit-identical output
• WASM-clean: cargo build --target wasm32-unknown-unknown with no changes
• 258 tests — unit, integration, proptest invariants, doc-tests — plus criterion benchmarks (see Testing)

Dependencies

Runtime: zero. Only std. No nalgebra, no argmin, no libm, no FFI. Every optimiser and solver — Nelder-Mead, Levenberg-Marquardt, linear least-squares, Brent — is hand-rolled from its primary source.

License and supply-chain policy is enforced via cargo-deny (deny.toml). No copyleft dependencies.

Algorithms

All implemented from primary paper sources. No ports from Python, no reading existing Rust crates.

Documentation

• MATH.md — Full mathematical derivations for every algorithm
• CHANGELOG.md — Release history
• SECURITY.md — Vulnerability disclosure policy

License

Apache License 2.0. See LICENSE and NOTICE.

Copyright 2026 Regit.io — Nicolas Koenig

Part of Regit OS — the operating system for investment products. From Luxembourg.