Explicit Rational Formulae for Bachelier (Normal) Implied Volatility

Battle Count: 78

Highly relevant for quantitative trading in interest rate derivatives, swaption markets, and any environment where normal (Bachelier) volatility is the market convention. The explicit formulae eliminate iterative root-finding, enabling faster calibration, risk computation, and large-scale grid calculations. Particularly relevant for EUR, USD, GBP, JPY rate markets and commodity markets with negative prices. The ~3x speedup over Jäckel's method with near-machine-epsilon accuracy makes it production-ready for latency-sensitive applications.

The formulae use only arithmetic operations and a single logarithm in the OTM routing variable. The branch structure is straightforward (ITM/near-ATM vs. three OTM zones). Coefficient tables are provided. The main complexity lies in correctly implementing the branch logic, the erfcx-based reference price for validation, and choosing between standard Horner, FMA, or compensated Horner evaluation. The LFK-2026C variant is simpler (fewer coefficients in ITM branch). Overall, this is a well-specified numerical algorithm that is straightforward to implement in any language.

The paper provides complete coefficient tables in appendices (Appendices H, I, J) for all rational functions. The methodology is fully specified with branch boundaries, polynomial degrees, and evaluation algorithms (Horner, FMA, compensated Horner). Timing benchmarks are provided on a specific CPU (Intel Core Ultra 5 245KF, Java 25.0.3). However, no code repository is explicitly linked, and the reference price implementation (Jäckel's) is described but not provided as code. The test methodology (random sampling of d in [0,35], 40M cases per bucket) is well-described.