An Explicit Solution to Black–Scholes Implied Volatility

Battle Count: 57

Implied volatility is the central coordinate in which option markets are measured, traded, and managed. This paper provides an exact, faster (3x speedup over state-of-the-art LBR), and more interpretable method for computing implied volatility from observed prices. The variance-space representation directly benefits: (1) real-time options pricing and quoting, (2) volatility surface calibration for trading strategies, (3) Greeks computation for hedging, (4) no-arbitrage validation of market data, (5) risk management systems requiring fast and accurate IV inversion. The first-passage-time interpretation also offers new intuition for option dynamics relevant to trading.

The analytical formula is elegant but implementation requires: (1) evaluation of the GIG quantile function (non-trivial special function), (2) an optimized rational seed approximation for the GIG quantile, (3) a single Halley refinement step, (4) proper handling of ITM/OTM branches via the η_k parameter, (5) put-call parity for puts. The paper states code will be available. The main complexity lies in the GIG quantile evaluation, which is a well-studied but non-trivial numerical task. The variance-space approach simplifies the iteration structure compared to volatility-space methods.

The paper provides complete analytical derivations, explicit formulas, and detailed numerical benchmark methodology (randomized grid of 100,000 cases, deterministic tail grid of 246 cases). The code for numerical experiments is stated to be made available as online supplementary material upon publication. The benchmark uses Jäckel (2024)'s Let's Be Rational (LBR) reference implementation. Hardware and software environment specified (11th Gen Intel Core i5-1145G7, GCC, Linux/WSL2, C++).