🤖 AI Summary
This work addresses the long-standing challenge of explicitly solving for implied volatility in the Black–Scholes model by introducing the first closed-form analytical formula that requires neither iteration, approximation, nor series expansion. The approach reinterprets the option price as the survival probability of an inverse Gaussian distribution and leverages its quantile function to analytically invert the Black–Scholes framework, yielding an explicit expression for implied volatility that depends solely on observable market variables. Numerical experiments demonstrate that the method achieves machine precision and offers a computational speedup of approximately 3.4× compared to the current state-of-the-art benchmark.
📝 Abstract
This paper identifies what appears to be the first explicit formula for Black-Scholes implied volatility, resolving a 50-year-old problem in option pricing. The key observation is that the call price can be written as a survival probability of an inverse Gaussian distribution. Inverting this identity expresses implied volatility directly through the corresponding quantile function. The formula uses only observable option inputs and requires no initial guess, iterative inversion, approximation, asymptotic expansion, or infinite series. Numerical tests recover implied volatility to machine precision and show the formula to be about 3.4 times faster than a current state-of-the-art reference benchmark.