The Role of Erfcx and Erfc in the Black-Scholes Formula

This note compares special-function choices at the level that matters to the implied volatility solvers : the normalized Black beta price, not standalone erfcx(x) . For x = log(F/K) <= 0 and total volatility s = sigma sqrt(T) , the beta-space OTM call price is B(x, s) = 0.5 * (exp(x/2) * erfc(q1) - exp(-x/2) * erfc(q2)) q1 = -(x/s + s/2) / sqrt(2) q2 = -(x/s - s/2) / sqrt(2) The corresponding forward-normalized OTM-call price is c = B(x, s) / sqrt(ex) , with ex = exp(x) .