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) . &#xA; For x = log(F/K) <= 0 and total volatility s = sigma sqrt(T) , the beta-space OTM call price is &#xA; B(x, s) = 0.5 * (exp(x/2) * erfc(q1) - exp(-x/2) * erfc(q2))&#xA; q1 = -(x/s + s/2) / sqrt(2)&#xA; q2 = -(x/s - s/2) /…