// Calculate the Black Scholes European put option Delta
double BS_Put_Option_Delta(double S, double K, double r, double v, double T)
// Parameters: (S = Current Stock Price, K = Strike Price, r = Risk-Free Rate, v = Volatility of Stock Price, T = Time to Maturity)
{ return Normal_CDF(d_j(1, S, K, r, v, T)) - 1; }

const double Pi = 3.14159265359;
// Standard Normal cumulative distribution function
double Normal_CDF(const double & x)
{
double k = 1.0/(1.0 + 0.2316419 * x);
double k_sum = k * (0.319381530 + k * (-0.356563782 + k * (1.781477937 + k * (-1.821255978 + 1.330274429 * k))));
if (x >= 0.0)
{ return (1.0 - (1.0/(pow(2*Pi,0.5)))*exp(-0.5*x*x) * k_sum); }
else
{ return 1.0 - Normal_CDF(-x); }
}

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