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Advanced Mathematics for Trading Research

$C = S_0 N(d_1) - K e^{-rT} N(d_2)$

$d_1 = \frac{\ln(S_0 / K) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}$

// Black-Scholes formula for European call option
double BlackScholesCall(double S, double K, double T, double r, double sigma) {
    double d1 = (log(S / K) + (r + 0.5 * sigma * sigma) * T) / (sigma * sqrt(T));
    double d2 = d1 - sigma * sqrt(T);
    return S * NormalCDF(d1) - K * exp(-r * T) * NormalCDF(d2);
}

$C - P = S_0 - K e^{-rT}$

// Put-Call Parity relation
double PutCallParity(double callPrice, double S, double K, double r, double T) {
    return callPrice - S + K * exp(-r * T);
}

$N(x) = \frac{1}{2} \operatorname{erfc}\left(-\frac{x}{\sqrt{2}}\right)$

// Approximate CDF for the standard normal distribution
double NormalCDF(double x) {
    return 0.5 * erfc(-x / sqrt(2));
}