// Bisection Algorithm for Black Scholes Implied Volatility =================================================================================
public double BisecBSIV(OpSet settings, double K, double T, double a, double b, double MktPrice, double Tol, int MaxIter)
{
BlackScholesPrice BS = new BlackScholesPrice();
double S = settings.S;
double rf = settings.r;
double q = settings.q;
string PutCall = settings.PutCall;
double lowCdif = MktPrice - BS.BlackScholes(S, K, T, rf, q, a, PutCall);
double highCdif = MktPrice - BS.BlackScholes(S, K, T, rf, q, b, PutCall);
double BSIV = 0.0;
double midP;
if (lowCdif * highCdif > 0.0)
{
BSIV = -1.0;
}
else
{
for (int x = 0; x <= MaxIter; x++)
{
midP = (a + b) / 2.0;
double midCdif = MktPrice - BS.BlackScholes(S, K, T, rf, q, midP, PutCall);
if (Math.Abs(midCdif) < Tol)
{
break;
}
else
{
if (midCdif > 0.0)
{
a = midP;
}
else
{
b = midP;
}
}
BSIV = midP;
}
}
return(BSIV);
}
public double BGMApproxPrice(HParam param, OpSet settings, double K, double T)
{
double kappa = param.kappa;
double theta = param.theta;
double sigma = param.sigma;
double v0 = param.v0;
double rho = param.rho;
double lambda = 0.0;
double S = settings.S;
double rf = settings.r;
double q = settings.q;
string PutCall = settings.PutCall;
// Log Spot price
double x = Math.Log(settings.S);
// Integrated variance
double wT = (v0 - theta) * (1 - Math.Exp(-kappa * T)) / kappa + theta * T;
double y = wT;
// Black Scholes Put Price
BlackScholesPrice BS = new BlackScholesPrice();
double g = Math.Pow(y, -0.5) * (-x + Math.Log(K) - (rf - q) * T) - 0.5 * Math.Sqrt(y);
double f = Math.Pow(y, -0.5) * (-x + Math.Log(K) - (rf - q) * T) + 0.5 * Math.Sqrt(y);
double BSPut = K * Math.Exp(-rf * T) * BS.NormCDF(f) - S * Math.Exp(-q * T) * BS.NormCDF(g);
// Shortcut notation
double k = kappa;
double kT = kappa * T;
double ekT = Math.Exp(k * T);
double ekTm = Math.Exp(-k * T);
// Coefficients for the expansion
double a1T = (rho * sigma * ekTm / k / k) * (v0 * (-kT + ekT - 1.0) + theta * (kT + ekT * (kT - 2.0) + 2.0));
double a2T = (rho * rho * sigma * sigma * ekTm / 2.0 / (k * k * k)) * (v0 * (-kT * (kT + 2.0) + 2.0 * ekT - 2.0) + theta * (2.0 * ekT * (kT - 3.0) + kT * (kT + 4.0) + 6.0));
double b0T = (sigma * sigma * Math.Exp(-2.0 * kT) / 4.0 / (k * k * k)) * (v0 * (-4.0 * ekT * kT + 2.0 * Math.Exp(2.0 * kT) - 2.0) + theta * (4.0 * ekT * (kT + 1.0) + Math.Exp(2.0 * kT) * (2.0 * kT - 5.0) + 1.0));
double b2T = a1T * a1T / 2.0;
// Normal pdf, phi(f) and phi(g)
double pi = Math.PI;
double phif = Math.Exp(-f * f / 2.0) / Math.Sqrt(2.0 * pi);
double phig = Math.Exp(-g * g / 2.0) / Math.Sqrt(2.0 * pi);
// Derivatives of f and g
double fx = -Math.Pow(y, -0.5);
double fy = -0.5 / y * g;
double gx = fx;
double gy = -0.5 / y * f;
// The cdf PHI(f) and PHI(g)
double PHIf = BS.NormCDF(f);
double PHIg = BS.NormCDF(g);
// Derivatives of the pdf phi(f)
double phifx = Math.Pow(y, -0.5) * f * phif;
double phify = 0.5 / y * f * g * phif;
// Derivatives of the cdf PHI(f)
double PHIfxy = 0.5 * Math.Pow(y, -1.5) * phif * (1.0 - f * g);
double PHIfx2y = 0.5 * Math.Pow(y, -2.0) * phif * (2.0 * f + g - f * f * g);
double PHIfy2 = 0.5 * Math.Pow(y, -2.0) * phif * (g + f / 2.0 - f * g * g / 2.0);
double PHIfx2y2 = 0.5 * ((Math.Pow(y, -2.0) * phify - 2.0 * Math.Pow(y, -3.0) * phif) * (2.0 * f + g - f * f * g) +
Math.Pow(y, -2.0) * phif * (2.0 * fy + gy - 2.0 * f * fy * g - f * f * gy));
// Derivatives of the pdf phi(g)
double phigx = Math.Pow(y, -0.5) * g * phig;
double phigy = 0.5 / y * f * g * phig;
// Derivatives of cdf PHI(g)
double PHIgx = -phig *Math.Pow(y, -0.5);
double PHIgy = -0.5 * f * phig / y;
double PHIgxy = 0.5 * Math.Pow(y, -1.5) * phig * (1.0 - f * g);
double PHIgx2y = 0.5 * Math.Pow(y, -2.0) * phig * (2.0 * g + f - g * g * f);
double PHIgy2 = 0.5 * Math.Pow(y, -2.0) * phig * (f + g / 2.0 - g * f * f / 2.0);
double PHIgxy2 = 0.5 * Math.Pow(y, -2.0) * (phigx * (f + g / 2.0 - f * f * g / 2.0) + phig * (fx + gx / 2.0 - f * fx * g / 2.0 - f * f * gx / 2.0));
double PHIgx2y2 = 0.5 * ((Math.Pow(y, -2.0) * phigy - 2.0 * Math.Pow(y, -3.0) * phig) * (2.0 * g + f - g * g * f) +
Math.Pow(y, -2.0) * phig * (2.0 * gy + fy - 2.0 * g * gy * f - g * g * fy));
// Derivatives of Black-Scholes Put
double dPdxdy = K * Math.Exp(-rf * T) * PHIfxy - Math.Exp(-q * T) * S * (PHIgy + PHIgxy);
double dPdx2dy = K * Math.Exp(-rf * T) * PHIfx2y - Math.Exp(-q * T) * S * (PHIgy + 2.0 * PHIgxy + PHIgx2y);
double dPdy2 = K * Math.Exp(-rf * T) * PHIfy2 - Math.Exp(-q * T) * S * PHIgy2;
double dPdx2dy2 = K * Math.Exp(-rf * T) * PHIfx2y2 - Math.Exp(-q * T) * S * (PHIgy + 2.0 * PHIgxy + PHIgx2y + PHIgy2 + 2.0 * PHIgxy2 + PHIgx2y2);
// Benhamou, Gobet, Miri expansion
double Put = BSPut + a1T * dPdxdy + a2T * dPdx2dy + b0T * dPdy2 + b2T * dPdx2dy2;
// Return the put or the call by put-call parity
if (PutCall == "P")
{
return(Put);
}
else
{
return(Put - K * Math.Exp(-rf * T) + S * Math.Exp(-q * T));
}
}
