// Bisection Algorithm for Black Scholes Implied Volatility =================================================================================
public double BisecBSIV(string PutCall, double S, double K, double rf, double q, double T, double a, double b, double MktPrice, double Tol, int MaxIter)
{
BlackScholesPrice BS = new BlackScholesPrice();
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 VarianceSwap(double[] KCI, double[] CallVI, double[] KPI, double[] PutVI, double S, double T, double rf, double q)
{
// Demeterfi et al (1999) fair strike of a variance swap
// INPUTS
// KCI = Grid of call strikes
// CallVI = Interpolated call implied vol along the grid KCI
// KPI = Grid of put strikes
// PutVI = Interpolated put implied vol along the grid KPI
// S = Spot price
// T = Maturity
// rf = interest rate
// q = dividend yield
// Do the required calculations on calls.
// Rename CallK and CallV for convenience.
int n = CallVI.Length;
// Take ATM as the boundary point
double Sb = S;
// Start the replication algorithm
double[] Temp = new double[n - 1];
double[] CallWeight = new double[n - 1];
double[] CallValue = new double[n - 1];
double[] CallContrib = new double[n - 1];
BlackScholesPrice BS = new BlackScholesPrice();
for (int k = 0; k <= n - 2; k++)
{
Temp[k] = (f(KCI[k + 1], Sb, T) - f(KCI[k], Sb, T)) / (KCI[k + 1] - KCI[k]);
if (k == 0)
{
CallWeight[0] = Temp[0];
}
CallValue[k] = BS.BlackScholes(S, KCI[k], T, rf, q, CallVI[k], "C");
if (k > 0)
{
CallWeight[k] = Temp[k] - Temp[k - 1];
}
CallContrib[k] = CallValue[k] * CallWeight[k];
}
double Pi1 = CallContrib.Sum();
// Do the calculations on puts. Flip the Vectors for Convenience
n = PutVI.Length;
Array.Reverse(KPI);
Array.Reverse(PutVI);
double[] Temp2 = new double[n - 1];
double[] PutWeight = new double[n - 1];
double[] PutValue = new double[n - 1];
double[] PutContrib = new double[n - 1];
for (int k = 0; k <= n - 2; k++)
{
Temp2[k] = (f(KPI[k + 1], Sb, T) - f(KPI[k], Sb, T)) / (KPI[k] - KPI[k + 1]);
if (k == 0)
{
PutWeight[0] = Temp2[0];
}
PutValue[k] = BS.BlackScholes(S, KPI[k], T, rf, q, PutVI[k], "P");
if (k > 0)
{
PutWeight[k] = Temp2[k] - Temp2[k - 1];
}
PutContrib[k] = PutValue[k] * PutWeight[k];
}
double Pi2 = PutContrib.Sum();
// Total cost of the portfolio
double Pi_CP = Pi1 + Pi2;
// Results of the replication
// Estimate of fair variance
double Kvar = 2.0 / T * (rf * T - (S / Sb * Math.Exp(rf * T) - 1.0) - Math.Log(Sb / S)) + Math.Exp(rf * T) * Pi_CP;
return(Kvar);
}