// 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);
}
// Parameters for the shortest maturity appear first in the vectors theta, sigma, and rho.
// Parameters for the longest maturity appear last.
public double BGMApproxPriceTD(double kappa, double v0, double[] theta, double[] sigma, double[] rho, OPSet settings, double K, double[] T)
{
int NT = T.Length;
double[] a1T = new double[NT];
double[] a2T = new double[NT];
double[] b0T = new double[NT];
double[] b2T = new double[NT];
// First set of coefficients
a1T[0] = -rho[0] * sigma[0] * (2.0 * theta[0] * Math.Exp(kappa * T[0]) + v0 * T[0] * kappa + v0 - theta[0] * T[0] * kappa * Math.Exp(kappa * T[0]) - theta[0] * kappa * T[0] - 2.0 * theta[0] - v0 * Math.Exp(kappa * T[0])) / Math.Exp(kappa * T[0]) / Math.Pow(kappa, 2.0);
a2T[0] = -0.5 * rho[0] * rho[0] * sigma[0] * sigma[0] * (Math.Pow(kappa, 2.0) * v0 * Math.Pow(T[0], 2.0) + 6.0 * theta[0] * Math.Exp(kappa * T[0]) + 2.0 * v0 * T[0] * kappa + 2.0 * v0 - Math.Pow(kappa, 2.0) * theta[0] * Math.Pow(T[0], 2.0) - 4.0 * theta[0] * kappa * T[0] - 2.0 * theta[0] * T[0] * kappa * Math.Exp(kappa * T[0]) - 6.0 * theta[0] - 2.0 * v0 * Math.Exp(kappa * T[0])) / Math.Exp(kappa * T[0]) / Math.Pow(kappa, 3.0);
b0T[0] = -0.25 * sigma[0] * sigma[0] * (5.0 * theta[0] * Math.Pow(Math.Exp(kappa * T[0]), 2.0) + 4.0 * Math.Exp(kappa * T[0]) * v0 * T[0] * kappa - 4.0 * theta[0] * T[0] * kappa * Math.Exp(kappa * T[0]) - 2.0 * theta[0] * T[0] * kappa * Math.Pow(Math.Exp(kappa * T[0]), 2.0) + 2.0 * v0 - theta[0] - 2.0 * v0 * Math.Pow(Math.Exp(kappa * T[0]), 2.0) - 4.0 * theta[0] * Math.Exp(kappa * T[0])) / Math.Pow(Math.Exp(kappa * T[0]), 2.0) / Math.Pow(kappa, 3.0);
b2T[0] = a1T[0] * a1T[0] / 2.0;
double A1 = 0.0, A2 = 0.0, B0 = 0.0;
int j;
// Coefficients under multiple maturities
if (NT >= 2)
{
for (int t = 0; t <= NT - 2; t++)
{
// Coefficients for a1T[t+1]
j = 0;
A1 += rho[j] * sigma[j] * (-Math.Exp(-kappa * T[t]) + Math.Exp(-kappa * T[t + 1])) * (theta[j] - v0 * T[j] * kappa + theta[j] * T[j] * kappa - theta[j] * Math.Exp(kappa * T[j])) / Math.Pow(kappa, 2.0);
for (j = 1; j <= t; j++)
{
A1 += -rho[j] * sigma[j] * (-Math.Exp(-kappa * T[t]) + Math.Exp(-kappa * T[t + 1])) * (-v0 * T[j - 1] * kappa + theta[j] * T[j - 1] * kappa - theta[j] * Math.Exp(kappa * T[j - 1]) + v0 * T[j] * kappa - theta[j] * T[j] * kappa + theta[j] * Math.Exp(kappa * T[j])) / Math.Pow(kappa, 2.0);
}
j = t + 1;
A1 += -rho[j] * sigma[j] * (-v0 * Math.Exp(kappa * T[t + 1]) - v0 * T[t] * kappa * Math.Exp(kappa * T[t]) + theta[j] * Math.Exp(kappa * T[t + 1]) + theta[j] * T[t] * kappa * Math.Exp(kappa * T[t]) + theta[j] * T[t] * kappa * Math.Exp(kappa * (T[t] + T[t + 1])) - theta[j] * Math.Exp(2.0 * kappa * T[t]) + Math.Exp(kappa * T[t]) * v0 - Math.Exp(kappa * (T[t] + T[t + 1])) * theta[j] * T[t + 1] * kappa + theta[j] * Math.Exp(kappa * (T[t] + T[t + 1])) + Math.Exp(kappa * T[t]) * v0 * kappa * T[t + 1] - theta[j] * Math.Exp(kappa * T[t]) - Math.Exp(kappa * T[t]) * theta[j] * kappa * T[t + 1]) / Math.Pow(kappa, 2.0) * Math.Exp(-kappa * (T[t] + T[t + 1]));
a1T[t + 1] = a1T[t] + A1;
// Coefficients for a2T[t+1]
j = 0;
A2 += -Math.Pow(rho[j], 2.0) * Math.Pow(sigma[j], 2.0) * (Math.Exp(-kappa * T[t]) + Math.Exp(-kappa * T[t + 1]) * T[t] * kappa - Math.Exp(-kappa * T[t + 1]) - Math.Exp(-kappa * T[t + 1]) * kappa * T[t + 1]) * (theta[j] - v0 * T[j] * kappa + theta[j] * T[j] * kappa - theta[j] * Math.Exp(kappa * T[j])) / Math.Pow(kappa, 3.0)
+ 1.0 / 2.0 * Math.Pow(rho[j], 2.0) * Math.Pow(sigma[j], 2.0) * (-Math.Exp(-kappa * T[t]) + Math.Exp(-kappa * T[t + 1])) * (2.0 * kappa * theta[j] * T[t] + 2.0 * theta[j] - 2.0 * v0 * T[t] * T[j] * Math.Pow(kappa, 2.0) + v0 * Math.Pow(T[j], 2.0) * Math.Pow(kappa, 2.0) + 2.0 * theta[j] * T[t] * T[j] * Math.Pow(kappa, 2.0) - theta[j] * Math.Pow(T[j], 2.0) * Math.Pow(kappa, 2.0) - 2.0 * theta[j] * T[t] * Math.Exp(kappa * T[j]) * kappa + 2.0 * theta[j] * Math.Exp(kappa * T[j]) * kappa * T[j] - 2.0 * theta[j] * Math.Exp(kappa * T[j])) / Math.Pow(kappa, 3.0);
for (j = 1; j <= t; j++)
{
A2 += Math.Pow(rho[j], 2.0) * Math.Pow(sigma[j], 2.0) * (Math.Exp(-kappa * T[t]) + Math.Exp(-kappa * T[t + 1]) * T[t] * kappa - Math.Exp(-kappa * T[t + 1]) - Math.Exp(-kappa * T[t + 1]) * kappa * T[t + 1]) * (-v0 * T[j - 1] * kappa + theta[j] * T[j - 1] * kappa - theta[j] * Math.Exp(kappa * T[j - 1]) + v0 * T[j] * kappa - theta[j] * T[j] * kappa + theta[j] * Math.Exp(kappa * T[j])) / Math.Pow(kappa, 3.0)
- 1.0 / 2.0 * Math.Pow(rho[j], 2.0) * Math.Pow(sigma[j], 2.0) * (-Math.Exp(-kappa * T[t]) + Math.Exp(-kappa * T[t + 1])) * (-2.0 * v0 * T[t] * T[j - 1] * Math.Pow(kappa, 2.0) + v0 * Math.Pow(T[j - 1], 2.0) * Math.Pow(kappa, 2.0) + 2.0 * theta[j] * T[t] * T[j - 1] * Math.Pow(kappa, 2.0) - theta[j] * Math.Pow(T[j - 1], 2.0) * Math.Pow(kappa, 2.0) - 2.0 * theta[j] * T[t] * Math.Exp(kappa * T[j - 1]) * kappa + 2.0 * theta[j] * Math.Exp(kappa * T[j - 1]) * kappa * T[j - 1] - 2.0 * theta[j] * Math.Exp(kappa * T[j - 1]) + 2.0 * v0 * T[t] * T[j] * Math.Pow(kappa, 2.0) - v0 * Math.Pow(T[j], 2.0) * Math.Pow(kappa, 2.0) - 2.0 * theta[j] * T[t] * T[j] * Math.Pow(kappa, 2.0) + theta[j] * Math.Pow(T[j], 2.0) * Math.Pow(kappa, 2.0) + 2.0 * theta[j] * T[t] * Math.Exp(kappa * T[j]) * kappa - 2.0 * theta[j] * Math.Exp(kappa * T[j]) * kappa * T[j] + 2.0 * theta[j] * Math.Exp(kappa * T[j])) / Math.Pow(kappa, 3.0);
}
j = t + 1;
A2 += 1.0 / 2.0 * Math.Pow(rho[j], 2.0) * Math.Pow(sigma[j], 2.0) * (2.0 * v0 * Math.Exp(kappa * T[t + 1]) - v0 * Math.Pow(T[t], 2.0) * Math.Pow(kappa, 2.0) * Math.Exp(kappa * T[t]) + 2.0 * v0 * T[t] * kappa * Math.Exp(kappa * T[t]) + 2.0 * v0 * Math.Pow(kappa, 2.0) * T[t + 1] * T[t] * Math.Exp(kappa * T[t]) - 2.0 * theta[j] * Math.Exp(kappa * T[t + 1]) + theta[j] * Math.Pow(T[t], 2.0) * Math.Pow(kappa, 2.0) * Math.Exp(kappa * T[t]) - 2.0 * theta[j] * T[t] * kappa * Math.Exp(kappa * T[t]) - 2.0 * theta[j] * Math.Pow(kappa, 2.0) * T[t + 1] * T[t] * Math.Exp(kappa * T[t]) - 2.0 * theta[j] * T[t] * kappa * Math.Exp(kappa * (T[t] + T[t + 1])) - 2.0 * theta[j] * Math.Exp(2.0 * kappa * T[t]) * kappa * T[t] + 4.0 * theta[j] * Math.Exp(2.0 * kappa * T[t]) + 2.0 * theta[j] * T[t + 1] * Math.Exp(2.0 * kappa * T[t]) * kappa - 2.0 * Math.Exp(kappa * T[t]) * v0 - 2.0 * Math.Exp(kappa * T[t]) * v0 * kappa * T[t + 1] + 2.0 * Math.Exp(kappa * (T[t] + T[t + 1])) * theta[j] * T[t + 1] * kappa - Math.Exp(kappa * T[t]) * Math.Pow(kappa, 2.0) * v0 * Math.Pow(T[t + 1], 2.0) - 4.0 * theta[j] * Math.Exp(kappa * (T[t] + T[t + 1])) + Math.Exp(kappa * T[t]) * Math.Pow(kappa, 2.0) * theta[j] * Math.Pow(T[t + 1], 2.0) + 2.0 * theta[j] * Math.Exp(kappa * T[t]) + 2.0 * Math.Exp(kappa * T[t]) * theta[j] * kappa * T[t + 1]) / Math.Pow(kappa, 3.0) * Math.Exp(-kappa * (T[t] + T[t + 1]));
a2T[t + 1] = a2T[t] + A2;
// Coefficients for b0T[t+1]
j = 0;
B0 += -1.0 / 4.0 * Math.Pow(sigma[j], 2.0) * (-Math.Exp(-2.0 * kappa * T[t]) + 2.0 * Math.Exp(-kappa * (T[t] + T[t + 1])) - Math.Exp(-2.0 * kappa * T[t + 1])) * (-2.0 * v0 + theta[j] + 2.0 * v0 * Math.Exp(kappa * T[j]) - 2.0 * theta[j] * Math.Exp(kappa * T[j]) + theta[j] * Math.Exp(2.0 * kappa * T[j])) / Math.Pow(kappa, 3.0)
+ 1.0 / 2.0 * Math.Pow(sigma[j], 2.0) * (-theta[j] * Math.Exp(kappa * T[t + 1]) + theta[j] * Math.Exp(kappa * T[t]) + 2.0 * v0 * Math.Exp(kappa * T[t + 1]) - 2.0 * Math.Exp(kappa * T[t]) * v0 - 2.0 * theta[j] * Math.Exp(kappa * (T[t] + T[t + 1])) + 2.0 * theta[j] * Math.Exp(2.0 * kappa * T[t]) + 2.0 * Math.Exp(kappa * (T[t] + T[t + 1])) * v0 * T[j] * kappa - 2.0 * Math.Exp(kappa * (T[t + 1] + T[j])) * v0 - 2.0 * Math.Exp(kappa * (T[t] + T[t + 1])) * theta[j] * T[j] * kappa + 2.0 * Math.Exp(kappa * (T[t + 1] + T[j])) * theta[j] + 2.0 * Math.Exp(kappa * (T[t] + T[t + 1] + T[j])) * theta[j] - Math.Exp(kappa * (T[t + 1] + 2.0 * T[j])) * theta[j] - 2.0 * Math.Exp(2.0 * kappa * T[t]) * v0 * T[j] * kappa + 2.0 * Math.Exp(kappa * (T[t] + T[j])) * v0 + 2.0 * Math.Exp(2.0 * kappa * T[t]) * theta[j] * T[j] * kappa - 2.0 * Math.Exp(kappa * (T[t] + T[j])) * theta[j] - 2.0 * Math.Exp(kappa * (2.0 * T[t] + T[j])) * theta[j] + Math.Exp(kappa * (T[t] + 2.0 * T[j])) * theta[j]) * Math.Exp(-kappa * (2.0 * T[t] + T[t + 1])) / Math.Pow(kappa, 3.0);
for (j = 1; j <= t; j++)
{
B0 += -1.0 / 4.0 * Math.Pow(sigma[j], 2.0) * (-Math.Exp(-2.0 * kappa * T[t]) + 2.0 * Math.Exp(-kappa * (T[t] + T[t + 1])) - Math.Exp(-2.0 * kappa * T[t + 1])) * (-2.0 * v0 * Math.Exp(kappa * T[j - 1]) + 2.0 * theta[j] * Math.Exp(kappa * T[j - 1]) - theta[j] * Math.Exp(2.0 * kappa * T[j - 1]) + 2.0 * v0 * Math.Exp(kappa * T[j]) - 2.0 * theta[j] * Math.Exp(kappa * T[j]) + theta[j] * Math.Exp(2.0 * kappa * T[j])) / Math.Pow(kappa, 3.0)
+ 1.0 / 2.0 * Math.Pow(sigma[j], 2.0) * (Math.Exp(-kappa * T[t]) - Math.Exp(-kappa * T[t + 1])) * (-2.0 * v0 * T[j - 1] * kappa * Math.Exp(kappa * T[t]) + 2.0 * v0 * Math.Exp(kappa * T[j - 1]) + 2.0 * theta[j] * T[j - 1] * kappa * Math.Exp(kappa * T[t]) - 2.0 * theta[j] * Math.Exp(kappa * T[j - 1]) - 2.0 * theta[j] * Math.Exp(kappa * (T[j - 1] + T[t])) + theta[j] * Math.Exp(2.0 * kappa * T[j - 1]) + 2.0 * v0 * T[j] * kappa * Math.Exp(kappa * T[t]) - 2.0 * v0 * Math.Exp(kappa * T[j]) - 2.0 * theta[j] * T[j] * kappa * Math.Exp(kappa * T[t]) + 2.0 * theta[j] * Math.Exp(kappa * T[j]) + 2.0 * Math.Exp(kappa * (T[t] + T[j])) * theta[j] - theta[j] * Math.Exp(2.0 * kappa * T[j])) * Math.Exp(-kappa * T[t]) / Math.Pow(kappa, 3.0);
}
j = t + 1;
B0 += -1.0 / 4.0 * Math.Pow(sigma[j], 2.0) * (-2.0 * v0 * Math.Exp(2.0 * kappa * T[t + 1]) - 4.0 * v0 * T[t] * kappa * Math.Exp(kappa * (T[t] + T[t + 1])) + 2.0 * v0 * Math.Exp(2.0 * kappa * T[t]) + 2.0 * theta[j] * Math.Exp(2.0 * kappa * T[t + 1]) + 4.0 * theta[j] * T[t] * kappa * Math.Exp(kappa * (T[t] + T[t + 1])) - 2.0 * theta[j] * Math.Exp(2.0 * kappa * T[t]) + 2.0 * theta[j] * T[t] * kappa * Math.Exp(kappa * (T[t] + 2.0 * T[t + 1])) - 4.0 * theta[j] * Math.Exp(kappa * (2.0 * T[t] + T[t + 1])) + theta[j] * Math.Exp(3 * kappa * T[t]) + 3.0 * theta[j] * Math.Exp(kappa * (T[t] + 2.0 * T[t + 1])) + 4.0 * Math.Exp(kappa * (T[t] + T[t + 1])) * v0 * kappa * T[t + 1] - 4.0 * Math.Exp(kappa * (T[t] + T[t + 1])) * theta[j] * T[t + 1] * kappa - 2.0 * Math.Exp(kappa * (T[t] + 2.0 * T[t + 1])) * theta[j] * T[t + 1] * kappa) / Math.Pow(kappa, 3.0) * Math.Exp(-kappa * (T[t] + 2.0 * T[t + 1]));
b0T[t + 1] = b0T[t] + B0;
// Coefficients for b2T[t+1]
b2T[t + 1] = a1T[t + 1] * a1T[t + 1] / 2.0;
}
}
// Coefficients for the expansion are the last ones in the iterations
double A1T = a1T[NT - 1];
double A2T = a2T[NT - 1];
double B0T = b0T[NT - 1];
double B2T = b2T[NT - 1];
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[NT - 1]) * (1 - Math.Exp(-kappa * T[NT - 1])) / kappa + theta[NT - 1] * T[NT - 1];
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[NT - 1]) - 0.5 * Math.Sqrt(y);
double f = Math.Pow(y, -0.5) * (-x + Math.Log(K) - (rf - q) * T[NT - 1]) + 0.5 * Math.Sqrt(y);
double BSPut = K * Math.Exp(-rf * T[NT - 1]) * BS.NormCDF(f) - S * Math.Exp(-q * T[NT - 1]) * BS.NormCDF(g);
// 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[NT - 1]) * PHIfxy - Math.Exp(-q * T[NT - 1]) * S * (PHIgy + PHIgxy);
double dPdx2dy = K * Math.Exp(-rf * T[NT - 1]) * PHIfx2y - Math.Exp(-q * T[NT - 1]) * S * (PHIgy + 2.0 * PHIgxy + PHIgx2y);
double dPdy2 = K * Math.Exp(-rf * T[NT - 1]) * PHIfy2 - Math.Exp(-q * T[NT - 1]) * S * PHIgy2;
double dPdx2dy2 = K * Math.Exp(-rf * T[NT - 1]) * PHIfx2y2 - Math.Exp(-q * T[NT - 1]) * 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[NT - 1]) + S * Math.Exp(-q * T[NT - 1]));
}
}
static void Main(string[] args)
{
// Classes
BGMPrice BGM = new BGMPrice();
BisectionAlgo BA = new BisectionAlgo();
// 32-point Gauss-Laguerre Abscissas and weights
double[] X = new Double[32];
double[] W = new Double[32];
using (TextReader reader = File.OpenText("../../GaussLaguerre32.txt"))
for (int k = 0; k <= 31; k++)
{
string text = reader.ReadLine();
string[] bits = text.Split(' ');
X[k] = double.Parse(bits[0]);
W[k] = double.Parse(bits[1]);
}
// DJIA ETF (ticker DIA) put data
double[,] PutIV = new double[13, 4] {
{ 0.1962, 0.1947, 0.2019, 0.2115 }, { 0.1910, 0.1905, 0.1980, 0.2082 },
{ 0.1860, 0.1861, 0.1943, 0.2057 }, { 0.1810, 0.1812, 0.1907, 0.2021 },
{ 0.1761, 0.1774, 0.1871, 0.2000 }, { 0.1718, 0.1743, 0.1842, 0.1974 },
{ 0.1671, 0.1706, 0.1813, 0.1950 }, { 0.1644, 0.1671, 0.1783, 0.1927 },
{ 0.1645, 0.1641, 0.1760, 0.1899 }, { 0.1661, 0.1625, 0.1743, 0.1884 },
{ 0.1701, 0.1602, 0.1726, 0.1862 }, { 0.1755, 0.1610, 0.1716, 0.1846 },
{ 0.1796, 0.1657, 0.1724, 0.1842 }
};
double[] K = new double[] { 124.0, 125.0, 126.0, 127.0, 128.0, 129.0, 130.0, 131.0, 132.0, 133.0, 134.0, 135.0, 136.0 };
double[] T = new double[] { 37.0 / 365.0, 72.0 / 365.0, 135.0 / 365.0, 226.0 / 365.0, };
double[,] MktIV = PutIV;
double S = 129.14;
double r = 0.0010;
double q = 0.0068;
int NT = 4;
int NK = 13;
// Create the market prices
BlackScholesPrice BS = new BlackScholesPrice();
string[,] PutCall = new string[NK, NT];
double[,] MktPrice = new double[NK, NT];
for (int t = 0; t < NT; t++)
{
for (int k = 0; k < NK; k++)
{
PutCall[k, t] = "P";
MktPrice[k, t] = BS.BlackScholes(S, K[k], T[t], r, q, MktIV[k, t], PutCall[k, t]);
}
}
// Bounds on the parameter estimates
double e = 1e-3;
double kappaU = 20.0; double kappaL = e;
double thetaU = 3.0; double thetaL = e;
double sigmaU = 10.0; double sigmaL = e;
double v0U = 3.0; double v0L = e;
double rhoU = 0.99; double rhoL = -0.99;
int N = 2 + 3 * NT;
double[] ub = new double[N];
double[] lb = new double[N];
for (int j = 0; j <= N; j++)
{
ub[0] = kappaU; lb[0] = kappaL;
ub[1] = v0U; lb[1] = v0L;
for (int k = 1; k <= NT; k++)
{
ub[3 * k - 1] = thetaU; lb[3 * k - 1] = thetaL;
ub[3 * k] = sigmaU; lb[3 * k - 1] = sigmaL;
ub[3 * k + 1] = rhoU; lb[3 * k + 1] = rhoL;
}
}
// Starting values for the parameters
// Parameter order is [kappa v0 | theta[0] sigma[0] rho[0] | theta[1] sigma[1] rho[1] | etc...];
double kappaS = 2.0;
double thetaS = 0.1;
double sigmaS = 1.2;
double v0S = 0.05;
double rhoS = -0.5;
double[,] s = new double[N, N + 1];
for (int j = 0; j <= N; j++)
{
s[0, j] = kappaS + BS.RandomNum(-0.01, 0.01) * kappaS;
s[1, j] = v0S + BS.RandomNum(-0.01, 0.01) * v0S;
for (int k = 1; k <= NT; k++)
{
s[3 * k - 1, j] = thetaS + BS.RandomNum(-0.01, 0.01) * thetaS;
s[3 * k, j] = sigmaS + BS.RandomNum(-0.01, 0.01) * sigmaS;
s[3 * k + 1, j] = rhoS + BS.RandomNum(-0.01, 0.01) * rhoS;
}
}
// Structure for the option settings
OPSet opset;
opset.S = S;
opset.r = r;
opset.q = q;
opset.trap = 1;
opset.PutCall = "P";
// Structure for the market data
MktData data;
data.MktIV = MktIV;
data.MktPrice = MktPrice;
data.K = K;
data.T = T;
data.PutCall = PutCall;
// Structure for the objective function settings
OFSet ofset;
ofset.opset = opset;
ofset.data = data;
ofset.X = X;
ofset.W = W;
ofset.ObjFunction = 4;
ofset.lb = lb;
ofset.ub = ub;
// Structure for the Nelder Mead algorithm settings
NMSet nmset;
nmset.ofset = ofset;
nmset.MaxIters = 5000;
nmset.Tolerance = 1e-8;
nmset.N = N;
// Run the Nelder Mead Algorithm
NelderMeadAlgo NM = new NelderMeadAlgo();
ObjectiveFunction OF = new ObjectiveFunction();
double[] B = NM.NelderMead(OF.f, nmset, s);
int NB = B.Length;
// Separate the parameter vector into different vectors
double kappa = B[0];
double v0 = B[1];
double[] theta = new double[NT];
double[] sigma = new double[NT];
double[] rho = new double[NT];
for (int k = 0; k <= NT - 1; k++)
{
theta[k] = B[3 * k + 2];
sigma[k] = B[3 * k + 3];
rho[k] = B[3 * k + 4];
}
// Bisection algorithm settings
double a = 0.01;
double b = 4.0;
double Tol = 1e-5;
int MaxIter = 2500;
// Find the implied volatilities
double[,] ModelPrice = new double[NK, NT];
double[,] ModelIV = new double[NK, NT];
double Error = 0.00;
List <double> MatList = new List <double>();
List <double> thetaList = new List <double>();
List <double> sigmaList = new List <double>();
List <double> rhoList = new List <double>();
double[] Mat, Theta, Sigma, Rho;
for (int t = 0; t < NT; t++)
{
// Stack the parameters
MatList.Add(T[t]);
thetaList.Add(theta[t]);
sigmaList.Add(sigma[t]);
rhoList.Add(rho[t]);
// Convert to arrays
Mat = MatList.ToArray();
Theta = thetaList.ToArray();
Sigma = sigmaList.ToArray();
Rho = rhoList.ToArray();
for (int k = 0; k < NK; k++)
{
ModelPrice[k, t] = BGM.BGMApproxPriceTD(kappa, v0, Theta, Sigma, Rho, opset, K[k], Mat);
ModelIV[k, t] = BA.BisecBSIV(opset, K[k], T[t], a, b, ModelPrice[k, t], Tol, MaxIter);
Error += Math.Pow(ModelIV[k, t] - MktIV[k, t], 2.0) / Convert.ToDouble(NT * NK);
}
}
// Output the results
Console.WriteLine("----------------------------------------------------------");
Console.WriteLine(" kappa v0 theta sigma rho");
Console.WriteLine("----------------------------------------------------------");
for (int k = 0; k <= NT - 1; k++)
{
Console.WriteLine("{0,10:F5} {1,10:F5} {2,10:F5} {3,10:F5} {4,10:F5}", kappa, v0, theta[k], sigma[k], rho[k]);
}
Console.WriteLine("----------------------------------------------------------");
Console.WriteLine("Value of objective function {0:F5} ", B[NB - 2]);
Console.WriteLine("Number of iterations required {0:0} ", B[NB - 1]);
Console.WriteLine("----------------------------------------------------------");
Console.WriteLine(" ");
Console.WriteLine("Model/Market implied volatilities");
Console.WriteLine("----------------------------------------------------------");
Console.WriteLine("Strike Mat1 Mat2 Mat3 Mat4");
Console.WriteLine("----------------------------------------------------------");
for (int k = 0; k <= NK - 1; k++)
{
Console.WriteLine("{0,5:0} {1,10:F5} {2,10:F5} {3,10:F5} {4,10:F5}", K[k], ModelIV[k, 0], ModelIV[k, 1], ModelIV[k, 2], ModelIV[k, 3]);
Console.WriteLine("{0,5:0} {1,10:F5} {2,10:F5} {3,10:F5} {4,10:F5}", K[k], MktIV[k, 0], MktIV[k, 1], MktIV[k, 2], MktIV[k, 3]);
Console.WriteLine("----------------------------------------------------------");
}
Console.WriteLine(" ");
Console.WriteLine("IV Estimation error {0,5:E5}", Error);
Console.WriteLine(" ");
}