// Series I expansion
public double SeriesICall(double S, double K, double rf, double q, double T, double v0, double rho, double theta, double kappa, double sigma)
{
// The "J" integrals
double J1 = J(rho, theta, kappa, T, v0, 1);
double J3 = J(rho, theta, kappa, T, v0, 3);
double J4 = J(rho, theta, kappa, T, v0, 4);
// Time average of the deterministic volatility
double v = theta + (v0 - theta) * (1.0 - Math.Exp(-kappa * T)) / (kappa * T);
// X and Z required for the ratios of Black Scholes derivatives
double X = Math.Log(S * Math.Exp((rf - q) * T) / K);
double Z = v * T;
// The ratios of Black Scholes derivatives
double R20 = R(2, 0, X, Z, T);
double R11 = R(1, 1, X, Z, T);
double R12 = R(1, 2, X, Z, T);
double R22 = R(2, 2, X, Z, T);
// Black Scholes call price evaluated at v
BlackScholes BS = new BlackScholes();
double c = BS.BSC(S, K, rf, q, v, T);
// Black Scholes vega evaluated at v
double cv = BS.BSV(S, K, rf, q, v, T);
// Series I volatility of volatility expansion for the implied volatility
return(c + sigma / T * J1 * R11 * cv
+ sigma * sigma * (J3 * R20 / T / T + J4 * R12 / T + J1 * J1 * R22 / T / T / 2) * cv);
}
// Series II expansion
public double[] SeriesIICall(double S, double K, double rf, double q, double T, double v0, double rho, double theta, double kappa, double sigma)
{
// The "J" integrals
double J1 = J(rho, theta, kappa, T, v0, 1);
double J3 = J(rho, theta, kappa, T, v0, 3);
double J4 = J(rho, theta, kappa, T, v0, 4);
// Time average of the deterministic volatility
double v = theta + (v0 - theta) * (1.0 - Math.Exp(-kappa * T)) / (kappa * T);
// X and Z required for the ratios of Black Scholes derivatives
double X = Math.Log(S * Math.Exp((rf - q) * T) / K);
double Z = v * T;
// The ratios of Black Scholes derivatives
double R20 = R(2, 0, X, Z, T);
double R11 = R(1, 1, X, Z, T);
double R12 = R(1, 2, X, Z, T);
double R22 = R(2, 2, X, Z, T);
// Series II volatility of volatility expansion for the implied volatility
double iv = v + sigma / T * J1 * R11
+ sigma * sigma * (J3 * R20 / T / T + J4 * R12 / T + 0.5 / T / T * J1 * J1 * (R22 - R11 * R11 * R20));
// Series II call price
BlackScholes BS = new BlackScholes();
double Price = BS.BSC(S, K, rf, q, iv, T);
// The series volatility
double ivx = Math.Sqrt(iv);
// Return the price and the volatility
double[] output = new double[2] {
Price, ivx
};
return(output);
}
private void GenerateLewisTable_MouseClick(object sender, MouseEventArgs e)
{
double S = 100.0; // Spot Price
double T = 0.25; // Maturity in Years
double rf = 0.0; // Interest Rate
double q = 0.0; // Dividend yield
string PutCall = "C"; // "P"ut or "Call"
int trap = 1; // 1="Little Trap" characteristic function
OpSet opset = new OpSet();
opset.S = S;
opset.T = T;
opset.r = rf;
opset.q = q;
opset.PutCall = PutCall;
opset.trap = trap;
double kappa = 4.0; // Heston Parameter
double theta = 0.09 / 4.0; // Heston Parameter
double sigma = 0.1; // Heston Parameter: Volatility of Variance
double v0 = 0.0225; // Heston Parameter: Current Variance
double rho = -0.5; // Heston Parameter: Correlation
double lambda = 0.0; // Heston Parameter
HParam param = new HParam();
param.kappa = kappa;
param.theta = theta;
param.sigma = sigma;
param.v0 = v0;
param.rho = rho;
param.lambda = lambda;
int NK = 7;
double[] K = new double[7] // Strikes
{
70.0, 80.0, 90.0, 100.0, 110.0, 120.0, 130.0
};
// Time average of the deterministic variance and volatility
double vol = param.theta + (param.v0 - param.theta) * (1.0 - Math.Exp(-param.kappa * T)) / (param.kappa * T);
double IV = Math.Sqrt(vol);
// Classes
BlackScholes BS = new BlackScholes();
NewtonCotes NC = new NewtonCotes();
Lewis L = new Lewis();
BisectionAlgo BA = new BisectionAlgo();
// Table 3.3.1 on Page 81 of Lewis (2001)
double[] SeriesIPrice = new double[NK]; // Series I price and implied vol
double[] IVI = new double[NK];
double[] SeriesIIPrice = new double[NK]; // Series II price and implied vol
double[] IVII = new double[NK];
double[] HPrice = new double[NK]; // Exact Heston price and implied vol
double[] IVe = new double[NK];
double[] BSPrice = new double[NK]; // Black Scholes price
// Bisection algorithm settings
double lo = 0.01;
double hi = 2.0;
double Tol = 1.0e-10;
int MaxIter = 10000;
// Newton Cotes settings
int method = 3;
double a = 1.0e-10;
double b = 100.0;
int N = 10000;
double[] SeriesII = new double[2];
for (int k = 0; k <= NK - 1; k++)
{
// Generate the prices and implied vols
opset.K = K[k];
SeriesIPrice[k] = L.SeriesICall(S, K[k], rf, q, T, v0, rho, theta, kappa, sigma);
IVI[k] = BA.BisecBSIV(PutCall, S, K[k], rf, q, T, lo, hi, SeriesIPrice[k], Tol, MaxIter);
SeriesII = L.SeriesIICall(S, K[k], rf, q, T, v0, rho, theta, kappa, sigma);
SeriesIIPrice[k] = SeriesII[0];
IVII[k] = SeriesII[1];
HPrice[k] = NC.HestonPriceNewtonCotes(param, opset, method, a, b, N);
IVe[k] = BA.BisecBSIV(PutCall, S, K[k], rf, q, T, lo, hi, HPrice[k], Tol, MaxIter);
BSPrice[k] = BS.BSC(S, K[k], rf, q, vol, T);
// Write to the List Boxes
Strike.Items.Add(K[k]);
ExactPrice.Items.Add(Math.Round(HPrice[k], 6));
IVExact.Items.Add(Math.Round(100 * IVe[k], 2));
SeriesICall.Items.Add(Math.Round(SeriesIPrice[k], 6));
IV1.Items.Add(Math.Round(100 * IVI[k], 2));
SeriesIICall.Items.Add(Math.Round(SeriesIIPrice[k], 6));
IV2.Items.Add(Math.Round(100 * IVII[k], 2));
BlackScholesPrice.Items.Add(Math.Round(BSPrice[k], 6));
IVBS.Items.Add(Math.Round(100 * IV, 2));
}
}