public double DHQuadExpSim(DHParam param, double S0, double Strike, double Mat, double r, double q, int T, int N, string PutCall) { RandomNumber RN = new RandomNumber(); NormInverse NI = new NormInverse(); double kappa1 = param.kappa1; double theta1 = param.theta1; double sigma1 = param.sigma1; double v01 = param.v01; double rho1 = param.rho1; double kappa2 = param.kappa2; double theta2 = param.theta2; double sigma2 = param.sigma2; double v02 = param.v02; double rho2 = param.rho2; // Time increment double dt = Mat / T; // Required quantities double K01 = -rho1 * kappa1 * theta1 * dt / sigma1; double K11 = dt / 2.0 * (kappa1 * rho1 / sigma1 - 0.5) - rho1 / sigma1; double K21 = dt / 2.0 * (kappa1 * rho1 / sigma1 - 0.5) + rho1 / sigma1; double K31 = dt / 2.0 * (1.0 - rho1 * rho1); double K02 = -rho2 * kappa2 * theta2 * dt / sigma2; double K12 = dt / 2.0 * (kappa2 * rho2 / sigma2 - 0.5) - rho2 / sigma2; double K22 = dt / 2.0 * (kappa2 * rho2 / sigma2 - 0.5) + rho2 / sigma2; double K32 = dt / 2.0 * (1.0 - rho2 * rho2); // Initialize the variance and stock processes double[,] V1 = new double[T, N]; double[,] V2 = new double[T, N]; double[,] S = new double[T, N]; // Starting values for the variance and stock processes for (int k = 0; k <= N - 1; k++) { S[0, k] = S0; // Spot price V1[0, k] = v01; // Heston v0 initial variance V2[0, k] = v02; // Heston v0 initial variance } // Generate the stock and volatility paths double m1, s1, phi1, p1, U1, b1, a1, Zv1; double m2, s2, phi2, p2, U2, b2, a2, Zv2; double beta1, beta2, B1, B2, logS; for (int k = 0; k <= N - 1; k++) { for (int t = 1; t <= T - 1; t++) { m1 = theta1 + (V1[t - 1, k] - theta1) * Math.Exp(-kappa1 * dt); s1 = V1[t - 1, k] * sigma1 *sigma1 *Math.Exp(-kappa1 *dt) / kappa1 * (1.0 - Math.Exp(-kappa1 * dt)) + theta1 * sigma1 * sigma1 / 2.0 / kappa1 * Math.Pow(1.0 - Math.Exp(-kappa1 * dt), 2.0); phi1 = s1 / (m1 * m1); p1 = (phi1 - 1.0) / (phi1 + 1.0); U1 = RN.RandomNum(0.0, 1.0); if (phi1 < 0.5) { b1 = Math.Sqrt(2.0 / phi1 - 1.0 + Math.Sqrt(2.0 / phi1 * (2.0 / phi1 - 1.0))); a1 = m1 / (1.0 + b1 * b1); Zv1 = NI.normICDF(U1); V1[t, k] = a1 * (b1 + Zv1) * (b1 + Zv1); } else if (phi1 >= 0.5) { if (U1 <= p1) { V1[t, k] = 0.0; } else if (U1 > p1) { beta1 = (1.0 - p1) / m1; V1[t, k] = Math.Log((1.0 - p1) / (1 - U1)) / beta1; } } m2 = theta2 + (V2[t - 1, k] - theta2) * Math.Exp(-kappa2 * dt); s2 = V2[t - 1, k] * sigma2 *sigma2 *Math.Exp(-kappa2 *dt) / kappa2 * (1.0 - Math.Exp(-kappa2 * dt)) + theta2 * sigma2 * sigma2 / 2.0 / kappa2 * Math.Pow(1.0 - Math.Exp(-kappa2 * dt), 2.0); phi2 = s2 / (m2 * m2); p2 = (phi2 - 1.0) / (phi2 + 1.0); U2 = RN.RandomNum(0.0, 1.0); if (phi2 < 0.5) { b2 = Math.Sqrt(2.0 / phi2 - 1.0 + Math.Sqrt(2.0 / phi2 * (2.0 / phi2 - 1.0))); a2 = m2 / (1.0 + b2 * b2); Zv2 = NI.normICDF(U2); V2[t, k] = a2 * (b2 + Zv2) * (b2 + Zv2); } else if (phi2 >= 0.5) { if (U2 <= p2) { V2[t, k] = 0.0; } else if (U2 > p2) { beta2 = (1.0 - p2) / m2; V2[t, k] = Math.Log((1.0 - p2) / (1.0 - U2)) / beta2; } } // Predictor-Corrector for the stock price B1 = RN.RandomNorm(); B2 = RN.RandomNorm(); logS = Math.Log(Math.Exp(-r * Convert.ToDouble(t) * dt) * S[t - 1, k]) + K01 + K11 * V1[t - 1, k] + K21 * V1[t, k] + Math.Sqrt(K31 * (V1[t, k] + V1[t - 1, k])) * B1 + K02 + K12 * V2[t - 1, k] + K22 * V2[t, k] + Math.Sqrt(K32 * (V2[t, k] + V2[t - 1, k])) * B2; S[t, k] = Math.Exp(logS) * Math.Exp(r * Convert.ToDouble(t + 1) * dt); } } // Terminal stock prices double[] ST = new double[N]; for (int k = 0; k <= N - 1; k++) { ST[k] = S[T - 1, k]; } // Payoff vectors double[] Payoff = new double[N]; for (int k = 0; k <= N - 1; k++) { if (PutCall == "C") { Payoff[k] = Math.Max(ST[k] - Strike, 0.0); } else if (PutCall == "P") { Payoff[k] = Math.Max(Strike - ST[k], 0.0); } } // Simulated price EulerAlfonsiSimulation EA = new EulerAlfonsiSimulation(); double SimPrice = Math.Exp(-r * Mat) * EA.VMean(Payoff); return(SimPrice); }
// Alfonsi function public double AlfonsiV(HParam param, double vt, double dt) { RandomNumber RN = new RandomNumber(); double kappa = param.kappa; double theta = param.theta; double sigma = param.sigma; double v0 = param.v0; double rho = param.rho; double phi = (1.0 - Math.Exp(-kappa * dt / 2.0)) / kappa; double S = (sigma * sigma / 4.0 - theta * kappa); double E = Math.Exp(kappa * dt / 2.0); double K2 = 0.0; double U = 0.0; double Y = 0.0; double newV = 0.0; if (sigma * sigma > 4.0 * kappa * theta) { K2 = E * (S * phi + Math.Pow((Math.Sqrt(E * S * phi) + sigma / 2.0 * Math.Sqrt(3.0 * dt)), 2)); } else { K2 = 0; } if (vt >= K2) { U = RN.RandomNum(0.0, 1.0); if (U <= 1.0 / 6.0) { Y = Math.Sqrt(3.0); } else if (U <= 1.0 / 3.0) { Y = -Math.Sqrt(3.0); } else { Y = 0.0; } phi = (1.0 - Math.Exp(-kappa * dt / 2.0)) / kappa; S = (theta * kappa - sigma * sigma / 4.0); E = Math.Exp(-kappa * dt / 2.0); newV = E * Math.Pow((Math.Sqrt(S * phi + E * vt) + sigma / 2.0 * Math.Sqrt(dt) * Y), 2) + S * phi; } else { double[] u = CIRmoments(param, vt, dt); double u1 = u[0]; double u2 = u[1]; double Pi = 0.5 - 0.5 * Math.Sqrt(1 - u1 * u1 / u2); U = RN.RandomNum(0.0, 1.0); if (U <= Pi) { newV = u1 / 2.0 / Pi; } else if (U > Pi) { newV = u1 / 2.0 / (1.0 - Pi); } } return(newV); }