// Objective function ===========================================================================
public double f(double[] param, double tau, double[] tau0, double[,] param0, MktData data, double S, double[] K, double r, double q, int trap,
double[] X, double[] W, int LossFunction, double[] lb, double[] ub)
{
HestonPriceTD HPTD = new HestonPriceTD();
BisectionAlgo BA = new BisectionAlgo();
double[] MktIV = data.MktIV;
double[] MktPrice = data.MktPrice;
string[] PutCall = data.PutCall;
int NK = PutCall.Length;
int NT = 1;
HParam param2 = new HParam();
param2.kappa = param[0];
param2.theta = param[1];
param2.sigma = param[2];
param2.v0 = param[3];
param2.rho = param[4];
// Settings for the Bisection algorithm
double a = 0.001;
double b = 3.0;
double Tol = 1e5;
int MaxIter = 10000;
// Initialize the model price and model implied vol vectors, and the objective function value
double[] ModelPrice = new double[NK];
double[] ModelIV = new double[NK];
double Vega = 0.0;
double Error = 0.0;
double pi = Math.PI;
double kappaLB = lb[0]; double kappaUB = ub[0];
double thetaLB = lb[1]; double thetaUB = ub[1];
double sigmaLB = lb[2]; double sigmaUB = ub[2];
double v0LB = lb[3]; double v0UB = ub[3];
double rhoLB = lb[4]; double rhoUB = ub[4];
if ((param2.kappa <= kappaLB) || (param2.theta <= thetaLB) || (param2.sigma <= sigmaLB) || (param2.v0 <= v0LB) || (param2.rho <= rhoLB) ||
(param2.kappa >= kappaUB) || (param2.theta >= thetaUB) || (param2.sigma >= sigmaUB) || (param2.v0 >= v0UB) || (param2.rho >= rhoUB))
{
Error = 1e50;
}
else
{
for (int k = 0; k <= NK - 1; k++)
{
ModelPrice[k] = HPTD.MNPriceGaussLaguerre(param2, param0, tau, tau0, S, K[k], r, q, PutCall[k], trap, X, W);
switch (LossFunction)
{
case 1:
// MSE Loss Function
Error += Math.Pow(ModelPrice[k] - MktPrice[k], 2);
break;
case 2:
// RMSE Loss Function
Error += Math.Pow(ModelPrice[k] - MktPrice[k], 2) / MktPrice[k];
break;
case 3:
// IVMSE Loss Function
ModelIV[k] = BA.BisecBSIV(PutCall[k], S, K[k], r, q, tau, a, b, ModelPrice[k], Tol, MaxIter);
Error += Math.Pow(ModelIV[k] - MktIV[k], 2);
break;
case 4:
// IVRMSE Christoffersen, Heston, Jacobs proxy
double d = (Math.Log(S / K[k]) + (r - q + MktIV[k] * MktIV[k] / 2.0) * tau) / MktIV[k] / Math.Sqrt(tau);
double NormPDF = Math.Exp(-0.5 * d * d) / Math.Sqrt(2.0 * pi);
Vega = S * NormPDF * Math.Sqrt(tau);
Error += Math.Pow(ModelPrice[k] - MktPrice[k], 2) / (Vega * Vega);
break;
}
}
}
return(Error); // / Convert.ToDouble(NT*NK);
}
static void Main(string[] args)
{
// Classes
BlackScholesPrice BS = new BlackScholesPrice();
BisectionAlgo BA = new BisectionAlgo();
NelderMeadAlgo NM = new NelderMeadAlgo();
ObjectiveFunction OF = new ObjectiveFunction();
HestonPriceTD HPTD = new HestonPriceTD();
// 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]);
}
int NT = 4;
int NK = 13;
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[13] {
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[4] {
0.1014, 0.1973, 0.3699, 0.6192
};
string[,] PutCall = new string[13, 4];
double[,] MktPrice = new double[13, 4];
double Spot = 129.14;
double r = 0.0010;
double q = 0.0068;
int trap = 1;
int LossFunction = 2;
for (int t = 0; t <= NT - 1; t++)
{
for (int k = 0; k <= NK - 1; k++)
{
PutCall[k, t] = "P";
MktPrice[k, t] = BS.BlackScholes(Spot, K[k], T[t], r, q, PutIV[k, t], PutCall[k, t]);
}
}
// Create the maturity increments
double[] tau = new double[4];
tau[0] = T[0];
for (int t = 1; t <= NT - 1; t++)
{
tau[t] = T[t] - T[t - 1];
}
// Starting values and upper bounds
double e = 1e-5;
double[] lb = new double[5] {
e, e, e, e, -0.999
}; // Lower bound on the estimates
double[] ub = new double[5] {
20.0, 2.0, 2.0, 3.0, 0.999
}; // Upper bound on the estimates
// Starting values (vertices) in vector form. Add random increment about each starting value
int N = 5;
double kappaS = 4.0;
double thetaS = 0.1;
double sigmaS = 1.5;
double v0S = 0.04;
double rhoS = -0.30;
double[,] s = new double[N, N + 1];
for (int j = 0; j <= N; j++)
{
s[0, j] = kappaS + BA.RandomNum(-0.01, 0.01);
s[1, j] = thetaS + BA.RandomNum(-0.01, 0.01);
s[2, j] = sigmaS + BA.RandomNum(-0.01, 0.01);
s[3, j] = v0S + BA.RandomNum(-0.01, 0.01);
s[4, j] = rhoS + BA.RandomNum(-0.01, 0.01);
}
// Arrays for old maturities and parameters, and current parameters (old parameters, but stacked)
ArrayList OldTau = new ArrayList();
double[,] param0 = new double[NT - 1, 5];
double[,] paramTD = new double[NT, 5];
// Nelder Mead settings
int MaxIters = 1000;
double Tolerance = 1e-5;
// Market data
MktData data = new MktData();
double[] MKIV = new double[NK];
double[] MKPR = new double[NK];
string[] PC = new string[NK];
// Step-by-step parameter estimates
double[] B = new double[6];
double[] ObjectiveFun = new double[NT];
double[] NumIteration = new double[NT];
// First maturity parameter estimates ===================================================================
double[] tau0 = { 0.0 };
for (int k = 0; k <= NK - 1; k++)
{
MKIV[k] = PutIV[k, 0];
MKPR[k] = MktPrice[k, 0];
PC[k] = PutCall[k, 0];
}
data.MktIV = MKIV;
data.PutCall = PC;
data.MktPrice = MKPR;
B = NM.NelderMead(OF.f, N, MaxIters, Tolerance, s, tau[0], tau0, param0, data, Spot, K, r, q, trap, X, W, LossFunction, lb, ub);
Console.WriteLine("Finished parameter estimate set 1");
for (int k = 0; k <= 4; k++)
{
paramTD[0, k] = B[k];
}
ObjectiveFun[0] = B[5];
NumIteration[0] = B[6];
// Remaining Maturity estimates and updated starting values ==================================================
for (int mat = 1; mat <= NT - 1; mat++)
{
OldTau.Add(tau[mat - 1]);
for (int k = 0; k <= 4; k++)
{
param0[mat - 1, k] = B[k];
}
for (int k = 0; k <= NK - 1; k++)
{
MKIV[k] = PutIV[k, mat];
MKPR[k] = MktPrice[k, mat];
PC[k] = PutCall[k, mat];
}
data.MktIV = MKIV;
data.PutCall = PC;
data.MktPrice = MKPR;
for (int j = 0; j <= N; j++)
{
s[0, j] = B[0] + BA.RandomNum(-0.01, 0.01);
s[1, j] = B[1] + BA.RandomNum(-0.01, 0.01);
s[2, j] = B[2] + BA.RandomNum(-0.01, 0.01);
s[3, j] = B[3] + BA.RandomNum(-0.01, 0.01);
s[4, j] = B[4] + BA.RandomNum(-0.01, 0.01);
}
B = NM.NelderMead(OF.f, N, MaxIters, Tolerance, s, tau[mat], tau0, param0, data, Spot, K, r, q, trap, X, W, LossFunction, lb, ub);
Console.WriteLine("Finished parameter estimate set {0}", mat + 1);
for (int k = 0; k <= 4; k++)
{
paramTD[mat, k] = B[k];
}
ObjectiveFun[mat] = B[5];
NumIteration[mat] = B[6];
}
// Fit the prices and implied volatilities ====================================================================
// Bisection algorithm settings
double a = 0.001;
double b = 5.0;
double Tol = 1e-5;
int MaxIter = 5000;
ArrayList OldTau00 = new ArrayList();
double[] tau00 = { 0.0 };
double[,] ModelPrice = new double[NK, NT];
double[,] ModelIV = new double[NK, NT];
Array.Clear(param0, 0, 15);
// First maturity
HParam param = new HParam();
param.kappa = paramTD[0, 0];
param.theta = paramTD[0, 1];
param.sigma = paramTD[0, 2];
param.v0 = paramTD[0, 3];
param.rho = paramTD[0, 4];
for (int k = 0; k <= NK - 1; k++)
{
ModelPrice[k, 0] = HPTD.MNPriceGaussLaguerre(param, param0, tau[0], tau00, Spot, K[k], r, q, PutCall[k, 0], trap, X, W);
ModelIV[k, 0] = BA.BisecBSIV(PutCall[k, 0], Spot, K[k], r, q, T[0], a, b, ModelPrice[k, 0], Tol, MaxIter);
}
// Remaining maturity
for (int t = 1; t <= NT - 1; t++)
{
OldTau00.Add(tau[t - 1]);
param.kappa = paramTD[t, 0];
param.theta = paramTD[t, 1];
param.sigma = paramTD[t, 2];
param.v0 = paramTD[t, 3];
param.rho = paramTD[t, 4];
for (int j = 0; j <= 4; j++)
{
param0[t - 1, j] = paramTD[t - 1, j];
}
for (int k = 0; k <= NK - 1; k++)
{
ModelPrice[k, t] = HPTD.MNPriceGaussLaguerre(param, param0, tau[t], tau00, Spot, K[k], r, q, PutCall[k, t], trap, X, W);
ModelIV[k, t] = BA.BisecBSIV(PutCall[k, t], Spot, K[k], r, q, T[t], a, b, ModelPrice[k, t], Tol, MaxIter);
}
}
// Mean Square Implied Volatility Error
double Error = 0.0;
for (int t = 0; t <= NT - 1; t++)
{
for (int k = 0; k <= NK - 1; k++)
{
Error += Math.Pow(ModelIV[k, t] - PutIV[k, t], 2);
}
}
Error /= (Convert.ToDouble(NT) * Convert.ToDouble(NK));
// Output the estimation result
Console.WriteLine("-----------------------------------------------------------------------------");
Console.WriteLine("Market/Model implied volatilities");
Console.WriteLine(" Mat1 Mat2 Mat3 Mat4");
for (int k = 0; k <= NK - 1; k++)
{
Console.WriteLine("--------------------------------------------- Strike {0} Market/Model", K[k]);
Console.WriteLine("{0,10:F4} {1,10:F4} {2,10:F4} {3,10:F4}", PutIV[k, 0], PutIV[k, 1], PutIV[k, 2], PutIV[k, 3]);
Console.WriteLine("{0,10:F4} {1,10:F4} {2,10:F4} {3,10:F4}", ModelIV[k, 0], ModelIV[k, 1], ModelIV[k, 2], ModelIV[k, 3]);
}
Console.WriteLine("-----------------------------------------------------------------------------");
Console.WriteLine("Model Price ");
for (int k = 0; k <= NK - 1; k++)
{
Console.WriteLine("{0,10:0} {1,10:F4} {2,10:F4} {3,10:F4} {4,10:F4}", K[k], ModelPrice[k, 0], ModelPrice[k, 1], ModelPrice[k, 2], ModelPrice[k, 3]);
}
Console.WriteLine("-----------------------------------------------------------------------------");
Console.WriteLine("Results of time dependent parameter estimation");
Console.WriteLine(" ");
Console.WriteLine("Maturity kappa theta sigma v0 rho ObjecFun NumIterations");
Console.WriteLine("-----------------------------------------------------------------------------");
for (int t = 0; t <= NT - 1; t++)
{
Console.WriteLine("{0} {1,10:F4} {2,7:F4} {3,7:F4} {4,7:F4} {5,8:F4} {6,8:F4} {7,7:0}",
T[t], paramTD[t, 0], paramTD[t, 1], paramTD[t, 2], paramTD[t, 3], paramTD[t, 4], ObjectiveFun[t], NumIteration[t]);
}
Console.WriteLine("-----------------------------------------------------------------------------");
Console.WriteLine("IVMSE from time dependent estimates {0,10:E5}", Error);
Console.WriteLine("-----------------------------------------------------------------------------");
}