// 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);
}
static void Main(string[] args)
{
// Classes
BisectionAlgo BA = new BisectionAlgo();
BlackScholesPrice BS = new BlackScholesPrice();
HestonPrice HP = new HestonPrice();
ElicesAlgo EA = new ElicesAlgo();
NelderMeadAlgo NM = new NelderMeadAlgo();
ObjectiveFunction OF = new ObjectiveFunction();
// 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]);
}
}
// Bounds on the parameter estimates
// kappa theta sigma v0 rho
double e = 1.0e-2;
double[] lb = new double[5] {
e, e, e, -0.99, e
};
double[] ub = new double[5] {
10.0, 3.0, 10.0, 0.99, 3.0
};
// Read in SP500 implied volatilities
const int NT = 4;
const int NK = 13;
double[][] MktIV = new double[NT][];
MktIV[0] = new double[NK] {
0.1962, 0.1910, 0.1860, 0.1810, 0.1761, 0.1718, 0.1671, 0.1644, 0.1645, 0.1661, 0.1701, 0.1755, 0.1796
};
MktIV[1] = new double[NK] {
0.1947, 0.1905, 0.1861, 0.1812, 0.1774, 0.1743, 0.1706, 0.1671, 0.1641, 0.1625, 0.1602, 0.1610, 0.1657
};
MktIV[2] = new double[NK] {
0.2019, 0.1980, 0.1943, 0.1907, 0.1871, 0.1842, 0.1813, 0.1783, 0.1760, 0.1743, 0.1726, 0.1716, 0.1724
};
MktIV[3] = new double[NK] {
0.2115, 0.2082, 0.2057, 0.2021, 0.2000, 0.1974, 0.1950, 0.1927, 0.1899, 0.1884, 0.1862, 0.1846, 0.1842
};
double[] K = new double[NK] {
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[NT] {
37.0 / 365.0, 72.0 / 365.0, 135.0 / 365.0, 226.0 / 365.0
};
// PutCall identifier
string PutCall = "P";
// Maturity increments
double[] tau = new double[NT];
tau[0] = T[0];
for (int t = 1; t <= NT - 1; t++)
{
tau[t] = T[t] - T[t - 1];
}
// Option settings
OPSet opsettings;
opsettings.S = 129.14;
opsettings.r = 0.0010;
opsettings.q = 0.0068;
opsettings.trap = 1;
// Obtain the market prices for Elices, jagged array, and Heston (two dimensional array)
double[][] MktPrice = new double[4][];
double[,] MktPriceH = new double[NT, NK];
double[,] MktIVH = new double[NT, NK];
for (int t = 0; t <= NT - 1; t++)
{
MktPrice[t] = new double[13];
for (int k = 0; k <= NK - 1; k++)
{
MktPrice[t][k] = BS.BlackScholes(opsettings.S, K[k], T[t], opsettings.r, opsettings.q, MktIV[t][k], PutCall);
MktPriceH[t, k] = MktPrice[t][k];
MktIVH[t, k] = MktIV[t][k];
}
}
// Settings for the Elices objective function
OFSetE ofsetE = new OFSetE();
ofsetE.opsettings = opsettings;
ofsetE.X = X;
ofsetE.W = W;
ofsetE.LossFunction = 1;
ofsetE.lb = lb;
ofsetE.ub = ub;
// Elices Settings for the Market data
MktData MktData = new MktData();
MktData.K = K;
MktData.PutCall = PutCall;
ofsetE.data = MktData;
// Settings for the Heston objective function
OFSetH ofsetH = new OFSetH();
ofsetH.opsettings = opsettings;
ofsetH.X = X;
ofsetH.W = W;
ofsetH.LossFunction = ofsetE.LossFunction;
ofsetH.lb = ofsetE.lb;
ofsetH.ub = ofsetE.ub;
ofsetH.MktPrice = MktPriceH;
ofsetH.MktIV = MktIVH;
ofsetH.K = K;
ofsetH.T = T;
ofsetH.PutCall = PutCall;
// Parameter initial values
double kappaS = 2.00;
double thetaS = 0.10;
double sigmaS = 1.20;
double v0S = 0.03;
double rhoS = -0.80;
// Heston and Elices starting values (vertices) in vector form. Add random increment about each starting value
double[,] startH = new double[5, 6];
double[,] startE = new double[4, 5];
for (int j = 0; j <= 5; j++)
{
startH[0, j] = kappaS + BA.RandomNum(-0.05, 0.05) * kappaS;
startH[1, j] = thetaS + BA.RandomNum(-0.05, 0.05) * thetaS;
startH[2, j] = sigmaS + BA.RandomNum(-0.05, 0.05) * sigmaS;
startH[3, j] = rhoS + BA.RandomNum(-0.05, 0.05) * rhoS;
startH[4, j] = v0S + BA.RandomNum(-0.05, 0.05) * v0S;
}
for (int j = 0; j <= 4; j++)
{
startE[0, j] = startH[0, j];
startE[1, j] = startH[1, j];
startE[2, j] = startH[2, j];
startE[3, j] = startH[3, j];
}
// Settings for the Nelder Mead algorithm
NMSet nmsettings = new NMSet();
nmsettings.MaxIters = 50; // Maximum number of iterations
nmsettings.Tolerance = 1e-20; // Tolerance on best and worst function values
nmsettings.ofsettingsE = ofsetE; // Settings for the Elices objective function
nmsettings.ofsettingsH = ofsetH; // Settings for the Heston objective function
// Heston Parameter Estimation all maturities ===========================================================================================
nmsettings.choice = "Heston";
int N = 5;
nmsettings.N = N;
double[] ParamH = new double[7];
ParamH = NM.NelderMead(OF.f, nmsettings, startH);
// Elices Parameter Estimation maturity by maturity ====================================================================================
nmsettings.choice = "Elices";
N = 4;
nmsettings.N = N;
double v0 = ParamH[4];
ofsetE.v0 = v0;
double[][] ParamTD = new double[4][];
double[] B = new double[6];
double[] LossFunction = new double[NT];
// Initialize maturities
List <double> MatsList = new List <double>();
double[] Mats;
MatsList.Add(tau[0]);
MatsList.Add(tau[1]);
Mats = MatsList.ToArray();
// Loop through the maturities, estimating parameters
for (int t = 0; t <= NT - 1; t++)
{
ofsetE.t = t;
ofsetE.data.MktIV = MktIV[t]; // Vector of market Implied vol
ofsetE.data.MktPrice = MktPrice[t]; // Vector of market prices
ofsetE.paramfixed = ParamTD; // Fixed parameters
if (t >= 1)
{
for (int j = 0; j <= N; j++) // Starting values are last period's estimates
{
startE[0, j] = B[0] + BA.RandomNum(-0.01, 0.01) * B[0];
startE[1, j] = B[1] + BA.RandomNum(-0.01, 0.01) * B[1];
startE[2, j] = B[2] + BA.RandomNum(-0.01, 0.01) * B[2];
startE[3, j] = B[3] + BA.RandomNum(-0.01, 0.01) * B[3];
}
}
if (t >= 2)
{
MatsList.Add(tau[t]);
Mats = MatsList.ToArray();
}
ofsetE.T = Mats;
nmsettings.ofsettingsE = ofsetE;
B = NM.NelderMead(OF.f, nmsettings, startE);
ParamTD[t] = new double[4] {
B[0], B[1], B[2], B[3]
};
LossFunction[t] = B[4];
Console.WriteLine("------ Finished maturity {0:F0} ------", ofsetE.t);
}
// Write the parameters
Console.WriteLine("Heston statict parameters");
Console.WriteLine("------------------------------------------------------------------");
Console.WriteLine(" kappa theta sigma rho v0 LossFunction");
Console.WriteLine("------------------------------------------------------------------");
Console.WriteLine("{0,10:F5} {1,10:F5} {2,10:F5} {3,10:F5} {4,10:F5} {5,10:F5}",
ParamH[0], ParamH[1], ParamH[2], ParamH[3], ParamH[4], ParamH[5]);
Console.WriteLine("------------------------------------------------------------------");
Console.WriteLine(" ");
Console.WriteLine("Elices Time dependent parameters");
Console.WriteLine("-------------------------------------------------------------");
Console.WriteLine(" kappa theta sigma rho LossFunction");
Console.WriteLine("-------------------------------------------------------------");
for (int t = 0; t <= NT - 1; t++)
{
Console.WriteLine("{0,10:F5} {1,10:F5} {2,10:F5} {3,10:F5} {4,10:F5}",
ParamTD[t][0], ParamTD[t][1], ParamTD[t][2], ParamTD[t][3], LossFunction[t]);
}
Console.WriteLine("-------------------------------------------------------------");
// Fitting Elices prices and implied vols ==========================================================================================================
// Matrices for prices, parameters, and fixed parameters
double[,] PriceE = new double[NK, NT];
double[] param = new double[4];
double[][] paramfixed = new double[NT - 1][];
paramfixed[0] = ParamTD[0];
// Define the dynamic array for the maturities and assign the first two maturities
List <double> MatList = new List <double>();
double[] Mat;
MatList.Add(tau[0]);
MatList.Add(tau[1]);
Mat = MatList.ToArray();
// Elices Implied volatilities
double[,] IVE = new double[NK, NT];
double a = 0.01;
double b = 3.0;
double tol = 1.0e-5;
int MaxIter = 1000;
double ElicesIVRMSE = 0.0;
// Find the prices and implied vols
for (int t = 0; t <= NT - 1; t++)
{
for (int j = 0; j <= 3; j++)
{
param[j] = ParamTD[t][j];
}
if (t == 0)
{
for (int k = 0; k <= NK - 1; k++)
{
PriceE[k, 0] = EA.ElicesPrice(PutCall, opsettings.S, K[k], Mat, opsettings.r, opsettings.q, param, v0, opsettings.trap, X, W);
}
}
if (t == 1)
{
for (int k = 0; k <= NK - 1; k++)
{
PriceE[k, 1] = EA.ElicesPrice(PutCall, opsettings.S, K[k], Mat, opsettings.r, opsettings.q, param, v0, opsettings.trap, X, W, paramfixed);
}
}
if (t >= 2)
{
MatList.Add(tau[t]);
Mat = MatList.ToArray();
paramfixed[t - 1] = ParamTD[t - 1];
for (int k = 0; k <= NK - 1; k++)
{
PriceE[k, t] = EA.ElicesPrice(PutCall, opsettings.S, K[k], Mat, opsettings.r, opsettings.q, param, v0, opsettings.trap, X, W, paramfixed);
}
}
for (int k = 0; k <= NK - 1; k++)
{
IVE[k, t] = BA.BisecBSIV(PutCall, opsettings.S, K[k], opsettings.r, opsettings.q, T[t], a, b, PriceE[k, t], tol, MaxIter);
ElicesIVRMSE += Math.Pow(IVE[k, t] - MktIV[t][k], 2.0);
}
}
// Fitting Heston prices and implied vols ==========================================================================================================
double[,] PriceH = new double[NK, NT];
double[,] IVH = new double[NK, NT];
HParam ParamTI = new HParam();
ParamTI.kappa = ParamH[0];
ParamTI.sigma = ParamH[1];
ParamTI.theta = ParamH[2];
ParamTI.rho = ParamH[3];
ParamTI.v0 = ParamH[4];
double HestonIVRMSE = 0.0;
for (int k = 0; k <= NK - 1; k++)
{
for (int t = 0; t <= NT - 1; t++)
{
PriceH[k, t] = HP.HestonPriceGaussLaguerre(ParamTI, ofsetH.opsettings, K[k], T[t], PutCall, X, W);
IVH[k, t] = BA.BisecBSIV(PutCall, opsettings.S, K[k], opsettings.r, opsettings.q, T[t], a, b, PriceH[k, t], tol, MaxIter);
HestonIVRMSE += Math.Pow(IVH[k, t] - MktIVH[t, k], 2.0);
}
}
Console.WriteLine("Heston IVRMSE {0,2:E5}", HestonIVRMSE);
Console.WriteLine("Elices IVRMSE {0,2:E5}", ElicesIVRMSE);
}