// Bisection Algorithm for Black Scholes Implied Volatility =================================================================================
public double BisectionMSIV(double S, double K, double r, double q, double T, double a, double b, double MktPrice, double Tol, int MaxIter, double B, double dt)
{
MSPrices MS = new MSPrices();
double lowPrice = MS.MSPriceBS(S, K, T, a, r, q, MaxIter, Tol, B, dt);
double highPrice = MS.MSPriceBS(S, K, T, b, r, q, MaxIter, Tol, B, dt);
double lowCdif = MktPrice - lowPrice;
double highCdif = MktPrice - highPrice;
double y = 0.0;
double midP;
if (lowCdif * highCdif > 0.0)
{
y = -1.0;
}
else
{
for (int x = 0; x <= MaxIter; x++)
{
midP = (a + b) / 2.0;
double midCdif = MktPrice - MS.MSPriceBS(S, K, T, midP, r, q, MaxIter, Tol, B, dt);
if (Math.Abs(midCdif) < Tol)
{
break;
}
else
{
if (midCdif > 0.0)
{
a = midP;
}
else
{
b = midP;
}
}
y = midP;
}
}
return(y);
}
// Objective function ===========================================================================
public double f(double[] param, OFSet ofsettings)
{
// Option price settings
double S = ofsettings.opsettings.S;
double r = ofsettings.opsettings.r;
double q = ofsettings.opsettings.q;
double T = ofsettings.opsettings.T;
int trap = ofsettings.opsettings.trap;
// Market data
double[] MktIV = ofsettings.data.MktIV;
int NK = MktIV.Length;
// MS settings
double dt = ofsettings.mssettings.dt;
double a = ofsettings.mssettings.a;
double b = ofsettings.mssettings.b;
double tol = ofsettings.mssettings.tol;
int MaxIter = ofsettings.mssettings.MaxIter;
// Optimizatin settings
double[] lb = ofsettings.lb;
double[] ub = ofsettings.ub;
double[] X = ofsettings.X;
double[] W = ofsettings.W;
HParam param2 = new HParam();
param2.kappa = param[0];
param2.theta = param[1];
param2.sigma = param[2];
param2.v0 = param[3];
param2.rho = param[4];
param2.lambda = 0.0;
// Settings for the Bisection algorithm
a = 0.01;
b = 3.0;
double B = 3.0;
// 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[] Error = new double[NK];
double SumError = 0.0;
// Parameter bounds
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];
// Classes
MSPrices MS = new MSPrices();
BisectionImpliedVol BA = new BisectionImpliedVol();
// Penalty for inadmissible parameter values
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))
{
SumError = 1.0e50;
}
// Penalty for inadmissible implied vol
else
{
Console.WriteLine("----------------------------------------");
Console.WriteLine("ModelPrice ImpliedVol MktVol Error");
Console.WriteLine("----------------------------------------");
for (int k = 0; k <= NK - 1; k++)
{
double Strike = ofsettings.data.K[k];
ofsettings.opsettings.K = Strike;
double[] output = new double[6];
output = MS.MSPriceHeston(param2, ofsettings.opsettings, ofsettings.mssettings);
ModelPrice[k] = Math.Max(0.01, output[2]);
ModelIV[k] = BA.BisectionMSIV(S, Strike, r, q, T, a, b, ModelPrice[k], tol, MaxIter, B, dt);
if (ModelIV[k] == -1.0)
{
Error[k] = 1.0e50;
}
else
{
Error[k] = Math.Pow(ModelIV[k] - MktIV[k], 2.0);
}
Console.WriteLine("{0,7:F4} {1,10:F4} {2,10:F4} {3,10:F6}", ModelPrice[k], ModelIV[k], MktIV[k], Error[k]);
SumError += Error[k];
}
}
return(SumError);
}
static void Main(string[] args)
{
// 32 point Gauss Laguerre
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 = 1e-5;
double[] lb = new double[5] {
e, e, e, e, -0.99
};
double[] ub = new double[5] {
20.0, 2.0, 5.0, 2.0, 0.99
};
//// IBM put prices May 7, 2010
double S = 122.10;
//double[,] MktPrice = new double[8,5]{
// { 0.130, 0.620, 1.275, 2.950, 4.750},
// { 0.260, 0.955, 1.830, 3.925, 6.075},
// { 0.485, 1.500, 2.610, 5.200, 7.625},
// { 0.995, 2.445, 3.775, 6.800, 9.375},
// { 2.155, 3.900, 5.475, 8.800, 11.550},
// { 4.525, 6.225, 7.775, 11.225, 13.975},
// { 8.375, 9.525, 10.850, 14.125, 16.775},
// {13.075, 13.600, 14.575, 17.425, 19.900}};
// Use the first set of prices, for the first maturity
double[] MktPrice = new double[8] {
0.130, 0.260, 0.485, 0.995, 2.155, 4.525, 8.375, 13.075
};
double[] K = new double[8] {
100.0, 105.0, 110.0, 115.0, 120.0, 125.0, 130.0, 135.0
};
double r = 0.015;
double q = 0.01;
//double[] T = new double[5] {0.0384,0.1151, 0.1918, 0.4411, 0.7096};
double T = 0.0384;
int NK = 8;
// Bisection algorithm
double a = 0.01;
double b = 2.00;
double Tol = 1.0e-5;
int MaxIter = 1000;
double B = 2.0;
double dt = 1.0e-10;
// Implied volatilities
BisectionImpliedVol BA = new BisectionImpliedVol();
double[] MktIV = new double[NK];
for (int k = 0; k <= NK - 1; k++)
{
MktIV[k] = BA.BisectionMSIV(S, K[k], r, q, T, a, b, MktPrice[k], Tol, MaxIter, B, dt);
}
MktData MktData = new MktData();
MktData.MktIV = MktIV;
MktData.K = K;
MktData.PutCall = "P";
OpSet opsettings = new OpSet();
opsettings.S = S;
opsettings.T = T;
opsettings.r = r;
opsettings.q = q;
opsettings.PutCall = "P";
opsettings.trap = 1;
MSSet mssettings = new MSSet();
mssettings.method = 3;
mssettings.A = 0.0001;
mssettings.B = 100.0;
mssettings.N = 3000;
mssettings.dt = 1.0e-10;
mssettings.tol = 1.0e-5;
mssettings.MaxIter = 1000;
mssettings.NumTerms = 3;
mssettings.yinf = 1.0e4;
mssettings.a = -10.0;
mssettings.b = +10.0;
OFSet ofsettings = new OFSet();
ofsettings.opsettings = opsettings;
ofsettings.mssettings = mssettings;
ofsettings.data = MktData;
ofsettings.X = X;
ofsettings.W = W;
ofsettings.lb = lb;
ofsettings.ub = ub;
// Settings for the Nelder Mead algorithm
NMSet nmsettings;
nmsettings.N = 5; // Number of Heston parameters
nmsettings.MaxIters = 20; // Maximum number of iterations
nmsettings.Tolerance = 1e-6; // Tolerance on best and worst function values
nmsettings.ofsettings = ofsettings;
// Starting values (vertices) in vector form. Add random increment about each starting value
NelderMeadAlgo NM = new NelderMeadAlgo();
double kappaS = 20.00;
double thetaS = 0.036;
double sigmaS = 3.9;
double v0S = 0.15;
double rhoS = -0.46;
int N = nmsettings.N;
double[,] x = new double[N, N + 1];
for (int j = 0; j <= N; j++)
{
x[0, j] = kappaS + NM.RandomNum(-0.01, 0.01) * kappaS;
x[1, j] = thetaS + NM.RandomNum(-0.01, 0.01) * thetaS;
x[2, j] = sigmaS + NM.RandomNum(-0.01, 0.01) * sigmaS;
x[3, j] = v0S + NM.RandomNum(-0.01, 0.01) * v0S;
x[4, j] = rhoS + NM.RandomNum(-0.01, 0.01) * rhoS;
}
// Obtain the parameter estimates
ObjectiveFunction OF = new ObjectiveFunction();
double[] BB = NM.NelderMead(OF.f, nmsettings, x);
HParam paramEst = new HParam();
paramEst.kappa = BB[0];
paramEst.theta = BB[1];
paramEst.sigma = BB[2];
paramEst.v0 = BB[3];
paramEst.rho = BB[4];
paramEst.lambda = 0.0;
// Obtain the fitted implied vols
MSPrices MS = new MSPrices();
double[] ModelIV = new double[NK];
double[] ModelPrice = new double[NK];
double[] output = new double[6];
double IVMSE = 0.0;
for (int k = 0; k <= NK - 1; k++)
{
opsettings.K = K[k];
output = MS.MSPriceHeston(paramEst, opsettings, mssettings);
ModelPrice[k] = output[2];
ModelIV[k] = BA.BisectionMSIV(S, K[k], r, q, T, a, b, ModelPrice[k], Tol, MaxIter, B, dt);
IVMSE += Math.Pow(ModelIV[k] - MktIV[k], 2.0);
}
// Output the estimation result
Console.WriteLine(" ");
Console.WriteLine("Parameter Estimates --------------------");
Console.WriteLine(" ");
Console.WriteLine("kappa = {0:F5}", BB[0]);
Console.WriteLine("theta = {0:F5}", BB[1]);
Console.WriteLine("sigma = {0:F5}", BB[2]);
Console.WriteLine("v0 = {0:F5}", BB[3]);
Console.WriteLine("rho = {0:F5}", BB[4]);
Console.WriteLine(" ");
Console.WriteLine("Value of the objective function is {0:E5}", BB[5]);
Console.WriteLine("Number of iterations required {0:0}", BB[6]);
Console.WriteLine(" ");
Console.WriteLine("----------------------------------------");
Console.WriteLine(" ");
Console.WriteLine("Strike MktIV ModelIV");
Console.WriteLine("-------------------------");
for (int k = 0; k <= NK - 1; k++)
{
Console.WriteLine("{0,3:F0} {1,12:F5} {2,12:F5}", K[k], MktIV[k], ModelIV[k]);
}
}