// 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
MiscFunctions MF = new MiscFunctions();
DiffEvoAlgo DE = new DiffEvoAlgo();
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]);
}
// Option settings
OPSet opset;
opset.S = 137.14;
opset.r = 0.0010;
opset.q = 0.0068;
opset.trap = 1;
// Bisection algorithm settings
double a = 0.01;
double b = 3.0;
double Tol = 1e-5;
int MaxIter = 1000;
int ObjectiveFun = 1;
// Read in SP500 implied volatilities
int NT = 4;
int NK = 7;
double[,] MktIV = new Double[7, 4]
{
{ 0.2780, 0.2638, 0.2532, 0.2518 },
{ 0.2477, 0.2402, 0.2364, 0.2369 },
{ 0.2186, 0.2158, 0.2203, 0.2239 },
{ 0.1878, 0.1930, 0.2047, 0.2098 },
{ 0.1572, 0.1712, 0.1894, 0.1970 },
{ 0.1334, 0.1517, 0.1748, 0.1849 },
{ 0.1323, 0.1373, 0.1618, 0.1736 }
};
double[] K = new Double[7] {
120.0, 125.0, 130.0, 135.0, 140.0, 145.0, 150.0
};
double[] T = new Double[4] {
0.123287671232877, 0.268493150684932, 0.715068493150685, 0.953424657534247
};
// PutCall identifiers
string[,] PutCall = new String[NK, NT];
for (int k = 0; k <= NK - 1; k++)
{
for (int t = 0; t <= NT - 1; t++)
{
PutCall[k, t] = "C";
}
}
// Obtain the market prices
BlackScholesPrice BS = new BlackScholesPrice();
double[,] MktPrice = new Double[NK, NT];
for (int k = 0; k <= NK - 1; k++)
{
for (int t = 0; t <= NT - 1; t++)
{
MktPrice[k, t] = BS.BlackScholes(opset.S, K[k], T[t], opset.r, opset.q, MktIV[k, t], PutCall[k, t]);
}
}
// Place the market data in the structure
MktData data = new MktData();
data.MktIV = MktIV;
data.MktPrice = MktPrice;
data.K = K;
data.T = T;
data.PutCall = PutCall;
// Estimation bounds
double e = 1e-5;
double kappaL = e; double kappaU = 10;
double thetaL = e; double thetaU = 5;
double sigmaL = e; double sigmaU = 5;
double v0L = e; double v0U = 1;
double rhoL = -.9; double rhoU = 0;
double[] ub = new double[5] {
kappaU, thetaU, sigmaU, v0U, rhoU
};
double[] lb = new double[5] {
kappaL, thetaL, sigmaL, v0L, rhoL
};
// Objective function settings;
OFSet ofset;
ofset.opset = opset;
ofset.data = data;
ofset.X = X;
ofset.W = W;
ofset.LossFunction = 4; // Choice of loss function 1=MSE, 2=RMSE, 3=IVMSE, 4=Christoffersen et al.
ofset.lb = lb;
ofset.ub = ub;
ofset.CF = "Heston"; // Choice of c.f. "Heston" or "Attari"
// Settings for the Differential Evolution algorithm
DEParam ParamLim = new DEParam();
ParamLim.ub = ub;
ParamLim.lb = lb;
ParamLim.NG = 500;
ParamLim.NP = 75;
ParamLim.F = 0.5;
ParamLim.CR = 0.8;
// Run the differential evolution algorithm
HParam DEparam = DE.HestonDE(ParamLim, opset, data, ObjectiveFun, a, b, Tol, MaxIter, X, W, ofset.CF);
// Starting values (vertices) in vector form. Add random increment about each starting value
double kappaS = 9;
double thetaS = 0.05;
double sigmaS = 0.3;
double v0S = 0.05;
double rhoS = -0.8;
int N = 5;
double[,] s = new double[N, N + 1];
for (int j = 0; j <= N; j++)
{
s[0, j] = kappaS + MF.RandomNum(-0.10, 0.10);
s[1, j] = thetaS + MF.RandomNum(-0.01, 0.01);
s[2, j] = sigmaS + MF.RandomNum(-0.05, 0.05);
s[3, j] = v0S + MF.RandomNum(-0.01, 0.01);
s[4, j] = rhoS + MF.RandomNum(-0.05, 0.05);
}
// Nelder Mead settings
NMSet nmset;
nmset.ofset = ofset;
nmset.N = 5; // Number of Heston parameters
nmset.MaxIters = 1000; // Maximum number of iterations
nmset.Tolerance = 1e-4; // Tolerance on best and worst function values
// Run the Nelder Mead algorithm
double[] B = NM.NelderMead(OF.f, nmset, s);
HParam NMparam = new HParam();
NMparam.kappa = B[0];
NMparam.theta = B[1];
NMparam.sigma = B[2];
NMparam.v0 = B[3];
NMparam.rho = B[4];
// Calculate IVMSE under both parameter estimates
double[,] NMPrice = new double[NK, NT];
double[,] DEPrice = new double[NK, NT];
double[,] NMIV = new double[NK, NT];
double[,] DEIV = new double[NK, NT];
double NMIVMSE = 0.0;
double DEIVMSE = 0.0;
double S = opset.S;
double r = opset.r;
double q = opset.q;
int trap = opset.trap;
HestonPrice HP = new HestonPrice();
Bisection BA = new Bisection();
for (int t = 0; t < NT; t++)
{
for (int k = 0; k < NK; k++)
{
NMPrice[k, t] = HP.HestonPriceGaussLaguerre(NMparam, S, K[k], r, q, T[t], trap, PutCall[k, t], X, W);
NMIV[k, t] = BA.BisecBSIV(PutCall[k, t], S, K[k], r, q, T[t], a, b, NMPrice[k, t], Tol, MaxIter);
NMIVMSE += Math.Pow(MktIV[k, t] - NMIV[k, t], 2) / Convert.ToDouble(NT * NK);
DEPrice[k, t] = HP.HestonPriceGaussLaguerre(DEparam, S, K[k], r, q, T[t], trap, PutCall[k, t], X, W);
DEIV[k, t] = BA.BisecBSIV(PutCall[k, t], S, K[k], r, q, T[t], a, b, DEPrice[k, t], Tol, MaxIter);
DEIVMSE += Math.Pow(MktIV[k, t] - DEIV[k, t], 2) / Convert.ToDouble(NT * NK);
}
}
// Output the result
Console.WriteLine(" ");
Console.WriteLine("Parameter Estimates --------------------");
Console.WriteLine(" Differential Evolution Nelder Mead");
Console.WriteLine("kappa {0,10:F4} {1,25:F4}", DEparam.kappa, NMparam.kappa);
Console.WriteLine("theta {0,10:F4} {1,25:F4}", DEparam.theta, NMparam.theta);
Console.WriteLine("sigma {0,10:F4} {1,25:F4}", DEparam.sigma, NMparam.sigma);
Console.WriteLine("v0 {0,10:F4} {1,25:F4}", DEparam.v0, NMparam.v0);
Console.WriteLine("rho {0,10:F4} {1,25:F4}", DEparam.rho, NMparam.rho);
Console.WriteLine(" ");
Console.WriteLine("IV MSE ---------------------------------");
Console.WriteLine("Differential Evolution IVMSE {0:E8}", DEIVMSE);
Console.WriteLine("Nelder Mead IVMSE {0:E8}", NMIVMSE);
Console.WriteLine(" ");
}