// Golden Section search form Heston
public double GoldenSearchMS(double a, double b, double tol, int MaxIter, double K, HParam param, double theta, double r, double q, double T, int NumTerms)
{
MSExpansionHeston MS = new MSExpansionHeston();
double x1, x2, x3, x4, f1, f2, f3, f4;
double GR = (Math.Sqrt(5.0) - 1.0) / 2.0;
int k = 0;
while (Math.Abs(b - a) > tol)
{
k = k + 1;
x1 = a;
x2 = a + (1.0 - GR) * (b - a);
x3 = a + GR * (b - a);
x4 = b;
f1 = -MS.MSPutHeston(x1, theta, K, param, r, q, T, NumTerms);
f2 = -MS.MSPutHeston(x2, theta, K, param, r, q, T, NumTerms);
f3 = -MS.MSPutHeston(x3, theta, K, param, r, q, T, NumTerms);
f4 = -MS.MSPutHeston(x4, theta, K, param, r, q, T, NumTerms);
if ((f1 > f2) & (f2 < f3))
{
b = x3;
}
else if ((f2 > f3) & (f3 < f4))
{
a = x2;
}
if (k > MaxIter)
{
break;
}
}
return((a + b) / 2.0);
}
public double[] MSPriceHeston(HParam param, OpSet opset, MSSet msset)
{
// param = Heston parameters
// opset = option settings
// msset = Medvedev-Scaillet expansion settings
// Extract quantities from the option settings
double K = opset.K;
double S = opset.S;
double v0 = param.v0;
double T = opset.T;
double r = opset.r;
double q = opset.q;
// Extract quantities from the MS structure
int N = msset.N;
double a = msset.a;
double b = msset.b;
double A = msset.A;
double B = msset.B;
double dt = msset.dt;
double tol = msset.tol;
int MaxIter = msset.MaxIter;
int NumTerms = msset.NumTerms;
double yinf = msset.yinf;
int method = msset.method;
// Closed-form European put
NewtonCotes NC = new NewtonCotes();
double EuroPutClosed = NC.HestonPriceNewtonCotes(param, opset, method, A, B, N);
// Moneyness
double theta = Math.Log(K / S) / Math.Sqrt(v0) / Math.Sqrt(T);
// Find the barrier level
// Bisection method to find y
// a = Math.Max(1.25,0.95*theta);
// b = 3.0;
// double y = BisectionMS(a,b,theta,K,param,r,q,T,NumTerms,tol,MaxIter,dt);
// Golden section search method to find y
GoldenSearch GS = new GoldenSearch();
a = 1.5;
b = 2.5;
double y = GS.GoldenSearchMS(a, b, tol, MaxIter, K, param, theta, r, q, T, NumTerms);
if (y < theta)
{
y = theta;
}
// Find the early exercise premium
MSExpansionHeston MS = new MSExpansionHeston();
double AmerPutMS = MS.MSPutHeston(y, theta, K, param, r, q, T, NumTerms);
double EuroPutMS = MS.MSPutHeston(yinf, theta, K, param, r, q, T, NumTerms);
double EEP = AmerPutMS - EuroPutMS;
// Control variate American put
double AmerPut = EuroPutClosed + EEP;
// Output the results
double[] output = new double[6];
output[0] = EuroPutClosed;
output[1] = AmerPutMS;
output[2] = AmerPut; // Medvedev and Scaillet recommend this one.
output[3] = EEP;
output[4] = theta;
output[5] = y;
return(output);
}
public double MSGreeksFD(HParam param, OpSet opset, int method, double A, double B, int N, double hi, double tol, int MaxIter, int NumTerms, double yinf, string Greek)
{
MSExpansionHeston MS = new MSExpansionHeston();
double[] output = new double[6];
double AmerPut, AmerPutp, AmerPutm;
double AmerPutpp, AmerPutpm, AmerPutmp, AmerPutmm;
double S = opset.S;
double v0 = param.v0;
double T = opset.T;
// Define the finite difference increments
double ds = opset.S * 0.005;
double dt = opset.T * 0.005;
double dv = param.v0 * 0.005;
if (Greek == "price")
{
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPut = output[2];
return(AmerPut);
}
else if ((Greek == "delta") || (Greek == "gamma"))
{
opset.S = S + ds;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutp = output[2];
opset.S = S - ds;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutm = output[2];
if (Greek == "gamma")
{
opset.S = S;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPut = output[2];
return((AmerPutp - 2.0 * AmerPut + AmerPutm) / ds / ds);
}
else
{
return((AmerPutp - AmerPutm) / 2.0 / ds);
}
}
else if (Greek == "theta")
{
opset.T = T + dt;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutp = output[2];
opset.T = T - dt;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutm = output[2];
return(-(AmerPutp - AmerPutm) / 2.0 / dt);
}
else if ((Greek == "vega1") || (Greek == "volga"))
{
param.v0 = v0 + dv;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutp = output[2];
param.v0 = v0 - dv;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutm = output[2];
double Vega1 = (AmerPutp - AmerPutm) / 2.0 / dv * 2.0 * Math.Sqrt(v0);
if (Greek == "volga")
{
param.v0 = v0;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPut = output[2];
double dC2 = (AmerPutp - 2.0 * AmerPut + AmerPutm) / dv / dv;
return(4.0 * Math.Sqrt(v0) * (dC2 * Math.Sqrt(v0) + Vega1 / 4.0 / v0));
}
else
{
return(Vega1);
}
}
else if (Greek == "vanna")
{
opset.S = S + ds;
param.v0 = v0 + dv;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutpp = output[2];
param.v0 = v0 - dv;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutpm = output[2];
opset.S = S - ds;
param.v0 = v0 + dv;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutmp = output[2];
param.v0 = v0 - dv;
output = MS.MSPrice(param, opset, method, A, B, N, hi, tol, MaxIter, NumTerms, yinf);
AmerPutmm = output[2];
return((AmerPutpp - AmerPutpm - AmerPutmp + AmerPutmm) / 4.0 / dv / ds * 2.0 * Math.Sqrt(v0));
}
else
{
return(0.0);
}
}
// Derivative of the MS approximation
public double MSPutHestonDiff(double y, double theta, double Strike, HParam param, double r, double q, double T, int NumTerms, double dy)
{
MSExpansionHeston MS = new MSExpansionHeston();
return((MS.MSPutHeston(y + dy, theta, Strike, param, r, q, T, NumTerms) - MS.MSPutHeston(y - dy, theta, Strike, param, r, q, T, NumTerms)) / 2.0 / dy);
}
