public static SymmetricMatrix CreateSymmetricHilbertMatrix(int n)
{
SymmetricMatrix H = new SymmetricMatrix(n);
H.Fill((r, c) => 1.0 / (r + c + 1));
return(H);
}
public void RealEigenvalueOrdering()
{
int d = 10;
Random rng = new Random(d + 1);
SymmetricMatrix A = new SymmetricMatrix(d);
A.Fill((int r, int c) => - 1.0 + 2.0 * rng.NextDouble());
RealEigensystem E = A.Eigensystem();
for (int i = 0; i < E.Dimension; i++)
{
Console.WriteLine(E.Eigenvalue(i));
}
E.Sort(OrderBy.ValueAscending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(E.Eigenvalue(i - 1) <= E.Eigenvalue(i));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
E.Sort(OrderBy.ValueDescending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(E.Eigenvalue(i - 1) >= E.Eigenvalue(i));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
E.Sort(OrderBy.MagnitudeAscending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(Math.Abs(E.Eigenvalue(i - 1)) <= Math.Abs(E.Eigenvalue(i)));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
E.Sort(OrderBy.MagnitudeDescending);
for (int i = 1; i < E.Dimension; i++)
{
Assert.IsTrue(Math.Abs(E.Eigenvalue(i - 1)) >= Math.Abs(E.Eigenvalue(i)));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, E.Eigenvector(i), E.Eigenvalue(i)));
}
}
public void RealEigenvalueOrdering()
{
int d = 10;
Random rng = new Random(d + 1);
SymmetricMatrix A = new SymmetricMatrix(d);
A.Fill((int r, int c) => - 1.0 + 2.0 * rng.NextDouble());
RealEigendecomposition E = A.Eigendecomposition();
RealEigenpairCollection pairs = E.Eigenpairs;
pairs.Sort(OrderBy.ValueAscending);
for (int i = 1; i < pairs.Count; i++)
{
Assert.IsTrue(pairs[i - 1].Eigenvalue <= pairs[i].Eigenvalue);
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, pairs[i].Eigenvector, pairs[i].Eigenvalue));
}
pairs.Sort(OrderBy.ValueDescending);
for (int i = 1; i < pairs.Count; i++)
{
Assert.IsTrue(pairs[i - 1].Eigenvalue >= pairs[i].Eigenvalue);
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, pairs[i].Eigenvector, pairs[i].Eigenvalue));
}
pairs.Sort(OrderBy.MagnitudeAscending);
for (int i = 1; i < pairs.Count; i++)
{
Assert.IsTrue(Math.Abs(pairs[i - 1].Eigenvalue) <= Math.Abs(pairs[i].Eigenvalue));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, pairs[i].Eigenvector, pairs[i].Eigenvalue));
}
pairs.Sort(OrderBy.MagnitudeDescending);
for (int i = 1; i < pairs.Count; i++)
{
Assert.IsTrue(Math.Abs(pairs[i - 1].Eigenvalue) >= Math.Abs(pairs[i].Eigenvalue));
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(A, pairs[i].Eigenvector, pairs[i].Eigenvalue));
}
}
public double VIXfuture_TruncatedChlsky(double T, Func <double, double> epsilon)
{
DateTime start = DateTime.Now;
//StreamWriter sw = new StreamWriter("C:\\Users\\Marouane\\Desktop\\M2IF\\rough volatility\\rBergomiFile.txt");
//Grid
int N = 100;
int S = 7;
Grid grid = new Grid(T, T + Delta, N);
#region Correlation "Correl Matrix Construction"
// Small Covariance Matrix t_0 , ... , t_7
SymmetricMatrix covM_Small = new SymmetricMatrix(S);
//the full Covariance Matrix
SymmetricMatrix covM = new SymmetricMatrix(N + 1);
//Setting for integrale approximations
EvaluationSettings settings = new EvaluationSettings();
settings.AbsolutePrecision = 1.0E-7;
settings.RelativePrecision = 0.0;
// Small Covariance Calcul t_0,.....,t_7
for (int i = 0; i < S; i++)
{
covM_Small[i, i] = (Math.Pow(grid.t(i), 2 * H) - Math.Pow(grid.t(i) - T, 2 * H)) / (2 * H);
for (int j = 0; j < i; j++)
{
Func <double, double> covfunc = t => Math.Pow((grid.t(j) - t) * (grid.t(i) - t), H - 0.5);
IntegrationResult integresult = FunctionMath.Integrate(covfunc, Meta.Numerics.Interval.FromEndpoints(0.0, T), settings);
covM_Small[i, j] = integresult.Value;
}
}
// full COrrelation Calcul
for (int i = 0; i <= N; i++)
{
covM[i, i] = (Math.Pow(grid.t(i), 2 * H) - Math.Pow(grid.t(i) - T, 2 * H)) / (2 * H);
for (int j = 0; j < i; j++)
{
Func <double, double> covfunc = t => Math.Pow((grid.t(j) - t) * (grid.t(i) - t), H - 0.5);
IntegrationResult integresult = FunctionMath.Integrate(covfunc, Meta.Numerics.Interval.FromEndpoints(0.0, T), settings);
covM[i, j] = integresult.Value;
}
}
Func <int, int, double> corrf_Small = (i, j) => covM_Small[i, j] / (Math.Sqrt(covM_Small[i, i] * covM_Small[j, j]));
SymmetricMatrix Correl_Small = new SymmetricMatrix(S);
Correl_Small.Fill(corrf_Small);
Func <int, int, double> corrf = (i, j) => covM[i, j] / (Math.Sqrt(covM[i, i] * covM[j, j]));
SymmetricMatrix Correl = new SymmetricMatrix(N + 1);
Correl.Fill(corrf);
#endregion
CholeskyDecomposition cholesky = Correl_Small.CholeskyDecomposition();
SquareMatrix choleskyCorrel_Small = new SquareMatrix(S);
choleskyCorrel_Small = cholesky.SquareRootMatrix();
GaussianSimulator simulator = new GaussianSimulator();
double VIX = 0.0;
var MC = 1.0E5;
for (int mc = 1; mc < MC; mc++)
{
ColumnVector GaussianVector = new ColumnVector(S);
// Simulating Volterra at 8 first steps on [T, T+Delta]
for (int i = 0; i < S; i++)
{
GaussianVector[i] = simulator.Next();
}
ColumnVector Volterra_small = choleskyCorrel_Small * GaussianVector;
//Adjusting the variance of Volterra Processus
for (int i = 0; i < S; i++)
{
Volterra_small[i] = Volterra_small[i] * Math.Sqrt((Math.Pow(grid.t(i), 2 * H) - Math.Pow(grid.t(i) - grid.t(0), 2 * H)) / (2 * H));
}
// Simulating VOlterra on t_8 ... t_N with the truncated Formula
double[] Volterra = new double[N + 1];
for (int i = 0; i <= N; i++)
{
if (i < S)
{
Volterra[i] = Volterra_small[i];
}
//Tranceted Formula
else
{
Volterra[i] = Math.Sqrt(covM[i, i]) * (Correl[i, i - 1] * Volterra[i - 1] / Math.Sqrt(covM[i - 1, i - 1]) + Math.Sqrt(1 - Math.Pow(Correl[i, i - 1], 2)) * simulator.Next());
}
}
double VIX_ = VIX_Calculate(epsilon, grid, Volterra);
VIX += VIX_;
}
//sw.Close();
VIX /= MC;
TimeSpan dur = DateTime.Now - start;
return(VIX);
}