/// <summary>
///
/// </summary>
/// <param name="timeGrid">Time discretisation used.</param>
/// <param name="factors">Number of factors in the model.</param>
/// <param name="weights">Precomputed swaption weights.</param>
/// <param name="correlation">A homogeneous forward correlation matrix.</param>
public InterestRateVolatilities(PedersenTimeGrid timeGrid, int factors, SwaptionWeights weights, DenseMatrix correlation)
{
_volFixed = false;
_timeGrid = timeGrid;
_factors = factors;
_weights = weights;
_correlation = correlation;
_volatility = new Matrix[_timeGrid.MaxExpiry];
for (int i = 0; i < _timeGrid.MaxExpiry; i++)
{
_volatility[i] = new Matrix(_timeGrid.MaxTenor, _factors);
}
_storedImpliedVol = new double[_timeGrid.MaxExpiry][];
_storedImpliedVolSq = new double[_timeGrid.MaxExpiry][];
_storedImpliedVolTerm = new double[_timeGrid.MaxExpiry][][];
for (int i = 0; i < _timeGrid.MaxExpiry; i++)
{
_storedImpliedVol[i] = new double[_timeGrid.MaxTenor - i];
_storedImpliedVolSq[i] = new double[_timeGrid.MaxTenor - i];
_storedImpliedVolTerm[i] = new double[_timeGrid.MaxTenor - i][];
for (int j = 0; j < _timeGrid.MaxTenor - i; j++)
{
_storedImpliedVolTerm[i][j] = new double[i + 1];
}
}
}
private DenseMatrix SetXi(int exp, Vector x)
{
var cor = _economy.Correlation;
var cov = new Matrix(_param.NumberOfTenors, _param.NumberOfTenors);
for (int i = 0; i < _param.NumberOfTenors; i++)
{
for (int j = 0; j <= i; j++)
{
cov[i, j] = cor[i, j] * x[i] * x[j];
}
}
var eigenValueDecomposition = new EigenvalueDecomposition(cov);
var baseVector = eigenValueDecomposition.RealEigenvalues.ToList();
var eval = baseVector.GetRange(_param.NumberOfTenors - _param.NumberOfFactors, _param.NumberOfTenors - 1 + _param.NumberOfFactors);//Select the sub-vector starting at the index i and with length y
var baseMatrix = eigenValueDecomposition.EigenVectors;
var eigenVectors = baseMatrix.GetMatrix(0, baseMatrix.RowCount - 1, _param.NumberOfTenors - _param.NumberOfFactors, _param.NumberOfTenors - 1);
var xiSection = new DenseMatrix(_param.UnderlyingTenor, _param.NumberOfFactors);
for (int j = 0; j < _param.NumberOfFactors; j++)
{
eval[j] = Math.Sqrt(eval[j]);
for (int i = 0; i < _param.NumberOfTenors; i++)
{
var temp = eval[j] * eigenVectors[i, j];
for (int n = _param.Tenor[i]; n < Math.Min(_param.Tenor[i + 1], _param.UnderlyingTenor - _param.Expiry[exp]); n++)
{
xiSection[n, j] = temp;
}
}
}
return(xiSection);
}
/// <summary>
/// Computes the implied volatility of swaption with given expiry and tenor, or a caplet with given expiry,
/// with the volatility at one particular expiry time bucket replaced with temporary values.
/// A caplet implied volatility is equivalent to that of a swaption with the tenor length of 1.
/// Saves computation time when checking the effect of a simple perturbation.
/// </summary>
/// <param name="expiry"></param>
/// <param name="tenor"></param>
/// <param name="newRow">Indicating the expiry time bucket to be replaced.</param>
/// <param name="replacedRow">New interest rate volatility values for the expiry time bucket.</param>
/// <returns></returns>
public double FindImpliedVolatility(int expiry, int tenor, Matrix newRow, int replacedRow)
{
int exp = expiry - 1;
int ten = tenor - 1;
double result;
if (exp >= _timeGrid.ExpiryGrid[replacedRow])
{
result = _storedImpliedVolSq[exp][ten];
for (int i = _timeGrid.ExpiryGrid[replacedRow]; i < Math.Min(_timeGrid.ExpiryGrid[replacedRow + 1], expiry); i++)
{
var tempVector = new DenseVector(_factors);
for (int j = 0; j <= ten; j++)
{
tempVector = tempVector + _weights.Get(expiry, tenor, j + 1) * newRow.RowD(j + exp - i);
}
double tempSum = tempVector * tempVector;
result += tempSum - _storedImpliedVolTerm[exp][ten][i];
}
result = Math.Sqrt(result / expiry);
}
else
{
result = _storedImpliedVol[exp][ten];
}
return(result);
}
/// <summary>
/// Computes the volatilities of a particular expiry time bucket.
/// </summary>
/// <param name="row"></param>
/// <param name="inputVol"></param>
/// <returns></returns>
public Matrix SingleRowOfVolatilities(int row, Vector inputVol)
{
//create covariance
var covariance = new Matrix(_timeGrid.TenorCount, _timeGrid.TenorCount);
for (int i = 0; i < _timeGrid.TenorCount; i++)
{
for (int j = 0; j <= i; j++)
{
covariance[i, j] = _correlation[i, j] * inputVol[i] * inputVol[j];
}
}
//principal component analysis
var eigenValueDecomposition = new EigenvalueDecomposition(covariance);
var baseVector = eigenValueDecomposition.RealEigenvalues.ToList();
var eval = baseVector.GetRange(_timeGrid.TenorCount - 1 - _factors, _factors);//Select the sub-vector starting at the index i and with length y
var baseMatrix = eigenValueDecomposition.EigenVectors;
var eigenVectors = baseMatrix.GetMatrix(0, baseMatrix.RowCount - 1, _timeGrid.TenorCount - _factors, _timeGrid.TenorCount - 1);
var resultVol = new Matrix(_timeGrid.MaxTenor, _factors);
for (int j = 0; j < _factors; j++)
{
eval[j] = Math.Sqrt(eval[j]);
for (int i = 0; i < _timeGrid.TenorCount; i++)
{
var temp = eval[j] * eigenVectors[i, j];
for (int n = _timeGrid.TenorGrid[i]; n < Math.Min(_timeGrid.TenorGrid[i + 1], _timeGrid.MaxTenor - _timeGrid.ExpiryGrid[row]); n++)
{
resultVol[n, j] = temp;
}
}
}
return(resultVol);
}
private double SmoothV(Matrix <double> xMatrix)
{
double total = 0;
for (int i = 0; i < _param.NumberOfExpiries - 1; i++)
{
for (int j = 0; j < _param.NumberOfTenors; j++)
{
double temp = Math.Log(xMatrix[i, j] / xMatrix[i + 1, j]);
total += temp * temp;
}
}
total = total * _vFactor;
return(total);
}
private void PopulateXi(Matrix <double> xMatrix)
{
for (int i = 0; i < _param.NumberOfExpiries; i++)
{
var gm = SetXi(i, (DenseVector)xMatrix.Row(i));
for (int j = _param.Expiry[i]; j < _param.Expiry[i + 1]; j++)
{
for (int k = 0; k < _param.UnderlyingTenor - j; k++)
{
for (int l = 0; l < _param.NumberOfFactors; l++)
{
_economy.Xi[j][k, l] = gm[k, l];
}
}
//economy.Xi[j] = (GeneralMatrix)gm.Clone();
}
}
}
/// <summary>
/// Computes the multi-factored volatilities given a particular input volatility grid, using principal component analysis.
/// (See Pedersen, M. B. (1998) Calibrating Libor market models. Working paper, Financial Research Department, SimCorp.)
/// </summary>
/// <param name="inputVolMat"></param>
public void Populate(Matrix inputVolMat)
{
if (!_volFixed)
{
for (int i = 0; i < _timeGrid.ExpiryCount; i++)
{
var currentRow = SingleRowOfVolatilities(i, inputVolMat.RowD(i));
for (int j = _timeGrid.ExpiryGrid[i]; j < _timeGrid.ExpiryGrid[i + 1]; j++)
{
for (int k = 0; k < _timeGrid.MaxTenor - j; k++)
{
for (int l = 0; l < _factors; l++)
{
_volatility[j][k, l] = currentRow[k, l];
}
}
}
}
}
else
{
throw new Exception("Cannot alter finalised volatility.");
}
}
