private static double Alpha(Bergomi2FModel b2F)
{
var correl = Correl(b2F);
var thetaVect = new[] { 1.0 - b2F.Theta, b2F.Theta };
return(1.0 / Math.Sqrt(correl.BilinearProd(thetaVect, thetaVect)));
}
public static double[] AtmfSkewApprox(Bergomi2FModel b2F, double[] maturities)
{
var alpha = Alpha(b2F);
double coeffX = b2F.Nu * alpha * (1.0 - b2F.Theta) * b2F.RhoSX;
double coeffY = b2F.Nu * alpha * b2F.Theta * b2F.RhoSY;
return(maturities.Map(t => coeffX * DoubleExpInt(b2F.K1, t) + coeffY * DoubleExpInt(b2F.K2, t)));
}
public static double[,] FwdVolInstantCovariance(Bergomi2FModel b2F, double[] fwdVolStart, double[] fwdVolEnd)
{
Debug.Assert(fwdVolStart.Length == fwdVolEnd.Length);
var correl = Correl(b2F);
var varSwapDeformations = VarSwapDeformation(b2F, fwdVolStart, fwdVolEnd);
return(ArrayUtils.CartesianProd(varSwapDeformations, varSwapDeformations,
(varSwapDef1, varSwapDef2) => correl.BilinearProd(varSwapDef1, varSwapDef2)));
}
public static double[] FwdVolInstantVol(Bergomi2FModel b2F, double[] fwdVolStart, double[] fwdVolEnd)
{
Debug.Assert(fwdVolStart.Length == fwdVolEnd.Length);
var correl = Correl(b2F);
var varSwapDeformations = VarSwapDeformation(b2F, fwdVolStart, fwdVolEnd);
return(EnumerableUtils.For(0, fwdVolStart.Length, i =>
{
var varSwapDef = varSwapDeformations[i];
return Math.Sqrt(correl.BilinearProd(varSwapDef, varSwapDef));
}));
}
private static double[][] VarSwapDeformation(Bergomi2FModel b2F, double[] fwdVolStart, double[] fwdVolEnd)
{
Debug.Assert(fwdVolStart.Length == fwdVolEnd.Length);
var initCurve = b2F.Xi.Integral(0.0);
var factor1 = (b2F.Xi * RrFunctions.Exp(-b2F.K1)).Integral(0.0);
var factor2 = (b2F.Xi * RrFunctions.Exp(-b2F.K2)).Integral(0.0);
var alpha = Alpha(b2F);
return(EnumerableUtils.For(0, fwdVolStart.Length, i =>
{
double volMatStart = fwdVolStart[i];
double volMatEnd = fwdVolEnd[i];
double initFwdVariance = initCurve.Eval(volMatEnd) - initCurve.Eval(volMatStart);
double def1 = factor1.Eval(volMatEnd) - factor1.Eval(volMatStart);
double def2 = factor2.Eval(volMatEnd) - factor2.Eval(volMatStart);
return new[] { (1.0 - b2F.Theta) * def1, b2F.Theta * def2 }.Mult(b2F.Nu * alpha / initFwdVariance);
}));
}
private static double[,] Correl(Bergomi2FModel b2F)
{
return(new[, ] {
{ 1.0, b2F.RhoXY }, { b2F.RhoXY, 1.0 }
});
}
