private double GetFirstDerivativeToStrike(double t, double k)
{
double forwardPrice = PriceBump.BumpForward(k);
double backwardPrice = PriceBump.BumpBackward(k);
return((ImpliedVol.GetValue(t, forwardPrice) - ImpliedVol.GetValue(t, backwardPrice)) / (forwardPrice - backwardPrice));
}
// below three functions assume time t and strike k is in good region
private double GetFirstDerivativeToTime(double t, double k)
{
double forwardTime = TimeBump.BumpForward(t);
double backwardTime = TimeBump.BumpBackward(t);
return((ImpliedVol.GetValue(forwardTime, k) - ImpliedVol.GetValue(backwardTime, k)) / (forwardTime - backwardTime));
}
private double GetSecondDerivativeToStrike(double t, double k)
{
// there is possibility that the bump sizes are different between forward and backward.
// we use the minimum of them to calculate the second order derivative.
double bumpsize = PriceBump.MinBump(k);
return((ImpliedVol.GetValue(t, k + bumpsize) + ImpliedVol.GetValue(t, k - bumpsize) - 2 * ImpliedVol.GetValue(t, k)) /
(bumpsize * bumpsize));
}
private double GetLocalVolFromImpliedVol(double t, double k)
{
var forwardPrice = ImpliedVol.GetForwardPrice(t);
var sigmaImpl = ImpliedVol.GetValue(t, k);
var firstOrderToT = GetFirstDerivativeToTime(t, k);
var firstOrderToK = GetFirstDerivativeToStrike(t, k);
var secondOrderToK = GetSecondDerivativeToStrike(t, k);
var d1 = (Math.Log(forwardPrice / k) + 0.5 * sigmaImpl * sigmaImpl * t) / (sigmaImpl * Math.Sqrt(t));
var sigmaLSquare = (Math.Pow(sigmaImpl, 2) + 2 * sigmaImpl * t * firstOrderToT) /
(Math.Pow((1 + d1 * Math.Sqrt(t) * k * firstOrderToK), 2) +
sigmaImpl * t * k * k * (secondOrderToK - d1 * Math.Sqrt(t) * Math.Pow(firstOrderToK, 2)));
return(sigmaLSquare >= 0
? Math.Sqrt(sigmaLSquare)
: 0.0);
}
