public static IInterpolator1D GenerateCDF(this IVolSurface surface, int numSamples, DateTime expiry, double fwd, bool returnInverse = false)
{
var deltaKLow = 0.0000001;
var deltaKHi = 0.9999999;
var kStepD = (deltaKHi - deltaKLow) / (numSamples + 3);
var deltaKBump = deltaKLow / 10;
var t = surface.OriginDate.CalculateYearFraction(expiry, DayCountBasis.Act365F);
var x = new double[numSamples + 2];
var y = new double[numSamples + 2];
for (var i = 0; i < x.Length; i++)
{
var deltaKNew = deltaKLow + i * kStepD;
var mStrike = deltaKNew - deltaKBump / 2;
var pStrike = deltaKNew + deltaKBump / 2;
var mStrikeVol = surface.GetVolForDeltaStrike(mStrike, t, fwd);
var mk = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -mStrike, 0, t, mStrikeVol);
var pStrikeVol = surface.GetVolForDeltaStrike(pStrike, t, fwd);
var pk = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -pStrike, 0, t, pStrikeVol);
if (i == 0)
{
x[0] = mk / 2.0;
y[0] = 0;
continue;
}
if (i == x.Length - 1)
{
x[i] = pk * 2;
y[i] = 1;
continue;
}
var dkAbs = (pk - mk);
var pPut = BlackFunctions.BlackPV(fwd, pk, 0, t, pStrikeVol, OptionType.P);
var mPut = BlackFunctions.BlackPV(fwd, mk, 0, t, mStrikeVol, OptionType.P);
var digital = (pPut - mPut) / dkAbs;
y[i] = digital;
x[i] = (mk + pk) / 2.0;
}
return(returnInverse ?
InterpolatorFactory.GetInterpolator(y, x, Interpolator1DType.LinearFlatExtrap) :
InterpolatorFactory.GetInterpolator(x, y, Interpolator1DType.LinearFlatExtrap));
}
public static IInterpolator1D GeneratePDF(this IVolSurface surface, int numSamples, DateTime expiry, double fwd)
{
var deltaK = fwd * 0.0001;
var t = surface.OriginDate.CalculateYearFraction(expiry, DayCountBasis.Act365F);
var lowStrikeVol = surface.GetVolForDeltaStrike(0.0001, t, fwd);
var lowStrike = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.0001, 0, t, lowStrikeVol);
var hiStrikeVol = surface.GetVolForDeltaStrike(0.9999, t, fwd);
var hiStrike = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.9999, 0, t, hiStrikeVol);
var x = new double[numSamples + 2];
var y = new double[numSamples + 2];
var k = lowStrike;
var kStep = (hiStrike - lowStrike) / numSamples;
for (var i = 0; i < x.Length; i++)
{
if (i == 0)
{
x[0] = lowStrike / 2.0;
y[0] = 0;
continue;
}
if (i == x.Length - 1)
{
x[i] = k * 2;
y[i] = 0;
continue;
}
var volLow = surface.GetVolForAbsoluteStrike(k - deltaK, t, fwd);
var callLow = BlackFunctions.BlackPV(fwd, k - deltaK, 0, t, volLow, OptionType.C);
var volMid = surface.GetVolForAbsoluteStrike(k, t, fwd);
var callMid = BlackFunctions.BlackPV(fwd, k, 0, t, volMid, OptionType.C);
var volHi = surface.GetVolForAbsoluteStrike(k + deltaK, t, fwd);
var callHi = BlackFunctions.BlackPV(fwd, k + deltaK, 0, t, volHi, OptionType.C);
var digitalLo = (callLow - callMid) / deltaK;
var digitalHi = (callMid - callHi) / deltaK;
y[i] = (digitalLo - digitalHi) / deltaK;
x[i] = k;
k += kStep;
}
var firstPass = InterpolatorFactory.GetInterpolator(x, y, Interpolator1DType.LinearFlatExtrap);
var totalY = ((IIntegrableInterpolator)firstPass).DefiniteIntegral(x.First(), x.Last());
y = y.Select(v => v / totalY).ToArray();
return(InterpolatorFactory.GetInterpolator(x, y, Interpolator1DType.LinearFlatExtrap));
}
public static IInterpolator1D GenerateCDF2(this IVolSurface surface, int numSamples, DateTime expiry, double fwd, bool returnInverse = false, double strikeScale = 1.0, bool logStrikes = false)
{
var premInterp = GeneratePremiumInterpolator(surface, numSamples, expiry, fwd, OptionType.P);
var t = surface.OriginDate.CalculateYearFraction(expiry, DayCountBasis.Act365F);
var x = new double[numSamples];
var y = new double[numSamples];
var deltaKLow = 0.0000000001;
var deltaKHi = 0.9999999999;
var kStepD = (deltaKHi - deltaKLow) / numSamples;
for (var i = 0; i < x.Length; i++)
{
var deltaKNew = -deltaKLow - i * kStepD;
var newStrikeVol = surface.GetVolForDeltaStrike(-deltaKNew, t, fwd);
var k = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, deltaKNew, 0, t, newStrikeVol);
var digital = premInterp.FirstDerivative(k);
y[i] = digital;
x[i] = k * strikeScale;
if (logStrikes)
{
x[i] = Log(x[i]);
}
}
return(returnInverse ?
InterpolatorFactory.GetInterpolator(y, x, Interpolator1DType.MonotoneCubicSpline) :
InterpolatorFactory.GetInterpolator(x, y, Interpolator1DType.MonotoneCubicSpline));
}
public void Finish(IFeatureCollection collection)
{
if (!_timesteps.IsComplete)
{
return;
}
//drifts and vols...
_drifts = new Vector <double> [_timesteps.TimeStepCount];
_lvInterps = new IInterpolator1D[_timesteps.TimeStepCount - 1];
var strikes = new double[_timesteps.TimeStepCount][];
var atmVols = new double[_timesteps.TimeStepCount];
for (var t = 0; t < strikes.Length; t++)
{
var fwd = _forwardCurve(_timesteps.Times[t]);
atmVols[t] = _surface.GetVolForDeltaStrike(0.5, _timesteps.Times[t], fwd);
if (_timesteps.Times[t] == 0)
{
strikes[t] = new double[] { fwd };
continue;
}
else
{
var nStrikes = 200;
var strikeStep = 1.0 / nStrikes;
strikes[t] = new double[nStrikes];
for (var k = 0; k < strikes[t].Length; k++)
{
var deltaK = -(strikeStep + strikeStep * k);
strikes[t][k] = Options.BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, deltaK, 0, _timesteps.Times[t], atmVols[t]);
}
}
}
var lvSurface = Options.LocalVol.ComputeLocalVarianceOnGridFromCalls(_surface, strikes, _timesteps.Times, _forwardCurve);
for (var t = 0; t < _lvInterps.Length; t++)
{
_lvInterps[t] = InterpolatorFactory.GetInterpolator(strikes[t], lvSurface[t], t == 0 ? Interpolator1DType.DummyPoint : Interpolator1DType.LinearFlatExtrap);
}
var prevSpot = _forwardCurve(0);
for (var t = 1; t < _drifts.Length; t++)
{
var fxAtmVol = _adjSurface == null ? 0.0 : _adjSurface.GetForwardATMVol(0, _timesteps.Times[t]);
var driftAdj = _adjSurface == null ? 1.0 : Exp(atmVols[t] * fxAtmVol * _timesteps.Times[t] * _correlation);
var spot = _forwardCurve(_timesteps.Times[t]) * driftAdj;
_drifts[t] = new Vector <double>(Log(spot / prevSpot) / _timesteps.TimeSteps[t]);
prevSpot = spot;
}
_isComplete = true;
}
public static IInterpolator1D GeneratePremiumInterpolator(this IVolSurface surface, int numSamples, double t, double fwd, OptionType cp)
{
var lowStrikeVol = surface.GetVolForDeltaStrike(0.001, t, fwd);
var lowStrike = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.001, 0, t, lowStrikeVol);
var hiStrikeVol = surface.GetVolForDeltaStrike(0.999, t, fwd);
var hiStrike = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.999, 0, t, hiStrikeVol);
var x = new double[numSamples + 2];
var y = new double[numSamples + 2];
var k = lowStrike;
var kStep = (hiStrike - lowStrike) / (numSamples + 1.0);
var vol = surface.GetVolForAbsoluteStrike(k / 10, t, fwd);
var call = BlackFunctions.BlackPV(fwd, k / 10, 0, t, vol, cp);
y[0] = call;
x[0] = k / 10;
for (var i = 0; i < x.Length - 1; i++)
{
vol = surface.GetVolForAbsoluteStrike(k, t, fwd);
call = BlackFunctions.BlackPV(fwd, k, 0, t, vol, cp);
y[i + 1] = call;
x[i + 1] = k;
k += kStep;
}
vol = surface.GetVolForAbsoluteStrike(k * 10, t, fwd);
call = BlackFunctions.BlackPV(fwd, k * 10, 0, t, vol, cp);
y[x.Length - 1] = call;
x[x.Length - 1] = k * 10;
return(InterpolatorFactory.GetInterpolator(x, y, Interpolator1DType.MonotoneCubicSpline));
}
public void Finish(FeatureCollection collection)
{
if (!_timesteps.IsComplete)
{
return;
}
//drifts and vols...
_drifts = new double[_timesteps.TimeStepCount];
_lvInterps = new IInterpolator1D[_timesteps.TimeStepCount - 1];
var strikes = new double[_timesteps.TimeStepCount][];
for (var t = 0; t < strikes.Length; t++)
{
strikes[t] = new double[98];
var fwd = _forwardCurve(_timesteps.Times[t]);
var atmVol = _surface.GetVolForDeltaStrike(fwd, _timesteps.Times[t], fwd);
for (var k = 0; k < strikes[t].Length; k++)
{
var deltaK = -(0.01 + 0.01 * k);
strikes[t][k] = Options.BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, deltaK, 0, _timesteps.Times[t], atmVol);
}
}
var lvSurface = Options.LocalVol.ComputeLocalVarianceOnGrid(_surface, strikes, _timesteps.Times, _forwardCurve);
for (var t = 0; t < _lvInterps.Length; t++)
{
_lvInterps[t] = InterpolatorFactory.GetInterpolator(strikes[t], lvSurface[t], t == 0 ? Interpolator1DType.DummyPoint : Interpolator1DType.LinearFlatExtrap);
}
var prevSpot = _forwardCurve(0);
for (var t = 1; t < _drifts.Length; t++)
{
var spot = _forwardCurve(_timesteps.Times[t]);
_drifts[t] = System.Math.Log(spot / prevSpot) / _timesteps.TimeSteps[t];
prevSpot = spot;
}
_isComplete = true;
}
public static IInterpolator1D GenerateCompositeSmileB(this IVolSurface surface, IVolSurface fxSurface, int numSamples, DateTime expiry, double fwdAsset, double fwdFx, double correlation, bool strikesInDeltaSpace = false)
{
var t = surface.OriginDate.CalculateYearFraction(expiry, DayCountBasis.Act365F);
var fxInv = new InverseFxSurface("fxInv", fxSurface as IATMVolSurface, null);
var atmFx = fxSurface.GetVolForDeltaStrike(0.5, t, fwdFx);
var atmA = surface.GetVolForDeltaStrike(0.5, t, fwdAsset);
var compoFwd = fwdAsset * fwdFx;
var atmCompo = Sqrt(atmFx * atmFx + atmA * atmA + 2.0 * correlation * atmA * atmFx);
var lowK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(compoFwd, -0.01, 0, t, atmCompo);
var hiK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(compoFwd, -0.99, 0, t, atmCompo);
//var cdfInvFx = fxSurface.GenerateCDF2(numSamples * 10, expiry, fwdFx, true);
//var cdfInvAsset = surface.GenerateCDF2(numSamples * 10, expiry, fwdAsset, true);
//var yFx = new Func<double, double>(z => cdfInvFx.Interpolate(Statistics.NormSDist(z)));
//var yAsset = new Func<double, double>(z => cdfInvAsset.Interpolate(Statistics.NormSDist(z)));
var fxCDFCache = new Dictionary <double, double>();
var assetCDFCache = new Dictionary <double, double>();
var yFx = new Func <double, double>(z =>
{
if (fxCDFCache.TryGetValue(z, out var K))
{
return(K);
}
K = fxInv.InverseCDF(expiry, 1.0 / fwdFx, Statistics.NormSDist(z));
fxCDFCache.Add(z, K);
return(K);
});
var yAsset = new Func <double, double>(z =>
{
if (assetCDFCache.TryGetValue(z, out var K))
{
return(K);
}
K = surface.InverseCDF(expiry, fwdAsset, Statistics.NormSDist(z));
assetCDFCache.Add(z, K);
return(K);
});
//var fxCDFCache = new Dictionary<double, double>();
//var assetCDFCache = new Dictionary<double, double>();
//var putFx = fxInv.GeneratePremiumInterpolator(numSamples * 10, expiry, 1.0/fwdFx, OptionType.P);
//var putAsset = surface.GeneratePremiumInterpolator(numSamples * 10, expiry, fwdAsset, OptionType.P);
//var yFx = new Func<double, double>(z =>
//{
// if (fxCDFCache.TryGetValue(z, out var K)) return K;
// K = InverseCDF(putFx, t, 1.0/fwdFx, Statistics.NormSDist(z));
// fxCDFCache.Add(z, K);
// return K;
//});
//var yAsset = new Func<double, double>(z =>
//{
// if (assetCDFCache.TryGetValue(z, out var K)) return K;
// K = InverseCDF(putAsset, t, fwdAsset, Statistics.NormSDist(z));
// var kl = assetCDFCache.Keys.ToList();
// var closerIx = kl.BinarySearch(z);
// var keyIx = ~closerIx;
// if (closerIx < 0 && z < 0 && kl.Count > keyIx)
// {
// if (assetCDFCache[kl[keyIx]] < K)
// K = assetCDFCache[kl[keyIx]];
// }
// assetCDFCache.Add(z, K);
// return K;
//});
var payoff = new Func <double, double, double, double>((z1, z2, kQ) => Max(kQ * yFx(z2) - yAsset(z1), 0));
var integrand = new Func <double, double, double, double>((z1, z2, kQ) => payoff(z1, z2, kQ) * BivariateNormal.PDF(z1, z2, -correlation));
var kStep = (hiK - lowK) / numSamples;
var ks = Enumerable.Range(0, numSamples).Select(kk => lowK + kk * kStep).ToArray();
var premiums = new double[ks.Length];
var vols = new double[ks.Length];
for (var i = 0; i < ks.Length; i++)
{
var ik = new Func <double, double, double>((z1, z2) => integrand(z1, z2, ks[i]));
var pk = Integration.TwoDimensionalGaussLegendre(ik, -5, 5, -5, 5, 16);
//var pk = Integration.TwoDimensionalSimpsons(ik, -5, 5, -5, 5, 100);
pk *= fwdFx;
var volK = BlackFunctions.BlackImpliedVol(compoFwd, ks[i], 0.0, t, pk, OptionType.P);
vols[i] = volK;
premiums[i] = pk;
}
if (strikesInDeltaSpace)
{
ks = ks.Select((ak, ix) => - BlackFunctions.BlackDelta(compoFwd, ak, 0.0, t, vols[ix], OptionType.P)).ToArray();
}
return(InterpolatorFactory.GetInterpolator(ks, vols, Interpolator1DType.CubicSpline));
}
public static IInterpolator1D GenerateCompositeSmile(this IVolSurface surface, IVolSurface fxSurface, int numSamples, DateTime expiry, double fwdAsset, double fwdFx, double rho, bool strikesInDeltaSpace = false)
{
var t = surface.OriginDate.CalculateYearFraction(expiry, DayCountBasis.Act365F);
var atmFx = fxSurface.GetVolForDeltaStrike(0.5, t, fwdFx);
var atmA = surface.GetVolForDeltaStrike(0.5, t, fwdAsset);
var compoFwd = fwdAsset / fwdFx;
var atmCompo = Sqrt(atmFx * atmFx + atmA * atmA + 2.0 * rho * atmA * atmFx);
var lowK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(compoFwd, -0.01, 0, t, atmCompo);
var hiK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(compoFwd, -0.99, 0, t, atmCompo);
var nuA = Sqrt(t) * atmA;
var nuFx = Sqrt(t) * atmFx;
var cdfFx = new Func <double, double>(k => fxSurface.CDF(expiry, fwdFx, Exp(k)));
var cdfA = new Func <double, double>(k => surface.CDF(expiry, fwdAsset, Exp(k)));
var fxCDFCache = new Dictionary <double, double>();
var assetCDFCache = new Dictionary <double, double>();
var yFx = new Func <double, double>(z =>
{
if (fxCDFCache.TryGetValue(z, out var K))
{
return(K);
}
K = Log(fxSurface.InverseCDF(expiry, fwdFx, Statistics.NormSDist(z)));
fxCDFCache.Add(z, K);
return(K);
});
var yA = new Func <double, double>(z =>
{
if (assetCDFCache.TryGetValue(z, out var K))
{
return(K);
}
K = Log(surface.InverseCDF(expiry, fwdAsset, Statistics.NormSDist(z)));
assetCDFCache.Add(z, K);
return(K);
});
//var zfxS = new Func<double, double, double>((zA, K) => Statistics.NormInv(Max(1e-18, Min(1.0 - 1e-18, cdfFx(yA(zA) - Log(K))))));
//var zAs = new Func<double, double, double>((zFx, K) => Statistics.NormInv(Max(1e-18, Min(1.0 - 1e-18, cdfA(yFx(zFx) + Log(K))))));
var zfxS = new Func <double, double, double>((zA, K) => Statistics.NormInv(cdfFx(yA(zA) - Log(K))));
var zAs = new Func <double, double, double>((zFx, K) => Statistics.NormInv(cdfA(yFx(zFx) + Log(K))));
var d = -1.0;
var p2 = 1.0 / Sqrt(2.0 * PI);
//var I1 = new Func<double, double, double>((zA, K) =>
//p2*Exp(yA(zA) - (nuA * zA - nuA * nuA / 2)) * Statistics.NormSDist(d * (zfxS(zA, K) - rho * zA) / Sqrt(1 - rho * rho)) * Exp(-(zA - nuA) * (zA - nuA) / 2.0)
// );
//var I2 = new Func<double, double, double>((zFx, K) =>
//p2*Exp(yFx(zFx) - (nuFx * zFx - nuFx * nuFx / 2)) * Statistics.NormSDist(-d * (zAs(zFx, K) - rho * zFx) / Sqrt(1 - rho * rho)) * Exp(-(zFx - nuFx) * (zFx - nuFx) / 2.0)
// );
var I1 = new Func <double, double, double>((zA, K) =>
p2 * Exp(yA(zA)) * Statistics.NormSDist(d * (zfxS(zA, K) - rho * zA) / Sqrt(1 - rho * rho)) * Exp(-(zA * zA) / 2.0)
);
var I2 = new Func <double, double, double>((zFx, K) =>
p2 * Exp(yFx(zFx)) * Statistics.NormSDist(-d * (zAs(zFx, K) - rho * zFx) / Sqrt(1 - rho * rho)) * Exp(-(zFx * zFx) / 2.0)
);
var kStep = (hiK - lowK) / numSamples;
var ks = Enumerable.Range(0, numSamples).Select(kk => lowK + kk * kStep).ToArray();
var premiums = new double[ks.Length];
var vols = new double[ks.Length];
for (var i = 0; i < ks.Length; i++)
{
var I1k = new Func <double, double>(z => I1(z, ks[i]));
var I2k = new Func <double, double>(z => I2(z, ks[i]));
//var i1 = Integration.GaussLegendre(I1k, -5, 5, 16);
//var i2 = Integration.GaussLegendre(I2k, -5, 5, 16);
var i1 = Integration.SimpsonsRule(I1k, -5, 5, numSamples);
var i2 = Integration.SimpsonsRule(I2k, -5, 5, numSamples);
var pk = d * (i1 - ks[i] * i2);
pk /= fwdFx;
var volK = BlackFunctions.BlackImpliedVol(compoFwd, ks[i], 0.0, t, pk, OptionType.P);
vols[i] = volK;
premiums[i] = pk;
}
if (strikesInDeltaSpace)
{
ks = ks.Select((ak, ix) => - BlackFunctions.BlackDelta(compoFwd, ak, 0.0, t, vols[ix], OptionType.P)).ToArray();
}
return(InterpolatorFactory.GetInterpolator(ks, vols, Interpolator1DType.CubicSpline));
}