public void SABRSurfaceRRBF()
{
//flat surface
var origin = new DateTime(2017, 02, 07);
var expiry = origin.AddYears(1);
var t = (expiry - origin).TotalDays / 365.0;
var fwd = 1.5;
var vol = 0.32;
var rr = new[] { new[] { 0.02, 0.03 } };
var bf = new[] { new[] { 0.005, 0.007 } };
Func <double, double> fwdCurve = (tt => { return(fwd); });
var surface = new SabrVolSurface(origin, new[] { vol }, new[] { expiry }, new[] { 0.25, 0.1 }, rr, bf, new[] { 100.0 }, WingQuoteType.Arithmatic, AtmVolType.ZeroDeltaStraddle, Math.Interpolation.Interpolator1DType.Linear);
var gSurface = new RiskyFlySurface(origin, new[] { vol }, new[] { expiry }, new[] { 0.25, 0.1 }, rr, bf, new[] { 100.0 }, WingQuoteType.Arithmatic, AtmVolType.ZeroDeltaStraddle, Math.Interpolation.Interpolator1DType.Linear, Math.Interpolation.Interpolator1DType.Linear);
var atmK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, 0.5, 0.0, t, vol);
Assert.Equal(vol, surface.GetVolForAbsoluteStrike(atmK, expiry, fwd), 2);
var v25c = gSurface.GetVolForDeltaStrike(0.75, expiry, fwd);
var v25p = gSurface.GetVolForDeltaStrike(0.25, expiry, fwd);
var k25c = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, 0.25, 0.0, t, v25c);
var k25p = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.25, 0.0, t, v25p);
var t25c = surface.GetVolForAbsoluteStrike(k25c, expiry, fwd);
var t25p = surface.GetVolForAbsoluteStrike(k25p, expiry, fwd);
Assert.Equal(rr[0][0], t25c - t25p, 2);
}
public double GetVolForAbsoluteStrike(double strike, double maturity, double forward)
{
if (StrikeType == StrikeType.Absolute)
{
var interpForStrike = InterpolatorFactory.GetInterpolator(ExpiriesDouble,
_interpolators.Select(x => x.Interpolate(strike)).ToArray(),
TimeInterpolatorType);
return(interpForStrike.Interpolate(maturity));
}
else
{
var fwd = forward;
var cp = strike < 0 ? OptionType.Put : OptionType.Call;
Func <double, double> testFunc = (deltaK =>
{
var interpForStrike = InterpolatorFactory.GetInterpolator(ExpiriesDouble,
_interpolators.Select(x => x.Interpolate(deltaK)).ToArray(),
TimeInterpolatorType);
var vol = interpForStrike.Interpolate(maturity);
var absK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, deltaK, 0, maturity, vol);
return(absK - strike);
});
var solvedStrike = Qwack.Math.Solvers.Brent.BrentsMethodSolve(testFunc, 0.000000001, 0.999999999, 1e-8);
var interpForSolvedStrike = InterpolatorFactory.GetInterpolator(ExpiriesDouble,
_interpolators.Select(x => x.Interpolate(solvedStrike)).ToArray(),
TimeInterpolatorType);
return(interpForSolvedStrike.Interpolate(maturity));
}
}
private double GetDeltaStrikeForAbs(double fwd, double strike, double maturity)
{
var cp = strike < 0 ? OptionType.Put : OptionType.Call;
Func <double, double> testFunc = (deltaK =>
{
var interpForStrike = InterpolatorFactory.GetInterpolator(ExpiriesDouble, ExpiriesDouble.Select(e => GetVolForDeltaStrike(deltaK, e, fwd)).ToArray(), TimeInterpolatorType);
var vol2 = interpForStrike.Interpolate(maturity);
var absK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, deltaK, 0, maturity, vol2);
return(absK - strike);
});
var solvedStrike = -Math.Solvers.Brent.BrentsMethodSolve(testFunc, -0.99999999999, -0.00000000001, 1e-8);
if (solvedStrike == 0.00000000001 || solvedStrike == 0.99999999999) //out of bounds
{
var upperK = testFunc(-0.00000000001);
var lowerK = testFunc(-0.99999999999);
if (Abs(upperK - fwd) < Abs(lowerK - fwd))
{
solvedStrike = 0.00000000001;
}
else
{
solvedStrike = 0.99999999999;
}
}
return(solvedStrike);
}
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 double GetVolForAbsoluteStrike(double strike, double maturity, double forward)
{
var key = $"{strike:f6}~{maturity:f3}~{forward:f6}";
if (_allowCaching && _absVolCache.TryGetValue(key, out var vol))
{
return(vol);
}
if (StrikeType == StrikeType.Absolute)
{
var interpForStrike = InterpolatorFactory.GetInterpolator(ExpiriesDouble,
_interpolators.Select(x => x.Interpolate(strike)).ToArray(),
TimeInterpolatorType);
vol = interpForStrike.Interpolate(maturity);
}
else
{
var fwd = forward;
var cp = strike < 0 ? OptionType.Put : OptionType.Call;
Func <double, double> testFunc = (deltaK =>
{
var dkModified = FlatDeltaSmileInExtreme ? Min(1.0 - FlatDeltaPoint, Max(deltaK, FlatDeltaPoint)) : deltaK;
var interpForStrike = InterpolatorFactory.GetInterpolator(ExpiriesDouble,
_interpolators.Select(x => x.Interpolate(-dkModified)).ToArray(),
TimeInterpolatorType);
var vol2 = interpForStrike.Interpolate(maturity);
var absK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, deltaK, 0, maturity, vol2);
return(absK - strike);
});
var hiK = FlatDeltaSmileInExtreme ? 1.0 - FlatDeltaPoint : 0.999999999;
var loK = FlatDeltaSmileInExtreme ? FlatDeltaPoint : 0.000000001;
var solvedStrike = -Math.Solvers.Brent.BrentsMethodSolve(testFunc, -hiK, -loK, 1e-12);
if (solvedStrike == loK || solvedStrike == hiK) //out of bounds
{
var upperK = testFunc(-loK);
var lowerK = testFunc(-hiK);
if (Abs(upperK - fwd) < Abs(lowerK - fwd))
{
solvedStrike = loK;
}
else
{
solvedStrike = hiK;
}
}
var interpForSolvedStrike = InterpolatorFactory.GetInterpolator(ExpiriesDouble,
_interpolators.Select(x => x.Interpolate(solvedStrike)).ToArray(),
TimeInterpolatorType);
vol = interpForSolvedStrike.Interpolate(maturity);
}
if (_allowCaching)
{
_absVolCache[key] = vol;
}
return(vol);
}
public static IInterpolator1D GenerateCompositeSmileBasic(this IVolSurface surface, IVolSurface fxSurface, int numSamples, DateTime expiry, double fwdAsset, double fwdFx, double correlation, bool deltaStrikeOutput = false)
{
var deltaKa = fwdAsset * 0.00001;
var deltaKfx = fwdFx * 0.00001;
var t = surface.OriginDate.CalculateYearFraction(expiry, DayCountBasis.Act365F);
var atmFx = fxSurface.GetVolForAbsoluteStrike(fwdFx, t, fwdFx);
var atmA = surface.GetVolForAbsoluteStrike(fwdAsset, t, fwdAsset);
var compoFwd = fwdAsset * fwdFx;
var atmCompo = Sqrt(atmFx * atmFx + atmA * atmA + 2.0 * correlation * atmA * atmFx);
var lowK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(compoFwd, -0.0001, 0, t, atmCompo);
var hiK = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(compoFwd, -0.9999, 0, t, atmCompo);
var lowKA = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwdAsset, -0.0001, 0, t, atmA);
var hiKA = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwdAsset, -0.9999, 0, t, atmA);
var x = new double[numSamples];
var y = new double[numSamples];
var k = lowK;
var kStep = (hiK - lowK) / numSamples;
var kStepA = (hiKA - lowKA) / numSamples;
var kStepFx = (hiK / hiKA - lowK / lowKA) / numSamples;
for (var i = 0; i < x.Length; i++)
{
x[i] = k;
var kA = lowKA;
var totalP = 0.0;
for (var j = 0; j < numSamples; j++)
{
var kFx = k / kA;
var volFx = fxSurface.GetVolForAbsoluteStrike(kFx, t, fwdFx);
var volA = surface.GetVolForAbsoluteStrike(kA, t, fwdAsset);
var volC = Sqrt(volFx * volFx + volA * volA + 2.0 * correlation * volA * volFx);
var fxBucketLow = kFx - deltaKfx / 2.0;
var fxBucketHi = kFx + deltaKfx / 2.0;
var assetBucketLow = kA - deltaKa / 2.0;
var assetBucketHi = kA + deltaKa / 2.0;
var weight = 1.0;
y[i] += weight * volC;
totalP += weight;
kA += kStepA;
}
y[i] /= totalP;
k += kStep;
}
if (deltaStrikeOutput)
{
x = x.Select((q, ix) => - BlackFunctions.BlackDelta(compoFwd, q, 0.0, t, y[ix], OptionType.P)).ToArray();
}
return(InterpolatorFactory.GetInterpolator(x, y, Interpolator1DType.LinearFlatExtrap));
}
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 void DeltaStrikeMappingFacts()
{
var t = 1.0;
var k = 100;
var f = 100;
var vol = 0.32;
var rf = 0.0;
var cp = OptionType.P;
//black forward delta matches
var delta = BlackFunctions.BlackDelta(f, k, rf, t, vol, cp);
var absolute = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(f, delta, rf, t, vol);
Assert.Equal(k, absolute, 10);
}
public static IInterpolator1D GenerateCDF(this IInterpolator1D smile, int numSamples, double t, double fwd)
{
var deltaK = fwd * 0.0001;
var atmVol = smile.Interpolate(fwd);
var lowStrike = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.0001, 0, t, atmVol);
var hiStrike = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.9999, 0, t, atmVol);
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] = k / 2.0;
y[0] = 0;
continue;
}
if (i == x.Length - 1)
{
x[i] = k * 2;
y[i] = 1;
continue;
}
var volLow = smile.Interpolate(k - deltaK / 2.0);
var putLow = BlackFunctions.BlackPV(fwd, k - deltaK / 2.0, 0, t, volLow, OptionType.P);
var volHi = smile.Interpolate(k + deltaK / 2.0);
var putHi = BlackFunctions.BlackPV(fwd, k + deltaK / 2.0, 0, t, volHi, OptionType.P);
var digital = (putHi - putLow) / deltaK;
y[i] = digital;
x[i] = k;
k += kStep;
}
return(InterpolatorFactory.GetInterpolator(x, y, Interpolator1DType.Linear));
}
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 SolveSmileMarket()
{
var valDate = new DateTime(2018, 07, 28);
var expDate = valDate.AddDays(365);
var tExp = (expDate - valDate).TotalDays / 365.0;
var fwd = 1000;
double[] strikes = { 0.25, 0.5, 0.75 };
var atmConstraint = new ATMStraddleConstraint
{
ATMVolType = AtmVolType.ZeroDeltaStraddle,
MarketVol = 0.32
};
var smile25d = new RRBFConstraint
{
Delta = 0.25,
FlyVol = 0.01,
RisykVol = 0.02,
WingQuoteType = WingQuoteType.Market
};
var s = new AssetSmileSolver();
if (IsCoverageOnly)
{
s.Tollerance = 1;
}
var smile = s.Solve(atmConstraint, new[] { smile25d }, valDate, expDate, fwd, strikes, Interpolator1DType.Linear);
if (!IsCoverageOnly)
{
Assert.Equal(atmConstraint.MarketVol, smile[1], 8);
}
var surface = new GridVolSurface(valDate, strikes, new[] { expDate }, new[] { smile }, StrikeType.ForwardDelta, Interpolator1DType.Linear, Interpolator1DType.Linear, DayCountBasis.Act365F);
//reprice market RR structrure off smile, premium must match
var marketVolC = atmConstraint.MarketVol + smile25d.FlyVol + 0.5 * smile25d.RisykVol;
var marketVolP = atmConstraint.MarketVol + smile25d.FlyVol - 0.5 * smile25d.RisykVol;
var marketKC25 = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, 0.25, 0, tExp, marketVolC);
var marketKP25 = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.25, 0, tExp, marketVolP);
var marketC25FV = BlackFunctions.BlackPV(fwd, marketKC25, 0, tExp, marketVolC, OptionType.C);
var marketP25FV = BlackFunctions.BlackPV(fwd, marketKP25, 0, tExp, marketVolP, OptionType.P);
var marketRR = marketC25FV - marketP25FV;
var volC25d = surface.GetVolForAbsoluteStrike(marketKC25, expDate, fwd);
var volP25d = surface.GetVolForAbsoluteStrike(marketKP25, expDate, fwd);
var call25FV = BlackFunctions.BlackPV(fwd, marketKC25, 0, tExp, volC25d, OptionType.C);
var put25FV = BlackFunctions.BlackPV(fwd, marketKP25, 0, tExp, volP25d, OptionType.P);
var smileRR = call25FV - put25FV;
if (!IsCoverageOnly)
{
Assert.Equal(marketRR, smileRR, 8);
}
//reprice market BF structrure off smile, premium must match
var marketVolBF = atmConstraint.MarketVol + smile25d.FlyVol;
var marketKBFC25 = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, 0.25, 0, tExp, marketVolBF);
var marketKBFP25 = BlackFunctions.AbsoluteStrikefromDeltaKAnalytic(fwd, -0.25, 0, tExp, marketVolBF);
var marketBFC25FV = BlackFunctions.BlackPV(fwd, marketKBFC25, 0, tExp, marketVolBF, OptionType.C);
var marketBFP25FV = BlackFunctions.BlackPV(fwd, marketKBFP25, 0, tExp, marketVolBF, OptionType.P);
var marketBF = marketBFC25FV + marketBFP25FV;
var volCBF25d = surface.GetVolForAbsoluteStrike(marketKBFC25, expDate, fwd);
var volPBF25d = surface.GetVolForAbsoluteStrike(marketKBFP25, expDate, fwd);
var callBF25FV = BlackFunctions.BlackPV(fwd, marketKBFC25, 0, tExp, volCBF25d, OptionType.C);
var putBF25FV = BlackFunctions.BlackPV(fwd, marketKBFP25, 0, tExp, volPBF25d, OptionType.P);
var smileBF = callBF25FV + putBF25FV;
if (!IsCoverageOnly)
{
Assert.Equal(marketBF, smileBF, 8);
}
}
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));
}
