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 double InverseCDFex(this IVolSurface surface, double t, double fwd, double p)
{
var deltaK = fwd * 1e-10;
var lowGuess = fwd / 2;
var highGuess = fwd * 2;
var targetFunc = new Func <double, double>(k =>
{
var volM = surface.GetVolForAbsoluteStrike(k - deltaK, t, fwd);
var volP = surface.GetVolForAbsoluteStrike(k + deltaK, t, fwd);
var pvM = BlackFunctions.BlackPV(fwd, k - deltaK, 0.0, t, volM, OptionType.P);
var pvP = BlackFunctions.BlackPV(fwd, k + deltaK, 0.0, t, volP, OptionType.P);
var digi = (pvP - pvM) / (2 * deltaK);
//var digi = BlackFunctions.BlackDigitalPV(fwd, k, 0, t, surface.GetVolForAbsoluteStrike(k, t, fwd), OptionType.P);
return(p - digi);
});
var breakCount = 0;
while (targetFunc(lowGuess) < 0)
{
// highGuess = lowGuess*2.0;
lowGuess /= 2.0;
breakCount++;
if (breakCount == 10)
{
return(lowGuess);
}
}
breakCount = 0;
while (targetFunc(highGuess) > 0)
{
//lowGuess = highGuess/2.0;
highGuess *= 2.0;
breakCount++;
if (breakCount == 10)
{
return(highGuess);
}
}
var b = Math.Solvers.Brent.BrentsMethodSolve(targetFunc, lowGuess, highGuess, 1e-8);
//var b = Math.Solvers.Newton1D.MethodSolve2(targetFunc, fwd, 1e-6, 1000, fwd * 0.00001);
if (double.IsInfinity(b) || double.IsNaN(b))
{
throw new Exception("Invalid strike found");
}
//if (b==lowGuess || b==highGuess)
// throw new Exception("Strike outside of bounds");
return(b);
}
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 double[][] ComputeLocalVarianceOnGridFromCalls(this IVolSurface VanillaSurface, double[][] strikes, double[] timeSteps, Func <double, double> forwardFunc)
{
var numberOfTimesteps = timeSteps.Length;
var deltaK = 0.001 * forwardFunc(timeSteps[0]);
var lvGrid = new double[numberOfTimesteps - 1][];
var fwds = timeSteps.Select(t => forwardFunc(t)).ToArray();
//ParallelUtils.Instance.For(1, numberOfTimesteps, 1, it =>
for (var it = 1; it < numberOfTimesteps; it++)
{
var T = timeSteps[it];
var T1 = timeSteps[it - 1];
var fwd = fwds[it];
var fwdtm1 = fwds[it - 1];
var rmq = Log(fwd / fwdtm1) / (T - T1);
var numberOfStrikes = strikes[it - 1].Length;
var cInterp = VanillaSurface.GeneratePremiumInterpolator(numberOfStrikes * 2, T, fwd, OptionType.C);
lvGrid[it - 1] = new double[numberOfStrikes];
if (numberOfStrikes > 1)
{
for (var ik = 0; ik < numberOfStrikes; ik++)
{
var K = strikes[it][ik];
var V = VanillaSurface.GetVolForAbsoluteStrike(K, T, fwd);
var C = BlackFunctions.BlackPV(fwd, K, 0.0, T, V, OptionType.C);
var Vtm1 = VanillaSurface.GetVolForAbsoluteStrike(K, T1, fwdtm1);
//var dcdt = -BlackFunctions.BlackTheta(fwd, K, 0.0, T, V, OptionType.C);
var dcdt = -(BlackFunctions.BlackPV(fwdtm1, K, 0.0, T1, Vtm1, OptionType.C) - C) / (T - T1);
var dcdk = cInterp.FirstDerivative(K);
var d2cdk2 = cInterp.SecondDerivative(K);
var localVariance = d2cdk2 == 0 ? V * V : (dcdt - rmq * (C - K * dcdk)) / (0.5 * K * K * d2cdk2);
lvGrid[it - 1][ik] = localVariance;
}
}
else
{
var K = strikes[it][0];
var V = VanillaSurface.GetVolForAbsoluteStrike(K, T, fwd);
lvGrid[it - 1][0] = V * V;
}
}//, false).Wait();
return(lvGrid);
}
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 static double StrikeForPV(double targetPV, double forward, double knownAverage, IVolSurface volSurface, DateTime evalDate, DateTime avgStartDate, DateTime avgEndDate, double riskFree, OptionType callPut)
{
var minStrike = forward / 100.0;
var maxStrike = forward * 100.0;
var volDate = avgStartDate.Average(avgEndDate);
Func <double, double> testFunc = (absK =>
{
var vol = volSurface.GetVolForAbsoluteStrike(absK, volDate, forward);
var pv = PV(forward, knownAverage, vol, absK, evalDate, avgStartDate, avgEndDate, riskFree, callPut);
return(targetPV - pv);
});
var solvedStrike = Math.Solvers.Brent.BrentsMethodSolve(testFunc, minStrike, maxStrike, 1e-8);
return(solvedStrike);
}
public static double StrikeForPV(double targetPV, double[] forwards, DateTime[] fixingDates, IVolSurface volSurface, DateTime evalDate, DateTime payDate, double riskFree, OptionType callPut)
{
var minStrike = forwards.Min() / 100.0;
var maxStrike = forwards.Max() * 100.0;
var ixMin = Array.BinarySearch(fixingDates, evalDate);
if (ixMin < 0)
{
ixMin = ~ixMin;
}
Func <double, double> testFunc = (absK =>
{
var vols = fixingDates.Select((d, ix) => ix >= ixMin ? volSurface.GetVolForAbsoluteStrike(absK, d, forwards[ix]) : 0.0).ToArray();
var pv = PV(forwards, fixingDates, evalDate, payDate, vols, absK, riskFree, callPut);
return(targetPV - pv);
});
var solvedStrike = Math.Solvers.Brent.BrentsMethodSolve(testFunc, minStrike, maxStrike, 1e-8);
return(solvedStrike);
}
public static double[][] ComputeLocalVarianceOnGrid(this IVolSurface VanillaSurface, double[][] strikes, double[] timeSteps, Func <double, double> forwardFunc)
{
var numberOfTimesteps = timeSteps.Length;
var deltaK = 0.001 * forwardFunc(timeSteps[0]);
var lvGrid = new double[numberOfTimesteps - 1][];
var Ss = new double[numberOfTimesteps];
for (var i = 0; i < Ss.Length; i++)
{
Ss[i] = forwardFunc(timeSteps[i]);
}
for (var it = 1; it < numberOfTimesteps; it++)
{
var numberOfStrikes = strikes[it - 1].Length;
lvGrid[it - 1] = new double[numberOfStrikes];
double K, V, S, Td, dwdT, localVariance, TdSq, T, T1;
double V_t1, V_Kp2, V_Km2, K_tm1, St;
double y, yPlus, yPlus2, yMinus, yMinus2, dwdY, dwdY_p, dwdY_m, d2wd2Y, w, w_t1, w_kPlus2, w_kMinus2; //, Y1, Y2;
T = timeSteps[it];
T1 = timeSteps[it - 1];
Td = T - T1;
TdSq = Sqrt(Td);
S = Ss[it];
St = Ss[it - 1];
var fwd = forwardFunc(T);
for (var ik = 0; ik < numberOfStrikes; ik++)
{
K = strikes[it][ik];
K_tm1 = K * St / S;
y = Log(K / S);
yPlus = Log((K + deltaK) / S);
yPlus2 = Log((K + deltaK * 2) / S);
yMinus = Log((K - deltaK) / S);
yMinus2 = Log((K - deltaK * 2) / S);
//Y1 = Log(Sqrt(K * K + 2 * deltaK * K) / S);
//Y2 = Log(Sqrt(K * K - 2 * deltaK * K) / S);
V = VanillaSurface.GetVolForAbsoluteStrike(K, T, fwd);
w = V * V * T;
V_t1 = VanillaSurface.GetVolForAbsoluteStrike(K_tm1, T1, fwd);
w_t1 = V_t1 * V_t1 * T1;
V_Kp2 = VanillaSurface.GetVolForAbsoluteStrike(K + deltaK * 2, T, fwd);
w_kPlus2 = V_Kp2 * V_Kp2 * T;
V_Km2 = VanillaSurface.GetVolForAbsoluteStrike(K - deltaK * 2, T, fwd);
w_kMinus2 = V_Km2 * V_Km2 * T;
dwdT = (w - w_t1) / Td;
dwdY_m = (w - w_kMinus2) / (y - yMinus2);
dwdY_p = (w_kPlus2 - w) / (yPlus2 - y);
dwdY = (dwdY_m + dwdY_p) / 2;
//d2wd2Y = (dwdY_p - dwdY_m) / (Y1 - Y2);
d2wd2Y = (dwdY_p - dwdY_m) / (yPlus - yMinus);
localVariance = dwdT / (1 - y / w * dwdY + 0.25 * (-0.25 - 1 / w + (y * y / (w))) * dwdY * dwdY + 0.5 * d2wd2Y);
lvGrid[it - 1][ik] = localVariance;
}
}
return(lvGrid);
}