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)); }