public virtual DeformedSurface localVolatilityFromPrice(Surface callPriceSurface, double spot, System.Func <double, double> interestRate, System.Func <double, double> dividendRate)
{
System.Func <DoublesPair, ValueDerivatives> func = (DoublesPair x) =>
{
double t = x.First;
double k = x.Second;
double r = interestRate(t);
double q = dividendRate(t);
double price = callPriceSurface.zValue(t, k);
DoubleArray priceSensi = callPriceSurface.zValueParameterSensitivity(t, k).Sensitivity;
double divT = FIRST_DERIV.differentiate(u => callPriceSurface.zValue(u, k)).apply(t);
DoubleArray divTSensi = FIRST_DERIV_SENSI.differentiate(u => callPriceSurface.zValueParameterSensitivity(u.get(0), k).Sensitivity).apply(DoubleArray.of(t)).column(0);
double divK = FIRST_DERIV.differentiate(l => callPriceSurface.zValue(t, l)).apply(k);
DoubleArray divKSensi = FIRST_DERIV_SENSI.differentiate(l => callPriceSurface.zValueParameterSensitivity(t, l.get(0)).Sensitivity).apply(DoubleArray.of(k)).column(0);
double divK2 = SECOND_DERIV.differentiate(l => callPriceSurface.zValue(t, l)).apply(k);
DoubleArray divK2Sensi = SECOND_DERIV_SENSI.differentiateNoCross(l => callPriceSurface.zValueParameterSensitivity(t, l.get(0)).Sensitivity).apply(DoubleArray.of(k)).column(0);
double var = 2d * (divT + q * price + (r - q) * k * divK) / (k * k * divK2);
if (var < 0d)
{
throw new System.ArgumentException("Negative variance");
}
double localVol = Math.Sqrt(var);
double factor = 1d / (localVol * k * k * divK2);
DoubleArray localVolSensi = divTSensi.multipliedBy(factor).plus(divKSensi.multipliedBy((r - q) * k * factor)).plus(priceSensi.multipliedBy(q * factor)).plus(divK2Sensi.multipliedBy(-0.5 * localVol / divK2));
return(ValueDerivatives.of(localVol, localVolSensi));
};
SurfaceMetadata metadata = DefaultSurfaceMetadata.builder().xValueType(ValueType.YEAR_FRACTION).yValueType(ValueType.STRIKE).zValueType(ValueType.LOCAL_VOLATILITY).surfaceName(SurfaceName.of("localVol_" + callPriceSurface.Name)).build();
return(DeformedSurface.of(metadata, callPriceSurface, func));
}
public virtual DeformedSurface localVolatilityFromImpliedVolatility(Surface impliedVolatilitySurface, double spot, System.Func <double, double> interestRate, System.Func <double, double> dividendRate)
{
System.Func <DoublesPair, ValueDerivatives> func = (DoublesPair x) =>
{
double t = x.First;
double k = x.Second;
double r = interestRate(t);
double q = dividendRate(t);
double vol = impliedVolatilitySurface.zValue(t, k);
DoubleArray volSensi = impliedVolatilitySurface.zValueParameterSensitivity(t, k).Sensitivity;
double divT = FIRST_DERIV.differentiate(u => impliedVolatilitySurface.zValue(u, k)).apply(t);
DoubleArray divTSensi = FIRST_DERIV_SENSI.differentiate(u => impliedVolatilitySurface.zValueParameterSensitivity(u.get(0), k).Sensitivity).apply(DoubleArray.of(t)).column(0);
double localVol;
DoubleArray localVolSensi = DoubleArray.of();
if (k < SMALL)
{
localVol = Math.Sqrt(vol * vol + 2 * vol * t * (divT));
localVolSensi = volSensi.multipliedBy((vol + t * divT) / localVol).plus(divTSensi.multipliedBy(vol * t / localVol));
}
else
{
double divK = FIRST_DERIV.differentiate(l => impliedVolatilitySurface.zValue(t, l)).apply(k);
DoubleArray divKSensi = FIRST_DERIV_SENSI.differentiate(l => impliedVolatilitySurface.zValueParameterSensitivity(t, l.get(0)).Sensitivity).apply(DoubleArray.of(k)).column(0);
double divK2 = SECOND_DERIV.differentiate(l => impliedVolatilitySurface.zValue(t, l)).apply(k);
DoubleArray divK2Sensi = SECOND_DERIV_SENSI.differentiateNoCross(l => impliedVolatilitySurface.zValueParameterSensitivity(t, l.get(0)).Sensitivity).apply(DoubleArray.of(k)).column(0);
double rq = r - q;
double h1 = (Math.Log(spot / k) + (rq + 0.5 * vol * vol) * t) / vol;
double h2 = h1 - vol * t;
double den = 1d + 2d * h1 * k * divK + k * k * (h1 * h2 * divK * divK + t * vol * divK2);
double var = (vol * vol + 2d * vol * t * (divT + k * rq * divK)) / den;
if (var < 0d)
{
throw new System.ArgumentException("Negative variance");
}
localVol = Math.Sqrt(var);
localVolSensi = volSensi.multipliedBy(localVol * k * h2 * divK * (1d + 0.5 * k * h2 * divK) / vol / den + 0.5 * localVol * Math.Pow(k * h1 * divK, 2) / vol / den + (vol + divT * t + rq * t * k * divK) / (localVol * den) - 0.5 * divK2 * localVol * k * k * t / den).plus(divKSensi.multipliedBy((vol * t * rq * k / localVol - localVol * k * h1 * (1d + k * h2 * divK)) / den)).plus(divTSensi.multipliedBy(vol * t / (localVol * den))).plus(divK2Sensi.multipliedBy(-0.5 * vol * localVol * k * k * t / den));
}
return(ValueDerivatives.of(localVol, localVolSensi));
};
SurfaceMetadata metadata = DefaultSurfaceMetadata.builder().xValueType(ValueType.YEAR_FRACTION).yValueType(ValueType.STRIKE).zValueType(ValueType.LOCAL_VOLATILITY).surfaceName(SurfaceName.of("localVol_" + impliedVolatilitySurface.Name)).build();
return(DeformedSurface.of(metadata, impliedVolatilitySurface, func));
}
public virtual void flatVolTest()
{
double constantVol = 0.15;
ConstantSurface impliedVolSurface = ConstantSurface.of("impliedVol", constantVol);
System.Func <double, double> zeroRate = (double?x) =>
{
return(0.05d);
};
System.Func <double, double> zeroRate1 = (double?x) =>
{
return(0.02d);
};
double[] strikes = new double[] { 90d, 100d, 115d };
foreach (double strike in strikes)
{
foreach (double time in TEST_TIMES)
{
DeformedSurface localVolSurface = CALC.localVolatilityFromImpliedVolatility(impliedVolSurface, 100d, zeroRate, zeroRate1);
assertEquals(localVolSurface.zValue(time, strike), constantVol);
}
}
}
