public void BracketTest(Func <double, double> f, double x1, double x2)
{
double xa, xb;
RootUtils.Bracket(f, x1, x2, out xa, out xb);
Assert.IsTrue(f(xa) * f(xb) < 0.0);
}
public static double ComputeQuantile(double forward, double maturity, Func <double, double> vol, double proba)
{
double s = vol(forward) * Math.Sqrt(maturity);
double kGuess = forward * Math.Exp(s * (NormalDistribution.FastCumulativeInverse(proba) + 0.5 * s));
Func <double, double> cumErr = m => Cumulative(forward, maturity, vol, forward * Math.Exp(m)) - proba;
double m1, m2;
RootUtils.Bracket(cumErr, Math.Log(kGuess / forward), 0.0, out m1, out m2);
var mQuantile = RootUtils.Brenth(cumErr, m1, m2, forward * 1.0e-5, DoubleUtils.MachineEpsilon);
return(forward * Math.Exp(mQuantile));
}
/// <summary>
/// Implied volatility from option price. (Inversion of the two step quadrature formula.)
/// </summary>
/// <param name="maturity">option maturity</param>
/// <param name="strike">option strike</param>
/// <param name="price">option price </param>
/// <param name="q">option type : 1 for call, -1 for put</param>
/// <returns></returns>
public double ImpliedVol(double maturity, double strike, double price, double q)
{
//Proxy using Lehman Formula
double proxyFwd, proxyDk;
affineDivUtils.LehmanProxy(maturity, spot, out proxyFwd, out proxyDk);
double lehmanTargetVol = BlackScholesOption.ImpliedVol(price, proxyFwd, strike + proxyDk, maturity, q);
var pricer = new BsDivPrice(maturity, strike, spot, affineDivUtils, quadPoints, quadWeights);
Func <double, double> volToLehmanVolErr =
v => BlackScholesOption.ImpliedVol(pricer.Price(v, q), proxyFwd, strike + proxyDk, maturity, q) - lehmanTargetVol;
//Solve
double v1 = lehmanTargetVol;
double err1 = volToLehmanVolErr(v1);
double v2 = lehmanTargetVol - err1;
double err2 = volToLehmanVolErr(v2);
double v3 = v1 - err1 * (v2 - v1) / (err2 - err1);
if (Math.Abs(v3 - v2) < v2 * 1.0e-13)
{
return(v3);
}
double err3 = volToLehmanVolErr(v3);
double v4 = RootUtils.TrinomRoot(v1, v2, v3, err1, err2, err3);
if (Math.Abs(v4 - v3) < v3 * 1.0e-13)
{
return(v4);
}
double err4 = volToLehmanVolErr(v4);
double v5 = RootUtils.TrinomRoot(v2, v3, v4, err2, err3, err4);
return(v5);
}
public double[] CalibrateVol(double[] maturities, double[] targetPrices, double[] strikes, double[] optionTypes)
{
Contract.Requires(EnumerableUtils.IsSorted(maturities));
Contract.Requires(maturities.Length == strikes.Length &&
strikes.Length == targetPrices.Length &&
targetPrices.Length == optionTypes.Length);
double[] variances = new double[maturities.Length + 1];
double[] varPillars = new[] { 0.0 }.Concat(maturities).ToArray();
double[] calibVols = new double[maturities.Length];
for (int step = 0; step < maturities.Length; step++)
{
var maturity = maturities[step];
var strike = strikes[step];
var targetPrice = targetPrices[step];
var q = optionTypes[step];
//Proxy using Lehman Formula
double proxyFwd, proxyDk;
affineDivUtils.LehmanProxy(maturity, spot, out proxyFwd, out proxyDk);
double lehmanTargetVol = BlackScholesOption.ImpliedVol(targetPrice, proxyFwd, strike + proxyDk, maturity, q);
var pricer = new BsDivPrice(maturities[step], strikes[step], spot, affineDivUtils, quadPoints, quadWeights);
Func <double, double> volToLehmanVolErr = v =>
{
variances[1 + step] = v * v * maturity;
var varFunc = RrFunctions.LinearInterpolation(varPillars, variances);
Func <double, double> volFunc = t => Math.Sqrt(varFunc.Eval(t) / t);
var price = pricer.Price(volFunc, q);
return(BlackScholesOption.ImpliedVol(price, proxyFwd, strike + proxyDk, maturity, q) - lehmanTargetVol);
};//TODO use a cache
//Bracket & Solve
double v1 = lehmanTargetVol;
double v2;
if (step == 0)
{
v2 = lehmanTargetVol - volToLehmanVolErr(lehmanTargetVol);
}
else
{
var volIfZeroVolOnStep = Math.Sqrt(calibVols[step - 1] * calibVols[step - 1] * maturities[step - 1] / maturities[step]);
var minError = volToLehmanVolErr(volIfZeroVolOnStep);
if (minError > 0.0) //saturation case
{
calibVols[step] = volIfZeroVolOnStep;
variances[1 + step] = volIfZeroVolOnStep * volIfZeroVolOnStep * maturity;
continue;
}
v2 = volIfZeroVolOnStep;
}
if (!RootUtils.Bracket(volToLehmanVolErr, v1, v2, out v1, out v2))
{
throw new Exception("Failed to inverse vol");
}
var impliedVol = RootUtils.Brenth(volToLehmanVolErr, v1, v2, 1.0e-10, 2.0 * DoubleUtils.MachineEpsilon, 10);
calibVols[step] = impliedVol;
variances[1 + step] = impliedVol * impliedVol * maturity;
}
return(calibVols);
}
