A volatility surface and Heston's calibration

The DAX options market of 5 July 2002, thirteen strikes by eight expiries, as QuantLib's tests calibrate Heston's model to it (from A. Sepp's "Pricing European-Style Options under Jump Diffusion Processes with Stochastic Volatility"). The calibrated parameters match QuantLib's on the same data. The sample, VolatilityAndHeston.cs, carries QuantLib's notice.

The surface

The quoted Black volatilities as a surface by expiry and strike, BlackVarianceSurface, which interpolates in variance:

Date today = Context.EvaluationDate;
Date[] expiries = [.. Days.Select(d => today + d)];
var vols = new Matrix(Strikes.Count, Days.Count);
for (int s = 0; s < Strikes.Count; ++s)
{
    for (int m = 0; m < Days.Count; ++m)
    {
        vols[s, m] = Vols[s, m];
    }
}

var surface = new BlackVarianceSurface(today, new TARGET(), expiries, Strikes, vols, new Actual365Fixed());

Read back at three of its strikes, it gives the quotes:

Expiry, days	Vol at 4,000	Vol at 4,400	Vol at 5,000
13	45.41%	37.26%	33.43%
41	38.69%	33.96%	29.59%
75	34.92%	31.08%	27.05%
165	31.49%	27.81%	25.40%
256	29.63%	27.88%	25.04%
345	29.26%	27.22%	24.64%
524	28.19%	26.61%	24.48%
703	28.00%	26.86%	24.62%

Between the nodes it interpolates in variance, bilinearly by default.

Heston's calibration

A HestonModelHelper per quote, each priced by the semi-analytic engine, and Levenberg-Marquardt minimizing the vols' errors:

Date today = Context.EvaluationDate;
DayCounter dayCounter = new Actual365Fixed();
var dates = new List<Date> { today };
var rates = new List<double> { 0.0357 };
double[] zeroRates = [0.0357, 0.0349, 0.0341, 0.0355, 0.0359, 0.0368, 0.0386, 0.0401];
for (int i = 0; i < Days.Count; ++i)
{
    dates.Add(today + Days[i]);
    rates.Add(zeroRates[i]);
}

var riskFree = new ZeroCurve(dates, rates, dayCounter);
var dividend = new FlatForward(today, 0.0, dayCounter);

var helpers = new List<HestonModelHelper>();
for (int s = 0; s < Strikes.Count; ++s)
{
    for (int m = 0; m < Days.Count; ++m)
    {
        var maturity = new Period((int)((Days[m] + 3) / 7.0), TimeUnit.Weeks); // to the nearest week
        helpers.Add(new HestonModelHelper(maturity, new TARGET(), Spot, Strikes[s], Vols[s, m], riskFree, dividend,
            Context, BlackCalibrationHelperCalibrationErrorType.ImpliedVolError)
            .SetPricingEngine<HestonModel>((option, context, model) =>
                AnalyticHestonEngine.Calculate(option, context, model, 64)));
    }
}

var start = new HestonModel(new HestonProcess(riskFree, dividend, Spot, 0.1, 1.0, 0.1, 0.5, -0.5));
HestonModel calibrated = start.Calibrate(helpers, new LevenbergMarquardt(1e-8, 1e-8, 1e-8),
    new EndCriteria(400, 40, 1.0e-8, 1.0e-8, 1.0e-8), Context).Model;

The calibrated model:

Parameter	Calibrated
θ, the long-run variance	0.0745912
κ, the mean reversion	15.6627
σ, the volatility of variance	3.36192
ρ, the correlation	-0.511492
v₀, the initial variance	0.195662
Sum of squared vol errors, in vol points²	177.233

Calibrate returns a new model with its record (the criterion that ended the optimization, the problem's values, the function evaluations), and leaves the starting model as it was.