Class Garch11

Namespace

NanoQL.Models

Assembly

NanoQL.dll

The GARCH(1,1) volatility model: σ²ₜ = ω + α u²ₜ₋₁ + β σ²ₜ₋₁, ω = γ V_L with γ = 1 − α − β and V_L the long-term variance. Volatilities are assumed to be expressed on an annual basis.

public sealed class Garch11 : VolatilityCompositor

Inheritance

object

VolatilityCompositor

Garch11

Inherited Members

object.GetType()

object.ToString()

object.Equals(object)

object.Equals(object, object)

object.ReferenceEquals(object, object)

object.GetHashCode()

Remarks

Immutable: a model built from a series is calibrated at construction, and each Calibrate returns a new model with the receiver's mode, where QuantLib's change the object. The statics are QuantLib's: Calculate(TimeSeries<double>, double, double, double), ToR2(ReadOnlySpan<double>), CostFunction(ReadOnlySpan<double>, double, double, double) and CalibrateR2's six overloads, each returning the parameters and the minimization's results that QuantLib writes through its references and returns.

Documented divergences:

• A calibration whose minimizer doesn't converge is refused, "GARCH(1,1) calibration did not converge: optimizer stopped with end criterion …", which the mode-based calibrations catch and fall back to their initial guess: QuantLib's master (#2844), which 1.44 doesn't check (Divergences/GarchConvergenceTests.cs).
• The cost's gradient is the gradient of its value, the recursive ∂σ²ₜ/∂θ = (1, u²ₜ₋₁, σ²ₜ₋₁) + β ∂σ²ₜ₋₁/∂θ, where QuantLib's takes the direct term alone, 0.13 to 0.80 times the true gradient; QuantLib's own paths, the Simplex and the ACF fit, never read it (Divergences/GarchGradientTests.cs).
• The cost refuses a conditional variance that isn't positive, which only the overloads that subtract the mean from the squared returns can produce, where QuantLib takes its log, NaN (Divergences/GarchDemeanedVarianceTests.cs).
• A series of fewer than two points is refused by Calculate(TimeSeries<double>, double, double, double), where QuantLib reads before its start.
• The mode-based initial guesses fit the autocorrelations by NonLinearLeastSquare on the conjugate gradient, whose returned point is the last one its line search accepted (Divergences/LineSearchReturnTests.cs), so the Simplex starts a hair from QuantLib's guess and the calibrated parameters end within its tolerance, 1e-8, of QuantLib's, the log-likelihood the same.
• The second initial guess (the ACF fit's, which GammaGuess and BestOfTwo take) sets α = γ − β, as the first does, where the fit fails or its guess misses the constraint; QuantLib leaves α there as its caller's, an uninitialized element of its guess array (garch.cpp:318-368, :435-439), so its value is undefined.

Constructors

Garch11(TimeSeries<double>, Garch11Mode)

Initializes a new instance of the Garch11 class calibrated to a series.

public Garch11(TimeSeries<double> qs, Garch11Mode mode = Garch11Mode.BestOfTwo)

Parameters

qs TimeSeries<double>

The series of returns.

mode Garch11Mode

The initial guess.

Exceptions

ArgumentNullException

qs is null.

ArgumentOutOfRangeException

mode isn't a Garch11Mode member.

NanoQLException

The series is too short or constant.

Garch11(double, double, double)

Initializes a new instance of the Garch11 class from its parameters.

public Garch11(double a, double b, double vl)

Parameters

a double

α.

b double

β.

vl double

The long-term variance V_L.

Properties

Alpha

Gets α.

public double Alpha { get; }

Property Value

double

Beta

Gets β.

public double Beta { get; }

Property Value

double

LogLikelihood

Gets the log-likelihood of the calibration, 0 for a model built from its parameters.

public double LogLikelihood { get; }

Property Value

double

LtVol

Gets the long-term variance V_L.

public double LtVol { get; }

Property Value

double

Mode

Gets the initial guess the model's calibrations take.

public Garch11Mode Mode { get; }

Property Value

Garch11Mode

Omega

Gets ω = γ V_L.

public double Omega { get; }

Property Value

double

Methods

Calculate(TimeSeries<double>)

The volatilities the model gives on the series.

public override TimeSeries<double> Calculate(TimeSeries<double> volatilitySeries)

Parameters

volatilitySeries TimeSeries<double>

The series.

Returns

TimeSeries<double>

The volatility series.

Exceptions

ArgumentNullException

volatilitySeries is null.

ArgumentException

The series has fewer than two points.

Calculate(TimeSeries<double>, double, double, double)

The model's volatilities on a series of returns: at each date after the first, the square root of the variance the returns before it give, and one more at the last date plus its spacing.

public static TimeSeries<double> Calculate(TimeSeries<double> quoteSeries, double alpha, double beta, double omega)

Parameters

quoteSeries TimeSeries<double>

The series of returns.

alpha double

α.

beta double

β.

omega double

ω.

Returns

TimeSeries<double>

The volatility series.

Exceptions

ArgumentNullException

quoteSeries is null.

ArgumentException

The series has fewer than two points.

Calibrate(TimeSeries<double>)

The model calibrated to the series.

public override Garch11 Calibrate(TimeSeries<double> volatilitySeries)

Parameters

volatilitySeries TimeSeries<double>

The series.

Returns

Garch11

A new model, or this one where calibration changes nothing.

Exceptions

ArgumentNullException

volatilitySeries is null.

NanoQLException

The series is too short or constant.

Calibrate(TimeSeries<double>, OptimizationMethod, EndCriteria)

The model calibrated by its mode's initial guess and an optimization method.

public Garch11 Calibrate(TimeSeries<double> quoteSeries, OptimizationMethod method, EndCriteria endCriteria)

Parameters

quoteSeries TimeSeries<double>

The series of returns.

method OptimizationMethod

The optimization method.

endCriteria EndCriteria

Its end criteria.

Returns

Garch11

A new model, with this one's mode.

Exceptions

ArgumentNullException

An argument is null.

NanoQLException

The series is too short or constant.

Calibrate(TimeSeries<double>, OptimizationMethod, EndCriteria, ReadOnlySpan<double>)

The model calibrated from an initial guess by an optimization method.

public Garch11 Calibrate(TimeSeries<double> quoteSeries, OptimizationMethod method, EndCriteria endCriteria, ReadOnlySpan<double> initialGuess)

Parameters

quoteSeries TimeSeries<double>

The series of returns.

method OptimizationMethod

The optimization method.

endCriteria EndCriteria

Its end criteria.

initialGuess ReadOnlySpan<double>

The initial ω, α and β.

Returns

Garch11

A new model, with this one's mode.

Exceptions

ArgumentNullException

An argument is null.

NanoQLException

The minimizer doesn't converge.

ArgumentException

initialGuess holds fewer than three values, where QuantLib reads past its end.

CalibrateR2(Garch11Mode, ReadOnlySpan<double>, double)

Calibrates GARCH for r² by a mode's initial guess and QuantLib's Simplex.

public static Garch11Calibration CalibrateR2(Garch11Mode mode, ReadOnlySpan<double> r2, double meanR2)

Parameters

mode Garch11Mode

The initial guess.

r2 ReadOnlySpan<double>

The squared returns.

meanR2 double

Their mean.

Returns

Garch11Calibration

The parameters and the minimization's results, null where it fell back to its guess.

Exceptions

ArgumentOutOfRangeException

mode isn't a Garch11Mode member.

NanoQLException

The series is too short or constant.

CalibrateR2(Garch11Mode, ReadOnlySpan<double>, double, OptimizationMethod, EndCriteria)

Calibrates GARCH for r² by a mode's initial guess and an optimization method.

public static Garch11Calibration CalibrateR2(Garch11Mode mode, ReadOnlySpan<double> r2, double meanR2, OptimizationMethod method, EndCriteria endCriteria)

Parameters

mode Garch11Mode

The initial guess.

r2 ReadOnlySpan<double>

The squared returns.

meanR2 double

Their mean.

method OptimizationMethod

The optimization method.

endCriteria EndCriteria

Its end criteria.

Returns

Garch11Calibration

The parameters and the minimization's results, null where it fell back to its guess.

Exceptions

ArgumentNullException

method or endCriteria is null.

ArgumentOutOfRangeException

mode isn't a Garch11Mode member.

NanoQLException

The series is too short or constant.

CalibrateR2(ReadOnlySpan<double>, OptimizationMethod, Constraint, EndCriteria, ReadOnlySpan<double>)

Calibrates GARCH for r² from an initial guess, by an optimization method under a constraint.

public static Garch11Calibration CalibrateR2(ReadOnlySpan<double> r2, OptimizationMethod method, Constraint constraints, EndCriteria endCriteria, ReadOnlySpan<double> initialGuess)

Parameters

r2 ReadOnlySpan<double>

The squared returns.

method OptimizationMethod

The optimization method.

constraints Constraint

The constraint.

endCriteria EndCriteria

Its end criteria.

initialGuess ReadOnlySpan<double>

The initial ω, α and β.

Returns

Garch11Calibration

The parameters and the minimization's results.

Exceptions

ArgumentNullException

An argument is null.

NanoQLException

The minimizer doesn't converge (#2844).

ArgumentException

initialGuess holds fewer than three values, where QuantLib reads past its end.

CalibrateR2(ReadOnlySpan<double>, OptimizationMethod, EndCriteria, ReadOnlySpan<double>)

Calibrates GARCH for r² from an initial guess, by an optimization method under the constraint ω > 0, α, β ≥ 0, α + β < 1 − 1e-8.

public static Garch11Calibration CalibrateR2(ReadOnlySpan<double> r2, OptimizationMethod method, EndCriteria endCriteria, ReadOnlySpan<double> initialGuess)

Parameters

r2 ReadOnlySpan<double>

The squared returns.

method OptimizationMethod

The optimization method.

endCriteria EndCriteria

Its end criteria.

initialGuess ReadOnlySpan<double>

The initial ω, α and β.

Returns

Garch11Calibration

The parameters and the minimization's results.

Exceptions

ArgumentNullException

method or endCriteria is null.

NanoQLException

The minimizer doesn't converge.

ArgumentException

initialGuess holds fewer than three values, where QuantLib reads past its end.

CalibrateR2(ReadOnlySpan<double>, double, OptimizationMethod, Constraint, EndCriteria, ReadOnlySpan<double>)

Calibrates GARCH for r² less its mean from an initial guess, by an optimization method under a constraint.

public static Garch11Calibration CalibrateR2(ReadOnlySpan<double> r2, double meanR2, OptimizationMethod method, Constraint constraints, EndCriteria endCriteria, ReadOnlySpan<double> initialGuess)

Parameters

r2 ReadOnlySpan<double>

The squared returns.

meanR2 double

Their mean, subtracted from each.

method OptimizationMethod

The optimization method.

constraints Constraint

The constraint.

endCriteria EndCriteria

Its end criteria.

initialGuess ReadOnlySpan<double>

The initial ω, α and β.

Returns

Garch11Calibration

The parameters and the minimization's results.

Exceptions

ArgumentNullException

An argument is null.

NanoQLException

The minimizer doesn't converge, or the cost reaches a conditional variance that isn't positive.

ArgumentException

initialGuess holds fewer than three values, where QuantLib reads past its end.

CalibrateR2(ReadOnlySpan<double>, double, OptimizationMethod, EndCriteria, ReadOnlySpan<double>)

Calibrates GARCH for r² less its mean from an initial guess, by an optimization method under the constraint ω > 0, α, β ≥ 0, α + β < 1 − 1e-8.

public static Garch11Calibration CalibrateR2(ReadOnlySpan<double> r2, double meanR2, OptimizationMethod method, EndCriteria endCriteria, ReadOnlySpan<double> initialGuess)

Parameters

r2 ReadOnlySpan<double>

The squared returns.

meanR2 double

Their mean, subtracted from each.

method OptimizationMethod

The optimization method.

endCriteria EndCriteria

Its end criteria.

initialGuess ReadOnlySpan<double>

The initial ω, α and β.

Returns

Garch11Calibration

The parameters and the minimization's results.

Exceptions

ArgumentNullException

method or endCriteria is null.

NanoQLException

The minimizer doesn't converge, or the cost reaches a conditional variance that isn't positive.

ArgumentException

initialGuess holds fewer than three values, where QuantLib reads past its end.

CostFunction(ReadOnlySpan<double>, double, double, double)

The model's cost on a series of returns: Σ(ln σ²ₜ + u²ₜ/σ²ₜ)/(2N), 0 on an empty series.

public static double CostFunction(ReadOnlySpan<double> values, double alpha, double beta, double omega)

Parameters

values ReadOnlySpan<double>

The returns.

alpha double

α.

beta double

β.

omega double

ω.

Returns

double

The cost, the negated log-likelihood.

Forecast(double, double)

The variance forecast after a return r at a variance σ², γ V_L + α r² + β σ².

public double Forecast(double r, double sigma2)

Parameters

r double

The return.

sigma2 double

The variance.

Returns

double

The next variance.

ToR2(ReadOnlySpan<double>)

The squared returns and their mean, QuantLib's to_r2.

public static (ImmutableArray<double> R2, double MeanR2) ToR2(ReadOnlySpan<double> values)

Parameters

values ReadOnlySpan<double>

The returns.

Returns

(ImmutableArray<double> R2, double MeanR2)

The squared returns, and their running mean.