Class LIBORMarketModelWithTenorRefinement

  • All Implemented Interfaces:
    TermStructureModel, ProcessModel

    public class LIBORMarketModelWithTenorRefinement
    extends AbstractProcessModel
    implements TermStructureModel
    Implements a discretized Heath-Jarrow-Morton model / LIBOR market model with dynamic tenor refinement, see https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2884699.

    In its default case the class specifies a multi-factor LIBOR market model, that is \( L_{j} = \frac{1}{T_{j+1}-T_{j}} ( exp(Y_{j}) - 1 ) \), where \[ dY_{j} = \mu_{j} dt + \lambda_{1,j} dW_{1} + \ldots + \lambda_{m,j} dW_{m} \]
    The model uses an AbstractLIBORCovarianceModel for the specification of 1,j,...,λm,j) as a covariance model. See ProcessModel for details on the implemented interface

    The model uses an AbstractLIBORCovarianceModel as a covariance model. If the covariance model is of type AbstractLIBORCovarianceModelParametric a calibration to swaptions can be performed.
    Note that λ may still depend on L (through a local volatility model).
    The simulation is performed under spot measure, that is, the numeraire is \( N(T_{i}) = \prod_{j=0}^{i-1} (1 + L(T_{j},T_{j+1};T_{j}) (T_{j+1}-T_{j})) \). The map properties allows to configure the model. The following keys may be used:
    • liborCap: An optional Double value applied as a cap to the LIBOR rates. May be used to limit the simulated valued to prevent values attaining POSITIVE_INFINITY and numerical problems. To disable the cap, set liborCap to Double.POSITIVE_INFINITY.

    The main task of this class is to calculate the risk-neutral drift and the corresponding numeraire given the covariance model. The calibration of the covariance structure is not part of this class.
    Version:
    1.2
    Author:
    Christian Fries
    See Also:
    The interface for numerical schemes., The interface for models provinding parameters to numerical schemes., https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2884699
    • Constructor Detail

      • LIBORMarketModelWithTenorRefinement

        public LIBORMarketModelWithTenorRefinement​(TimeDiscretization[] liborPeriodDiscretizations,
                                                   Integer[] numberOfDiscretizationIntervalls,
                                                   AnalyticModel analyticModel,
                                                   ForwardCurve forwardRateCurve,
                                                   DiscountCurve discountCurve,
                                                   TermStructureCovarianceModel covarianceModel,
                                                   CalibrationProduct[] calibrationProducts,
                                                   Map<String,​?> properties)
                                            throws CalculationException
        Creates a model for given covariance. Creates a discretized Heath-Jarrow-Morton model / LIBOR market model with dynamic tenor refinement, see https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2884699.
        If calibrationItems in non-empty and the covariance model is a parametric model, the covariance will be replaced by a calibrate version of the same model, i.e., the LIBOR Market Model will be calibrated.
        The map properties allows to configure the model. The following keys may be used:
        • liborCap: An optional Double value applied as a cap to the LIBOR rates. May be used to limit the simulated valued to prevent values attaining POSITIVE_INFINITY and numerical problems. To disable the cap, set liborCap to Double.POSITIVE_INFINITY.
        • calibrationParameters: Possible values:
          • Map<String,Object> a parameter map with the following key/value pairs:
            • accuracy: Double specifying the required solver accuracy.
            • maxIterations: Integer specifying the maximum iterations for the solver.
        Parameters:
        liborPeriodDiscretizations - A vector of tenor discretizations of the interest rate curve into forward rates (tenor structure), finest first.
        numberOfDiscretizationIntervalls - A vector of number of periods to be taken from the liborPeriodDiscretizations.
        analyticModel - The associated analytic model of this model (containing the associated market data objects like curve).
        forwardRateCurve - The initial values for the forward rates.
        discountCurve - The discount curve to use. This will create an LMM model with a deterministic zero-spread discounting adjustment.
        covarianceModel - The covariance model to use.
        calibrationProducts - The vector of calibration items (a union of a product, target value and weight) for the objective function sum weight(i) * (modelValue(i)-targetValue(i).
        properties - Key value map specifying properties like measure and stateSpace.
        Throws:
        CalculationException - Thrown if the valuation fails, specific cause may be available via the cause() method.
    • Method Detail

      • getNumeraire

        public RandomVariable getNumeraire​(MonteCarloProcess process,
                                           double time)
                                    throws CalculationException
        Return the numeraire at a given time. The numeraire is provided for interpolated points. If requested on points which are not part of the tenor discretization, the numeraire uses a linear interpolation of the reciprocal value. See ISBN 0470047224 for details.
        Specified by:
        getNumeraire in interface ProcessModel
        Parameters:
        time - Time time t for which the numeraire should be returned N(t).
        process - The discretization process generating this model. The process provides call backs for TimeDiscretization and allows calls to getProcessValue for timeIndices less or equal the given one.
        Returns:
        The numeraire at the specified time as RandomVariable
        Throws:
        CalculationException - Thrown if the valuation fails, specific cause may be available via the cause() method.
      • getInitialState

        public RandomVariable[] getInitialState​(MonteCarloProcess process)
        Description copied from interface: ProcessModel
        Returns the initial value of the state variable of the process Y, not to be confused with the initial value of the model X (which is the state space transform applied to this state value.
        Specified by:
        getInitialState in interface ProcessModel
        Parameters:
        process - The discretization process generating this model. The process provides call backs for TimeDiscretization and allows calls to getProcessValue for timeIndices less or equal the given one.
        Returns:
        The initial value of the state variable of the process Y(t=0).
      • getDrift

        public RandomVariable[] getDrift​(MonteCarloProcess process,
                                         int timeIndex,
                                         RandomVariable[] realizationAtTimeIndex,
                                         RandomVariable[] realizationPredictor)
        Return the complete vector of the drift for the time index timeIndex, given that current state is realizationAtTimeIndex. The drift will be zero for rates being already fixed. The method currently provides the drift for either Measure.SPOT or Measure.TERMINAL - depending how the model object was constructed. For Measure.TERMINAL the j-th entry of the return value is the random variable \[ \mu_{j}^{\mathbb{Q}^{P(T_{n})}}(t) \ = \ - \mathop{\sum_{l\geq j+1}}_{l\leq n-1} \frac{\delta_{l}}{1+\delta_{l} L_{l}(t)} (\lambda_{j}(t) \cdot \lambda_{l}(t)) \] and for Measure.SPOT the j-th entry of the return value is the random variable \[ \mu_{j}^{\mathbb{Q}^{N}}(t) \ = \ \sum_{m(t) < l\leq j} \frac{\delta_{l}}{1+\delta_{l} L_{l}(t)} (\lambda_{j}(t) \cdot \lambda_{l}(t)) \] where \( \lambda_{j} \) is the vector for factor loadings for the j-th component of the stochastic process (that is, the diffusion part is \( \sum_{k=1}^m \lambda_{j,k} \mathrm{d}W_{k} \)). Note: The scalar product of the factor loadings determines the instantaneous covariance. If the model is written in log-coordinates (using exp as a state space transform), we find \(\lambda_{j} \cdot \lambda_{l} = \sum_{k=1}^m \lambda_{j,k} \lambda_{l,k} = \sigma_{j} \sigma_{l} \rho_{j,l} \). If the model is written without a state space transformation (in its orignial coordinates) then \(\lambda_{j} \cdot \lambda_{l} = \sum_{k=1}^m \lambda_{j,k} \lambda_{l,k} = L_{j} L_{l} \sigma_{j} \sigma_{l} \rho_{j,l} \).
        Specified by:
        getDrift in interface ProcessModel
        Parameters:
        process - The discretization process generating this model. The process provides call backs for TimeDiscretization and allows calls to getProcessValue for timeIndices less or equal the given one.
        timeIndex - Time index i for which the drift should be returned μ(ti).
        realizationAtTimeIndex - Time current forward rate vector at time index i which should be used in the calculation.
        realizationPredictor - The given realization at timeIndex+1 or null if no predictor is available.
        Returns:
        The drift vector μ(ti) as RandomVariableFromDoubleArray[]
        See Also:
        The calculation of the drift is consistent with the calculation of the numeraire in getNumeraire., The factor loading \( \lambda_{j,k} \).
      • getFactorLoading

        public RandomVariable[] getFactorLoading​(MonteCarloProcess process,
                                                 int timeIndex,
                                                 int componentIndex,
                                                 RandomVariable[] realizationAtTimeIndex)
        Description copied from interface: ProcessModel
        This method has to be implemented to return the factor loadings, i.e. the coefficient vector
        λj = (λ1,j, ..., λm,j) such that X = f(Y) and
        dYj = μj dt + λ1,j dW1 + ... + λm,j dWm
        in an m-factor model. Here j denotes index of the component of the resulting process.
        Specified by:
        getFactorLoading in interface ProcessModel
        Parameters:
        process - The discretization process generating this model. The process provides call backs for TimeDiscretization and allows calls to getProcessValue for timeIndices less or equal the given one.
        timeIndex - The time index (related to the model times discretization).
        componentIndex - The index j of the driven component.
        realizationAtTimeIndex - The realization of X at the time corresponding to timeIndex (in order to implement local and stochastic volatlity models).
        Returns:
        The factor loading for given factor and component.
      • applyStateSpaceTransform

        public RandomVariable applyStateSpaceTransform​(MonteCarloProcess process,
                                                       int timeIndex,
                                                       int componentIndex,
                                                       RandomVariable randomVariable)
        Description copied from interface: ProcessModel
        Applies the state space transform fi to the given state random variable such that Yi → fi(Yi) =: Xi.
        Specified by:
        applyStateSpaceTransform in interface ProcessModel
        Parameters:
        process - The discretization process generating this model. The process provides call backs for TimeDiscretization and allows calls to getProcessValue for timeIndices less or equal the given one.
        timeIndex - The time index (related to the model times discretization).
        componentIndex - The component index i.
        randomVariable - The state random variable Yi.
        Returns:
        New random variable holding the result of the state space transformation.
      • applyStateSpaceTransformInverse

        public RandomVariable applyStateSpaceTransformInverse​(MonteCarloProcess process,
                                                              int timeIndex,
                                                              int componentIndex,
                                                              RandomVariable randomVariable)
        Description copied from interface: ProcessModel
        Applies the inverse state space transform f-1i to the given random variable such that Xi → f-1i(Xi) =: Yi.
        Specified by:
        applyStateSpaceTransformInverse in interface ProcessModel
        Parameters:
        process - The discretization process generating this model. The process provides call backs for TimeDiscretization and allows calls to getProcessValue for timeIndices less or equal the given one.
        timeIndex - The time index (related to the model times discretization).
        componentIndex - The component index i.
        randomVariable - The state random variable Xi.
        Returns:
        New random variable holding the result of the state space transformation.
      • getRandomVariableForConstant

        public RandomVariable getRandomVariableForConstant​(double value)
        Description copied from interface: ProcessModel
        Return a random variable initialized with a constant using the models random variable factory.
        Specified by:
        getRandomVariableForConstant in interface ProcessModel
        Parameters:
        value - The constant value.
        Returns:
        A new random variable initialized with a constant value.
      • getForwardRate

        public RandomVariable getForwardRate​(MonteCarloProcess process,
                                             double time,
                                             double periodStart,
                                             double periodEnd)
        Description copied from interface: TermStructureModel
        Returns the time \( t \) forward rate on the models forward curve. Note: It is guaranteed that the random variable returned by this method is \( \mathcal{F}_{t} ) \)-measurable.
        Specified by:
        getForwardRate in interface TermStructureModel
        Parameters:
        process - The discretization process generating this model. The process provides call backs for TimeDiscretization and allows calls to getProcessValue for timeIndices less or equal the given one.
        time - The evaluation time.
        periodStart - The period start of the forward rate.
        periodEnd - The period end of the forward rate.
        Returns:
        The forward rate.
      • getNumberOfComponents

        public int getNumberOfComponents()
        Description copied from interface: ProcessModel
        Returns the number of components
        Specified by:
        getNumberOfComponents in interface ProcessModel
        Returns:
        The number of components
      • getNumberOfFactors

        public int getNumberOfFactors()
        Description copied from interface: ProcessModel
        Returns the number of factors m, i.e., the number of independent Brownian drivers.
        Specified by:
        getNumberOfFactors in interface ProcessModel
        Returns:
        The number of factors.
      • getNumberOfLibors

        public int getNumberOfLibors()
      • getAnalyticModel

        public AnalyticModel getAnalyticModel()
        Description copied from interface: TermStructureModel
        Return the associated analytic model, a collection of market date object like discount curve, forward curve and volatility surfaces.
        Specified by:
        getAnalyticModel in interface TermStructureModel
        Returns:
        The associated analytic model.
      • getCovarianceModel

        public TermStructureCovarianceModel getCovarianceModel()
        Returns the term structure covariance model.
        Returns:
        the term structure covariance model.