Class BlackScholesModel

  • All Implemented Interfaces:
    ProcessModel

    public class BlackScholesModel
    extends AbstractProcessModel
    This class implements a Black Scholes Model, that is, it provides the drift and volatility specification and performs the calculation of the numeraire (consistent with the dynamics, i.e. the drift). The model is \[ dS = r S dt + \sigma S dW, \quad S(0) = S_{0}, \] \[ dN = r N dt, \quad N(0) = N_{0}, \] The class provides the model of S to an MonteCarloProcess via the specification of \( f = exp \), \( \mu = r - \frac{1}{2} \sigma^2 \), \( \lambda_{1,1} = \sigma \), i.e., of the SDE \[ dX = \mu dt + \lambda_{1,1} dW, \quad X(0) = \log(S_{0}), \] with \( S = f(X) \). See MonteCarloProcess for the notation.
    Version:
    1.0
    Author:
    Christian Fries
    See Also:
    The interface for numerical schemes., The interface for models provinding parameters to numerical schemes.
    • Constructor Detail

      • BlackScholesModel

        public BlackScholesModel​(RandomVariable initialValue,
                                 RandomVariable riskFreeRate,
                                 RandomVariable volatility,
                                 RandomVariableFactory randomVariableFactory)
        Create a Black-Scholes specification implementing AbstractProcessModel.
        Parameters:
        initialValue - Spot value.
        riskFreeRate - The risk free rate.
        volatility - The log volatility.
        randomVariableFactory - The random variable factory used to create random variables from constants.
      • BlackScholesModel

        public BlackScholesModel​(double initialValue,
                                 double riskFreeRate,
                                 double volatility,
                                 RandomVariableFactory randomVariableFactory)
        Create a Monte-Carlo simulation using given time discretization.
        Parameters:
        initialValue - Spot value.
        riskFreeRate - The risk free rate.
        volatility - The log volatility.
        randomVariableFactory - The random variable factory used to create random variables from constants.
      • BlackScholesModel

        public BlackScholesModel​(double initialValue,
                                 double riskFreeRate,
                                 double volatility)
        Create a Black-Scholes model from given parameters.
        Parameters:
        initialValue - Spot value.
        riskFreeRate - The risk free rate.
        volatility - The log volatility.
    • Method Detail

      • 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.
        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)
        Description copied from interface: ProcessModel
        This method has to be implemented to return the drift, i.e. the coefficient vector
        μ = (μ1, ..., μn) 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. Since the model is provided only on a time discretization, the method may also (should try to) return the drift as \( \frac{1}{t_{i+1}-t_{i}} \int_{t_{i}}^{t_{i+1}} \mu(\tau) \mathrm{d}\tau \).
        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).
        realizationAtTimeIndex - The given realization at timeIndex
        realizationPredictor - The given realization at timeIndex+1 or null if no predictor is available.
        Returns:
        The drift or average drift from timeIndex to timeIndex+1, i.e. \( \frac{1}{t_{i+1}-t_{i}} \int_{t_{i}}^{t_{i+1}} \mu(\tau) \mathrm{d}\tau \) (or a suitable approximation).
      • getFactorLoading

        public RandomVariable[] getFactorLoading​(MonteCarloProcess process,
                                                 int timeIndex,
                                                 int component,
                                                 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.
        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).
        component - 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.
        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.
        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.
      • getNumeraire

        public RandomVariable getNumeraire​(MonteCarloProcess process,
                                           double time)
        Description copied from interface: ProcessModel
        Return the numeraire at a given time index. Note: The random variable returned is a defensive copy and may be modified.
        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 time t for which the numeraire N(t) should be returned.
        Returns:
        The numeraire at the specified time as RandomVariable
      • getNumberOfComponents

        public int getNumberOfComponents()
        Description copied from interface: ProcessModel
        Returns the number of components
        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.
        Returns:
        The number of factors.
      • 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.
        Parameters:
        value - The constant value.
        Returns:
        A new random variable initialized with a constant value.
      • getCloneWithModifiedData

        public BlackScholesModel getCloneWithModifiedData​(Map<String,​Object> dataModified)
        Description copied from interface: ProcessModel
        Returns a clone of this model where the specified properties have been modified. Note that there is no guarantee that a model reacts on a specification of a properties in the parameter map dataModified. If data is provided which is ignored by the model no exception may be thrown.
        Parameters:
        dataModified - Key-value-map of parameters to modify.
        Returns:
        A clone of this model (or this model if no parameter was modified).
      • getInitialValue

        public RandomVariable[] getInitialValue​(MonteCarloProcess process)
        Return the initial value of this model.
        Overrides:
        getInitialValue in class AbstractProcessModel
        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 this model.
      • getRiskFreeRate

        public RandomVariable getRiskFreeRate()
        Returns the risk free rate parameter of this model.
        Returns:
        Returns the riskFreeRate.
      • getVolatility

        public RandomVariable getVolatility()
        Returns the volatility parameter of this model.
        Returns:
        Returns the volatility.