Module net.finmath.lib

Package net.finmath.montecarlo.assetderivativevaluation.models

Class MultiAssetBlackScholesModel

java.lang.Object

net.finmath.montecarlo.model.AbstractProcessModel

net.finmath.montecarlo.assetderivativevaluation.models.MultiAssetBlackScholesModel

All Implemented Interfaces:

ProcessModel

public class MultiAssetBlackScholesModel
extends AbstractProcessModel

This class implements a multi-asset Black Scholes model providing an AbstractProcessModel. The class can be used with an EulerSchemeFromProcessModel to create a Monte-Carlo simulation. The model can be specified by general factor loadings, that is, in the form \[ dS_{i} = r S_{i} dt + S_{i} \sum_{j=0}^{m-1} f{i,j} dW_{j}, \quad S_{i}(0) = S_{i,0}, \] \[ dN = r N dt, \quad N(0) = N_{0}. \] Alternatively, the model can be specifies by providing volatilities and correlations from which the factor loadings \( f_{i,j} \) are derived such that \[ \sum_{k=0}^{m-1} f{i,k} f{j,k} = \sigma_{i} \sigma_{j} \rho_{i,j} \] such that the effective model is \[ dS_{i} = r S_{i} dt + \sigma_{i} S_{i} dW_{i}, \quad S_{i}(0) = S_{i,0}, \] \[ dN = r N dt, \quad N(0) = N_{0}, \] \[ dW_{i} dW_{j} = \rho_{i,j} dt, \] Note that in case the model is used with an EulerSchemeFromProcessModel, the BrownianMotion used can have a correlation, which alters the simulation (which is admissible). The specification above hold, provided that the BrownianMotion used has independent components. The class provides the model of \( S_{i} \) to an MonteCarloProcess via the specification of \( f = exp \), \( \mu_{i} = r - \frac{1}{2} \sigma_{i}^2 \), \( \lambda_{i,j} = \sigma_{i} g_{i,j} \), i.e., of the SDE \[ dX_{i} = \mu_{i} dt + \sum_{j=0}^{m-1} \lambda_{i,j} dW_{j}, \quad X_{i}(0) = \log(S_{i,0}), \] with \( S = f(X) \). See MonteCarloProcess for the notation.

Version:

1.1

Author:

Christian Fries

See Also:

The interface for numerical schemes., The interface for models provinding parameters to numerical schemes.

Constructor Summary

Constructors

Constructor	Description
MultiAssetBlackScholesModel(double[] initialValues, double riskFreeRate, double[][] factorLoadings)	Create a multi-asset Black-Scholes model.
MultiAssetBlackScholesModel(double[] initialValues, double riskFreeRate, double[] volatilities, double[][] correlations)	Create a multi-asset Black-Scholes model.
MultiAssetBlackScholesModel(RandomVariableFactory randomVariableFactory, double[] initialValues, double riskFreeRate, double[][] factorLoadings)	Create a multi-asset Black-Scholes model.
MultiAssetBlackScholesModel(RandomVariableFactory randomVariableFactory, double[] initialValues, double riskFreeRate, double[] volatilities, double[][] correlations)	Create a multi-asset Black-Scholes model.

Method Summary

Modifier and Type	Method	Description
RandomVariable	applyStateSpaceTransform(MonteCarloProcess process, int timeIndex, int componentIndex, RandomVariable randomVariable)	Applies the state space transform fi to the given state random variable such that Yi → fi(Yi) =: Xi.
RandomVariable	applyStateSpaceTransformInverse(MonteCarloProcess process, int timeIndex, int componentIndex, RandomVariable randomVariable)	Applies the inverse state space transform f-1i to the given random variable such that Xi → f-1i(Xi) =: Yi.
MultiAssetBlackScholesModel	getCloneWithModifiedData(Map<String,Object> dataModified)	Returns a clone of this model where the specified properties have been modified.
double[][]	getCorrelationMatrix()	Returns the volatility parameters of this model.
RandomVariable[]	getDrift(MonteCarloProcess process, int timeIndex, RandomVariable[] realizationAtTimeIndex, RandomVariable[] realizationPredictor)	This method has to be implemented to return the drift, i.e.
RandomVariable[]	getFactorLoading(MonteCarloProcess process, int timeIndex, int component, RandomVariable[] realizationAtTimeIndex)	This method has to be implemented to return the factor loadings, i.e.
double[][]	getFactorLoadingMatrix()	Returns the factorLoadings parameters of this model.
RandomVariable[]	getInitialState(MonteCarloProcess process)	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.
int	getNumberOfComponents()	Returns the number of components
int	getNumberOfFactors()	Returns the number of factors m, i.e., the number of independent Brownian drivers.
RandomVariable	getNumeraire(MonteCarloProcess process, double time)	Return the numeraire at a given time index.
RandomVariable	getRandomVariableForConstant(double value)	Return a random variable initialized with a constant using the models random variable factory.
double	getRiskFreeRate()	Returns the risk free rate parameter of this model.
double[]	getVolatilityVector()	Returns the volatility parameters of this model.
String	toString()

Methods inherited from class net.finmath.montecarlo.model.AbstractProcessModel

getInitialValue, getReferenceDate

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

MultiAssetBlackScholesModel

public MultiAssetBlackScholesModel(RandomVariableFactory randomVariableFactory,
                                   double[] initialValues,
                                   double riskFreeRate,
                                   double[][] factorLoadings)

Create a multi-asset Black-Scholes model.

Parameters:

randomVariableFactory - The RandomVariableFactory used to construct model parameters as random variables.

initialValues - Spot values.

riskFreeRate - The risk free rate.

factorLoadings - The matrix of factor loadings, where factorLoadings[underlyingIndex][factorIndex] is the coefficient of the Brownian driver factorIndex used for the underlying underlyingIndex.

MultiAssetBlackScholesModel

public MultiAssetBlackScholesModel(RandomVariableFactory randomVariableFactory,
                                   double[] initialValues,
                                   double riskFreeRate,
                                   double[] volatilities,
                                   double[][] correlations)

Create a multi-asset Black-Scholes model.

Parameters:

randomVariableFactory - The RandomVariableFactory used to construct model parameters as random variables.

initialValues - Spot values.

riskFreeRate - The risk free rate.

volatilities - The log volatilities.

correlations - A correlation matrix.

MultiAssetBlackScholesModel

public MultiAssetBlackScholesModel(double[] initialValues,
                                   double riskFreeRate,
                                   double[][] factorLoadings)

Create a multi-asset Black-Scholes model.

Parameters:

initialValues - Spot values.

riskFreeRate - The risk free rate.

factorLoadings - The matrix of factor loadings, where factorLoadings[underlyingIndex][factorIndex] is the coefficient of the Brownian driver factorIndex used for the underlying underlyingIndex.

MultiAssetBlackScholesModel

public MultiAssetBlackScholesModel(double[] initialValues,
                                   double riskFreeRate,
                                   double[] volatilities,
                                   double[][] correlations)

Create a multi-asset Black-Scholes model.

Parameters:

initialValues - Spot values.

riskFreeRate - The risk free rate.

volatilities - The log volatilities.

correlations - A correlation matrix.

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, ..., μ_n) such that X = f(Y) and
dY_j = μ_j dt + λ_1,j dW₁ + ... + λ_m,j dW_m
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
λ_j = (λ_1,j, ..., λ_m,j) such that X = f(Y) and
dY_j = μ_j dt + λ_1,j dW₁ + ... + λ_m,j dW_m
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 f_i to the given state random variable such that Y_i → f_i(Y_i) =: X_i.

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 Y_i.

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^-1_i to the given random variable such that X_i → f^-1_i(X_i) =: Y_i.

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 X_i.

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

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.

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.

getCloneWithModifiedData

public MultiAssetBlackScholesModel 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).

toString

public String toString()

Overrides:

toString in class Object

getRiskFreeRate

public double getRiskFreeRate()

Returns the risk free rate parameter of this model.

Returns:

Returns the riskFreeRate.

getFactorLoadingMatrix

public double[][] getFactorLoadingMatrix()

Returns the factorLoadings parameters of this model.

Returns:

Returns the factorLoadings.

getVolatilityVector

public double[] getVolatilityVector()

Returns the volatility parameters of this model.

Returns:

Returns the volatilities.

getCorrelationMatrix

public double[][] getCorrelationMatrix()

Returns the volatility parameters of this model.

Returns:

Returns the volatilities.