## Class 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
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

All Methods
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, ..., 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).

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
• #### 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.

public double[][] getFactorLoadingMatrix()
Returns the factorLoadings parameters of this model.
Returns:
public double[] getVolatilityVector()
public double[][] getCorrelationMatrix()