Package net.finmath.montecarlo.assetderivativevaluation.models

Class MertonModel

java.lang.Object

net.finmath.montecarlo.model.AbstractProcessModel

net.finmath.montecarlo.assetderivativevaluation.models.MertonModel

All Implemented Interfaces:

ProcessModel

public class MertonModel
extends AbstractProcessModel

This class implements a Merton 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 \[ dS = \mu S dt + \sigma S dW + S dJ, \quad S(0) = S_{0}, \] \[ dN = r N dt, \quad N(0) = N_{0}, \] where \( W \) is Brownian motion and \( J \) is a jump process (compound Poisson process). The process \( J \) is given by \( J(t) = \sum_{i=1}^{N(t)} (Y_{i}-1) \), where \( \log(Y_{i}) \) are i.i.d. normals with mean \( a - \frac{1}{2} b^{2} \) and standard deviation \( b \). Here \( a \) is the jump size mean and \( b \) is the jump size std. dev. The model can be rewritten as \( S = \exp(X) \), where \[ dX = \mu dt + \sigma dW + dJ^{X}, \quad X(0) = \log(S_{0}), \] with \[ J^{X}(t) = \sum_{i=1}^{N(t)} \log(Y_{i}) \] with \( \mu = r - \frac{1}{2} \sigma^2 - (exp(a)-1) \lambda \). The class provides the model of S to an MonteCarloProcess via the specification of \( f = exp \), \( \mu = r - \frac{1}{2} \sigma^2 - (exp(a)-1) \lambda \), \( \lambda_{1,1} = \sigma, \lambda_{1,2} = a - \frac{1}{2} b^2, \lambda_{1,3} = b \), i.e., of the SDE \[ dX = \mu dt + \lambda_{1,1} dW + \lambda_{1,2} dN + \lambda_{1,3} Z dN, \quad X(0) = \log(S_{0}), \] with \( S = f(X) \). See MonteCarloProcess for the notation. For an example on the construction of the three factors \( dW \), \( dN \), and \( Z dN \) see MonteCarloMertonModel.

Version:

1.0

Author:

Christian Fries

See Also:

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

Constructor Summary

Constructors

Constructor	Description
MertonModel(double initialValue, double riskFreeRate, double volatility, double jumpIntensity, double jumpSizeMean, double jumpSizeStDev)	Create a Merton model.
MertonModel(double initialValue, double riskFreeRate, double volatility, double discountRate, double jumpIntensity, double jumpSizeMean, double jumpSizeStDev)	Create a Merton model.
MertonModel(double initialValue, DiscountCurve discountCurveForForwardRate, double volatility, DiscountCurve discountCurveForDiscountRate, double jumpIntensity, double jumpSizeMean, double jumpSizeStDev)	Create a Merton model.
MertonModel(MertonModelDescriptor descriptor)	Create the model from a descriptor.

Method Summary

Modifier and Type	Method	Description
RandomVariable	applyStateSpaceTransform(int componentIndex, RandomVariable randomVariable)	Applies the state space transform fi to the given state random variable such that Yi → fi(Yi) =: Xi.
RandomVariable	applyStateSpaceTransformInverse(int componentIndex, RandomVariable randomVariable)
ProcessModel	getCloneWithModifiedData(Map<String,Object> dataModified)	Returns a clone of this model where the specified properties have been modified.
RandomVariable[]	getDrift(int timeIndex, RandomVariable[] realizationAtTimeIndex, RandomVariable[] realizationPredictor)	This method has to be implemented to return the drift, i.e.
RandomVariable[]	getFactorLoading(int timeIndex, int componentIndex, RandomVariable[] realizationAtTimeIndex)	This method has to be implemented to return the factor loadings, i.e.
RandomVariable[]	getInitialState()	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.
double	getJumpIntensity()
double	getJumpSizeMean()
double	getJumpSizeStdDev()
int	getNumberOfComponents()	Returns the number of components
RandomVariable	getNumeraire(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()
double	getVolatility()

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

getInitialValue, getMonteCarloWeights, getNumberOfFactors, getProcess, getProcessValue, getReferenceDate, getTime, getTimeDiscretization, getTimeIndex, setProcess

Methods inherited from class java.lang.Object

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

Methods inherited from interface net.finmath.montecarlo.model.ProcessModel

getDrift, getFactorLoading, getInitialState, getNumeraire

Constructor Detail

MertonModel

public MertonModel(MertonModelDescriptor descriptor)

Create the model from a descriptor.

Parameters:

descriptor - A descriptor of the model.

MertonModel

public MertonModel(double initialValue,
                   DiscountCurve discountCurveForForwardRate,
                   double volatility,
                   DiscountCurve discountCurveForDiscountRate,
                   double jumpIntensity,
                   double jumpSizeMean,
                   double jumpSizeStDev)

Create a Merton model.

Parameters:

initialValue - \( S_{0} \) - spot - initial value of S

discountCurveForForwardRate - The curve specifying \( t \mapsto exp(- r^{\text{c}}(t) \cdot t) \) - with \( r^{\text{c}}(t) \) the risk free rate

volatility - The log volatility.

discountCurveForDiscountRate - The curve specifying \( t \mapsto exp(- r^{\text{d}}(t) \cdot t) \) - with \( r^{\text{d}}(t) \) the discount rate

jumpIntensity - The intensity parameter lambda of the compound Poisson process.

jumpSizeMean - The mean jump size of the normal distributes jump sizes of the compound Poisson process.

jumpSizeStDev - The standard deviation of the normal distributes jump sizes of the compound Poisson process.

MertonModel

public MertonModel(double initialValue,
                   double riskFreeRate,
                   double volatility,
                   double discountRate,
                   double jumpIntensity,
                   double jumpSizeMean,
                   double jumpSizeStDev)

Create a Merton model.

Parameters:

initialValue - Spot value.

riskFreeRate - The risk free rate.

volatility - The log volatility.

discountRate - The discount rate used in the numeraire.

jumpIntensity - The intensity parameter lambda of the compound Poisson process.

jumpSizeMean - The mean jump size of the normal distributes jump sizes of the compound Poisson process.

jumpSizeStDev - The standard deviation of the normal distributes jump sizes of the compound Poisson process.

MertonModel

public MertonModel(double initialValue,
                   double riskFreeRate,
                   double volatility,
                   double jumpIntensity,
                   double jumpSizeMean,
                   double jumpSizeStDev)

Create a Merton model.

Parameters:

initialValue - Spot value.

riskFreeRate - The risk free rate.

volatility - The log volatility.

jumpIntensity - The intensity parameter lambda of the compound Poisson process.

jumpSizeMean - The mean jump size of the normal distributes jump sizes of the compound Poisson process.

jumpSizeStDev - The standard deviation of the normal distributes jump sizes of the compound Poisson process.

Method Detail

getNumberOfComponents

public int getNumberOfComponents()

Description copied from interface: ProcessModel

Returns the number of components

Returns:

The number of components

applyStateSpaceTransform

public RandomVariable applyStateSpaceTransform(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:

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(int componentIndex,
                                                      RandomVariable randomVariable)

getInitialState

public RandomVariable[] getInitialState()

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.

Returns:

The initial value of the state variable of the process Y(t=0).

getNumeraire

public RandomVariable getNumeraire(double time)
                            throws CalculationException

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:

time - The time t for which the numeraire N(t) should be returned.

Returns:

The numeraire at the specified time as RandomVariableFromDoubleArray

Throws:

CalculationException - Thrown if the valuation fails, specific cause may be available via the cause() method.

getDrift

public RandomVariable[] getDrift(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:

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(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
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:

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.

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 ProcessModel 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).

getRiskFreeRate

public double getRiskFreeRate()

Returns:

the riskFreeRate

getVolatility

public double getVolatility()

Returns:

the volatility

getJumpIntensity

public double getJumpIntensity()

Returns:

the jumpIntensity

getJumpSizeMean

public double getJumpSizeMean()

Returns:

the jumpSizeMean

getJumpSizeStdDev

public double getJumpSizeStdDev()

Returns:

the jumpSizeStdDev