## Class 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
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, double riskFreeRate, double volatility, double discountRate, double jumpIntensity, double jumpSizeMean, double jumpSizeStDev, RandomVariableFactory randomVariableFactory)
Create a Merton model.
MertonModel​(double initialValue, DiscountCurve discountCurveForForwardRate, double volatility, DiscountCurve discountCurveForDiscountRate, 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, RandomVariableFactory randomVariableFactory)
Create a Merton model.
MertonModel​(MertonModelDescriptor descriptor)
Create the model from a descriptor.
MertonModel​(RandomVariable initialValue, DiscountCurve discountCurveForForwardRate, RandomVariable volatility, DiscountCurve discountCurveForDiscountRate, RandomVariable jumpIntensity, RandomVariable jumpSizeMean, RandomVariable jumpSizeStDev, RandomVariableFactory randomVariableFactory)
Create a Merton model.
MertonModel​(RandomVariable initialValue, RandomVariable riskFreeRate, RandomVariable volatility, RandomVariable discountRate, RandomVariable jumpIntensity, RandomVariable jumpSizeMean, RandomVariable jumpSizeStDev, RandomVariableFactory randomVariableFactory)
Create a Merton 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.
ProcessModel getCloneWithModifiedData​(Map<String,​Object> dataModified)
Returns a clone of this model where the specified properties have been modified.
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 componentIndex, RandomVariable[] realizationAtTimeIndex)
This method has to be implemented to return the factor loadings, i.e.
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.
RandomVariable getJumpIntensity()
RandomVariable getJumpSizeMean()
RandomVariable getJumpSizeStdDev()
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.
RandomVariable getRiskFreeRate()
RandomVariable getVolatility()
• ### 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, toString, wait, wait, wait
• ### Constructor Detail

• #### MertonModel

public MertonModel​(RandomVariable initialValue,
DiscountCurve discountCurveForForwardRate,
RandomVariable volatility,
DiscountCurve discountCurveForDiscountRate,
RandomVariable jumpIntensity,
RandomVariable jumpSizeMean,
RandomVariable jumpSizeStDev,
RandomVariableFactory randomVariableFactory)
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.
randomVariableFactory - The factory to be used to construct random variables.
• #### MertonModel

public MertonModel​(double initialValue,
DiscountCurve discountCurveForForwardRate,
double volatility,
DiscountCurve discountCurveForDiscountRate,
double jumpIntensity,
double jumpSizeMean,
double jumpSizeStDev,
RandomVariableFactory randomVariableFactory)
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.
randomVariableFactory - The factory to be used to construct random variables.
• #### MertonModel

public MertonModel​(RandomVariable initialValue,
RandomVariable riskFreeRate,
RandomVariable volatility,
RandomVariable discountRate,
RandomVariable jumpIntensity,
RandomVariable jumpSizeMean,
RandomVariable jumpSizeStDev,
RandomVariableFactory randomVariableFactory)
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.
randomVariableFactory - The factory to be used to construct random variables.
• #### MertonModel

public MertonModel​(double initialValue,
double riskFreeRate,
double volatility,
double discountRate,
double jumpIntensity,
double jumpSizeMean,
double jumpSizeStDev,
RandomVariableFactory randomVariableFactory)
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.
randomVariableFactory - The factory to be used to construct random variables.
• #### 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

• #### 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.
• #### 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).
• #### 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
• #### 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 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.
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:
• #### 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 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 RandomVariable getRiskFreeRate()
Returns:
the riskFreeRate
• #### getVolatility

public RandomVariable getVolatility()
Returns:
the volatility
• #### getJumpIntensity

public RandomVariable getJumpIntensity()
Returns:
the jumpIntensity
• #### getJumpSizeMean

public RandomVariable getJumpSizeMean()
Returns:
the jumpSizeMean
• #### getJumpSizeStdDev

public RandomVariable getJumpSizeStdDev()
Returns:
the jumpSizeStdDev