# Class HestonModel

java.lang.Object
net.finmath.montecarlo.model.AbstractProcessModel
net.finmath.montecarlo.assetderivativevaluation.models.HestonModel
All Implemented Interfaces:
ProcessModel

public class HestonModel extends AbstractProcessModel
This class implements a Heston 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 $dS(t) = r^{\text{c}} S(t) dt + \sqrt{V(t)} S(t) dW_{1}(t), \quad S(0) = S_{0},$ $dV(t) = \kappa ( \theta - V(t) ) dt + \xi \sqrt{V(t)} dW_{2}(t), \quad V(0) = \sigma^2,$ $dW_{1} dW_{2} = \rho dt$ $dN(t) = r^{\text{d}} N(t) dt, \quad N(0) = N_{0},$ where $$W$$ is a Brownian motion. The class provides the model of (S,V) to an MonteCarloProcess via the specification of $$f_{1} = exp , f_{2} = identity$$, $$\mu_{1} = r^{\text{c}} - \frac{1}{2} V^{+}(t) , \mu_{2} = \kappa ( \theta - V^{+}(t) )$$, $$\lambda_{1,1} = \sqrt{V^{+}(t)} , \lambda_{1,2} = 0 ,\lambda_{2,1} = \xi \sqrt{V^+(t)} \rho , \lambda_{2,2} = \xi \sqrt{V^+(t)} \sqrt{1-\rho^{2}}$$, i.e., of the SDE $dX_{1} = \mu_{1} dt + \lambda_{1,1} dW_{1} + \lambda_{1,2} dW_{2}, \quad X_{1}(0) = \log(S_{0}),$ $dX_{2} = \mu_{2} dt + \lambda_{2,1} dW_{1} + \lambda_{2,2} dW_{2}, \quad X_{2}(0) = V_{0} = \sigma^2,$ with $$S = f_{1}(X_{1}) , V = f_{2}(X_{2})$$. See MonteCarloProcess for the notation. Here $$V^{+}$$ denotes a truncated value of V. Different truncation schemes are available: FULL_TRUNCATION: $$V^{+} = max(V,0)$$, REFLECTION: $$V^{+} = abs(V)$$. The model allows to specify two independent rate for forwarding ($$r^{\text{c}}$$) and discounting ($$r^{\text{d}}$$). It thus allow for a simple modelling of a funding / collateral curve (via ($$r^{\text{d}}$$) and/or the specification of a dividend yield. The free parameters of this model are:
$$S_{0}$$
spot - initial value of S
$$r^{\text{c}}$$
the risk free rate
$$\sigma$$
the initial volatility level
$$r^{\text{d}}$$
the discount rate
$$\xi$$
the volatility of volatility
$$\theta$$
the mean reversion level of the stochastic volatility
$$\kappa$$
the mean reversion speed of the stochastic volatility
$$\rho$$
the correlation of the Brownian drivers
Version:
1.0
Author:
Christian Fries
• ## Nested Class Summary

Nested Classes
Modifier and Type
Class
Description
static enum
HestonModel.Scheme
Truncation schemes to be used in the calculation of drift and diffusion coefficients.
• ## Constructor Summary

Constructors
Constructor
Description
HestonModel(double initialValue, double riskFreeRate, double volatility, double discountRate, double theta, double kappa, double xi, double rho, HestonModel.Scheme scheme)
Create a Heston model.
HestonModel(double initialValue, double riskFreeRate, double volatility, double discountRate, double theta, double kappa, double xi, double rho, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create a Heston model.
HestonModel(double initialValue, double riskFreeRate, double volatility, double theta, double kappa, double xi, double rho, HestonModel.Scheme scheme)
Create a Heston model.
HestonModel(HestonModelDescriptor descriptor, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create the model from a descriptor.
HestonModel(RandomVariable initialValue, DiscountCurve discountCurveForForwardRate, RandomVariable volatility, DiscountCurve discountCurveForDiscountRate, RandomVariable theta, RandomVariable kappa, RandomVariable xi, RandomVariable rho, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create a Heston model.
HestonModel(RandomVariable initialValue, RandomVariable riskFreeRate, RandomVariable volatility, RandomVariable discountRate, RandomVariable theta, RandomVariable kappa, RandomVariable xi, RandomVariable rho, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create a Heston 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.
HestonModel
getCloneWithModifiedData(Map<String,Object> dataModified)
Returns a clone of this model where the specified properties have been modified.
DiscountCurve
getDiscountCurveForDiscountRate()

DiscountCurve
getDiscountCurveForForwardRate()

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.
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
getInitialValue()

RandomVariable
getKappa()

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
getRho()

RandomVariable
getRiskFreeRate()
Returns the risk free rate parameter of this model.
HestonModel.Scheme
getScheme()

RandomVariable
getTheta()

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

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 Details

• ### HestonModel

public HestonModel(HestonModelDescriptor descriptor, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create the model from a descriptor.
Parameters:
descriptor - A descriptor of the model.
scheme - The scheme.
randomVariableFactory - A random variable factory to be used for the parameters.
• ### HestonModel

public HestonModel(RandomVariable initialValue, DiscountCurve discountCurveForForwardRate, RandomVariable volatility, DiscountCurve discountCurveForDiscountRate, RandomVariable theta, RandomVariable kappa, RandomVariable xi, RandomVariable rho, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create a Heston model.
Parameters:
initialValue - $$S_{0}$$ - spot - initial value of S
discountCurveForForwardRate - the discount curve $$df^{\text{c}}$$ used to calculate the risk free rate $$r^{\text{c}}(t_{i},t_{i+1}) = \frac{\ln(\frac{df^{\text{c}}(t_{i})}{df^{\text{c}}(t_{i+1})}}{t_{i+1}-t_{i}}$$
volatility - $$\sigma$$ the initial volatility level
discountCurveForDiscountRate - the discount curve $$df^{\text{d}}$$ used to calculate the numeraire, $$r^{\text{d}}(t_{i},t_{i+1}) = \frac{\ln(\frac{df^{\text{d}}(t_{i})}{df^{\text{d}}(t_{i+1})}}{t_{i+1}-t_{i}}$$
theta - $$\theta$$ - the mean reversion level of the stochastic volatility
kappa - $$\kappa$$ - the mean reversion speed of the stochastic volatility
xi - $$\xi$$ - the volatility of volatility
rho - $$\rho$$ - the correlation of the Brownian drivers
scheme - The truncation scheme, that is, either reflection (V → abs(V)) or truncation (V → max(V,0)).
randomVariableFactory - The factory to be used to construct random variables.
• ### HestonModel

public HestonModel(RandomVariable initialValue, RandomVariable riskFreeRate, RandomVariable volatility, RandomVariable discountRate, RandomVariable theta, RandomVariable kappa, RandomVariable xi, RandomVariable rho, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create a Heston model.
Parameters:
initialValue - $$S_{0}$$ - spot - initial value of S
riskFreeRate - $$r^{\text{c}}$$ - the risk free rate
volatility - $$\sigma$$ the initial volatility level
discountRate - $$r^{\text{d}}$$ - the discount rate
theta - $$\theta$$ - the mean reversion level of the stochastic volatility
kappa - $$\kappa$$ - the mean reversion speed of the stochastic volatility
xi - $$\xi$$ - the volatility of volatility
rho - $$\rho$$ - the correlation of the Brownian drivers
scheme - The truncation scheme, that is, either reflection (V → abs(V)) or truncation (V → max(V,0)).
randomVariableFactory - The factory to be used to construct random variables.
• ### HestonModel

public HestonModel(double initialValue, double riskFreeRate, double volatility, double discountRate, double theta, double kappa, double xi, double rho, HestonModel.Scheme scheme, RandomVariableFactory randomVariableFactory)
Create a Heston model.
Parameters:
initialValue - Spot value.
riskFreeRate - The risk free rate.
volatility - The log volatility.
discountRate - The discount rate used in the numeraire.
theta - The longterm mean reversion level of V (a reasonable value is volatility*volatility).
kappa - The mean reversion speed.
xi - The volatility of the volatility (of V).
rho - The instantaneous correlation of the Brownian drivers (aka leverage).
scheme - The truncation scheme, that is, either reflection (V → abs(V)) or truncation (V → max(V,0)).
randomVariableFactory - The factory to be used to construct random variables..
• ### HestonModel

public HestonModel(double initialValue, double riskFreeRate, double volatility, double discountRate, double theta, double kappa, double xi, double rho, HestonModel.Scheme scheme)
Create a Heston model.
Parameters:
initialValue - Spot value.
riskFreeRate - The risk free rate.
volatility - The log volatility.
discountRate - The discount rate used in the numeraire.
theta - The longterm mean reversion level of V (a reasonable value is volatility*volatility).
kappa - The mean reversion speed.
xi - The volatility of the volatility (of V).
rho - The instantaneous correlation of the Brownian drivers (aka leverage).
scheme - The truncation scheme, that is, either reflection (V → abs(V)) or truncation (V → max(V,0)).
• ### HestonModel

public HestonModel(double initialValue, double riskFreeRate, double volatility, double theta, double kappa, double xi, double rho, HestonModel.Scheme scheme)
Create a Heston model.
Parameters:
initialValue - Spot value.
riskFreeRate - The risk free rate.
volatility - The log volatility.
theta - The longterm mean reversion level of V (a reasonable value is volatility*volatility).
kappa - The mean reversion speed.
xi - The volatility of the volatility (of V).
rho - The instantaneous correlation of the Brownian drivers (aka leverage).
scheme - The truncation scheme, that is, either reflection (V → abs(V)) or truncation (V → max(V,0)).
• ## Method Details

• ### getInitialState

public  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  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).

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:
• ### 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 HestonModel 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
• ### getInitialValue

public RandomVariable getInitialValue()
• ### 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.
• ### getDiscountCurveForForwardRate

public DiscountCurve getDiscountCurveForForwardRate()
• ### getDiscountCurveForDiscountRate

public DiscountCurve getDiscountCurveForDiscountRate()
• ### getTheta

public RandomVariable getTheta()
• ### getKappa

public RandomVariable getKappa()
• ### getXi

public RandomVariable getXi()
• ### getRho

public RandomVariable getRho()
• ### getScheme

public HestonModel.Scheme getScheme()