## Class HullWhiteModelWithDirectSimulation

• All Implemented Interfaces:
IndependentModelParameterProvider, LIBORModel, TermStructureModel, ProcessModel

public class HullWhiteModelWithDirectSimulation
extends AbstractProcessModel
implements LIBORModel
Implements a Hull-White model with time dependent mean reversion speed and time dependent short rate volatility. Note: This implementation is for illustrative purposes. For a numerically equivalent, more efficient implementation see HullWhiteModel. Please use HullWhiteModel for real applications.

Model Dynamics

The Hull-While model assumes the following dynamic for the short rate: $d r(t) = ( \theta(t) - a(t) r(t) ) d t + \sigma(t) d W(t) \text{,} \quad r(t_{0}) = r_{0} \text{,}$ where the function $$\theta$$ determines the calibration to the initial forward curve, $$a$$ is the mean reversion and $$\sigma$$ is the instantaneous volatility. The dynamic above is under the equivalent martingale measure corresponding to the numeraire $N(t) = \exp\left( \int_0^t r(\tau) \mathrm{d}\tau \right) \text{.}$ The main task of this class is to provide the risk-neutral drift and the volatility to the numerical scheme (given the volatility model), simulating $$r(t_{i})$$. The class then also provides and the corresponding numeraire and forward rates (LIBORs).

Time Discrete Model

Assuming piecewise constant coefficients (mean reversion speed $$a$$ and short rate volatility $$\sigma$$ the class specifies the drift and factor loadings as piecewise constant functions for an Euler-scheme. The class provides the exact Euler step for the short rate r. More specifically (assuming a constant mean reversion speed $$a$$ for a moment), considering $\Delta \bar{r}(t_{i}) = \frac{1}{t_{i+1}-t_{i}} \int_{t_{i}}^{t_{i+1}} d r(t)$ we find from $\exp(-a t) \ \left( \mathrm{d} \left( \exp(a t) r(t) \right) \right) \ = \ a r(t) + \mathrm{d} r(t) \ = \ \theta(t) \mathrm{d}t + \sigma(t) \mathrm{d}W(t)$ that $\exp(a t_{i+1}) r(t_{i+1}) - \exp(a t_{i}) r(t_{i}) \ = \ \int_{t_{i}}^{t_{i+1}} \left[ \exp(a t) \theta(t) \mathrm{d}t + \exp(a t) \sigma(t) \mathrm{d}W(t) \right]$ that is $r(t_{i+1}) - r(t_{i}) \ = \ -(1-\exp(-a (t_{i+1}-t_{i})) r(t_{i}) + \int_{t_{i}}^{t_{i+1}} \left[ \exp(-a (t_{i+1}-t)) \theta(t) \mathrm{d}t + \exp(-a (t_{i+1}-t)) \sigma(t) \mathrm{d}W(t) \right]$ Assuming piecewise constant $$\sigma$$ and $$\theta$$, being constant over $$(t_{i},t_{i}+\Delta t_{i})$$, we thus find $r(t_{i+1}) - r(t_{i}) \ = \ \frac{1-\exp(-a \Delta t_{i})}{a \Delta t_{i}} \left( ( \theta(t_{i}) - a \bar{r}(t_{i})) \Delta t_{i} \right) + \sqrt{\frac{1-\exp(-2 a \Delta t_{i})}{2 a \Delta t_{i}}} \sigma(t_{i}) \Delta W(t_{i})$ . In other words, the Euler scheme is exact if the mean reversion $$a$$ is replaced by the effective mean reversion $$\frac{1-\exp(-a \Delta t_{i})}{a \Delta t_{i}} a$$ and the volatility is replaced by the effective volatility $$\sqrt{\frac{1-\exp(-2 a \Delta t_{i})}{2 a \Delta t_{i}}} \sigma(t_{i})$$. In the calculations above the mean reversion speed is treated as a constants, but it is straight forward to see that the same holds for piecewise constant mean reversion speeds, replacing the expression $$a \ t$$ by $$\int_{0}^t a(s) \mathrm{d}s$$.

Calibration

The drift of the short rate is calibrated to the given forward curve using $\theta(t) = \frac{\partial}{\partial T} f(0,t) + a(t) f(0,t) + \phi(t) \text{,}$ where the function $$f$$ denotes the instantanenous forward rate and $$\phi(t) = \frac{1}{2} a \sigma^{2}(t) B(t)^{2} + \sigma^{2}(t) B(t) \frac{\partial}{\partial t} B(t)$$ with $$B(t) = \frac{1-\exp(-a t)}{a}$$.

Volatility Model

The Hull-White model is essentially equivalent to LIBOR Market Model where the forward rate normal volatility $$\sigma(t,T)$$ is given by $\sigma(t,T_{i}) \ = \ (1 + L_{i}(t) (T_{i+1}-T_{i})) \sigma(t) \exp(-a (T_{i}-t)) \frac{1-\exp(-a (T_{i+1}-T_{i}))}{a (T_{i+1}-T_{i})}$ (where $$\{ T_{i} \}$$ is the forward rates tenor time discretization (note that this is the normal volatility, not the log-normal volatility). Hence, we interpret both, short rate mean reversion speed and short rate volatility as part of the volatility model. The mean reversion speed and the short rate volatility have to be provided to this class via an object implementing ShortRateVolatilityModel.
Version:
1.2
Author:
Christian Fries
ShortRateVolatilityModel, HullWhiteModel
• ### Constructor Summary

Constructors
Constructor Description
HullWhiteModelWithDirectSimulation​(TimeDiscretization liborPeriodDiscretization, AnalyticModel analyticModel, ForwardCurve forwardRateCurve, DiscountCurve discountCurve, ShortRateVolatilityModel volatilityModel, Map<String,​?> properties)
Creates a Hull-White model which implements LIBORMarketModel.
• ### 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.
AnalyticModel getAnalyticModel()
Return the associated analytic model, a collection of market date object like discount curve, forward curve and volatility surfaces.
LIBORMarketModel getCloneWithModifiedData​(Map<String,​Object> dataModified)
Create a new object implementing LIBORModel, using the new data.
DiscountCurve getDiscountCurve()
Return the discount curve associated the forwards.
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.
ForwardCurve getForwardRateCurve()
Return the initial forward rate curve.
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.
double getIntegratedBondSquaredVolatility​(double time, double maturity)
RandomVariable getLIBOR​(MonteCarloProcess process, double time, double periodStart, double periodEnd)
Returns the time $$t$$ forward rate on the models forward curve.
RandomVariable getLIBOR​(MonteCarloProcess process, int timeIndex, int liborIndex)
Return the forward rate at a given timeIndex and for a given liborIndex.
double getLiborPeriod​(int timeIndex)
The period start corresponding to a given forward rate discretization index.
TimeDiscretization getLiborPeriodDiscretization()
The tenor time discretization of the forward rate curve.
int getLiborPeriodIndex​(double time)
Same as java.util.Arrays.binarySearch(liborPeriodDiscretization,time).
Map<String,​RandomVariable> getModelParameters()
Returns a map of independent model parameters of this model.
int getNumberOfComponents()
Returns the number of components
int getNumberOfFactors()
Returns the number of factors m, i.e., the number of independent Brownian drivers.
int getNumberOfLibors()
Get the number of LIBORs in the LIBOR discretization.
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 getShortRateConditionalVariance​(double time, double maturity)
Calculates the variance $$\mathop{Var}(r(t) \vert r(s) )$$, that is $$\int_{s}^{t} \sigma^{2}(\tau) \exp(-2 \cdot \int_{\tau}^{t} a(u) \mathrm{d}u ) \ \mathrm{d}\tau$$ where $$a$$ is the meanReversion and $$\sigma$$ is the short rate instantaneous volatility.
• ### 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
• ### Methods inherited from interface net.finmath.montecarlo.model.ProcessModel

getReferenceDate
• ### Methods inherited from interface net.finmath.montecarlo.interestrate.TermStructureModel

getForwardDiscountBond
• ### Constructor Detail

• #### HullWhiteModelWithDirectSimulation

public HullWhiteModelWithDirectSimulation​(TimeDiscretization liborPeriodDiscretization,
AnalyticModel analyticModel,
ForwardCurve forwardRateCurve,
DiscountCurve discountCurve,
ShortRateVolatilityModel volatilityModel,
Map<String,​?> properties)
Creates a Hull-White model which implements LIBORMarketModel.
Parameters:
liborPeriodDiscretization - The forward rate discretization to be used in the getLIBOR method.
analyticModel - The analytic model to be used (currently not used, may be null).
forwardRateCurve - The forward curve to be used (currently not used, - the model uses disocuntCurve only.
discountCurve - The disocuntCurve (currently also used to determine the forward curve).
volatilityModel - The volatility model specifying mean reversion and instantaneous volatility of the short rate.
properties - A map specifying model properties (currently not used, may be null).
• ### Method Detail

• #### getNumberOfComponents

public int getNumberOfComponents()
Description copied from interface: ProcessModel
Returns the number of components
Specified by:
getNumberOfComponents in interface ProcessModel
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.
Specified by:
getNumberOfFactors in interface ProcessModel
Returns:
The number of factors.
• #### 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.
Specified by:
applyStateSpaceTransform in interface ProcessModel
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.
Specified by:
applyStateSpaceTransformInverse in interface ProcessModel
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.
Specified by:
getInitialState in interface ProcessModel
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)
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.
Specified by:
getNumeraire in interface ProcessModel
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
Throws:
CalculationException - Thrown if the valuation fails, specific cause may be available via the cause() method.
• #### 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 $$\frac{1}{t_{i+1}-t_{i}} \int_{t_{i}}^{t_{i+1}} \mu(\tau) \mathrm{d}\tau$$.
Specified by:
getDrift in interface ProcessModel
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.
Specified by:
getFactorLoading in interface ProcessModel
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:
• #### 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.
Specified by:
getRandomVariableForConstant in interface ProcessModel
Parameters:
value - The constant value.
Returns:
A new random variable initialized with a constant value.
• #### getLIBOR

public RandomVariable getLIBOR​(MonteCarloProcess process,
double time,
double periodStart,
double periodEnd)
throws CalculationException
Description copied from interface: TermStructureModel
Returns the time $$t$$ forward rate on the models forward curve. Note: It is guaranteed that the random variable returned by this method is $$\mathcal{F}_{t} )$$-measurable.
Specified by:
getLIBOR in interface TermStructureModel
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 evaluation time.
periodStart - The period start of the forward rate.
periodEnd - The period end of the forward rate.
Returns:
The forward rate.
Throws:
CalculationException - Thrown if model fails to calculate the random variable.
• #### getLIBOR

public RandomVariable getLIBOR​(MonteCarloProcess process,
int timeIndex,
int liborIndex)
throws CalculationException
Description copied from interface: LIBORModel
Return the forward rate at a given timeIndex and for a given liborIndex.
Specified by:
getLIBOR in interface LIBORModel
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 (associated with Process.getTimeDiscretization().
liborIndex - The forward rate index (associated with LIBORModel.getLiborPeriodDiscretization().
Returns:
The forward rate.
Throws:
CalculationException - Thrown if calculation failed.
• #### getLiborPeriodDiscretization

public TimeDiscretization getLiborPeriodDiscretization()
Description copied from interface: LIBORModel
The tenor time discretization of the forward rate curve.
Specified by:
getLiborPeriodDiscretization in interface LIBORModel
Returns:
The tenor time discretization of the forward rate curve.
• #### getNumberOfLibors

public int getNumberOfLibors()
Description copied from interface: LIBORModel
Get the number of LIBORs in the LIBOR discretization.
Specified by:
getNumberOfLibors in interface LIBORModel
Returns:
The number of LIBORs in the LIBOR discretization
• #### getLiborPeriod

public double getLiborPeriod​(int timeIndex)
Description copied from interface: LIBORModel
The period start corresponding to a given forward rate discretization index.
Specified by:
getLiborPeriod in interface LIBORModel
Parameters:
timeIndex - The index corresponding to a given time (interpretation is start of period)
Returns:
The period start corresponding to a given forward rate discretization index.
• #### getLiborPeriodIndex

public int getLiborPeriodIndex​(double time)
Description copied from interface: LIBORModel
Same as java.util.Arrays.binarySearch(liborPeriodDiscretization,time). Will return a negative value if the time is not found, but then -index-1 corresponds to the index of the smallest time greater than the given one.
Specified by:
getLiborPeriodIndex in interface LIBORModel
Parameters:
time - The period start.
Returns:
The index corresponding to a given time (interpretation is start of period)
• #### getAnalyticModel

public AnalyticModel getAnalyticModel()
Description copied from interface: TermStructureModel
Return the associated analytic model, a collection of market date object like discount curve, forward curve and volatility surfaces.
Specified by:
getAnalyticModel in interface TermStructureModel
Returns:
The associated analytic model.
• #### getDiscountCurve

public DiscountCurve getDiscountCurve()
Description copied from interface: TermStructureModel
Return the discount curve associated the forwards.
Specified by:
getDiscountCurve in interface TermStructureModel
Returns:
the discount curve associated the forwards.
• #### getForwardRateCurve

public ForwardCurve getForwardRateCurve()
Description copied from interface: TermStructureModel
Return the initial forward rate curve.
Specified by:
getForwardRateCurve in interface TermStructureModel
Returns:
the forward rate curve
• #### getCloneWithModifiedData

public LIBORMarketModel getCloneWithModifiedData​(Map<String,​Object> dataModified)
Description copied from interface: LIBORModel
Create a new object implementing LIBORModel, using the new data.
Specified by:
getCloneWithModifiedData in interface LIBORModel
Specified by:
getCloneWithModifiedData in interface ProcessModel
Specified by:
getCloneWithModifiedData in interface TermStructureModel
Parameters:
dataModified - A map with values to be used in constructions (keys are identical to parameter names of the constructors).
Returns:
A new object implementing LIBORModel, using the new data.
• #### getShortRateConditionalVariance

public double getShortRateConditionalVariance​(double time,
double maturity)
Calculates the variance $$\mathop{Var}(r(t) \vert r(s) )$$, that is $$\int_{s}^{t} \sigma^{2}(\tau) \exp(-2 \cdot \int_{\tau}^{t} a(u) \mathrm{d}u ) \ \mathrm{d}\tau$$ where $$a$$ is the meanReversion and $$\sigma$$ is the short rate instantaneous volatility.
Parameters:
time - The parameter s in $$\int_{s}^{t} \sigma^{2}(\tau) \exp(-2 \cdot \int_{\tau}^{t} a(u) \mathrm{d}u ) \ \mathrm{d}\tau$$
maturity - The parameter t in $$\int_{s}^{t} \sigma^{2}(\tau) \exp(-2 \cdot \int_{\tau}^{t} a(u) \mathrm{d}u ) \ \mathrm{d}\tau$$
Returns:
The conditional variance of the short rate, $$\mathop{Var}(r(t) \vert r(s) )$$.
• #### getIntegratedBondSquaredVolatility

public double getIntegratedBondSquaredVolatility​(double time,
double maturity)
• #### getModelParameters

public Map<String,​RandomVariable> getModelParameters()
Description copied from interface: IndependentModelParameterProvider
Returns a map of independent model parameters of this model.
Specified by:
getModelParameters in interface IndependentModelParameterProvider
Returns:
Map of independent model parameters of this model.