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net.finmath.montecarlo.interestrate

## Class HullWhiteModel

• All Implemented Interfaces:
IndependentModelParameterProvider, HullWhiteModelInterface, LIBORModelInterface, TermStructureModelInterface, AbstractModelInterface

public class HullWhiteModel
extends AbstractModel
implements HullWhiteModelInterface, LIBORModelInterface
Implements a Hull-White model with time dependent mean reversion speed and time dependent short rate volatility.

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 joint distribution of $$(r,N)$$, where $$r$$ denotes the short rate and $$N$$ denotes the numeraire, following the scheme in ssrn.com/abstract=2737091. 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) (see ssrn.com/abstract=2737091 for details on the derivation). 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 ShortRateVolatilityModelInterface.
Version:
1.4
Author:
Christian Fries
See Also:
ShortRateVolatilityModelInterface, ssrn.com/abstract=2737091
• ### Constructor Detail

• #### HullWhiteModel

public HullWhiteModel(AbstractRandomVariableFactory randomVariableFactory,
TimeDiscretizationInterface liborPeriodDiscretization,
AnalyticModelInterface analyticModel,
ForwardCurveInterface forwardRateCurve,
DiscountCurveInterface discountCurve,
ShortRateVolatilityModelInterface volatilityModel,
Map<String,?> properties)
Creates a Hull-White model which implements LIBORMarketModelInterface.
Parameters:
randomVariableFactory - The factory to be used to construct random variables.
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).
• #### HullWhiteModel

public HullWhiteModel(TimeDiscretizationInterface liborPeriodDiscretization,
AnalyticModelInterface analyticModel,
ForwardCurveInterface forwardRateCurve,
DiscountCurveInterface discountCurve,
ShortRateVolatilityModelInterface volatilityModel,
Map<String,?> properties)
Creates a Hull-White model which implements LIBORMarketModelInterface.
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

• #### getReferenceDate

public LocalDateTime getReferenceDate()
Description copied from interface: AbstractModelInterface
Returns the model's date corresponding to the time discretization's $$t = 0$$. Note: Currently not all models provide a reference date. This will change in future versions.
Specified by:
getReferenceDate in interface AbstractModelInterface
Overrides:
getReferenceDate in class AbstractModel
Returns:
The model's date corresponding to the time discretization's $$t = 0$$.
• #### applyStateSpaceTransform

public RandomVariableInterface applyStateSpaceTransform(int componentIndex,
RandomVariableInterface randomVariable)
Description copied from interface: AbstractModelInterface
Applies the state space transform fi to the given state random variable such that Yi → fi(Yi) =: Xi.
Specified by:
applyStateSpaceTransform in interface AbstractModelInterface
Parameters:
componentIndex - The component index i.
randomVariable - The state random variable Yi.
Returns:
New random variable holding the result of the state space transformation.
• #### getInitialState

public RandomVariableInterface[] getInitialState()
Description copied from interface: AbstractModelInterface
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 AbstractModelInterface
Returns:
The initial value of the state variable of the process Y(t=0).
• #### getNumeraire

public RandomVariableInterface getNumeraire(double time)
throws CalculationException
Description copied from interface: AbstractModelInterface
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 AbstractModelInterface
Parameters:
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 RandomVariableInterface[] getDrift(int timeIndex,
RandomVariableInterface[] realizationAtTimeIndex,
RandomVariableInterface[] realizationPredictor)
Description copied from interface: AbstractModelInterface
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 AbstractModelInterface
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 RandomVariableInterface[] getFactorLoading(int timeIndex,
int componentIndex,
RandomVariableInterface[] realizationAtTimeIndex)
Description copied from interface: AbstractModelInterface
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 AbstractModelInterface
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.
• #### getNumberOfLibors

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

public double getLiborPeriod(int timeIndex)
Description copied from interface: LIBORModelInterface
The period start corresponding to a given forward rate discretization index.
Specified by:
getLiborPeriod in interface LIBORModelInterface
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: LIBORModelInterface
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 LIBORModelInterface
Parameters:
time - The period start.
Returns:
The index corresponding to a given time (interpretation is start of period)
• #### 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)
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Copyright © 2018 Christian P. Fries.

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