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HullWhite1F

Create Hull-White one-factor model

Description

The Hull-White one-factor model is specified using the zero curve, alpha, and sigma parameters.

Specifically, the HullWhite1F model is defined using the following equation:

dr=[θ(t)a(t)r]dt+σ(t)dW

where

dr is the change in the short-term interest rate over a small interval.

r is the short-term interest rate.

Θ(t) is a function of time determining the average direction in which r moves, chosen such that movements in r are consistent with today's zero coupon yield curve.

α is the mean reversion rate.

dt is a small change in time.

σ is the annual standard deviation of the short rate.

W is the Brownian motion.

Creation

Description

example

HW1F = HullWhite1F(ZeroCurve,Alpha,Sigma) creates a HullWhite1F (HW1F) object using the required arguments to set the Properties.

Properties

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Zero curve, specified as an output from IRDataCurve or a RateSpec that is obtained from intenvset. This is the zero curve used to evolve the path of future interest rates.

Data Types: object | struct

Mean reversion, specified either as a scalar or function handle which takes time as input and returns a scalar mean reversion value.

Data Types: double

Volatility, specified either as a scalar or function handle which takes time as input and returns a scalar mean volatility.

Data Types: double

Object Functions

simTermStructsSimulate term structures for Hull-White one-factor model

Examples

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Create a Hull-White one-factor model using an IRDataCurve.

Settle = datetime(2007,12,15);
CurveTimes = [1:5 7 10 20]';
ZeroRates = [.01 .018 .024 .029 .033 .034 .035 .034]';
CurveDates = daysadd(Settle,360*CurveTimes,1);
 
irdc = IRDataCurve('Zero',Settle,CurveDates,ZeroRates);
    
alpha = .1;
sigma = .01;
 
HW1F = HullWhite1F(irdc,alpha,sigma)
HW1F = 
  HullWhite1F with properties:

    ZeroCurve: [1x1 IRDataCurve]
        Alpha: @(t,V)inAlpha
        Sigma: @(t,V)inSigma

Use the simTermStructs method with the HullWhite1F model to simulate term structures.

SimPaths = simTermStructs(HW1F, 10,'nTrials',100);

Create a Hull-White one-factor model using a RateSpec.

Settle = datetime(2007,12,15);
CurveTimes = [1:5 7 10 20]';
ZeroRates = [.01 .018 .024 .029 .033 .034 .035 .034]';
CurveDates = daysadd(Settle,360*CurveTimes,1);
 
RateSpec = intenvset('Rates',ZeroRates,'EndDates',CurveDates,'StartDate',Settle);

alpha = .1;
sigma = .01;
 
HW1F = HullWhite1F(RateSpec,alpha,sigma)
HW1F = 
  HullWhite1F with properties:

    ZeroCurve: [1x1 IRDataCurve]
        Alpha: @(t,V)inAlpha
        Sigma: @(t,V)inSigma

Use the simTermStructs method with the HullWhite1F model to simulate term structures.

SimPaths = simTermStructs(HW1F, 10,'nTrials',100);

Define the zero curve data.

Settle = datetime(2016,4,4);
ZeroTimes = [3/12 6/12 1 5 7 10 20 30]';
ZeroRates = [0.033 0.034 0.035 0.040 0.042 0.044 0.048 0.0475]';
ZeroDates = datemnth(Settle,ZeroTimes*12);
RateSpec = intenvset('StartDates', Settle,'EndDates', ZeroDates, 'Rates', ZeroRates)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 2
             Disc: [8x1 double]
            Rates: [8x1 double]
         EndTimes: [8x1 double]
       StartTimes: [8x1 double]
         EndDates: [8x1 double]
       StartDates: 736424
    ValuationDate: 736424
            Basis: 0
     EndMonthRule: 1

Define the bond parameters.

Maturity = datemnth(Settle,12*5);
CouponRate = 0;

Define the Hull-White parameters.

alpha = .1;
sigma = .01;
HW1F = HullWhite1F(RateSpec,alpha,sigma)
HW1F = 
  HullWhite1F with properties:

    ZeroCurve: [1x1 IRDataCurve]
        Alpha: @(t,V)inAlpha
        Sigma: @(t,V)inSigma

Define the simulation parameters.

nTrials = 100;
nPeriods = 12*5;
deltaTime = 1/12;
SimZeroCurvePaths = simTermStructs(HW1F, nPeriods,'nTrials',nTrials,'deltaTime',deltaTime);
SimDates = datemnth(Settle,1:nPeriods);

Preallocate and initialize for the simulation.

SimBondPrice = zeros(nPeriods+1,nTrials);
SimBondPrice(1,:,:) = bondbyzero(RateSpec,CouponRate,Settle,Maturity);
SimBondPrice(end,:,:) = 100;

Compute the bond values for each simulation date and path, note that you can vectorize over the trial dimension.

for periodidx=1:nPeriods-1
    simRateSpec = intenvset('StartDate',SimDates(periodidx),'EndDates',...
        datemnth(SimDates(periodidx),ZeroTimes*12),'Rates',squeeze(SimZeroCurvePaths(periodidx+1,:,:)));
    SimBondPrice(periodidx+1,:) = bondbyzero(simRateSpec,CouponRate,SimDates(periodidx),Maturity);
end

plot([Settle SimDates],SimBondPrice)
datetick
ylabel('Bond Price')
xlabel('Simulation Dates')
title('Simulated Bond Price')

Figure contains an axes object. The axes object with title Simulated Bond Price, xlabel Simulation Dates, ylabel Bond Price contains 100 objects of type line.

Define the zero curve data.

Settle = datetime(2016,4,4);
ZeroTimes = [3/12 6/12 1 5 7 10 20 30]';
ZeroRates = [-0.01 -0.009 -0.0075 -0.003 -0.002 -0.001 0.002 0.0075]';
ZeroDates = datemnth(Settle,ZeroTimes*12);
RateSpec = intenvset('StartDates', Settle,'EndDates', ZeroDates, 'Rates', ZeroRates)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: 2
             Disc: [8x1 double]
            Rates: [8x1 double]
         EndTimes: [8x1 double]
       StartTimes: [8x1 double]
         EndDates: [8x1 double]
       StartDates: 736424
    ValuationDate: 736424
            Basis: 0
     EndMonthRule: 1

Define the bond parameters for the five bonds in the portfolio.

Maturity = datemnth(Settle,12*5);  % All bonds have the same maturity
CouponRate = [0.035;0.04;0.02;0.015;0.042];  % Different coupon rates for the bonds
nBonds = length(CouponRate);

Define the Hull-White parameters.

alpha = .1;
sigma = .01;
HW1F = HullWhite1F(RateSpec,alpha,sigma)
HW1F = 
  HullWhite1F with properties:

    ZeroCurve: [1x1 IRDataCurve]
        Alpha: @(t,V)inAlpha
        Sigma: @(t,V)inSigma

Define the simulation parameters.

nTrials = 1000;
nPeriods = 12*5;
deltaTime = 1/12;
SimZeroCurvePaths = simTermStructs(HW1F, nPeriods,'nTrials',nTrials,'deltaTime',deltaTime);
SimDates = datemnth(Settle,1:nPeriods);

Preallocate and initialize for the simulation.

SimBondPrice = zeros(nPeriods+1,nBonds,nTrials);
SimBondPrice(1,:,:) = repmat(bondbyzero(RateSpec,CouponRate,Settle,Maturity)',[1 1 nTrials]);
SimBondPrice(end,:,:) = 100;

[BondCF,BondCFDates,~,CFlowFlags] = cfamounts(CouponRate,Settle,Maturity);
BondCF(CFlowFlags == 4) = BondCF(CFlowFlags == 4) - 100;
SimBondCF = zeros(nPeriods+1,nBonds,nTrials);

Compute bond values for each simulation date and path. Note that you can vectorize over the trial dimension.

for periodidx=1:nPeriods
    if periodidx < nPeriods
        simRateSpec = intenvset('StartDate',SimDates(periodidx),'EndDates',...
            datemnth(SimDates(periodidx),ZeroTimes*12),'Rates',squeeze(SimZeroCurvePaths(periodidx+1,:,:)));
        SimBondPrice(periodidx+1,:,:) = bondbyzero(simRateSpec,CouponRate,SimDates(periodidx),Maturity);
    end
    
    simidx = SimDates(periodidx) == BondCFDates;
    SimCF = zeros(1,nBonds);
    SimCF(any(simidx,2)) = BondCF(simidx);
    ReinvestRate = 1 + SimZeroCurvePaths(periodidx+1,1,:);
    SimBondCF(periodidx+1,:,:) = bsxfun(@times,bsxfun(@plus,SimBondCF(periodidx,:,:),SimCF),ReinvestRate);
end

Compute the total return series.

TotalCF = SimBondPrice + SimBondCF;

Assume the bond portfolio is equally weighted and plot the simulated bond portfolio returns.

TotalCF = squeeze(sum(TotalCF,2));

TotRetSeries = bsxfun(@rdivide,TotalCF(2:end,:),TotalCF(1,:)) - 1;
plot(SimDates,TotRetSeries)
datetick
ylabel('Bond Portfolio Returns')
xlabel('Simulation Dates')
title('Simulated Bond Portfolio Returns')

Figure contains an axes object. The axes object with title Simulated Bond Portfolio Returns, xlabel Simulation Dates, ylabel Bond Portfolio Returns contains 1000 objects of type line.

More About

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References

[1] Brigo, D. and F. Mercurio. Interest Rate Models - Theory and Practice. Springer Finance, 2006.

[2] Hull, J. Options, Futures, and Other Derivatives. Prentice-Hall, 2011.

Version History

Introduced in R2013a