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D. Sevcovic, January 29, 2007 CALIBRATION OF INTEREST RATE MODELS - p. 1/40

A TWO PHASE METHOD FOR CALIBRATIONOF INTEREST RATE MODELS AND ITS

APPLICATION TO CENTRAL EUROPEANINTERBANK OFFER RATES

The first Bratislava-Vienna economic meetingMay 7, 2004 Bratislava

D. Sevcovic and A. Urbanova CsajkovaInstitute of Appl. Mathematics, Comenius University, 842 48 Bratislava, Slovak Republic

http://www.iam.fmph.uniba.sk/institute/sevcovic

l Term structuresl Interest rate modelsl Calibration of Interest rate modelsl Historical overviewl Contentl CIR interest rate modell Calibration methodl Nonlinear regressionl Numerical methodl Restricted maximum likelihoodl Nonlinear R2 ratiol Calibration resultsl Term structures descriptionl Calibration resultsl Conclusionsl Appendix


Term structures

What are Term structureand Interest rate models?The term structure is a functional dependence between thetime to maturity of a discount bond and its present price.



Term structures

Time evolution of Czech interbank offer rates (PRIBOR). Shortrate, bonds with 1 and 10 year maturities.

Time evolution of Slovak interbank offer rates (BRIBOR).Highly volatile short rate and rates of bonds with 1-yearmaturities.



Term structures

50 100 150 200 2500.052

0.054

0.056

0.058

0.06

50 100 150 200 2500.05

0.052

0.054

0.056

0.058

50 100 150 200 250

0.058

0.059

0.06

0.061

0.062

0.063

0.064

50 100 150 200 2500.054

0.056

0.058

0.06

0.062

50 100 150 200 250

0.05

0.055

0.06

0.065

0.07

0.075

0.08

50 100 150 200 250

0.055

0.06

0.065

0.07

Slovak interbank offer rates. Short rate, 1-week, 1-month,3-months, 6-months and 1-year interest rates on bonds. Bondswith larger maturities are usually less volatile compared toshorter ones.



Term structures

1 2 3 4 5

0.042

0.044

0.046

0.048

0.05

0.052

0.054

A typical shape of the term structure in some time. Bonds withlarger maturities often have higher interest rates.



Term structures

0.2 0.4 0.6 0.8 1

0.058

0.059

0.06

0.061

0.062

0.2 0.4 0.6 0.8 1

0.05

0.052

0.054

0.056

0.058

0.2 0.4 0.6 0.8 1

0.052

0.054

0.056

0.058

0.06

0.2 0.4 0.6 0.8 1

0.055

0.06

0.065

0.07

0.075

Real interbank offer rates. Bonds with larger maturities neednot have higher interest rates. (Previous three plots arecourtesy of Stehlikova, 2004).



Interest rate models

What are Interest rate models?Interest rate models describe the bond prices (or yields) as afunction of time to maturity, state variables like e.g.instantaneous interest rate as well as several modelparameters.



Interest rate models

Variety of models availablen Vasicek (1977)n Cox, Ingersoll, and Ross (1985)n Ho and Lee (1986)n Brennan and Schwartz (1982)n Hull and White (1990)

and many others ...



Calibration of Interest rate models

Why to calibrate Interest ratemodels?n estimated model parameters provide better insight in

analyzed term structuresn enable us to perform qualitative analysis of data and explain

some unobserved phenomena like e.g. identification ofmarket price of risk, long term interest rates etc



Historical overview

n Cox, Ingersoll, and Ross (1985) derived the interest ratemodel from basic assumption made on the form of theinstantaneous (short rate) interest rate.

n Brown and Dybvig (1986) applied the CIR model in order tointerpret nominal bond prices

n Chan, Karolyi, Longstaff and Sanders (1992) applied theGeneralized Method of Moments to identify modelparameters

n Pearson and Sun (1994) have shown failure of calibration ofa two-factor extension of the CIR model applied to portfoliosof US Treasury bills

n Nowman, Saltoglu (2003) applied Gaussian estimationmethods of Nowman (1997) to continuous time interest ratemodels



Historical overview

n Takahashi and Sato (2001) developed new methodology forestimation of general class of term structure models basedon a Monte Carlo filtering approach. The method wasapplied to LIBORs and interest rates swaps in the Japanesemarket

n Modern calibration methods are based on other interest ratederivatives like e.g. prices of caps and floors have beenproposed by Rebonato (1999) Such derivatives are still notavailable in most transitional countries

n Vojtek (2003) applied the Brace, Gatarek, Musiela model(1997) and various types of GARCH models in order tocalibrate interest rate models for term structures transitionaleconomies like Central European countries including Czechrepublic, Slovakia, Poland and Hungary



Content

n Cox, Ingersoll, and Ross interest model (CIR)n New two phase minmax calibration method based on CIR

modeln Evolution strategies used as optimization methodsn Calibration of Central European Interbank offer rates and

comparison to stable Western Europe term structuresn Conclusions



CIR interest rate model

The Cox-Ingersoll-Ross interest rate model is derived from abasic assumption made on the form of the stochastic processdriving the instantaneous interest rate rt, t [0, T ].

drt = ( rt)dt+ rtdwtwhere {wt, t 0}, denotes the standard Wiener process.Positive constants , and denote the adjustment speed ofreversion, the long term interest rate and volatility factor of theprocess, respectively.




0 5 10 15 20time

3.963.98

44.024.044.064.08

r

H

%

L

Simulation of a mean reverting process driven by theOrstein-Uhlenbeck mean-reverting process with parameters: = 1, = 0.2, = 0.04 within the time interval (0, 20).Time discretized version of the mean reverting equation hasthe form:

rt+1 rt = ( rt) + rttwith t N(0, 1) is normally distributed random variable.




In the CIR theory the price P = P (t, T, r) of a zero couponbond is assumed to be a function of the present time t [0, T ],expiration time T > 0 and the present value of the short rateinterest rate r = rt.

P = eR(t,T )(Tt)

where R(t, T ) is the yield on the bond at time t with maturity T ,rt = R(t, t).




Constructing a risk-less portfolio containing two bonds withdifferent maturities and applying stochastic differential calculus(It lemma) we can conclude that the price of the zero couponbond P = P (t, T, r) satisfies a backward parabolic partialdifferential equation

P

t+((r)r)P

r+1

22r

2P

r2rP = 0 , t (0, T ) , r > 0 .

subject to the terminal condition P (T, T, r) = 1 for any r > 0.

The parameter R represents the so-called market price ofrisk.




CIR model has an explicitsolution!

P (T , T, r) = A()eB()r, = T t [0, T ] ,

where

B() =2(e 1)

(+ + )(e 1) + 2 , A() =(e(++)/2

e 1 B()) 2

2

,

and =

(+ )2 + 22.

Four parameters , , , enter analytic formula




As it has been already pointed by Pearson and Sun (1994) theadjustment speed and the risk premium enter the analyticformula only in summation

+

four CIR parameters can be reduced to three essentialparameters fully describing the behavior of the functions A,Band thus the whole term structure




Three parameters reductionby introducing the following new parameters:

1. = e [0, 1]2. = ++2 [0, 1]3. = 22 R

where =

(+ )2 + 22.

Summarizing, in the CIR model the price of bonds and,consequently, the corresponding yield curve depend only onthree transformed parameters , and .



Calibration method

Two-phase minmax method1. we identify one dimensional curve of the CIR parameters by

minimizing the cost functional that mimics least squaresapproach in linear regression methods. The global minimumis attained at a unique point , , of transformed vars on a one dimensional curve of genuine CIR parameters, , ,

2. maximization of the likelihood function restricted to that onedimensional curve of global minimizers

The restricted maximum likelihood is attained in a unique point- desired estimation of the CIR model parameters.



Nonlinear regression

Nonlinear least squareminimizationWe minimize the weighted least square sum of differencesbetween real market yield curve interest rates and thosepredicted by CIR model.

U(, , ) =1

m

mj=1

1

n

ni=1

(Rij Rij)22j

n {Rij , j = 1, ...,m} is the yield curve of the length m at timei = 1, ..., n, with time j to maturity.

n {Rij , j = 1, ...,m} is the yield curve computed by using theCIR model with parameters , , .

n The over-night (short-rate) interest rate at time i = 1, ..., n, isdenoted by Ri0.




Nonlinear least squareminimizationvalues Rij can be calculated from the bond price - yield curverelationship:

AjeBjR

i0 = P = eR

ijj .

Rijj = BjRi0 lnAj and the cost functional U can berewritten as

U(, , ) =1

m

mj=1

((jE(Rj)BjE(R0) + lnAj)2 +D(jRj BjR0)




Nonlinear least squareminimizationthe cost function U attains a unique global minimum

U(, , ) = min(,,)

U(, , ) .

In terms of genuine CIR parameters:

(, , , ) (, , )the global minimum is attained on a -parameterized curve ofglobal minimizers (, , , ), R.



Numerical method

Nonlinear regression problem!n the function U = U(, , ) is nonlinear and it is not know

whether is convex or notn standard steepest descent gradient minimization method of

Newton-Kantorovich type fail to converge to a globalminimum of U

n we need a robust and efficient numerical methodunconditionally converging to a unique global minimum of U

we had to apple a variant of the evolution strategy (ES)globally converging algorithmEvolution strategy (ES) is the one of the most successfulstochastic algorithm which was invented to solve technicaloptimization problems (see e.g. Schwefel (1977, 1995, 1998).



Restricted maximum likelihood

Restricted likelihood functionGaussian estimation of parameters of the mean revertingprocess driving the short rate rt

lnL(, , ) = 12

nt=2

(ln v2t +

2tv2t

)

v2t =2

2

(1 e2) rt1, t = rt ert1 (1 e).

Combining information gained from minimization of the costfunctional U and likelihood parameter estimation restrictedmaximum likelihood function maximize lnL over a onedimensional curve (, , , ), R.




Restricted likelihood functionThe argument (, , ) of the restricted likelihood function

lnLr = lnL(, , ) = max

lnL(, , ) .

is now identified the resulting optimal values as estimated CIRparameters obtained by the two-phase minmax optimizationmethod.




Restricted maximum likelihoodratioThe ratio between the unrestricted maximum likelihood

lnLu = lnL(u, u, u) = max,,>0

lnL(, , ) .

and the restricted maximum likelihood lnLr is called themaximum likelihood ratio (MLR)

MLR =lnLr

lnLu.

n MLR 1n value of MLR close to 1 indicates that the restricted

maximum likelihood value lnLr is close to unrestricted one simple estimation of parameters of mean reversionequation is also suitable for calibration of the whole termstructure



Nonlinear R2 ratio

We introduce the nonlinear R2 ratio as follows:

R2 = 1 U(, , )U(1, 1, 1)

where (, , ) is the argument of the unique global minimum ofthe cost functional Un 0 R2 1n a value of R2 close to one indicate that U(, , ) U(1, 1, 1) the cost functional U is close to zero almost perfect matching of the CIR yield curve computedfor parameters (, , ) and that of the given real market dataset.



Calibration results

We present results of the two-phase calibration for termsstructures for various Central European countries andcomparison to stable western European financial markets.

n EURIBOR - Euro-zonen London inter-bank offer rates EURO-LIBOR and USD-LIBORn Slovakia (BRIBOR),n Hungary (BUBOR)n Czech republic (PRIBOR)n Poland (WIBOR)



Term structures description

n The data for overnight interest rate are available for BRIBOR,PRIBOR, WIBOR, BUBOR as well as for London Inter-bankOffer Rates (LIBOR).

n For the Euro-zone term structure EURIBOR is concerned werecall that there exists a substitute for the overnight ratewhich is called EONIA.

Calibration covers the period from the year 2001 up to 2003and is done on quarterly basis




n EURO-LIBOR and USD-LIBOR contains bonds withmaturities: 1 week, 1 up to 12 month, i.e. its length ism = 13.

n EURIBOR, in addition, contains 2 and 3 weeks maturities,m = 15.

n BRIBOR contains 1, 2 weeks and 1, 2, 3, 6, 9, 12 monthsmaturities

n BUBOR, PRIBOR, and WIBOR contain maturities: 1 and 2weeks, 1, 2, 3, 6, 9, 12 months, and then each quarter up to10 years maturity




Graphical description of overnight (short-rate) interest ratesand those of bond with longer maturity. Daily data are plottedfor EURO-LIBOR, BRIBOR and PRIBOR.



Calibration results

Results of calibration for term structures. Maximum likelihoodratio (a) and R2 ratio (b) for EURO-LIBOR and USD-LIBOR.



Calibration results

Maximum likelihood ratio and R2 ratio for EURO-LIBOR andcomparison to BRIBOR, PRIBOR and WIBOR. Estimatedparameters and (below).



Calibration results

Results of calibration for term structures with maturities up to10 years for BRIBOR, WIBOR and BUBOR.



Calibration results

Risk premium analysisRecall that the parameter represents the market price of riskin the CIR model

n The bond price P = P (t, T, r) satisfying PDE is given byformula P (t, T, r) = A(T t)eB(Tt)r. Thus rP = BP .

n Partial differential equation can be rewritten as

P

t+ ( r)P

r+

1

22r

2P

r2 rP = 0 , t (0, T ) , r > 0

where r = (1 B)r.n Multiplier 1 B can interpreted as the risk premium factor

and r as the expected rate of return on the bond.n We have r > r iff < 0



Calibration results

Comparison of risk premium factors 1 B1 for EURO-LIBORand BRIBOR (left) and EURO-LIBOR, WIBOR and BRIBOR(right).



Conclusions

n We have proposed a two-phase minmax optimizationmethod for calibration of the CIR one factor interest ratemodel. It is based on minimization of the cost functionaltogether with maximization of the likelihood functionrestricted to the set of minimizers.

n We have tested the calibration method various termstructures including both stable western inter-bank offerrates as well as those of emerging economies.

n Based on our results of calibration of the CIR one-factormodel we can state that the western European termstructure data are better described with CIR modelcompared to transitional economies represented by CentralEuropean countries



Conclusions

n The CIR model can applied to EURO-LIBOR, USD-LIBORand EURIBOR. Interestingly, to some extent, it could be alsoused for Czech PRIBOR term structure

n On the other hand, we can observe, at least partial, failure ofthe CIR model for other Central European term structures.

The paper as well as presentation and other material can bedownloaded from:

http://www.iam.fmph.uniba.sk/institute/sevcovic



Appendix

Term structure descriptive statistics

Tabeled calibration results

Term structuresTerm structuresTerm structuresTerm structuresTerm structuresInterest rate modelsInterest rate modelsCalibration of Interest rate modelsHistorical overviewHistorical overviewContentCIR interest rate modelCIR interest rate modelCIR interest rate modelCIR interest rate modelCIR interest rate modelCIR interest rate modelCIR interest rate modelCalibration methodNonlinear regressionNonlinear regressionNonlinear regressionNumerical methodRestricted maximum likelihoodRestricted maximum likelihoodRestricted maximum likelihoodNonlinear R2 ratioCalibration resultsTerm structures descriptionTerm structures descriptionTerm structures descriptionCalibration resultsCalibration resultsCalibration resultsCalibration resultsCalibration resultsConclusionsConclusionsAppendix

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