179179

ISSN 1518-3548

Are Interest Rate Options Important for the Assessment ofInterest Rate Risk?

Caio Almeida and José Vicente

December, 2008

Working Paper Series

ISSN 1518-3548

CGC 00.038.166/0001-05

Working Paper Series Brasília n. 179 Dec. 2008 p. 1-46

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Are Interest Rate Options Important for the

Assessment of Interest Rate Risk? �

Caio Almeiday Jos�e Vicente z

The Working Papers should not be reported as representing the viewsof the Banco Central do Brasil. The views expressed in the papers arethose of the author(s) and not necessarily re ect those of the BancoCentral do Brasil.

Abstract

Fixed income options contain substantial information on the price of in-

terest rate volatility risk. In this paper, we ask if those options will provide

information related to other moments of the objective distribution of interest

rates. Based on a dynamic term structure model, we �nd that interest rate op-

tions are useful for the identi�cation of interest rate quantiles. A three-factor

model with stochastic volatility is adopted and its adequacy to estimate Value

at Risk of zero coupon bonds is tested. We �nd signi�cant di�erence on the

quantitative assessment of risk when options are (or not) included in the esti-

mation process of the dynamic model. Statistical backtests indicate that bond

estimated risk is clearly more adequate when options are adopted, although

not yet completely satisfactory.

Keywords: Dynamic term structure models, Value at risk, Back-testing pro-

cedures, Feller processes.

JEL: C13, G12, G13.

�We thank comments and suggestions from seminar participants at the Sixth Brazilian Meetingof Finance, and XXVIII Brazilian Meeting of Econometrics. The �rst author acknowledges supportfrom CNPq-Brazil.

yGraduate School of Economics, Getulio Vargas Foundation. e-mail: calmeida@fgv.brzResearch Department, Central Bank of Brazil. e-mail: jose.valentim@bcb.gov.br

3

1 Introduction

It is undeniable that options contain invaluable information on investors preferencesand beliefs. Since the work of Breeden and Litzenberger (1978), a large numberof researchers have used options to extract Arrow-Debreu prices for a variety ofapplications in �nancial economics (see for instance, Rubinstein (1994), Jackwerthand Rubinstein (1996), Ait Sahalia and Lo (1998, 2000), and Jackwerth (2000)).

More recently, some authors have adopted joint datasets of options and underly-ing assets to better identify risk premium factors in dynamic arbitrage-free models1.In such cases, the non-linear richer structure of option payo�s helps to econometri-cally identify the risk neutral dynamics of underlying assets. A better identi�cationof risk neutral dynamics translates into more precise risk premia dynamics, suggest-ing that options data in the estimation of a dynamic model decrease the chances ofmispricing risks.

Motivated by the idea above, in this paper, we analyze if interest rate optionswill capture more general probabilistic properties of the objective distribution ofinterest rates other than its conditional mean. In particular, we ask how helpfulinterest rate options are to determine the quantiles of the distribution ofinterest rates. Knowledge of these quantities can be specially important to assessinterest rate risk, implying that our empirical results might be of importance to riskmanagers and portfolio managers.

In order to perform our tests, based on a dataset of bonds and interest rateoptions, we estimate two versions of a three-factor Cox et al. (1985) model (here-after CIR model). The �rst adopts only bonds data (bond version), while the sec-ond includes bonds and options data (option version) on the estimation process.The multi-factor CIR model was chosen due to its ability to generate conditionalprobabilities with strong potential to depart from normality (non-central chi-squaretransition probability). In addition, the stochastic volatility implied by the threedynamic factors increases model ability to react to changes in conditional volatilitiesof interest rates.

We adopt Value at Risk (VaR) as a metric to analyze the dynamic term structuremodel2. VaR for di�erent con�dence levels and time-frequencies is estimated. Its

1This is, for instance, the case of Pan (2002) and Eraker (2004), who estimate stochastic volatil-ity jump-di�usion processes based on joint options and equities data to analyze jump and volatilityrisk premium. See also Chernov and Ghysels (2000) and Johanes et al. (2007) for other papersbased on equities data. On a �xed income context, Almeida et al. (2006) show that a�ne mod-els estimated with joint bonds and interest rate options better predict excess bond returns thanmodels estimated based on only bonds data. Almeida and Vicente (2006) and Joslin (2007) adoptjoint datasets of �xed income options and interest rates to correctly identify interest rate volatilityrisk premium.

2Since the Bank of International Settlements (BIS) accord proposed in 1996, regulated banks arerequired to report VaR and also to control their levels of capital based on VaR. In addition, most�nancial institutions maintain a daily VaR control on their portfolios (see Du�e and Pan (2001)).However, despite being a very popular risk measure adopted in the �nancial industry, VaR is not acoherent measure of risk (see Artzner et al. (1999)), which means not preserving subadditivity. Onthe other hand, in a recent paper Garcia et al. (2007) show that proper conditioning information

4

statistical accuracy is veri�ed based on tests that control the expected number ofviolations (Kupiec (1995) and Seiler (2006)) and tests for the independence of vi-olations (Christo�ersen (1998) and Ljung-Box (1978)). Considering the number ofVaR violations, the bond version clearly overestimates risk. On the other hand, theperformance of the option version is comparable to that of two popular benchmarks,namely, two historical simulations based on respectively 125 and 250 observations ina moving-window. In what regards independence tests, both versions of the modelperform well but with performances clearly attributed to distinct reasons. The op-tion version provides acceptable results when tested via Christo�ersen's (1998) orLjung-Box's (1978) statistics. On its turn, the bond version presents even betterresults on the very same tests, but to the extent that it generates an unaccept-ably small number of violations. Such small number of violations would, at �rst,invalidate its use as a risk management tool.

Results reported above indicate that interest rate options aggregate relevantinformation for the process of interest rate risk management. The implications herewill be important for both capital allocation and internal risk control purposes. Forinstance, overestimation of risk directly implies an excessive capital allocation ascollateral for trades. It may also mislead possible analysis of relative performanceof traders 3.

Papers related to our work include Ait Sahalia and Lo (2000), and Egorov etal. (2006). Ait Sahalia and Lo (2000) propose an alternative way to estimate VaR,what they call economic VaR, which consists in obtaining the quantiles of the riskneutral distribution of returns extracted from option prices. Our approach consistsin combining avors of traditional VaR and economic VaR, in the sense that weestimate VaR under the objective probability measure but making use of informationcoming from option prices to estimate the dynamic model. Egorov et al. (2006) testthe ability of a�ne models to forecast out-of-sample conditional probability densitiesof bond yields. Note that while we include options on the estimation process to testtheir importance on interest rate quantile forecasting, Egorov et al. (2006) adoptonly bonds data to estimate the dynamic a�ne models on their forecasting exercises.

The rest of the paper is organized as follows. Section 2 describes the datasetadopted. Section 3 presents a theoretical description of the dynamic term structuremodel implemented. Section 4 introduces the VaR measure and all statistical back-testing procedures adopted. On Section 5, the results on the implementation of theCIR models are detailed and a comparison between the two versions is performed.Section 6 o�ers concluding remarks.

may restore VaR subadditivity.3Consider when the VaR of a particular instrument is overestimated. In this case, risk managers

could incorrectly conclude that �xed income traders buying that product are violating requiredrisk limits. Relative performance of traders operating in distinct markets would also be biased incases like that.

5

2 Data and Market Description

We adopt data on bonds and interest rate options from one of the largest �xed in-come markets among all emerging economies, the Brazilian market. This particulardata can potentially o�er interesting insights due to at least two reasons. First,the payo� structure of its interest rate option is rather appropriate to analyze howvolatility expectations may a�ect bond risk estimation. In this market, a call op-tion gives the right to cap the accumulated short-term rate between its trade andexpiration date. Therefore, it represents a bet on the expectation of the short-ratepath directly determining a value for the expected volatility for this short-rate path4.A second reason comes from an econometric point: In emerging markets, volatil-ity changes due to macroeconomic and political events usually happen in a muchhigher frequency than in developed markets. Thus, using options data from emerg-ing economies we might better econometrically identify the transmission channelfrom volatility information to the probability distribution of interest rates.

In the next two subsections we describe the above mentioned market and itscorresponding interest rate instruments.

2.1 ID-Futures and Reference Bonds

The One-Day Inter Bank Deposit Future Contract (ID-Future) with maturity T is afuture contract whose underlying asset is the accumulated daily ID rate5 capitalizedbetween the trading time t (t � T ) and T . The contract size corresponds to R$100,000.00 (one hundred thousand Brazilian Reals) discounted by the accumulatedrate negotiated between the buyer and the seller of the contract.

This contract is very similar to a zero coupon bond, except that it pays marginadjustments every day. Each daily cash ow is the di�erence between the settlementprice6 on the current day and the settlement price on the day before corrected bythe ID rate of the day before.

The Brazilian Mercantile and Futures Exchange (BM&F) is the entity that o�ersthe ID-Future. The number of authorized contract-maturity months is �xed byBM&F (on average, there are about twenty authorized contract-maturity monthsfor each day but only around ten are liquid). Contract-maturity months are the �rstfour months subsequent to the month in which a trade has been made and, afterthat, the months that initiate each following quarter. Expiration date is the �rstbusiness day of the contract-maturity month.

Corresponding to each ID-future there is a reference bond. Reference bonds arezero coupon bonds whose yields are equal to the accumulated daily ID rate implicit

4It will work like a usual U.S. cap on a swap, except that this swap would have a unique cash ow, paid at the maturity of the option.

5The ID rate is the average one-day inter bank borrowing/lending rate, calculated by CETIP(Central of Custody and Financial Settlement of Securities) every workday. The ID rate is ex-pressed in e�ective rate per annum, based on 252 business-days.

6The settlement price at time t of an ID-Future with maturity T is equal to R$ 100,000.00discounted by its closing price quotation.

6

in the settlement price of a corresponding ID-future (with the same maturity). Wewill adopt these reference bonds in our study as shall be seen in Section 2.3.

2.2 ID Index and its Option Market

The ID index (IDI) is de�ned as the accumulated ID rate. If we associate thecontinuously-compounded ID rate to the short term rate rt then

IDIt = IDI0 � eR t0 rudu: (1)

This index, computed on every workday by BM&F, has been �xed to the value of100000 points in January 2, 1997, and has actually been resettled to its initial valuea couple of times, most recently in January 2, 2003.

An IDI option with time of maturity T is an European option where the un-derlying asset is the IDI and whose payo� depends on IDIT . When the strike isK, the payo� of an IDI option is Lc(T ) = (IDIT �K)+ for a call and Lp(T ) =(K � IDIT )

+ for a put.BM&F is also the entity that provides the IDI call options. Strike prices (ex-

pressed in index points) and the number of authorized contract-maturity monthsare established by BM&F. Contract-maturity months can happen to be any month,and the expiration date is the �rst business day of the maturity month. Usually,there are 30 authorized series within each day, from which about a third are liquid.

2.3 Data

Data consists on time series of yields of ID-Futures for all di�erent liquid maturities,and values of IDI options for di�erent strikes and maturities. The data covers theperiod from August 09, 2004 to August 06, 2007.

BM&F maintains a daily historical database with the price and number of tradesof every ID-Future and IDI option that have been traded in any day. According toSection 2.1, a reference bond is a zero coupon bond with the same maturity andsettlement price of an ID-future. We adopt reference bond yields with �xed timesto maturity of 1, 63, 126, 189, 252, and 378 days7. We also adopt an at-the-moneyIDI call with time to maturity of 95 days. The prices of these options were obtainedvia an interpolation based on Black's implied volatilities8.

After excluding weekends, holidays, and no-trade workdays, there is a total of741 yields for each bond, and prices for at-the-money IDI call options.

7With the ID-Future database and a time series of ID interest rates, it is straightforward toestimate, by cubic interpolation, reference bonds interest rates for �xed maturities, and for alltrading days.

8Our procedure is similar to that adopted to calculate VIX implied volatilities from S&P 500index options for hypothetical at-the-money short-maturity (21 days) options.

7

3 The Model

Uncertainty in the economy is characterized by a �ltered probability space (; (Ft)t�0 ;F ;P)where (Ft)t�0 is a �ltration generated by a standard N -dimensional Brownian mo-

tion W P =�W P

1 ; : : : ;WPN

�de�ned on (;F ;P) (see Du�e (2001)). We assume

the existence of a pricing measure Q under which discounted security prices aremartingales with respect to (Ft)t�0.

A multi-factor CIR model directly represents the short-term rate r(t):

rt = �0 +NXn=1

Xn(t); (2)

where �0 is a constant and the dynamics of Xn is given by

dXn(t) = �n (�n �Xn(t)) dt+ �npXn(t)dW

Qn (t); n = 1; : : : ; N; (3)

where WQ =�WQ

1 ; : : : ;WQN

�represents an N -dimensional Brownian motion under

Q, and �n, �n and �n represent positive constants satisfying Feller's condition that2�n�n > �2n for all n.

Compatible with the studies by Cox et al. (1985) and Dai and Singleton (2000),we assume a simple functional form for the market price of risk process �X(t) =��X1 (t); : : : ; �

XN(t)

�, proportional to the instantaneous volatilities of the latent vari-

ables9:

�Xn (t) =�n�n

pXn(t); n = 1; : : : ; N: (4)

The connection between martingale probability measure Q and objective prob-ability measure P is obtained via Girsanov's Theorem

dW Pn (t) = dWQ

n (t)� �Xn (t)dt: (5)

In fact, substituing (5) in (3) we obtain the dynamics of Xn under the objectiveprobability measure:

dXn(t) = �n��n �Xn(t)

�dt+ �n

pXn(t)dW

Pn (t); n = 1; : : : ; N; (6)

where �n = �n � �n and �n =�n�n�n��n

.The probability density of Xn at time T under Q, conditional on its value at the

current time t, is given by (see Brigo and Mercurio (2001)):

fXn(T )jXn(t)(x) = cnf�2(�n;qn)(cnx); (7)

9Recently Cheredito et al. (2007) have proposed a functional form for the market prices ofa�ne models that generate more general conditional expectations under the objective probabilitymeasure. However, as we are working with independent CIR models the gain with their speci�cationin principle would be small. In addition, by specifying a richer class of models we would be favoringthe option version since in this case, econometric identi�cation becomes harder.

8

where

cn =4�n

�2n (1� e��n(T�t));

�n =4�n�n�2n

;

qn = cnXn(t)e��n(T�t);

and f�2(�;q)(�) is the density function of a noncentral chi-square variable with �degrees of freedom and non-centrality parameter q.

3.1 Bond Prices

The proposed model is in the class of a�ne term structure models (Du�e and Kan(1996))10. The price of a zero coupon bond is derived in Cox et al. (1985) under thesingle-factor case and in Brigo and Mercurio (2001), under the multi-factor case11:

P (t; T ) = eA(t;T )+B(t;T )0Xt ; (8)

where

A(t; T ) = ��0� +NXn=1

2�n�n�2n

log

2 ne

(�n+ n)�2

2 n + (�n + n) (e� n � 1)

!; (9)

n =p�2n + 2�2n, B(t; T ) is a column vector with

Bn(t; T ) =2 (e� n � 1)

2 n + (�n + n) (e� n � 1); (10)

in the nth position, and P (t; T ) denotes the time t price of a zero coupon bondpaying one monetary unit at time T .

3.2 IDI Option Prices

An IDI option is an interest rate asian option12 whose payo� is a path-dependentfunction of the instantaneous short-term rate. Theoretical pricing of such optionswas provided by Longsta� (1995), Leblanc and Scaillet (1998), and Dassios andNagaradjasarma (2003) under single factor interest rate models, and by Chacko andDas (2002) under general a�ne models.

In this section, we propose an alternative way of pricing these options under ourmulti-factor CIR model, which consists in an e�cient implementation of the Laplace

10It is interesting to note that a�ne models have been frequently adopted in empirical �xedincome studies. See, for instance, Dai and Singleton (2000, 2002), and Du�ee (2002), among manyothers.

11See also Jagannathan et al. (2003).12See Shreve (2004) for the de�nition and examples of asian options.

9

transform method of Chacko and Das (2002) via a smart numerical procedure pro-posed by Abatt and Whitt (1995).

Denote by c(t; T ) the time t price of a call option on the IDI index, with time to

maturity T , and strike price K. Let Y (t; T ) =R Ttrudu. Then

c(t; T ) = EQt

�e�Y (t;T )Lc(T )

�= EQ

t

�eY (0;t) �Ke�Y (t;T )

�

= IDItEQt

��1� K

IDIT

�+�

=Ry�log K

IDIt

(IDIt �Ke�y) fY jFt(y)dy;

(11)

where fY jFt(y) is the probability density of Y (t; T )jFt. In order to obtain the pricec(t; T ) it is necessary that we know the conditional distribution of Y (t; T ) under Q.When rt is a Gaussian process Y jFt is normally distributed, and by using simpleproperties of normal distributions a closed-form formula for the option price canbe promptly obtained. Considering the case of a non-central �2 distribution wherethere is no closed-form formula for fY jFt (and neither for the option price), one canstill calculate the Laplace Transform of Y (t; T )jFt

L (Y jFt) (s) = e���NYn=1

L (YnjFt) (s); (12)

where L (YnjFt) (s) = EQt

�e�sYn

�is the Laplace Transform of Yn(t; T ) =

R TtXn(u)du

that is given by13

L (YnjFt) (s) =

"2~ ne

(�n+~ n)�2

2~ n + (�n + ~ n) (e�~ n � 1)

# 2�n�n�2n

� e2(e�~ n�1)

2~ n+(�n+~ n)(e�~ n�1) ;

with ~ n = ~ n(s) =p�2n + 2�2ns.

It should be clear at this point that searching for e�cient numerical proceduresfor inversions of Laplace transforms becomes central to solve our pricing problem.In this work, we decided to adopt and implement the Abate and Whitt (1995)method due to its precision under the CIR model and to its small processing-time,a fundamental factor for an e�cient implementation of our dynamic model14.

Observe that after inverting the Laplace transform (Equation (12)), in princi-ple we would need to numerically integrate (IDIt �Ke�y) fY jFt(y) on the interval

13The Laplace Transform of Yn(t; T ) is just the time t bond price maturing at T when short termrate follows a one-dimensional CIR process with parameters (�n; s�n;

ps�n) starting from srt.

14Nagaradjasarma (2003) reviews the main literature approach to the problem of pricing interestrate asian options with the CIR model.

10

hlog K

IDIt;+1

�(see Equation (11)). Nevertheless, Dassios and Nagaradjasarma

(2003) provide a useful lemma where they show that the call price can be computedin a single step:

Lemma 1 Considering � � 0, denote by G(s; �) the inverse Laplace transform of

EQt

�e�(�+s)Y (t;T )

�s

=L (Y jFt) (�+ s)

s

with respect to s > 0. Hence

c(t; T ) = KG

�log

K

IDIt; 1

�� IDItG

�log

K

IDIt; 0

��KP (t; T ) + IDIt: (13)

Proof: See Dassios and Nagaradjasarma (2003).We make direct use of their lemma to speed up our IDI option pricing procedure.

4 Measuring Market Risk

4.1 Value at Risk

It is fundamental for a �nancial institution to perform an active control of its port-folio risk. Among a number of risk measures available, Value at Risk (VaR) hasbecome the most in uential one15. Despite its documented drawbacks16 it is largelyadopted by �nancial institutions as a tool to control risk, and by regulatory institu-tions considering capital requirements to cover market risk.

VaR is usually de�ned as the loss that is exceeded with some given probability,�, over a given horizon. If we denote by Mt the value of a portfolio at time t, thenthis de�nition can be translated to

P [Mt+HP �Mt � V aRt;HP (�)j Ft] = �; � 2 [0; 1];

where V aRt;HP (�) is the portfolio's VaR at time t for the horizon HP with con�-dence level 1� �.

Certainly, the widespread use of VaR-based risk management models resultsfrom the fact that it is a risk metric easy to interpret. Note, however, that theestimation process of VaR when the probability distribution of Mt+HP is not knownin closed-form, usually constitutes a task demanding high computational time. Onthe other hand, whenever that probability distribution is known, this task can betremendously simpli�ed, as in the case of a normal Mt+HP :

15For an overview of VaR see Du�e and Pan (2001).16VaR is not a sub-additive measure of risk. For instance, Kerkhof and Melenberg (2004) show

that for regulatory purposes, expected shortfall, which is a coherent measure of risk (Artzner etal. (1999)), produces better results than VaR. On the other hand, Garcia et al. (2007) indicatethat using appropriate conditioning variables usually is enough to restore VaR sub-additivity.

11

V aRt;HP (�) =Mt � � (Mt+HP )� � (Mt+HP ) ��1(�);

where � (Mt+HP ) is the mean of Mt+HP , � (Mt+HP ) is the standard deviation ofMt+HP and �(�) is the normal cumulative distribution function.

Unfortunately under our model, the distribution ofMt+HP is not known in closed-form, and as shown bellow we will make use of simulation techniques to estimateVaR.

4.2 Our experiment with VaR

In order to test the in uence of option prices on the VaR estimation of �xed incomeinstruments, we compute the VaR for zero coupon bonds with times to maturity ofsix and twelve months, respectively. We evaluate the VaR for two di�erent horizons,1-day and 10-day, and for four di�erent con�dence levels, 90%, 95%, 97.5%, and99%, which are the VaR parameters usually adopted by market participants andregulatory institutions.

VaR risk measures are computed for each version of the dynamic model, the bondversion, and the option version. We adopt a Monte Carlo Simulation, where simu-lated paths are obtained by using the fact that Xn(t+HP )jXn(t) is a multiple of anoncentral chi-square distribution (see Equation (7)). In order to generate randomnumbers chosen from the noncentral chi-square distribution we use a standard pro-cedure: We �rst generate random values drawn from a uniform distribution on theunit interval. Then invert a noncentral chi-square cumulative distribution functionevaluated at the previously obtained values. For each zero-coupon bond (6-month,12-moth), a total of 20.000 random paths for its price process were generated usingthe Antithetic Variable Technique (see Glasserman (2003)).

4.3 The Accuracy of a VaR Model

Under each version of the CIR model, we verify the accuracy of VaR estimation viabacktesting procedures. While there is a large number of backtesting methods todeal with validation of VaR models, we concentrate in particular on three importanttests proposed respectively by Kupiec (1995), Christo�ersen (1998) and Seiler (2006)(see Appendix A for details).

An accurate VaR model must jointly satis�ed two properties (see Campbell(2005)). First, the unconditional coverage property i.e., number of violations17 ison average equal one minus VaR con�dence level. Second the independence prop-erty, i.e., previous violations have no impact on the probability of futures violations.The Kupiec test veri�es only the former while Christo�ersen's approach tests thelatter.

The Kupiec and Christo�ersen tests are the tests most widely adopted by marketpractitioners, but they work only within single-period forecasting (1-day VaR). Un-

17A violation occurs when pro�t and losses (P&L) in a day is lower than VaR

12

der the case of a 10-day VaR, it is necessary to use a more elaborated test that willcomprise dependence among VaR violations. At this point, Seiler's (2006) methodallowed us to test the unconditional coverage property for VaR horizon greater thanone day.

Another important point to discuss is how general each test is when analyzingindependence of VaR violations. For instance, Christo�ersen's test considers onlyindependence between successive days (probability of violating tomorrow's VaR de-pends on whether or not today's VaR was violated). However, there are innumerousways for independence to fail18. In order to o�er a meaningful complement toChristo�ersen's test, we also examine dependence of violations up to lag 10 with theLjung-Box (1978) test.

5 Empirical Results

In order to evaluate the importance of options on the estimation of future interestrate quantiles, we adopt the three-factor CIR model introduced in Section 3. Themodel is estimated based on two distinct strategies. The �rst strategy (bond version)uses only ID-Futures, while the second (option version) includes both ID-Futuresand IDI calls in the estimation process. The estimation technique adopted here isbased on the Maximum Likelihood procedure proposed by Chen and Scott (1993).When implementing the bond version latent factors are obtained from ID referencebonds with maturities of 1, 126 and 252 days, while remaining bonds are pricedunder assumption of i.i.d. Gaussian errors. The option version uses the sameinformation set on ID reference bond yields adopted by the bond version, but inaddition it prices the Black implied volatility of a 95-day maturity at-the-moneycall under assumption of Gaussian pricing errors. Technical details regarding theestimation process are described in the Appendix B. Table 1 presents parametersvalues under the bond and option versions. Standard deviations are obtained bythe BHHH method19. All parameters whose ratio column presents bold values aresigni�cant at a 95% con�dence level.

Figure 1 shows the historical evolution of the ID reference bond yields withmaturities of 1, 126 and 256 days. In the beginning of the sample yields are increasingand the term structure is upward sloping (long-term rates higher than short-termrates). After July of 2005 yields decline over time and the shape of the term structurechanges to downward sloping due to an improvement in the Brazilian economicconditions in recent years. Figure 2 show that both dynamic versions of the modelkeep the pricing errors at small levels20. Figure 3 presents the time series of theinstantaneous implied volatility of the short-term rate under the two versions of the

18For instance, the likelihood of a violation tomorrow can depend on a violation happening aweek ago.

19See Davidson and MacKinnon (1993).20The bond version presents errors of less than 20 bps, while the option version presents slightly

higher errors but still less than 30 bps for most of the time series. Note that errors are smallerunder both versions in most recent periods, during the year of 2007.

13

model21. Note that volatility is much higher for the model without options, duringthe whole sample period. This overestimation of volatility will be one of the mainfactors contributing to the overestimation of VaR by the bond version of the model.

We estimate, for each model version, 1- and 10-day VaR's of zero-coupon bondswith time to maturities of 6- and 12- months, between August of 2004 and Augustof 2007. On each day, the present value of the portfolio is Mt = P (t; t + T ) withT = 0.5 or 1 years and its value at the end of the holding period is Mt+HP =P (t+HP; t+T +HP ) with HP = 1 or 10 days22. Figures 4 and 5 present the timeseries of the 95% VaR for 1- and 10- day VaR`s, and the corresponding P&L for bothzero coupon bonds. Qualitatively analyzing the pictures, we note that with a timeseries of 740 observations, for the 1-day VaR, we could expect on average around5% of the period (about 37 observations) to have the P&L value crossing the VaRvalue. Observing Figure 4 we see that VaR for the one year bond was crossed 29times under the option version, but only 8 times under the bond version. This is a�rst indication that the model without options would be overestimating VaR.

Table 2 con�rms with formal tests the qualitative results described above. Itrepresents the Kupiec tests with a size of 5% (type I error) for 1- and 10-day VaR'sestimated based on four di�erent con�dence levels (90%, 95%, 97.5% and 99%).These are usually adopted values, both by regulatory institutions as well as by in-vestment banks. Kupiec test for 10-day VaR were carry out ignoring dependenceof violations. As explained in Appendix B it is equivalent to a more conservativetest (a higher type I error). Each cell of this table presents the number of VaRviolations observed. The last column shows the number of VaR violations that wehave to observe to avoid rejection of the model at a 95% con�dence level. We alsopresent the number of violations when VaR is estimated by a Historical Simulation(HS) with respectively 125 or 250 days in a moving-window. HS is a very commonmethod for the estimation of VaR adopted by market practioners. Undoubtedly, itis the simplest method to compute VaR. The aim of comparing quantile forecastsof the dynamic model to HS is to show that the CIR model provides results whichare trustable according to a established benchmark. In other words, that the dy-namic model elected to study the contribution of options on VaR estimation is nota spurious model.

The option version clearly outperforms the bond version in terms of uncondi-tional coverage property: The bond version in fact consistently overestimates risk.If the bond version was adopted for capital allocation purposes it could be reallydemanding much more capital than the necessary to cover the risk of bond portfo-lios. On the other hand, the option version seems not to capture very well the 1-dayquantiles for the 6-month bond and the 10-day quantiles for the 1-year bond. In the

21The instantaneous implied volatility of the short-term rate is given byp�21X1(t) + �22X2(t) + �23X(t).22We could �x the future value as Mt+HP = P (t + HP; t + T ). But with this convention we

have to correct the present value over the holding period to calculate P&L of a portfolio. Since itdoes not add any important information related to the quantile forecast, we opt to use the simplerconvention described above.

14

former case this version overestimates risk while in the latter it underestimates risk.For a total of 16 tests, the bond version is not rejected in only 1 test while the optionversion is not rejected in 8 tests. It is interesting to note that the acceptance rateof the option version is similar to the one obtained by the two historical simulationbenchmarks, which are not rejected in respectively 9 times for the 125 days-windowHS, and 10 times for the 250 days-window HS.

Table 3 presents Seiler's test statistic for a 10-day VaR, and for both consideredbonds. When the test size is 5%, a model is not rejected if Seiler's test statistic islower than 1.96. Seiler's test is a simple way to take into account dependence dueto overlapping forecasting horizons. Observing Figure 5 we can see that the optionversion generates some clustered violations. This is a sign for existence of temporaldependence between observations. Note that the acceptance rates of all models aregreater when we use Seiler's test. This should be expected since under dependenceassumptions we have less information to test model's performance. Dependencemakes an e�ect which is equivalent to a decrease in sample size23. Formally, Seiler'stest has a low statistical power24. Notwithstanding the tremendous improvementin acceptance rate of the bond version, when we jointly analyze 1-day (Table 2)and 10-day (Table 3) containing VaR backtesting results, the option version stilloutperforms the bond version (the bond version is accepted in 7 tests while the optionversion is accepted in 9 tests). Summarizing, when we use the most conservativetest (Kupiec's test) the option version strongly outperforms the bond version, whilewhen we adopt an appropriate test for multiple period dependence but that presentslow power (Seiler's test), the option version still outperforms, but only by a mildadvantage.

Moving from the analysis of unconditional coverage property to the indepen-dence property we �rst examine the results of Christo�ersen's test. Table 4 reportsChristo�ersen's test statistic for a test size of 5%. A model is not rejected if thetest statistic is lower than 3.84. The bond version is statistically accepted in 14tests whereas the option version in 13 tests. In addition, both dynamic versionsoutperform historical simulations which do not show dependence of violations in6 tests (125 days-window) and 11 tests (250 days-windows). It is not surprisingthat the bond version performs extremely well in independence tests. Such goodperformance is deceiving since it is due to a bad characteristic of the bond version:It presents an extremely low number of violations, usually under the lower criticalvalue of the Kupiec test. This fact directly implies a very small probability thatit ends up exhibiting clustered violations. Indeed, from a practical point of view,the bond version should already have been discarded due to its tremendously weakperformance in unconditional coverage tests.

23We can easily translate Seiler's test statistic in terms of number of violations. For example,if we consider the 6-month bond and a test size of 2.5%, the critical values are around 0 and 38violations (the exact critical values depend on the tested model, but they do not vary much), whileKupiec's critical values are 11 and 27 violations.

24Low statistical power is a common feature on all multi-period backtesting procedures thatadopt small samples (see Dowd (2007) and Seiler (2006)).

15

Table 5 presents the Ljung-Box test statistic for a test considering an absenceof serial correlation in the hit sequence25 of VaR's of 6- and 12- month bonds. Weverify whether a group of autocorrelations between lags 1 and 10 (for the 1-dayVaR) and between lags 10 and 19 (for the 10-day VaR) are statistically di�erentfrom zero. A model is not rejected if the test statistic is lower than 18.31. Hereagain the important analysis is the comparison between option version and thecorresponding historical simulation benchmarks since the performance of the bondversion will be biased by its too small number of VaR violations. The option versionperforms well being statistically accepted in 9 out of 16 tests. On the order hand,historical simulation seems to have some kind of problem with clustered violations.In fact, under both historical simulation methods violations are statistically notautocorrelated in only 5 out of 16 cases.

In summary, it appears that options indeed contain information that improvethe ability of the dynamic term structure model to estimate VaR. Further analysisshould be done in future research by considering models that accommodate theexistence of jumps in data.

6 Conclusion

In this paper we test if interest rate options contain useful information that couldhelp on the econometric identi�cation of the quantiles of interest rate distributions.Identi�cation of these quantiles should be useful for risk and portfolio managementpurposes. Adopting a three-factor CIR dynamic term structure model, we comparethe performance of a version estimated with only bonds data (bond version) to aversion estimated based on both bonds and options data (option version). While thebond version strongly overestimates risk (formally measured by a Kupiec statisticalbacktest), the option version demonstrates to be more adequate, presenting a per-formance comparable to historical simulation benchmarks. Tests of independencebetween violations (Christo�ersen (1998) and Ljung-Box (1978)) con�rm that theoption version performs better than the historical simulation benchmarks (althoughthe bond version also performs well in these tests due to its inadequately small num-ber of VaR violations). Our results complement an existing literature that identi�esrisk premia based on the use of option data within dynamic models. Indeed, weshow that interest rate options also contain information related to higher momentsof interest rates that allow for a dynamic model to better estimate interest ratesquantiles. These results suggest that banks and other �nancial institutions mightconsider interesting and possibly useful to perform tests with integrated datasetson linear and non-linear interest rate instruments in order to better manage theirinterest rate risks.

25The hit sequence is a time series of binary variables that indicates at each time if a violationhas occurred or not. See Appendix A for more details.

16

References

[1] Abate, J. and W. Whitt (1995). Numerical Inversion of Laplace Transforms ofProbability Distributions. ORSA Journal on Computing, 7, 1, p. 36-43.

[2] Ait Sahalia Y. and A.W. Lo (1998). Nonparametric Estimation of State-PriceDensities Implicit in Financial Asset Prices. Journal of Finance, 2, 499-547.

[3] Ait Sahalia Y. and A.W. Lo (2000). Nonparametric Risk Management andImplied Risk Aversion. Journal of Econometrics, 94, 9-51.

[4] Almeida C.I.R. J. Graveline and S. Joslin (2006). Do Options Contain Informa-tion About Excess Bond Returns? Working Paper, Stanford Graduate Schoolof Business.

[5] Almeida C.I.R and J.V.M. Vicente (2006). Identifying Volatility Risk Premiafrom Fixed Income Asian Options. Working Paper, Graduate School of Eco-nomics, Getulio Vargas Foundation.

[6] Artzner P., F. Delbaen, J-M Eber, and D. Heath (1999). Coherent Measures ofRisk. Mathematical Finance, 9, 203-228.

[7] Breeden D.T. and R.H. Litzenberger (1978). Prices of State-Contingent ClaimsImplicit in Option Prices. Journal of Business, 51, 4, 621-651.

[8] Brigo D. and F. Mercurio (2001). Interest Rate Models: Theory and Practice,Springer Verlag.

[9] Campbell, S. (2005). A Review of Backtesting and Backtesting Procedures.Finance and Economics Discussion Series, 21, Board of Governors of the FederalReserve System (U.S.)

[10] Chen R.R. and L. Scott (1993). Maximum Likelihood Estimation for a Multi-factor Equilibrium Model of the Term Structure of Interest Rates. Journal ofFixed Income, 3, 14-31.

[11] Chacko G. and S. Das (2002). Pricing Interest Rate Derivatives: A Gen- eralApproach. Review of Financial Studies, 15, 1, 195-241.

[12] Cheridito P., D. Filipovic, and R. Kimmel (2007). Market Price of Risk Spec-i�cations for A�ne Models: Theory and Evidence. Journal of Financial Eco-nomics, 83, 1, 123-170.

[13] Chernov M. and E. Ghysels (2000). A Study Towards a Uni�ed Approach tothe Joint Estimation of Objective and Risk Neutral Measures for the Purposeof Options Valuation. Journal of Financial Economics, 56, 407-458.

[14] Christo�ersen P. (1998). Evaluating Interval Forecasts. International EconomicReview, 39, 841-862.

17

[15] Cox J. C., J.E. Ingersoll and S.A. Ross (1985). A Theory of the Term Structureof Interest Rates. Econometrica, 53, 385-407.

[16] Dai Q. and K. Singleton (2000). Speci�cation Analysis of A�ne Term StructureModels. Journal of Finance, LV, 5, 1943-1977.

[17] Dai Q. and K. Singleton (2002). Expectation Puzzles, Time-Varying Risk Pre-mia, and A�ne Models of the Term Structure. Journal of Financial Economics,63, 415-441.

[18] Dassios A. and J. Nagaradjasarma (2003). Pricing Asian Options On InterestRates in the CIR Model. Working Paper, Department of Statistics, LondonSchool of Economics.

[19] Davidson R. and J.G. MacKinnon (1993). Estimation and Inference in Econo-metrics, Oxford University Press.

[20] Dowd, K.(2007). Validating Multiple-Period Density Forecasting Models, Jour-nal of Forecasting, 26 (4), 251-270.

[21] Du�ee G. R. (2002). Term Premia and Interest Rates Forecasts in A�ne Mod-els. Journal of Finance, 57, 405-443.

[22] Du�e D. (2001). Dynamic Asset Pricing Theory, Princeton University Press.

[23] Du�e D. and R. Kan (1996). A Yield Factor Model of Interest Rates. Mathe-matical Finance, Vol. 6, 4, 379-406.

[24] Du�e D. and J. Pan (2001). An Overview of Value at Risk, Journal of Deriva-tives, 4, 3, 7-49.

[25] Egorov A.V., Y. Hong, and H. Li (2006). Validating Forecasts of the JointProbability Density of Bond Yields: can A�ne Models Beat the RandomWalk?Journal of Econometrics, 135, 255-284.

[26] Eraker B. (2004). Do Stock Prices and Volatility Jump? Reconciling Evidencefrom Spot and Option Prices. Journal of Finance, 59, 3, 1367-1404.

[27] Garcia R., E. Renault, and G. Tsafack (2007). Proper Conditioning for Coher-ent VaR in Portfolio Management. forthcoming at Management Science.

[28] Glasserman P. (2003).Monte Carlo Methods in Financial Engineering, SpringerVerlag.

[29] Jackwerth J.C. (2000). Recovering Risk Aversion from Option Prices and Re-alized Returns. Review of Financial Studies, 13, 433-451.

[30] Jackwerth J.C. and M. Rubinstein (1996). Recovering Probability Distributionsfrom Contemporary Security Prices. Journal of Finance, 51, 1611-1631.

18

[31] Johannes M., N. Polson, and J. Stroud (2007). Optimal Filtering of Jump-Di�usions: Extracting Latent States from Asset Prices. forthcoming at Reviewof Financial Studies.

[32] Joslin S. (2007). Pricing and Hedging Volatility Risk in Fixed Income Markets.Working Paper, Sloan School of Business, MIT.

[33] Kerkhof J. and B. Melenberg (2004). Backtesting for Risk-based RegulatoryCapital, Journal of Banking and Finance, 28, 8, 1845-1865.

[34] Kupiec P. (1995). Techniques for Verifying the Accuracy of Risk MeasurementModels. Journal of Derivatives, 3, 73-84.

[35] Leblanc B. and O. Scaillet (1998). Path Dependent Options on Yields in theA�ne Term Structure Model. Finance and Stochastics, 2, 349-367.

[36] Ljung, G. M. and G. E. P. Box (1978). On a Measure of Lack of Fit in TimeSeries Models. Biometrika, 65, 297-303.

[37] Nagaradjasarma J. (2003). Path-dependent Functionals of Constant Elasticityof Variance and Related Processes: Distributional Results and Applications inFinance, Ph. D Thesis, London School of Economics.

[38] Pan J. (2002). The Jump-Risk Premia Implicit in Options: Evidence from anIntegrated Time-Series Study. Journal of Financial Economics, 63, 3-50.

[39] Rubinstein M. (1994). Implied Binomial Trees. Journal of Finance, 49, 771-818.

[40] Seiler, D. (2006) Backtesting Multiple-Period Forecasting Models with Appli-cation to Credit Exposure Models, Master Thesis, University of Zurich.

[41] Shreve S. (2004). Stochastic Calculus for Finance II: Continuous-Time Models,Springer Finance Series.

19

Appendix A - Backtesting Procedures

In this appendix we describe the backtesting procedures proposed by Kupiec (1995),Christo�ersen (1998) and Seiler (2006), as well as the Portmanteau test developedby Ljung-Box (1978).

Suppose that we have a time series of VaRs with con�dence levels of 1� � anda time series of pro�ts and losses (P&L) on a portfolio over a �xed time interval.A violation occurs when P&L is lower than VaR. Each variable It(�) in the hitsequence is equal to zero if there is no violation at time t or equal to one if there isa violation at time t, viz:

It(�) =

�1 if P&Lt � V aRt�HP;t

0 else.

First we consider the single-period case (1-day ahead quantile forecasting). Christof-fersen (1998) points out that the problem of determining the accuracy of a VaRmodel is equivalent to the problem of determining whether the hit sequence satis�esthe following two properties:

1. Unconditional Coverage Property, which means that the average number of vio-lations is statistically equal to the expected one. Formally P [It+1(�) = 1j Ft] =�.

2. Independence Property, which means that violations are independently dis-tributed. In other words, the history of VaR violations does not contain infor-mation about future violations. Formally It(�) and It+k(�) are independentfor all t and k � 1.

The test developed by Kupiec (1995) examines only the unconditional coveragewhile Christo�ersen (1998) considers both the unconditional coverage and indepen-dence. The number of violations observed in a sample of size L is

NL(�) =LXt=1

It(�):

If the model is accurate, NL must present a binomial distribution. Using theLikelihood-Ratio test, the Kupiec's test statistic takes the form

2ln

�1� �

1� �

�L�NL(�) ���

�NL(�)!� �2(1); as L!1;

where � = NL(�)=L and �2(�) stands for a chi-squared distribution with � degreesof freedom.

The Kupiec test provides a necessary condition to classify a VaR model as ad-equate. But, not considering the arrival times of violations, it does not excludemodels that do not satisfy the independence property. In other words, the Kupiec

20

test can accept a model that creates successive violation clustering. In order to solvethis problem, Christo�ersen (1998) proposed a Markov test that takes into accountthe independence property. Within his test, if the probability of a violation in thecurrent period has no impact on the probability of violation in the next period, thenit must hold that

L10

L00 + L10=

L11

L01 + L11;

where Lij is the number of observations of It with value i followed by j, i; j = 0; 1.Again, the test statistic is given by the Likelihood-Ratio test:

2ln

�(1� �01)

L00�L0101 (1� �11)L10�L1111

(1� �)L00+L10�L01+L11

�� �2(1); as L!1; (14)

where �ij =LijPj Lij

and � = L01+L11L

. Note that this test does not depend on the true

coverage value �, and thus, only tests the independence property. Indeed, Christof-fersen (1998) also presents a test which simultaneously examines the unconditionalcoverage and the independence properties. We do not use this joint test because itdoes not disentangle which property has been violated.

Christo�ersen's test considers only dependence from day to day, what impliesthat the independence property is not fully tested. In order to examine the existenceof dependence with a time lag greater than one period, we can use the Ljung-Boxtest (see Ljung and Box (1978)). If i is the empirical autocorrelation of order i, totest i = 0 for the �rst K autocorrelations we can use the following statistic

L(L+ 2)KXi=1

2iL� i

� �2(K); as L!1: (15)

In order to validate the accuracy of a 10-day VaR we have to adjust the indepen-dence property, because the quantile forecasts are subject to common shocks. Thiscan be done by requiring that It(�) and It+k(�) are independent for all t and k � 10.This fact complicates the accuracy problem since the variables It are not presumedto be i.i.d. There are di�erent ways to handle the dependence of lag lower than 10(see Dowd (2007) and Seiler (2006)). The �rst alternative is ignoring dependence.Although it can appear to be a naive choice, ignoring dependence gives valuableinformation since it is equivalent to implementing a very conservative test26. Theproblem is that we can reject \good" models. In others words, under this alternativeprobably the type I error is higher than the true type I error. Seiler (2006) and Dowd(2007) propose di�erent tests that take into account dependence. Dowd (2007) usesa bootstrapping approach to yield an i.i.d. resample. On the other hand, Seiler'stest considers that the model under analysis captures the true autocorrelations of

26The correlation between It(�) and It+k(�) for k < 10 are expected to be positive. Thus, thevariance of NL when ignoring dependence is lower than the variance when correlations are takeninto account. Therefore the rejection region is larger if dependence is ignored.

21

lag lower than 10. Thus, based on an ARMA process, the dependence structure canbe estimated from the hit sequence observations. If L is large enough, we have

NL(�) � N (L�;LA);

where N represents a normal distribution and the correlation matrix A is estimatedusing only the hit sequences It(�) and It(0:5) (see Seiler (2006) for details aboutthis estimation procedure)27.

The adjusted independence property can be validated using the test in (14) withthe number of \00", \01", \10" and \11" in the hit sequence counted from 10 to 10days, that is

Lij = # f2 � t � LjIt(�) = i; It+10(�) = jg for i; j = 0; 1:

Finally, to test autocorrelation with a time lag greater than one period we usethe Ljung-Box test (15) with the summation extending from i = 10 to i = K + 9.

27We also experimented with the Bonferroni approach (see Dowd (2007)) to solve the problemof dependence, but the power of the resulting test was very low.

22

Appendix B - Maximum Likelihood Estimation

On this work, we adopt the Maximum Likelihood estimation procedure proposed byChen and Scott (1993). We observe the following reference ID bonds yields throughtime: rbt(1), rbt(63), rbt(126), rbt(189), rbt(252), and rbt(378). Let rb representsthe H � 7 matrix containing these ID bonds yields for the whole time series of Hpoints. Assume that model parameters are represented by vector � and that thedi�erence between times t� 1 and t is �t. From Equation (8), the relation betweenthe yield of a reference ID bond with maturity � and the state variables at time t is

R(t; �; �) = �0 �A(�; �)

��B(�; �)0

�X(t); (16)

where A(�; �) =PN

n=1 logAn(t; �), B(�; �) = � (B1(�; �); : : : ; BN(�; �))0 andX(t) =

(X1(t); : : : ; XN(t))0. In accordance with the empirical term structure literature, we

�x N = 3 and assume that the yields of the reference ID bond with maturities 1, 126and 252 days are observed without error. As the state vector is three-dimensional,knowledge of functions A(�; �) and B(�; �) allows us to solve a linear system toobtain the values of the state vector at each time t:

rbt(1) = �0 �A(1;�)

1� B(1;�)0

1X(t);

rbt(126) = �0 �A(126;�)

126� B(126;�)0

126X(t) and

rbt(252) = �0 �A(252;�)

252� B(252;�)0

252X(t):

(17)

For the reference ID bond with maturities 63, 189 and 378 days, we assume obser-vation with gaussian errors ut uncorrelated along time:

rbt(63) = �0 �A(63;�)

63� B(63;�)0

63X(t) + ut(63);

rbt(189) = �0 �A(189;�)

189� B(189;�)0

189X(t) + ut(189);

rbt(378) = �0 �A(378;�)

378� B(378;�)0

378X(t) + ut(378) and

(18)

When options are took account in the estimation procedure, in addition to yieldserrors, we have to consider options errors de�ned by

ucallt = ct � cst; (19)

where cst is the observed Black volatility for the call in the sample at time t and ctis the model Black volatility (see Section 3.2) to the same call at time t.

After extracting the corresponding state vector at the vector of parameters �,we can write the log-likelihood function of the reference ID bond yields as

L(rb; �) =PH

t=2 log p(X(t)jX(t� 1);�)�

�(H � 1) log jjacj � H�12

log jj � 12

PH

t=2 u0t

�1ut;

(20)

23

where:

1. jac =

2666664

�B(1;�)0

1

�B(126;�)0

126

�B(252;�)0

252

3777775 ;

2. represents the covariance matrix for ut, estimated using the sample covari-ance matrix of the ut's implied by the extracted state vector along time;

3. p(X(t)jX(t� 1);�) is the transition probability density from X(t� 1) to X(t)in the real world (that is, under the measure P). Since as the dynamics of Xin the real world is also a Feller process (see Equation (6)) and the Xn's areindependents we have

p(X(t)jX(t� 1);�) =3Y

n=1

fXn(t)jXn(t�1)(x)

�����x=Xn(t)

;

with fXn(t)jXn(t�1)(�) given by Equation (7) changing �n by �n and �n by �n.

4. u0t = [ut(63) ut(189) ut(378)] when we don't use options in the estimationprocedure and u0t =

�ut(63) ut(189) ut(378) u

callt

�when we use.

Our �nal objective is to estimate the vector of parameters � which maximizesfunction L(rb; �). In order to try to avoid possible local minima we use severaldi�erent starting values and search for the optimal point by making use of theNelder-Mead Simplex algorithm for non-linear functions optimization (implementedin the MATLAB fminsearch function) and the gradient-based optimization method(implemented in the MATLAB fminunc function).

24

ParameterBond Version Option Version

Value jValuejStd Error

Value jValuejStd Error

�1 11.1819 29.94 27.9618 26.83

�2 0.0003 0.30 0.0002 7.42

�3 2.0311 29.37 2.8889 19.79

�1 0.0291 37.06 0.0342 23.39

�2 4.7485 2.45 5.1643 34.02

�3 0.0430 40.35 0.0487 32.90

�1 0.1507 27.86 0.1367 14.84

�2 0.0518 36.81 0.0409 26.71

�3 0.1228 31.65 0.0650 9.38

�1 -14.0298 2.34 -22.5605 4.93

�2 -6.1960 1.32 -8.6856 2.60

�3 0.0296 0.00 0.037 0.00�0 0 - 0 -

Table 1: Parameters values and standard errors.

This table presents parameters values and standard errors for both versions

of the dynamic model. The model was estimated by Maximum Likelihood

adopting the methodology proposed by Chen and Scott (1993). For the

bond version, the 1-, 126-, and 252-days maturity ID reference bond were

priced exactly, and the other bonds were priced with i.i.d. Gaussian errors.

For the option version, the 1-, 126-, and 252-days maturity ID reference

bond were priced exactly, and the other bonds and the Black volatility

of the at-the-money IDI call maturing in 95 days were priced with i.i.d.

Gaussian errors. Standard errors were obtained by the BHHH method.

Bold value means that the parameter is signi�cant at a 95% con�dence

level.

25

Con�dence

Levels

1-day

VaR

6-month

zero

coupon

bond

12-m

onthszero

coupon

bond

Kupiec

Bond

Option

HS125

HS250

Bond

Option

HS125

HS250

90%

616

59

5015

5666

59

[59,90]

95%

211

32

228

29

40

27

[26,49]

97.5%

17

19

106

17

20

11

[11,27]

99%

15

11

45

10

11

6[3,13]

Con�dence

Levels

10-day

VaR

6-month

zero

coupon

bond

12-m

onthszero

coupon

bond

Kupiec

Bond

Option

HS125

HS250

Bond

Option

HS125

HS250

90%

960

67

4640

101

9458

[58,89]

95%

141

5518

1976

6026

[26,48]

97.5%

021

4217

751

3821

[11,27]

99%

07

2913

332

2818

[3,13]

Table

2:NumberofVaR

violationsin

theKupiectest.

ThistablepresentsthenumberofVaRviolationsforthe6-and12-m

onth

zero

couponbonds.AVaR

violationoccurswhen

P&Linaday

islowerthantheestimatedVaRforthisday.VaRwasestimated

fortwohorizonperiod(1

and10days)

andfourcon�dence

levels(90%,95%,97.5%

and99%).The

columnsrepresenttheteststatisticsforbo

ndversion,option

versionandhistoricalsimulationswith

125and250scenarios.ThelastcolumnrepresentstheKupieccriticalvalues

under

a5%

typeIerror.

Bold

values

meansthatthemodelpass

Kupiecvalidationtest.

26

Con�dence Levels6-month zero coupon bond

Bond Option HS 125 HS 25090% 2.94 0.59 0.29 1.16

95% 2.38 0.29 1.28 1.15

97.5% 1.81 0.26 2.42 0.12

99% 1.23 0.05 3.75 0.88

Con�dence Levels12-month zero coupon bond

Bond Option HS 125 HS 25090% 1.79 1.52 1.07 0.75

95% 1.39 3.16 1.75 0.77

97.5% 1.33 3.92 2.18 0.29

99% 0.86 5.07 3.88 1.97

Table 3: Seiler's test statistic for a 10-day VaR.

This table presents the Seiler test statistic for 10-day quantile forecast (10-

day VaR) of 6- and 12-month bonds. Seiler test takes in account dependence

between successive VaR violations. VaR was estimated for four di�erent

con�dence levels (90%, 95%, 97.5% and 99%). The columns represent the

test statistics for bond version, option version and historical simulations

with 125 and 250 scenarios, respectively. Under a 5% type I error, a model

is not rejected if Seiler's test statistic is lower than 1.96. Bold values means

that the model is not rejected in Seiler validation test.

27

Con�dence

Levels

1-day

VaR

6-month

zero

coupon

bond

12-m

onth

zero

coupon

bond

Bond

Option

HS125

HS250

Bond

Option

HS125

HS250

90%

0.10

0.87

5.81

10.82

0.62

5.17

13.22

5.81

95%

0.01

2.08

10.20

5.23

0.18

2.35

0.37

3.14

97.5%

0.00

0.13

2.87

2.43

0.10

0.80

0.33

2.08

99%

0.00

0.07

0.33

0.04

0.07

0.27

0.33

0.10

Con�dence

Levels

10-day

VaR

6-month

zero

coupon

bond

12-m

onth

zero

coupon

bond

Bond

Option

HS125

HS250

Bond

Option

HS125

HS250

90%

0.23

11.77

30.49

9.67

0.73

5.29

8.33

4.13

95%

0.00

1.06

25.73

0.56

1.03

3.82

12.03

2.35

97.5%

-1.26

11.68

0.48

0.14

0.22

8.48

0.72

99%

-0.14

14.05

0.29

0.03

2.69

1.61

0.51

Table

4:Christo�ersen'stest

statistic.

Thistablepresents

Christo�ersentest

statisticforthehitsequence

ofVaRofboth

bonds.

VaRwasestimatedfor1and10dayshorizonsandforcon�dence

levelsof90%,95%,97.5%,

and99%.

Thehit

sequence

isequalto

oneifthereis

aviolationandzero

otherwise.

Christo�ersentest

examines

theindependence

ofviolations.

Thecolumnsrepresentthe

test

statisticsforbo

ndversion,option

versionandhistoricalsimulationswith125and250

scenarios.

Adash

entrymeansthatis

impossible

toapply

Christo�ersentest

(because

L01=L10=L11=0).

Under

a5%

typeIerror,amodelisnotrejected

ifthetest

statistic

islower

than3.84.

Bold

values

meansthatthemodel

isnotrejected

inChristo�ersen

validationtest.

28

Con�dence

Levels

1-day

VaR

6-month

zero

coupon

bond

12-m

onth

zero

coupon

bond

Bond

Option

HS125

HS250

Bond

Option

HS125

HS250

90%

0.51

35.44

13.71

24.52

18.19

15.08

32.56

24.29

95%

0.06

14.51

44.19

36.19

40.28

10.59

15.02

47.76

97.5%

0.01

14.14

27.45

29.58

38.54

13.65

28.13

44.64

99%

0.01

0.35

5.99

135.48

56.77

44.62

14.60

39.07

Con�dence

Levels

10-day

VaR

6-month

zero

coupon

bond

12-m

onth

zero

coupon

bond

Bond

Option

HS125

HS250

Bond

Option

HS125

HS250

90%

1.21

90.85

180.37

51.75

11.82

27.79

206.21

70.33

95%

0.01

55.55

180.83

1.90

9.83

22.75

70.06

23.96

97.5%

-20.61

56.82

1.57

0.73

6.21

23.44

5.87

99%

-0.73

53.64

0.99

0.13

16.61

6.39

1.58

Table

5:Ljung-Boxtest

statistic.

ThistablepresentsLjung-Box

teststatisticforthehitsequence

ofVaRofboth

bonds.VaR

wasestimatedfor1and10dayshorizonsandforcon�dence

levelsof90%,95%,97.5%

and

99%.Thehitsequence

isequalto

oneifthereisaviolationandzero

otherwise.

Ljung-Box

test

examines

theabsence

ofserialcorrelationforthe�rsttenautocorrelationsafter

time

lagofone(forthe1-day

VaR)orten(forthe10-day

VaR).Thecolumnsrepresentthe

test

statisticsforbo

ndversion,option

versionandhistoricalsimulationswith125and250

scenarios.Adash

entrymeansthatisimpossibleto

applyLjung-Box

test.Under

a5%

type

Ierror,amodelisnotrejected

ifthetest

statisticislower

than18.31.Bold

values

means

thatthemodelisnotrejected

inLjung-Box

validationtest.

29

Figure 1: The term structure of Brazilian interest rates.

This �gure contains time series of the ID reference bond yields with time

to maturity of 1-, 126-, and 252-day between August 9, 2004 and August

6, 2007. ID reference bonds are zero coupon bonds obtained by cubic

interpolation from observed ID-Future rates.

30

Figure 2: Pricing errors for yields observed with error.

This �gure contains the time series, between August 9, 2004 and August

6, 2007, for the residuals of yields priced with error. Time to maturity

of those particular yields were 63, 189, and 378 days. Error terms were

estimated under the assumption of i.i.d. Gaussian distributions.

31

Figure 3: The instantaneous volatility of the short-term rate.

This �gure contains the instantaneous volatility of the short-term interest

rate implied by each model version, between August 9, 2004 and August

6, 2007. The instantaneous implied volatility of short-term rate is given byp�21X1(t) + �22X2(t) + �23X3(t).

32

Figure 4: P&L and 95% 1-day VaR for zero coupon-bonds.

This �gure shows the time series of 95% 1-day VaR and P&L for Brazilian

zero coupon bonds between August, 2004 and August, 2007. VaR were

estimated using both versions of the dynamic term structure model. The

top panel presents the evolution of the VaR for the 6-month bond and the

bottom panel presents the evolution of VaR for the 12-month bond.

33

Figure 5: P&L and 95% 10-day VaR for zero coupon-bonds.

This �gure shows the time series of 95% 10-day VaR and P&L for Brazilian

zero coupon bonds between August, 2004 and August, 2007. VaR were

estimated using both versions of the dynamic term structure model. The

top panel presents the evolution of the VaR for the 6-month bond and the

bottom panel presents the evolution of VaR for the 12-month bond.

34

35

Banco Central do Brasil

Trabalhos para Discussão Os Trabalhos para Discussão podem ser acessados na internet, no formato PDF,

no endereço: http://www.bc.gov.br

Working Paper Series Working Papers in PDF format can be downloaded from: http://www.bc.gov.br

1 Implementing Inflation Targeting in Brazil

Joel Bogdanski, Alexandre Antonio Tombini and Sérgio Ribeiro da Costa Werlang

Jul/2000

2 Política Monetária e Supervisão do Sistema Financeiro Nacional no Banco Central do Brasil Eduardo Lundberg Monetary Policy and Banking Supervision Functions on the Central Bank Eduardo Lundberg

Jul/2000

Jul/2000

3 Private Sector Participation: a Theoretical Justification of the Brazilian Position Sérgio Ribeiro da Costa Werlang

Jul/2000

4 An Information Theory Approach to the Aggregation of Log-Linear Models Pedro H. Albuquerque

Jul/2000

5 The Pass-Through from Depreciation to Inflation: a Panel Study Ilan Goldfajn and Sérgio Ribeiro da Costa Werlang

Jul/2000

6 Optimal Interest Rate Rules in Inflation Targeting Frameworks José Alvaro Rodrigues Neto, Fabio Araújo and Marta Baltar J. Moreira

Jul/2000

7 Leading Indicators of Inflation for Brazil Marcelle Chauvet

Sep/2000

8 The Correlation Matrix of the Brazilian Central Bank’s Standard Model for Interest Rate Market Risk José Alvaro Rodrigues Neto

Sep/2000

9 Estimating Exchange Market Pressure and Intervention Activity Emanuel-Werner Kohlscheen

Nov/2000

10 Análise do Financiamento Externo a uma Pequena Economia Aplicação da Teoria do Prêmio Monetário ao Caso Brasileiro: 1991–1998 Carlos Hamilton Vasconcelos Araújo e Renato Galvão Flôres Júnior

Mar/2001

11 A Note on the Efficient Estimation of Inflation in Brazil Michael F. Bryan and Stephen G. Cecchetti

Mar/2001

12 A Test of Competition in Brazilian Banking Márcio I. Nakane

Mar/2001

36

13 Modelos de Previsão de Insolvência Bancária no Brasil Marcio Magalhães Janot

Mar/2001

14 Evaluating Core Inflation Measures for Brazil Francisco Marcos Rodrigues Figueiredo

Mar/2001

15 Is It Worth Tracking Dollar/Real Implied Volatility? Sandro Canesso de Andrade and Benjamin Miranda Tabak

Mar/2001

16 Avaliação das Projeções do Modelo Estrutural do Banco Central do Brasil para a Taxa de Variação do IPCA Sergio Afonso Lago Alves Evaluation of the Central Bank of Brazil Structural Model’s Inflation Forecasts in an Inflation Targeting Framework Sergio Afonso Lago Alves

Mar/2001

Jul/2001

17 Estimando o Produto Potencial Brasileiro: uma Abordagem de Função de Produção Tito Nícias Teixeira da Silva Filho Estimating Brazilian Potential Output: a Production Function Approach Tito Nícias Teixeira da Silva Filho

Abr/2001

Aug/2002

18 A Simple Model for Inflation Targeting in Brazil Paulo Springer de Freitas and Marcelo Kfoury Muinhos

Apr/2001

19 Uncovered Interest Parity with Fundamentals: a Brazilian Exchange Rate Forecast Model Marcelo Kfoury Muinhos, Paulo Springer de Freitas and Fabio Araújo

May/2001

20 Credit Channel without the LM Curve Victorio Y. T. Chu and Márcio I. Nakane

May/2001

21 Os Impactos Econômicos da CPMF: Teoria e Evidência Pedro H. Albuquerque

Jun/2001

22 Decentralized Portfolio Management Paulo Coutinho and Benjamin Miranda Tabak

Jun/2001

23 Os Efeitos da CPMF sobre a Intermediação Financeira Sérgio Mikio Koyama e Márcio I. Nakane

Jul/2001

24 Inflation Targeting in Brazil: Shocks, Backward-Looking Prices, and IMF Conditionality Joel Bogdanski, Paulo Springer de Freitas, Ilan Goldfajn and Alexandre Antonio Tombini

Aug/2001

25 Inflation Targeting in Brazil: Reviewing Two Years of Monetary Policy 1999/00 Pedro Fachada

Aug/2001

26 Inflation Targeting in an Open Financially Integrated Emerging Economy: the Case of Brazil Marcelo Kfoury Muinhos

Aug/2001

27

Complementaridade e Fungibilidade dos Fluxos de Capitais Internacionais Carlos Hamilton Vasconcelos Araújo e Renato Galvão Flôres Júnior

Set/2001

37

28

Regras Monetárias e Dinâmica Macroeconômica no Brasil: uma Abordagem de Expectativas Racionais Marco Antonio Bonomo e Ricardo D. Brito

Nov/2001

29 Using a Money Demand Model to Evaluate Monetary Policies in Brazil Pedro H. Albuquerque and Solange Gouvêa

Nov/2001

30 Testing the Expectations Hypothesis in the Brazilian Term Structure of Interest Rates Benjamin Miranda Tabak and Sandro Canesso de Andrade

Nov/2001

31 Algumas Considerações sobre a Sazonalidade no IPCA Francisco Marcos R. Figueiredo e Roberta Blass Staub

Nov/2001

32 Crises Cambiais e Ataques Especulativos no Brasil Mauro Costa Miranda

Nov/2001

33 Monetary Policy and Inflation in Brazil (1975-2000): a VAR Estimation André Minella

Nov/2001

34 Constrained Discretion and Collective Action Problems: Reflections on the Resolution of International Financial Crises Arminio Fraga and Daniel Luiz Gleizer

Nov/2001

35 Uma Definição Operacional de Estabilidade de Preços Tito Nícias Teixeira da Silva Filho

Dez/2001

36 Can Emerging Markets Float? Should They Inflation Target? Barry Eichengreen

Feb/2002

37 Monetary Policy in Brazil: Remarks on the Inflation Targeting Regime, Public Debt Management and Open Market Operations Luiz Fernando Figueiredo, Pedro Fachada and Sérgio Goldenstein

Mar/2002

38 Volatilidade Implícita e Antecipação de Eventos de Stress: um Teste para o Mercado Brasileiro Frederico Pechir Gomes

Mar/2002

39 Opções sobre Dólar Comercial e Expectativas a Respeito do Comportamento da Taxa de Câmbio Paulo Castor de Castro

Mar/2002

40 Speculative Attacks on Debts, Dollarization and Optimum Currency Areas Aloisio Araujo and Márcia Leon

Apr/2002

41 Mudanças de Regime no Câmbio Brasileiro Carlos Hamilton V. Araújo e Getúlio B. da Silveira Filho

Jun/2002

42 Modelo Estrutural com Setor Externo: Endogenização do Prêmio de Risco e do Câmbio Marcelo Kfoury Muinhos, Sérgio Afonso Lago Alves e Gil Riella

Jun/2002

43 The Effects of the Brazilian ADRs Program on Domestic Market Efficiency Benjamin Miranda Tabak and Eduardo José Araújo Lima

Jun/2002

38

44 Estrutura Competitiva, Produtividade Industrial e Liberação Comercial no Brasil Pedro Cavalcanti Ferreira e Osmani Teixeira de Carvalho Guillén

Jun/2002

45 Optimal Monetary Policy, Gains from Commitment, and Inflation Persistence André Minella

Aug/2002

46 The Determinants of Bank Interest Spread in Brazil Tarsila Segalla Afanasieff, Priscilla Maria Villa Lhacer and Márcio I. Nakane

Aug/2002

47 Indicadores Derivados de Agregados Monetários Fernando de Aquino Fonseca Neto e José Albuquerque Júnior

Set/2002

48 Should Government Smooth Exchange Rate Risk? Ilan Goldfajn and Marcos Antonio Silveira

Sep/2002

49 Desenvolvimento do Sistema Financeiro e Crescimento Econômico no Brasil: Evidências de Causalidade Orlando Carneiro de Matos

Set/2002

50 Macroeconomic Coordination and Inflation Targeting in a Two-Country Model Eui Jung Chang, Marcelo Kfoury Muinhos and Joanílio Rodolpho Teixeira

Sep/2002

51 Credit Channel with Sovereign Credit Risk: an Empirical Test Victorio Yi Tson Chu

Sep/2002

52 Generalized Hyperbolic Distributions and Brazilian Data José Fajardo and Aquiles Farias

Sep/2002

53 Inflation Targeting in Brazil: Lessons and Challenges André Minella, Paulo Springer de Freitas, Ilan Goldfajn and Marcelo Kfoury Muinhos

Nov/2002

54 Stock Returns and Volatility Benjamin Miranda Tabak and Solange Maria Guerra

Nov/2002

55 Componentes de Curto e Longo Prazo das Taxas de Juros no Brasil Carlos Hamilton Vasconcelos Araújo e Osmani Teixeira de Carvalho de Guillén

Nov/2002

56 Causality and Cointegration in Stock Markets: the Case of Latin America Benjamin Miranda Tabak and Eduardo José Araújo Lima

Dec/2002

57 As Leis de Falência: uma Abordagem Econômica Aloisio Araujo

Dez/2002

58 The Random Walk Hypothesis and the Behavior of Foreign Capital Portfolio Flows: the Brazilian Stock Market Case Benjamin Miranda Tabak

Dec/2002

59 Os Preços Administrados e a Inflação no Brasil Francisco Marcos R. Figueiredo e Thaís Porto Ferreira

Dez/2002

60 Delegated Portfolio Management Paulo Coutinho and Benjamin Miranda Tabak

Dec/2002

39

61 O Uso de Dados de Alta Freqüência na Estimação da Volatilidade e do Valor em Risco para o Ibovespa João Maurício de Souza Moreira e Eduardo Facó Lemgruber

Dez/2002

62 Taxa de Juros e Concentração Bancária no Brasil Eduardo Kiyoshi Tonooka e Sérgio Mikio Koyama

Fev/2003

63 Optimal Monetary Rules: the Case of Brazil Charles Lima de Almeida, Marco Aurélio Peres, Geraldo da Silva e Souza and Benjamin Miranda Tabak

Feb/2003

64 Medium-Size Macroeconomic Model for the Brazilian Economy Marcelo Kfoury Muinhos and Sergio Afonso Lago Alves

Feb/2003

65 On the Information Content of Oil Future Prices Benjamin Miranda Tabak

Feb/2003

66 A Taxa de Juros de Equilíbrio: uma Abordagem Múltipla Pedro Calhman de Miranda e Marcelo Kfoury Muinhos

Fev/2003

67 Avaliação de Métodos de Cálculo de Exigência de Capital para Risco de Mercado de Carteiras de Ações no Brasil Gustavo S. Araújo, João Maurício S. Moreira e Ricardo S. Maia Clemente

Fev/2003

68 Real Balances in the Utility Function: Evidence for Brazil Leonardo Soriano de Alencar and Márcio I. Nakane

Feb/2003

69 r-filters: a Hodrick-Prescott Filter Generalization Fabio Araújo, Marta Baltar Moreira Areosa and José Alvaro Rodrigues Neto

Feb/2003

70 Monetary Policy Surprises and the Brazilian Term Structure of Interest Rates Benjamin Miranda Tabak

Feb/2003

71 On Shadow-Prices of Banks in Real-Time Gross Settlement Systems Rodrigo Penaloza

Apr/2003

72 O Prêmio pela Maturidade na Estrutura a Termo das Taxas de Juros Brasileiras Ricardo Dias de Oliveira Brito, Angelo J. Mont'Alverne Duarte e Osmani Teixeira de C. Guillen

Maio/2003

73 Análise de Componentes Principais de Dados Funcionais – uma Aplicação às Estruturas a Termo de Taxas de Juros Getúlio Borges da Silveira e Octavio Bessada

Maio/2003

74 Aplicação do Modelo de Black, Derman & Toy à Precificação de Opções Sobre Títulos de Renda Fixa Octavio Manuel Bessada Lion, Carlos Alberto Nunes Cosenza e César das Neves

Maio/2003

75 Brazil’s Financial System: Resilience to Shocks, no Currency Substitution, but Struggling to Promote Growth Ilan Goldfajn, Katherine Hennings and Helio Mori

Jun/2003

40

76 Inflation Targeting in Emerging Market Economies Arminio Fraga, Ilan Goldfajn and André Minella

Jun/2003

77 Inflation Targeting in Brazil: Constructing Credibility under Exchange Rate Volatility André Minella, Paulo Springer de Freitas, Ilan Goldfajn and Marcelo Kfoury Muinhos

Jul/2003

78 Contornando os Pressupostos de Black & Scholes: Aplicação do Modelo de Precificação de Opções de Duan no Mercado Brasileiro Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo, Antonio Carlos Figueiredo, Eduardo Facó Lemgruber

Out/2003

79 Inclusão do Decaimento Temporal na Metodologia Delta-Gama para o Cálculo do VaR de Carteiras Compradas em Opções no Brasil Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo, Eduardo Facó Lemgruber

Out/2003

80 Diferenças e Semelhanças entre Países da América Latina: uma Análise de Markov Switching para os Ciclos Econômicos de Brasil e Argentina Arnildo da Silva Correa

Out/2003

81 Bank Competition, Agency Costs and the Performance of the Monetary Policy Leonardo Soriano de Alencar and Márcio I. Nakane

Jan/2004

82 Carteiras de Opções: Avaliação de Metodologias de Exigência de Capital no Mercado Brasileiro Cláudio Henrique da Silveira Barbedo e Gustavo Silva Araújo

Mar/2004

83 Does Inflation Targeting Reduce Inflation? An Analysis for the OECD Industrial Countries Thomas Y. Wu

May/2004

84 Speculative Attacks on Debts and Optimum Currency Area: a Welfare Analysis Aloisio Araujo and Marcia Leon

May/2004

85 Risk Premia for Emerging Markets Bonds: Evidence from Brazilian Government Debt, 1996-2002 André Soares Loureiro and Fernando de Holanda Barbosa

May/2004

86 Identificação do Fator Estocástico de Descontos e Algumas Implicações sobre Testes de Modelos de Consumo Fabio Araujo e João Victor Issler

Maio/2004

87 Mercado de Crédito: uma Análise Econométrica dos Volumes de Crédito Total e Habitacional no Brasil Ana Carla Abrão Costa

Dez/2004

88 Ciclos Internacionais de Negócios: uma Análise de Mudança de Regime Markoviano para Brasil, Argentina e Estados Unidos Arnildo da Silva Correa e Ronald Otto Hillbrecht

Dez/2004

89 O Mercado de Hedge Cambial no Brasil: Reação das Instituições Financeiras a Intervenções do Banco Central Fernando N. de Oliveira

Dez/2004

41

90 Bank Privatization and Productivity: Evidence for Brazil Márcio I. Nakane and Daniela B. Weintraub

Dec/2004

91 Credit Risk Measurement and the Regulation of Bank Capital and Provision Requirements in Brazil – a Corporate Analysis Ricardo Schechtman, Valéria Salomão Garcia, Sergio Mikio Koyama and Guilherme Cronemberger Parente

Dec/2004

92

Steady-State Analysis of an Open Economy General Equilibrium Model for Brazil Mirta Noemi Sataka Bugarin, Roberto de Goes Ellery Jr., Victor Gomes Silva, Marcelo Kfoury Muinhos

Apr/2005

93 Avaliação de Modelos de Cálculo de Exigência de Capital para Risco Cambial Claudio H. da S. Barbedo, Gustavo S. Araújo, João Maurício S. Moreira e Ricardo S. Maia Clemente

Abr/2005

94 Simulação Histórica Filtrada: Incorporação da Volatilidade ao Modelo Histórico de Cálculo de Risco para Ativos Não-Lineares Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo e Eduardo Facó Lemgruber

Abr/2005

95 Comment on Market Discipline and Monetary Policy by Carl Walsh Maurício S. Bugarin and Fábia A. de Carvalho

Apr/2005

96 O que É Estratégia: uma Abordagem Multiparadigmática para a Disciplina Anthero de Moraes Meirelles

Ago/2005

97 Finance and the Business Cycle: a Kalman Filter Approach with Markov Switching Ryan A. Compton and Jose Ricardo da Costa e Silva

Aug/2005

98 Capital Flows Cycle: Stylized Facts and Empirical Evidences for Emerging Market Economies Helio Mori e Marcelo Kfoury Muinhos

Aug/2005

99 Adequação das Medidas de Valor em Risco na Formulação da Exigência de Capital para Estratégias de Opções no Mercado Brasileiro Gustavo Silva Araújo, Claudio Henrique da Silveira Barbedo,e Eduardo Facó Lemgruber

Set/2005

100 Targets and Inflation Dynamics Sergio A. L. Alves and Waldyr D. Areosa

Oct/2005

101 Comparing Equilibrium Real Interest Rates: Different Approaches to Measure Brazilian Rates Marcelo Kfoury Muinhos and Márcio I. Nakane

Mar/2006

102 Judicial Risk and Credit Market Performance: Micro Evidence from Brazilian Payroll Loans Ana Carla A. Costa and João M. P. de Mello

Apr/2006

103 The Effect of Adverse Supply Shocks on Monetary Policy and Output Maria da Glória D. S. Araújo, Mirta Bugarin, Marcelo Kfoury Muinhos and Jose Ricardo C. Silva

Apr/2006

42

104 Extração de Informação de Opções Cambiais no Brasil Eui Jung Chang e Benjamin Miranda Tabak

Abr/2006

105 Representing Roommate’s Preferences with Symmetric Utilities José Alvaro Rodrigues Neto

Apr/2006

106 Testing Nonlinearities Between Brazilian Exchange Rates and Inflation Volatilities Cristiane R. Albuquerque and Marcelo Portugal

May/2006

107 Demand for Bank Services and Market Power in Brazilian Banking Márcio I. Nakane, Leonardo S. Alencar and Fabio Kanczuk

Jun/2006

108 O Efeito da Consignação em Folha nas Taxas de Juros dos Empréstimos Pessoais Eduardo A. S. Rodrigues, Victorio Chu, Leonardo S. Alencar e Tony Takeda

Jun/2006

109 The Recent Brazilian Disinflation Process and Costs Alexandre A. Tombini and Sergio A. Lago Alves

Jun/2006

110 Fatores de Risco e o Spread Bancário no Brasil Fernando G. Bignotto e Eduardo Augusto de Souza Rodrigues

Jul/2006

111 Avaliação de Modelos de Exigência de Capital para Risco de Mercado do Cupom Cambial Alan Cosme Rodrigues da Silva, João Maurício de Souza Moreira e Myrian Beatriz Eiras das Neves

Jul/2006

112 Interdependence and Contagion: an Analysis of Information Transmission in Latin America's Stock Markets Angelo Marsiglia Fasolo

Jul/2006

113 Investigação da Memória de Longo Prazo da Taxa de Câmbio no Brasil Sergio Rubens Stancato de Souza, Benjamin Miranda Tabak e Daniel O. Cajueiro

Ago/2006

114 The Inequality Channel of Monetary Transmission Marta Areosa and Waldyr Areosa

Aug/2006

115 Myopic Loss Aversion and House-Money Effect Overseas: an Experimental Approach José L. B. Fernandes, Juan Ignacio Peña and Benjamin M. Tabak

Sep/2006

116 Out-Of-The-Money Monte Carlo Simulation Option Pricing: the Join Use of Importance Sampling and Descriptive Sampling Jaqueline Terra Moura Marins, Eduardo Saliby and Joséte Florencio dos Santos

Sep/2006

117 An Analysis of Off-Site Supervision of Banks’ Profitability, Risk and Capital Adequacy: a Portfolio Simulation Approach Applied to Brazilian Banks Theodore M. Barnhill, Marcos R. Souto and Benjamin M. Tabak

Sep/2006

118 Contagion, Bankruptcy and Social Welfare Analysis in a Financial Economy with Risk Regulation Constraint Aloísio P. Araújo and José Valentim M. Vicente

Oct/2006

43

119 A Central de Risco de Crédito no Brasil: uma Análise de Utilidade de Informação Ricardo Schechtman

Out/2006

120 Forecasting Interest Rates: an Application for Brazil Eduardo J. A. Lima, Felipe Luduvice and Benjamin M. Tabak

Oct/2006

121 The Role of Consumer’s Risk Aversion on Price Rigidity Sergio A. Lago Alves and Mirta N. S. Bugarin

Nov/2006

122 Nonlinear Mechanisms of the Exchange Rate Pass-Through: a Phillips Curve Model With Threshold for Brazil Arnildo da Silva Correa and André Minella

Nov/2006

123 A Neoclassical Analysis of the Brazilian “Lost-Decades” Flávia Mourão Graminho

Nov/2006

124 The Dynamic Relations between Stock Prices and Exchange Rates: Evidence for Brazil Benjamin M. Tabak

Nov/2006

125 Herding Behavior by Equity Foreign Investors on Emerging Markets Barbara Alemanni and José Renato Haas Ornelas

Dec/2006

126 Risk Premium: Insights over the Threshold José L. B. Fernandes, Augusto Hasman and Juan Ignacio Peña

Dec/2006

127 Uma Investigação Baseada em Reamostragem sobre Requerimentos de Capital para Risco de Crédito no Brasil Ricardo Schechtman

Dec/2006

128 Term Structure Movements Implicit in Option Prices Caio Ibsen R. Almeida and José Valentim M. Vicente

Dec/2006

129 Brazil: Taming Inflation Expectations Afonso S. Bevilaqua, Mário Mesquita and André Minella

Jan/2007

130 The Role of Banks in the Brazilian Interbank Market: Does Bank Type Matter? Daniel O. Cajueiro and Benjamin M. Tabak

Jan/2007

131 Long-Range Dependence in Exchange Rates: the Case of the European Monetary System Sergio Rubens Stancato de Souza, Benjamin M. Tabak and Daniel O. Cajueiro

Mar/2007

132 Credit Risk Monte Carlo Simulation Using Simplified Creditmetrics’ Model: the Joint Use of Importance Sampling and Descriptive Sampling Jaqueline Terra Moura Marins and Eduardo Saliby

Mar/2007

133 A New Proposal for Collection and Generation of Information on Financial Institutions’ Risk: the Case of Derivatives Gilneu F. A. Vivan and Benjamin M. Tabak

Mar/2007

134 Amostragem Descritiva no Apreçamento de Opções Européias através de Simulação Monte Carlo: o Efeito da Dimensionalidade e da Probabilidade de Exercício no Ganho de Precisão Eduardo Saliby, Sergio Luiz Medeiros Proença de Gouvêa e Jaqueline Terra Moura Marins

Abr/2007

44

135 Evaluation of Default Risk for the Brazilian Banking Sector Marcelo Y. Takami and Benjamin M. Tabak

May/2007

136 Identifying Volatility Risk Premium from Fixed Income Asian Options Caio Ibsen R. Almeida and José Valentim M. Vicente

May/2007

137 Monetary Policy Design under Competing Models of Inflation Persistence Solange Gouvea e Abhijit Sen Gupta

May/2007

138 Forecasting Exchange Rate Density Using Parametric Models: the Case of Brazil Marcos M. Abe, Eui J. Chang and Benjamin M. Tabak

May/2007

139 Selection of Optimal Lag Length inCointegrated VAR Models with Weak Form of Common Cyclical Features Carlos Enrique Carrasco Gutiérrez, Reinaldo Castro Souza and Osmani Teixeira de Carvalho Guillén

Jun/2007

140 Inflation Targeting, Credibility and Confidence Crises Rafael Santos and Aloísio Araújo

Aug/2007

141 Forecasting Bonds Yields in the Brazilian Fixed income Market Jose Vicente and Benjamin M. Tabak

Aug/2007

142 Crises Análise da Coerência de Medidas de Risco no Mercado Brasileiro de Ações e Desenvolvimento de uma Metodologia Híbrida para o Expected Shortfall Alan Cosme Rodrigues da Silva, Eduardo Facó Lemgruber, José Alberto Rebello Baranowski e Renato da Silva Carvalho

Ago/2007

143 Price Rigidity in Brazil: Evidence from CPI Micro Data Solange Gouvea

Sep/2007

144 The Effect of Bid-Ask Prices on Brazilian Options Implied Volatility: a Case Study of Telemar Call Options Claudio Henrique da Silveira Barbedo and Eduardo Facó Lemgruber

Oct/2007

145 The Stability-Concentration Relationship in the Brazilian Banking System Benjamin Miranda Tabak, Solange Maria Guerra, Eduardo José Araújo Lima and Eui Jung Chang

Oct/2007

146 Movimentos da Estrutura a Termo e Critérios de Minimização do Erro de Previsão em um Modelo Paramétrico Exponencial Caio Almeida, Romeu Gomes, André Leite e José Vicente

Out/2007

147 Explaining Bank Failures in Brazil: Micro, Macro and Contagion Effects (1994-1998) Adriana Soares Sales and Maria Eduarda Tannuri-Pianto

Oct/2007

148 Um Modelo de Fatores Latentes com Variáveis Macroeconômicas para a Curva de Cupom Cambial Felipe Pinheiro, Caio Almeida e José Vicente

Out/2007

149 Joint Validation of Credit Rating PDs under Default Correlation Ricardo Schechtman

Oct/2007

45

150 A Probabilistic Approach for Assessing the Significance of Contextual Variables in Nonparametric Frontier Models: an Application for Brazilian Banks Roberta Blass Staub and Geraldo da Silva e Souza

Oct/2007

151 Building Confidence Intervals with Block Bootstraps for the Variance Ratio Test of Predictability

Nov/2007

Eduardo José Araújo Lima and Benjamin Miranda Tabak

152 Demand for Foreign Exchange Derivatives in Brazil: Hedge or Speculation? Fernando N. de Oliveira and Walter Novaes

Dec/2007

153 Aplicação da Amostragem por Importância à Simulação de Opções Asiáticas Fora do Dinheiro Jaqueline Terra Moura Marins

Dez/2007

154 Identification of Monetary Policy Shocks in the Brazilian Market for Bank Reserves Adriana Soares Sales and Maria Tannuri-Pianto

Dec/2007

155 Does Curvature Enhance Forecasting? Caio Almeida, Romeu Gomes, André Leite and José Vicente

Dec/2007

156 Escolha do Banco e Demanda por Empréstimos: um Modelo de Decisão em Duas Etapas Aplicado para o Brasil Sérgio Mikio Koyama e Márcio I. Nakane

Dez/2007

157 Is the Investment-Uncertainty Link Really Elusive? The Harmful Effects of Inflation Uncertainty in Brazil Tito Nícias Teixeira da Silva Filho

Jan/2008

158 Characterizing the Brazilian Term Structure of Interest Rates Osmani T. Guillen and Benjamin M. Tabak

Feb/2008

159 Behavior and Effects of Equity Foreign Investors on Emerging Markets Barbara Alemanni and José Renato Haas Ornelas

Feb/2008

160 The Incidence of Reserve Requirements in Brazil: Do Bank Stockholders Share the Burden? Fábia A. de Carvalho and Cyntia F. Azevedo

Feb/2008

161 Evaluating Value-at-Risk Models via Quantile Regressions Wagner P. Gaglianone, Luiz Renato Lima and Oliver Linton

Feb/2008

162 Balance Sheet Effects in Currency Crises: Evidence from Brazil Marcio M. Janot, Márcio G. P. Garcia and Walter Novaes

Apr/2008

163 Searching for the Natural Rate of Unemployment in a Large Relative Price Shocks’ Economy: the Brazilian Case Tito Nícias Teixeira da Silva Filho

May/2008

164 Foreign Banks’ Entry and Departure: the recent Brazilian experience (1996-2006) Pedro Fachada

Jun/2008

165 Avaliação de Opções de Troca e Opções de Spread Européias e Americanas Giuliano Carrozza Uzêda Iorio de Souza, Carlos Patrício Samanez e Gustavo Santos Raposo

Jul/2008

46

166 Testing Hyperinflation Theories Using the Inflation Tax Curve: a case study Fernando de Holanda Barbosa and Tito Nícias Teixeira da Silva Filho

Jul/2008

167 O Poder Discriminante das Operações de Crédito das Instituições Financeiras Brasileiras Clodoaldo Aparecido Annibal

Jul/2008

168 An Integrated Model for Liquidity Management and Short-Term Asset Allocation in Commercial Banks Wenersamy Ramos de Alcântara

Jul/2008

169 Mensuração do Risco Sistêmico no Setor Bancário com Variáveis Contábeis e Econômicas Lucio Rodrigues Capelletto, Eliseu Martins e Luiz João Corrar

Jul/2008

170 Política de Fechamento de Bancos com Regulador Não-Benevolente: Resumo e Aplicação Adriana Soares Sales

Jul/2008

171 Modelos para a Utilização das Operações de Redesconto pelos Bancos com Carteira Comercial no Brasil Sérgio Mikio Koyama e Márcio Issao Nakane

Ago/2008

172 Combining Hodrick-Prescott Filtering with a Production Function Approach to Estimate Output Gap Marta Areosa

Aug/2008

173 Exchange Rate Dynamics and the Relationship between the Random Walk Hypothesis and Official Interventions Eduardo José Araújo Lima and Benjamin Miranda Tabak

Aug/2008

174 Foreign Exchange Market Volatility Information: an investigation of real-dollar exchange rate Frederico Pechir Gomes, Marcelo Yoshio Takami and Vinicius Ratton Brandi

Aug/2008

175 Evaluating Asset Pricing Models in a Fama-French Framework Carlos Enrique Carrasco Gutierrez and Wagner Piazza Gaglianone

Dec/2008

176 Fiat Money and the Value of Binding Portfolio Constraints Mário R. Páscoa, Myrian Petrassi and Juan Pablo Torres-Martínez

Dec/2008

177 Preference for Flexibility and Bayesian Updating Gil Riella

Dec/2008

178 An Econometric Contribution to the Intertemporal Approach of the Current Account Wagner Piazza Gaglianone and João Victor Issler

Dec/2008