Constructing the US interest rate volatility index

Transcription

1 Constructng the US nterest rate volatlty ndex Anouk G.P. Claes a, Marc J. K. De Ceuster b, Raquel López c,*, Elseo Navarro c,* a Louvan School of Management, Brussels Campus, Facultés Unverstares Sant-Lous, Belgum. b Faculty of Appled Economcs, Unversty of Antwerp, Belgum. c Department of Economc Analyss and Fnance, Unversty of Castlla-La Mancha, Span. E-mal addresses: claes@fusl.ac.be (A. Claes), marc.deceuster@ua.ac.be, (M. De Ceuster), raquel.lopez@uclm.es (R. López), elseo.navarro@uclm.es (E. Navarro). Ths verson: March 2010 Abstract Menton the word volatlty to most traders, and VIX comes to mnd. Surprsngly, much less attenton has been pad to the ntroducton of equvalent leadng ndcators of expected future volatlty n the fxed-ncome market. We suggest for the frst tme the constructon of an mpled volatlty ndex of forward nterest rates from the U.S. cap (floor) market based on the methodology developed n equty dervatves markets. From the results we notce that by September 2006 predctons regardng future nterest rate volatlty suddenly become more varable. That s, approxmately one year before the orgn of the current fnancal crss. Keywords: Impled volatlty, mpled volatlty ndexes, caps JEL Classfcaton: F31; G13 * Raquel and Elseo acknowledge the fnancal support provded by Mnstero de Educacón y Cenca grant ECO /ECON. Raquel acknowledges the fnancal support provded by Junta de Comundades de Castlla-La Mancha grant PCI

2 1. Introducton Volatlty s a basc feature of fnancal markets whose mportance of modellng and predctng s a growng research topc n modern fnance. As far as forecastng performance of future realzed volatlty (ex-post emprcal measure of daly return varablty) of asset prces s concerned, Poon and Granger (2003) show a revew of 93 papers focused on ths topc. Broadly speakng, there are two categores of methods wdely used n makng forecasts of future realzed volatlty: tme seres volatlty forecastng models (based on hstorcal prce nformaton) and opton-based volatlty forecasts (volatlty mpled from opton prces on a partcular underlyng). Overall, results from the prevous study suggest that forecasts based on mpled volatlty often beat forecasts based on hstorcal returns. Impled volatlty measures the market s assessment at tme t of the uncertanty regardng the future development of the asset underlyng an opton, as mpled volatltes are determned va the prces of traded optons wth a concrete tme to maturty, whch s the forecast horzon of the mpled volatlty of the asset. Thus, ths volatlty s forward lookng. Accordng to Got (2005), the sgnfcance of mpled volatlty as a ratonal forecast of future realzed volatlty and the nformaton content of mpled volatlty wth respect to hstorcal volatlty are two mportant research topcs n the academc lterature. In practce, these research topcs have been wdely exploted n stock markets through the constructon of mpled volatlty ndexes from optons on a partcular stock market ndex (see Flemng et al. (1995), Moraux et al. (1999), Bluhm and Yu (2001), Got (2002), and Corrado and Mller (2005)). However, we can hardly fnd research focused on these topcs n the fxed-ncome market. Thus, ths s one of the motvatons of ths paper. 2

3 The model-based methodology appled for the constructon of volatlty ndexes n the equty market conssts of a weghtng scheme of the mpled volatltes of a set of optons computed wthn the overall context of Black-Scholes (1973) opton prcng model or a smlar model. In partcular, mpled volatltes are weghted n such a way that the ndex represents the annualzed mpled volatlty of a partcular stock market ndex underlyng an at the money (ATM) opton wth constant tme to maturty (.e., constant forecast horzon of future expected volatlty). Thus, at any tme t the selecton of optons s done by takng as pont of reference the nearness of the tme to maturty and the strke of the traded optons to the constant tme to expraton establshed for the constructon of the ndex and to the ATM strke, respectccely. In practce, the process of constructon of the mpled volatlty ndexes s carred out by consderng that the value of the ATM strke s ether the spot prce (current value) or the forward prce of the stock market ndex on whch the opton s wrtten. 1 In 1993 the Chcago Board Optons Exchange (CBOE) ntroduced the frst mpled volatlty ndex on a stock market ndex usng data from optons on the S&P100 Index: the S&P 100 Volatlty Index (VIX). VIX very quckly became the benchmark rsk measure for stock market volatlty. It represents the mpled volatlty of an ATM synthetc S&P 100 opton wth constant tme to maturty (30 calendar days) at any pont n tme (detals regardng the constructon of VIX are avalable n Flemng et al. (1995)). 2 Followng the example of the CBOE, other optons markets ntroduced ther own volatlty ndexes n Europe. In 1994 the Deutsche Börse created a volatlty ndex for the German stock market: VDAX, from optons on the DAX Index (see Lyons (2005) for a detaled descrpton of the constructon process of VDAX). In 1997 the MONEP (Marché des Optons négocables de Pars) ntroduced VX1 and VX6 ndexes to 1 Black (1976) extended the Black-Scholes model to prce European optons on an asset n terms of the future (or forward) prce for a contract maturng at the same tme as the opton. 2 In 2003 the CBOE ntroduced the new VIX, computed from optons on the S&P 500 rather than the S&P

4 measure the uncertanty concernng the French stock market from optons on the CAC- 40 Index (see Moraux et al. (1999) for further detals about these ndexes). 3 Menton the word volatlty to most traders, and VIX comes to mnd. Surprsngly, much less attenton has been pad to the ntroducton of equvalent leadng ndcators of expected future volatlty n the fxed-ncome market. To the best of our knowledge, up to now, the Merrll Lynch Opton Volatlty Index (MOVE) and the Lehman Brothers Swapton Volatlty Index (LBPX), constructed as a weghted average of mpled volatltes of Treasury bond optons and a basket of lqud swaptons for dfferent terms to maturty of the underlyng nstrument: Treasury bonds and swaps, respectvely, have been the only attempts to measure expectatons of future volatlty from traded nterest rate dervatves. In ths study we suggest for the frst tme the constructon of a pure measure of the expected future volatlty referred to a partcular forward nterest rate (wth a concrete perod of reference) based on the model-based methodology appled n equty dervatves markets. The nterest rate volatlty ndex (IRVIX) s constructed from data of the U.S. cap (floor) market. Caps (floors) are portfolos of optons on nterest rates traded n the over-the-counter (OTC) nterest rate dervatves market, one of the most lqud OTC dervatves markets n the world. Informaton provded by the market conssts of mpled flat volatlty quotes of caps (floors), where mpled volatltes are computed by equallng the market prces of such dervatves and the Black (1976) model prce appled to all the caplets (floorlets) that compose the cap (floor) by assumng that the volatlty of forward nterest rates underlyng every opton s constant. Thus, flat volatltes do not enable us to know what the perod of reference of the underlyng forward nterest rate whose mpled volatlty has been estmated s. 3 An attempt to create mpled volatlty ndexes n the context of emergng markets can also be found n Skadopoulos (2004) for the Greek dervatves market. 4

5 We focus on ths tem n order to construct mpled volatlty ndexes of forward nterest rates wth a concrete perod of reference by usng spot volatltes recovered from flat volatlty quotes (.e., mpled volatltes of caplets (floorlets) wth a concrete perod of reference) and then applyng the methodology developed n equty markets. The nterest rate volatlty ndex constructed lke that ams to represent the annualzed mpled volatlty of the forward nterest rate underlyng an ATM caplet wth a fxed tme to maturty, whch s the forecast horzon of the expected future nterest rate volatlty. From our pont of vew, a wde range of applcatons can arse from the ntroducton of these ndexes. Next we consder some broad lnes of potental applcatons. On the one hand, analyzng whether nterest rate volatlty as measured by the ndex contans addtonal useful nformaton about the future state of economy to that broadly documented of the term structure of nterest rates s probably one of the most attractve applcatons. On the other hand, ths measure of forward nterest rate volatlty mght also be appled n the study of the mpact of monetary polcy on nterest rate volatlty, as well as used n the valuaton of more complex nterest rate dervatves such as swaptons. Fnally, one of the potental applcatons of the mpled volatlty ndexes s the possblty of ntroducng futures and optons on such ndexes, as occurred n the US after the launch of the VIX. On February 2006 optons on VIX began tradng on the CBOE, followng the prevous ntroducton of VIX futures on the CBOE Futures Exchange (CFE) n Accordng to Areal (2008), n practce these dervatves can be used n turn to create hedge strateges aganst changes n volatlty, or to speculate on changes n the market volatlty. In ths paper we analyze the behavour and statstcal propertes of four mpled volatltes ndexes coverng four dfferent forecast horzons over the perod from July 30, 2004 to January 30, The fact of the sample perod comprsng the orgn of the 5

6 current fnancal crss s especally relevant n order to vsualze the nformaton content of the ndexes as leadng ndcators of busness cycle. The structure of the paper s as follows. The next secton s focused on caps (floors) valuaton wthn the LIBOR Market Model (LMM) framework. In secton three we present the constructon process of volatlty ndexes n equty dervatves markets and how to mplement such methodology from cap (floor) market data. Secton four s amed at the descrpton of the database and the methodology appled for the constructon of the nterest rate volatlty ndex (IRVIX). In secton fve the behavour and statstcal propertes of the volatlty ndexes are analyzed. Fnally, secton sx provdes a summary of the study. 2. Caps and floors valuaton. The LIBOR Market Model and the Black formula A forward rate agreement (FRA) s the underlyng of one of the smplest nterest rate optons: the caplets (floorlets). A FRA can be defned as an agreement between two partes at tme t to exchange at tme T +τ an amount of money proportonal to the dfference between a strke, K, agreed upon at tme t, and the floatng nterest rate, R(T, T + τ), that resets at tme T. The proportonalty factor s gven by the product of the notonal prncpal, NP, and the tenor nterval, τ. The addtve sum of nterest rate optons on FRAs gves rse to caps (floors), one of the most popular nterest rate dervatves offered by fnancal nsttutons n the OTC market. Each opton composng the cap (floor) has the same strke and the same perod of lfe, tenor (tme dstance between floatng nterest rate resets), as the others, but a dfferent expraton date (the expraton date of an opton, T +τ, s the same as the exercse date, T, of the followng one). Typcally, the expraton dates for the caplets (floorlets) are on the same cycle as the frequency of the underlyng floatng rate (Longstaff et al., 2001). 6

7 Caps are desgned to hedge the nterest rate rsk created by the varablty of the floatng rate n some fnancal contracts where market partcpants pay cash flows ted to some floatng rate. Next we descrbe the way payoffs take place n a cap. On the frst reset date of the cap, the floatng rate of the contract s observed and compared to the strke. If the floatng rate s greater than the strke, then on the second reset date the seller of the cap pays the holder the dfference between the floatng rate and the strke multpled by the notonal prncpal and the tenor (f the floatng rate s less than the strke, there s no payoff from the cap). Thus, through the lfe of a cap, payments are done at the end of each tenor nterval although ts amount s fxed at the reset date (at the begnnng of the tenor nterval) when the nterest rate s observed. 4 Analogously to caps, a floor provdes a payoff when the nterest rate n some fnancal contract ted to a floatng rate falls below a certan rate. That s, the floor provdes nsurance aganst the nterest on the floatng rate of a contract fallng bellow a certan level. Next we show a bref overvew of the LIBOR Market Model (LMM) valuaton framework, whch leads to the Black (1976) prcng formula for caps (caplets) and floors (floorlets) used by market practtoners. As descrbed prevously, the payoff derved from a caplet at maturty, T +τ, s gven by 5 : + Payoff T+τ = NP [ f ( T T T + τ ) K ] τ, [1], Payoff T+τ =[ NP f ( T, T, T + τ ) τ NP τ K ] [2] + where f(t, T, T +τ) denotes the tme t forward rate applyng between T and T +τ, wth t pror to T. Notce that at reset, T, the forward rate s set by defnton to be equal to the correspondng nterest rate, R(T, T + τ): 4 Caps are usually defned so that the ntal floatng rate, even f t s greater than the cap rate, does not lead to a payoff on the frst reset date (Hull, 2009). 5 Analogous expressons for floorlets can easly be derved. 7

8 ( T T, T τ ) R( T T +τ ) f,, [3] + In order to obtan the prce of a caplet at tme t before T, we consder that the value of the forward rate f(t, T, T +τ) can be replcated from a portfolo of traded assets (see Díaz et al., 2009). The present value of ths portfolo π (t) s gven by assumng, for each unt of the notonal prncpal NP, a long poston on a zero coupon maturng at T and a short poston on a zero coupon bond maturng at T +τ. Thus, we have [ P( t, T ) P( t, T )] π ( t ) NP + τ [4] = Now t follows that f we renvest the prncpal payment of the shorter bond n the zero coupon bond maturng at T +τ, ths portfolo produces the same payoff as the floatng leg of the caplet: the frst term on the rght hand sde of Equaton (2). That s, [(1 + R( T, T + τ ) τ ) 1] = NP f ( T, T, T + τ τ π T + τ ) = NP ) [5] ( Then, by applyng a no-arbtrage argument, we should verfy the followng equalty: P ( t, T) P( t, T + τ ) = f ( t, T, T + τ ) τ P( t, T + τ ) [6] P( t, T ) P( t, T + τ ) f ( t, T, T + τ ) P( t, T + τ ) = τ [7] The LMM assumes that the forward nterest rate f(t, T, T +τ) follows a lognormal stochastc process and, therefore, under the forward measure Q, the arbtrage portfolo f ( t, T, T τ ) P( t, T + τ ) dscounted by the numerare P( t, T + τ ), + 8

9 that s, the forward rate f(t, T, T +τ), must follow a martngale (a zero drft-stochastc process): df ( t, T, T + τ ) = σ ( t, T )dz f ( t, T, T + τ ) [8] where dz s a standard Wener process. Concernng the volatlty functon, σ ( t, T ), the LMM approach s characterzed by mposng that the volatlty functons of the forward rates should be restrcted to beng determnstc functons of tme (Rebonato, 2002). Next, takng nto account that the forward nterest rate equals the expected future nterest rate n a world that s forward rsk neutral wth respect to a zero-coupon bond maturng at tme T +τ, P ( t, + τ ), we show the condtonal dstrbuton of the natural T logarthm of the forward rate after applyng Itô s lemma: ln[ f ( T, T, T + τ )] ~ G(ln[ f ( t, T, T + τ )] σ, Black ( T t); σ, Black ( T t)) [9] 2 where G( ) denotes the Gaussan dstrbuton, and the volatlty of changes (from t to T ) n the logarthm of the forward rate, σ, Black, can be understood as an average of the forward nstantaneous volatlty, σ ( t,t ), over the perod [t, T ]: T, Black t 2 2 σ ( T t) = σ ( u, T ) du [10] Fnally, when the payoff of the caplet s ntegrated over the log-normal dstrbuton, one recovers the market-standard Black formula for caplets valuaton at tme t pror to T : 9

10 [ f ( t, T, T + τ ) N( h ) K N( h )] P( t, + τ τ Caplet( t, T, τ, NP, σ, Black ) = NP 1 2 T ) [11] where h 1 ln = [ f ( t, T, T + τ ) / K ] σ, Black σ, 2 ( T t) Black ( T t) [12] and h 2 ln = [ f ( t, T, T + τ ) / K] σ, Black 1 2 σ, 2 ( T t) Black ( T t) [13] and P ( t, + τ ) denotes the value at tme t of a zero coupon bond payng 1 unt at tme T +τ. T It s now a smple step to compute the present value of a cap as the sum of the present values of ts caplets. That s, n Cap n = caplet ( σ ) [14] = 1 Note that the prce of the cap s computed by assumng that the volatlty of all the caplets that compose the cap s constant. Indeed, the market conventon for caps (floors) s to quote cap prces n terms of the mpled value of σ whch sets the Black model prce equal to the market prce (Longstaff et al., 2001). These volatltes are then referred to as flat volatltes. 3. Implementng the equty market methodology from cap market data In ths secton frst we descrbe the process appled n equty dervatves markets to construct mpled volatlty ndexes of stock market ndexes returns. In partcular, the 10

11 calculaton methodology of the US mpled volatlty ndex VIX s presented. Then, we show how to mplement the equty market methodology for the constructon of nterest rate volatlty ndexes from cap (floor) market data. The key dea under ths methodology s the selecton of a set of optons accordng to ther tme to maturty and strke n order to obtan at any pont n tme the market s assessment of the expected future volatlty of the asset underlyng an ATM opton wth constant tme to maturty (.e., constant forecast horzon of future volatlty). VIX s based on a weghtng scheme of the Black-Scholes mpled volatltes on eght nearest-to-the-money S&P 100 call and put optons at the two nearest maturtes to the constant tme to expraton establshed for the constructon of the ndex: 30 calendar days (22 tradng days). That s, two pars of call and put optons are selected for the nearby and second nearby optons. The value of the underlyng denotng the ATM strke s the current ndex level. Accordng to Poon and Granger (2003), snce dfferent mpled volatltes are recovered from optons wth the same tme to maturty but dfferent strkes, a decson has to be made about whch of these mpled volatltes should be used n order to acheve the best forecast of future realzed volatlty. In ths sense, the most common strategy conssts of selectng the mpled volatltes derved from ATM optons snce these are the most lqud optons and hence measurement errors are less probable to occur. In case that ATM optons were not avalable (whch s a very common stuaton when we are nterested n computng daly mpled volatltes along a concrete perod of tme), then nearest-to-the-money optons are used nstead through a weghtng scheme wth the am of obtanng a value for the mpled volatlty that s approxmately ATM. Thus, ths s the decson adopted n the constructon of mpled volatlty ndexes from optons on stock market ndexes. The weghtng process s carred out through three steps. Frst, the mpled volatltes of the four pars of call and put optons wthn the four categores of optons are averaged. Second, at each maturty, the two average volatltes at the two strkes 11

12 that straddle the spot level and are nearest to t are lnearly nterpolated to obtan ATM mpled volatltes. Fnally, the nearby and second nearby ATM volatltes are lnearly nterpolated to create a constant 30-calendar day (22-tradng day) mpled volatlty ndex, whch consttutes the VIX. The man dfferences across countres n the mplementaton of the methodology for the constructon of volatlty ndexes are those regardng the number and type of optons (call and/ or put optons, European or Amercan optons ) selected, and the value of the underlyng asset denotng the ATM strke of the opton: the spot prce (current value of the asset) or the forward prce of the asset. Implementng ths methodology to construct the nterest rate volatlty ndex from cap (floor) market data mples frst recoverng spot volatltes from flat volatlty quotes (.e., mpled volatltes of forward nterest rates underlyng a partcular caplet (floorlet)). Ths process s commonly known as strppng process. The mpled volatltes of caps (floors) nvolved n the strppng process must be chosen accordng to the selecton crtera establshed by the methodology appled n equty dervatves markets n order to obtan a measure of the expected future volatlty of the forward nterest rate underlyng an ATM caplet wth constant tme to maturty. Unlke stock markets, where mpled volatltes from optons can be computed at any tme before the expraton date of the opton, mpled volatltes of forward nterest rates recovered from caplets (floorlets) have a fxed tme to maturty: from t to T (exercse date of the opton). Thus, snce mpled forward rate volatltes have a constant term to maturty, the only crtera we must consder n order to mplement the equty market methodology wthn the strppng process s the strke. That s, the only crtera to select the flat volatlty quotes of caps (floors) nvolved n such process s the nearness of ther strkes to the ATM strke of a caplet (floorlet). 12

13 Accordng to the Black prcng formula, a caplet (floorlet) s sad to be ATM f the forward rate, f(t, T, T +τ), nvolved n such opton s equal to the strke of ts correspondng cap (floor). The value of the forward nterest rate s computed accordng to the followng formula: P( t, T ) 1 f ( t, T, + τ ) = 1 (, τ ) T [15] P t T + τ wth P t, T ) and P ( t, + τ ) denotng the values at tme t of two zero coupon bonds ( T payng 1 unt at maturty: T and T +τ, respectvely. The constructon process of the nterest rate volatlty ndex (IRVIX) s descrbed n detal n the next secton. 4. Data and methodology In conductng ths study, we use two types of data from the U.S. fxed-ncome market provded by Reuters: market-mpled flat volatltes of caps (floors) for dfferent strkes and terms to maturty 6, and the zero coupon curves (dscount factors bootstrapped from the most lqud rate nstruments that are avalable: a combnaton of deposts, lqud futures and nterest rate swaps). Daly data have been collected for the perod from July 30, 2004 to January 30, Flat mpled volatltes are recovered from caps (floors) wth maturtes from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15 to 20 years, and for the followng range of strkes: 0,01; 0,015; 0,02; 0,025; 0,03; 0,035; 0,04; 0,05, and 0,06. These strkes represent values above and below the ATM strkes for caps (floors). Accordng to Hull (2009), a cap 6 Informaton provded by Reuters conssts of flat volatlty quotes of caps/ floors: at a partcular strke and for a concrete term to maturty, traders may contract the same nstrument as a cap or a floor dependng on the crcumstances. 13

14 (floor) s sad to be ATM f the strke of such an nstrument equals the swap rate for a swap that has the same payment dates as the cap. Next we descrbe the process of constructon of the nterest rate volatlty ndex. As stated n the prevous secton, the process accordng to whch spot volatltes are recovered from flat volatlty quotes s commonly known as strppng process. The strppng process conssts of obtanng at any tme t the prce of the caplet caplet ( t, T, T + τ ) by subtractng the prces of two consecutve caps computed from flat volatlty quotes of the correspondng terms to maturty. 7 That s, Cap ( t, T τ ) Cap( t, T ) = Caplet( t, T, T + τ ) [16] + wth Cap ( t, + τ ) and Cap t, T ) denotng the prces of the caps maturng at T +τ T ( (expraton date of the caplet) and T (exercse date of the caplet), respectvely. Then, the mpled volatlty of the caplet, formula 8. σ ( t, T, T + τ ), s extracted from the Black prcng Implementng the equty market methodology for the constructon of the nterest rate volatlty ndex, IRVIX, frst mples computng at any tme t the value of the forward nterest rate, f(t, T, T +τ), that represents, accordng to the Black formula, the ATM strke of a caplet. As presented n Equaton (15): P( t, T ) 1 f ( t, T, + τ ) = 1 (, τ ) T [17] P t T + τ 7 Only caps notaton s used n the descrpton of the methodology. 8 US caps (floors) have a 3-month tenor. Notce that we need all the caps to have the same tenor n order to apply the requred nterpolaton technques to obtan ntermedate ponts along the term structure of nterest rate volatlty from avalable maturtes of caps. 14

15 Next, the two flat volatlty quotes nvolved n the strppng process, σ t, T ) and σ ( t, + τ ), are selected at the two strkes closest to the value of the forward rate: the T strke just above that level, strke out the money (OTM), and the strke just below that level, strke n the money (ITM), respectvely. 9 ( Notce that for maturtes dfferent from those quoted by the market for flat volatltes of caps, nterpolaton and extrapolaton technques must be appled. Followng the research work developed by Hernández, L.G. (2005), we nterpolate flat volatltes by usng cubc splnes n order to obtan smoother curves than by usng lnear nterpolaton (the market practce). Ths allows fttng better the typcal humped pattern for flat and spot volatltes as a functon of maturty. If at a gven date the number of avalable flat volatlty quotes for dfferent maturtes and for a partcular strke s less than sx then, f possble, we use lnear nterpolaton. 10 Then, from the two flat volatltes quotes selected at the two strkes closest to the value of the forward rate we can obtan the prces of the two caps mpled n the strppng process and, thus, the prces of the two nearest-to-the money caplets. Fnally, the two mpled volatltes of the caplet, σ ( t, T, T + τ ), recovered from the strke OTM (K A ) and the strke ITM (K B ), are lnearly nterpolated to create the nterest rate volatlty ndex, IRVIX, accordng to the followng expresson: IRVIX A Caplet K f ( t, T, T + τ ) f ( t, T, T + τ ) K + τ ) = σ B B Caplet ( t, T + ) + +, T τ σ A ( t, T A B K T τ ) A K K K K ( t, T, T K, B [18] 9 Note that volatlty quotes of ATM caps (the most lqud caps traded) can not be used as a ratonal approxmaton to the value of the forward rate f(t, T, T +τ) as flat cap volatltes quotes nvolved n the strppng process for the terms to maturty T and T +τ must have the same strke. And ATM cap volatltes have a dfferent strke for every term to maturty. 10 If at a partcular date, nether cubc splne nterpolaton nor lnear nterpolaton can be appled, the mpled volatlty of the caplet s computed as the one of the prevous day. 15

16 where IRVIX ( t, T, T + τ ) represents the annualzed (accordng to the actual/360 day count conventon) mpled volatlty of the forward nterest rate f(t, T, T +τ) underlyng an ATM caplet wth constant tme to maturty (from t to T, exercse date of the opton). Accordng to Flemng et al. (1995), the lnear nterpolaton of mpled volatltes from OTM and ITM optons to create an ATM mpled volatlty mplctly assumes that the volatlty smle s well approxmated by a lne. Thus, ths approxmaton s consdered reasonable when the nterpolaton s made for a small range of strkes. In ths case, the two strkes closest (above and below) to the ATM strke of a caplet for a concrete term to maturty. In ths study we daly construct four mpled volatlty ndexes of forward nterest rates for the followng tenor ntervals: 1 year to 1 year and 3 months (1Y, 1Y+3M), 1 year and 3 months to 1 year and 6 months (1Y+3M, 1Y+6M), 1 year and 6 months to 1 year and 9 months (1Y+6M, 1Y+9M), and 1 year and 9 months to 2 years (1Y+9M, 2Y). 11 That way, the mpled volatlty ndex IRVIX ( t,1y,1y + 3M ) measures the market s assessment at any tme t of the uncertanty regardng the 3-month nterest rate from t to t plus 1 year. 5. Emprcal analyss From July 30, 2004 to January 30, 2009 we analyze the evoluton and statstcal propertes of the four mpled volatlty ndexes that have been constructed: IRVIX ( t,1y,1y + 3M ), IRVIX ( t,1y + 3M,1Y + 6M ), IRVIX ( t,1y + 6M,1Y + 9M ), and IRVIX ( t,1y + 9M,2Y ). Fgures 1 to 4 plot the daly levels of the ndexes. Graphs of the seres n levels are provded n order to show more clearly the behavor and evoluton of the ndexes 11 These perods represent the four closest forecast horzons of future nterest rate volatlty. Accordng to Duarte et al. (2007), the most-lqud cap maturtes are one, two, three, four, fve, seven and ten years, thus, t s the ntenton of the authors constructng ndexes coverng these partcular forecast horzons. 16

17 over the perod, as t may help to better understand the statstcal propertes of the seres after beng transformed. Besdes, the nformaton content of the daly evoluton of the ndexes ncreases due to the sample perod comprsng the orgn of the current fnancal crss. Thus, next we analyze the behavour of the ndexes n levels durng the sample perod. [INSERT FIGURES 1-4] As suggested by the graphs, durng the sample perod the mpled volatlty ndexes are far from beng statonary. Moreover, the frst order autocorrelaton of 99% supports the dea that the seres appear to be near-random walk. We can observe that the trend n the evoluton of the four volatlty ndexes s qute smlar across the whole sample. At the begnnng of the sample perod the evoluton of the ndexes s characterzed by slght decreasng trend from August 2004 up to August From that date to approxmately September 2006, evoluton drawn by the volatlty ndexes shows a perod of maxmum stablty. Then, from September 2006 onwards the seres start to show frequent small-szed spkes. That s, approxmately one year before the orgn of the current fnancal crss, when the levels of the volatlty ndexes remarkably ncrease and larger (up and down) spkes are observed, predctons regardng future nterest rate volatlty become more varable. Thus, the emprcal evdence seems to suggest the exstence of a change n the behavor of the volatlty ndexes around September Wthn the context of the current fnancal crss, t seems to be a remarkable sgn of the potental applcaton of mpled volatlty ndexes of forward rates as leadng ndcators of busness cycle. In ths sense, n order to analyze the statstcal propertes of the ndexes we dvde the sample perod nto two subsamples: the frst subsample (from July 30,

18 to August 31, 2006) and the second subsample (from September 01, 2006 to January 30, 2009). Next, we mplement an analyss of the statstcal propertes of the ndexes based on frst dfferences (daly volatlty changes). Accordng to Flemng et al. (1995), the varable of nterest for academcs and practtoners s changes or nnovatons to expected volatlty as they want to know how changes n expected volatlty nfluence changes n securty valuaton. Table 1 shows the summary statstcs of the frst dfferences n the mpled volatlty ndexes over the whole sample (Panel A) and for the frst and second subsamples (Panel B and C, respectvely). [INSERT TABLE 1] The average value of daly changes n the volatlty ndexes reported for the second subsample s hgher than for the frst subsample. Furthermore, wthn the second subsample, the average future expected volatlty decreases over the forecast horzon. As well as the mean, the standard devaton (volatlty of volatlty) s hgher n the second subsample. Moreover, snce the seres of daly levels of the mpled volatlty ndex IRVIX ( t,1y + 9M,2Y ) evdences more frequent small-szed spkes over the tme, the volatlty of daly volatlty ndexes changes s hgher for the ndex wth furthest away forecast horzon n both subsamples. The seres show slght negatve skewness (except for the volatlty ndex maturng n 1 year and 9 months) and sgnfcant excess kurtoss (leptokurtoss). The frst order autocorrelaton coeffcents and the Augmented Dckey Fuller (ADF) test values are also provded. The autocorrelaton structure of the daly volatlty ndexes changes vares over the forecast horzon. Fnally, the values of the Augmented Dckey 18

19 Fuller (ADF) test evdence that the mpled volatlty ndexes are statonary n the frst dfferences. From the evdence reported regardng non-normalty n the frst dfferences of the ndexes we ntroduce a frst ln-dfference transformaton n the seres n levels. Daly evoluton of frst dfferences of ln mpled volatlty ndexes (.e., the day to day percentage change n the volatlty ndexes) s shown through Fgures 5 to 8. The graphs of the seres suggest agan the exstence of a change n the behavor of the seres around September [INSERT FIGURES 5 TO 8] Table 2 reports the statstcal propertes of the seres n frst ln-dfferences over the whole sample (Panel A) and for the frst and second subsamples (Panel B and C, respectvely). [INSERT TABLE 2] As expected, the excess kurtoss reported for the four ndexes n the frst dfferences has decreased, but t stll remans. Accordng to Dotss et al. (2007), the evdence of non-normalty may be attrbuted to the presence of jumps n mpled volatlty. Fgure 9 shows the emprcal dstrbuton of the seres n frst ln-dfferences. The frst order autocorrelaton coeffcents reveal a statstcally sgnfcant negatve autocorrelaton (expect for the volatlty ndex maturng n 1 year and 3 months). Ths degree of correlaton s stronger for the volatlty ndex maturng n 1 years and 9 months. The evdence of negatve autocorrelaton suggests the presence of mean reverson n the daly ln mpled volatlty ndexes changes. The same results (excess kurtoss and negatve frst-order autocorrelaton) are usually reported for most of the mpled volatlty ndexes ntroduced n stock markets (see Dotss et al. (2007) and 19

20 Konstantnd et al. (2008)). Fnally, the ADF test allows rejectng the null hypothess of a unt root n the seres. [INSERT FIGURE 9] 6. Summary and conclusons The model-based methodology appled for the constructon of volatlty ndexes n the equty market conssts of a weghtng scheme of the mpled volatltes of a set of optons computed wthn the overall context of Black-Scholes (1973) opton prcng model or a smlar model. In partcular, mpled volatltes are weghted n such a way that the ndex represents the annualzed mpled volatlty of a partcular stock market ndex underlyng an at the money (ATM) opton wth constant tme to maturty (.e., constant forecast horzon of future expected volatlty). VIX n the US, VDAX n Germany, and VX1 n France, are some benchmark rsk measures for stock market volatlty. In ths study we suggest for the frst tme the constructon of a pure measure of the expected future volatlty referred to a partcular forward nterest rate (wth a concrete perod of reference) based on the methodology appled n equty dervatves markets. The nterest rate volatlty ndex (IRVIX) s constructed from data of the U.S. cap (floor) market. Informaton provded by the market conssts of mpled flat volatlty quotes of caps (floors), where mpled volatltes are computed by equallng the market prces of such dervatves and the Black (1976) model prce appled to all the caplets (floorlets) that compose the cap (floor) by assumng that the volatlty of forward nterest rates underlyng every opton s constant. Thus, flat volatltes do not enable us to know what the perod of reference of the underlyng forward nterest rate whose mpled volatlty has been estmated s. 20

21 We focus on ths tem n order to construct mpled volatlty ndexes of forward nterest rates wth a concrete perod of reference by usng spot volatltes recovered from flat volatlty quotes (.e., mpled volatltes of caplets (floorlets) wth a concrete perod of reference) and then applyng the methodology developed n equty markets. The nterest rate volatlty ndex (IRVIX) constructed lke that ams to represent the annualzed mpled volatlty of the forward nterest rate underlyng an ATM caplet wth a fxed tme to maturty, whch s the forecast horzon of the expected future nterest rate volatlty. Some of the potental applcatons of the ndex are ncluded next. The analyss of the nformaton content of the volatlty ndexes as leadng ndcators of busness cycle s perhaps one of the most attractve applcatons. The ndexes mght also be appled for the study of the mpact of monetary polcy on nterest rate volatlty and for the valuaton of more complex nterest rate dervatves such as swaptons. Fnally, the volatlty ndexes mght gve rse to the ntroducton of futures and optons on such ndexes, as occurred n the US after the launch of the stock volatlty ndex VIX. Over the perod from July 30, 2004 to January 30, 2009 we daly construct four mpled volatlty ndexes of forward nterest rates for the followng tenor ntervals: 1 year to 1 year and 3 months (1Y, 1Y+3M), 1 year and 3 months to 1 year and 6 months (1Y+3M, 1Y+6M), 1 year and 6 months to 1 year and 9 months (1Y+6M, 1Y+9M), and 1 year and 9 months to 2 years (1Y+9M, 2Y). From the behavour of the four ndexes over the sample perod, we notce that from September 2006 onwards, after approxmately one year of maxmum stablty, the seres start to show frequent small-szed spkes. That s, predctons regardng future nterest rate volatlty become more varable approxmately one year before the orgn of the current fnancal crss. It seems to be a remarkable sgn of the potental applcaton of mpled volatlty ndexes of forward rates as leadng ndcators of busness cycle. 21

22 Fnally, the statstcal propertes of the seres after the ntroducton of a frst lndfference transformaton show excess kurtoss (leptokurtoss) and sgnfcant negatve frst-order autocorrelaton. The same evdence holds for most of the mpled volatlty ndexes n stock markets, where the non-normalty s sometmes attrbuted to the presence of jumps and the negatve frst-order autocorrelaton supports the modellng of mpled volatlty ndexes as mean revertng processes. 22

23 References Areal, N. (2008): FTSE-100 mpled volatlty ndex, Workng Paper Seres. Black, F. and Scholes, M. (1973): The prcng of optons and Corporate Labltes, The Journal of Poltcal Economy, Vol. 81, Nº 3, pp Black, F. (1976): The prcng of commodty contracts, Journal of Fnancal Economcs 3, Bluhm, H. and Yu, J. (2001): Forecastng volatlty: evdence from the German stock market, Economcs Workng Paper Seres (the Unversty of Auckland). Corrado, C. and Mller, Jr. T. (2005): The forecast qualty of CBOE mpled volatlty ndexes, The Journal of Future Markets, Vol. 25, Nº 4, pp Díaz, A., Meneu, V. and Navarro, E. (2009): Internatonal evdence on alternatve models of the term structure of volatltes, The Journal of Future markets, Vol. 29, Nº 7, pp Dotss, G., Psychoyos, D. and Skadopoulos, G. (2007): An emprcal comparson of contnuous-tme models of mpled volatlty ndces, Journal of Bankng and Fnance, Vol. 31, Nº 12, pp Duarte, J., Longstaff, F.A. and Yu, F. (2007): Rsk and Return n Fxed-ncome Arbtrage: Nckels n front of a streamroller?, Revew of Fnancal Studes, Vol. 20, Nº 3, pp Federal Home Loan Bank of Seattle (2007): What Counts: Q Flemng, J., Ostdek, B. and Whaley, R. (1995): Predctng stock market volatlty: A new measure, The Journal of Futures Markets, Vol. 15, No. 3, Got, P. (2002): The nformaton content of mpled volatlty ndexes for forecastng volatlty and market rsk, Workng Paper Seres. Got, P. (2005): Relatonshps Between Impled Volatlty Indexes and Stock Index Returns. Are mpled volatlty ndexes leadng ndcators?, The Journal of Portfolo Management 31,

24 Hernández, L.G. (2005): Prcng of Game Optons n a market wth stochastc nterest rates. Thess ( Hull, J. (2009): Optons, futures and other dervatves, New Jersey, Pearson Prentce Hall. Konstantnd, E., Skadopoulos, G. and Tzagkarak, E. (2008): Can the evoluton of mpled volatlty be forecasted? Evdence from European and US mpled volatlty ndces, Journal of Bankng and Fnance, 32, Longstaff, F., Santa-Clara P. and Schwartz, E. (2001): The Relatve Valuaton of Caps and Swaptons: Theory and Emprcal Evdence, The Journal of Fnance, Vol. LVI, 6, Lyons, L. (2005): Volatlty and ts Measurements: The Desgn of a Volatlty Index and the Executon of ts Hstorcal Tme Seres at the DEUTSCHE BÖRSE AG. Thess ( Volatlty_ and_ts_measurements.pdf). Moraux, F., Navatte, P. and Vlla, C. (1999): The Predctve Power of the French Market Volatlty Index: A Mult Horzons Study, European Fnance Revew 2: Poon, S-H. and Granger, C. (2003): Forecastng Volatlty n Fnancal Markets: A Revew, Journal of Economc Lterature, Vol. XLI, pp Rebonato, R. (2002): Modern Prcng of Interest-Rate Dervatves. The LIBOR Market Model and Beyond, New Jersey, Prnceton Unversty Press. Skadopoulos, G. (2004): The Greek mpled volatlty ndex: constructon and propertes, Appled Fnancal Economcs, Vol. 14, Nº 16,

25 FIGURE 1. Daly levels of the lmpled Volatlty Index perod from July 30, 2004 to January 30, IRVIX ( t,1y,1y + 3M ) durng the /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/2009 FIGURE 2. Daly levels of the Impled Volatlty Index IRVIX ( t,1y + 3M,1Y + 6M ) durng the perod from July 30, 2004 to January 30, /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/

26 FIGURE 3. Daly levels of the Impled Volatlty Index IRVIX ( t,1y + 6M,1Y + 9M ) durng the perod from July 30, 2004 to January 30, /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/2009 FIGURE 4. Daly levels of the Impled Volatlty Index IRVIX ( t,1y + 9M,2Y ) durng the perod from July 30, 2004 to January 30, /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/ /04/ /07/ /10/ /01/

27 FIGURE 5. Frst ln-dfferences n the Impled Volatlty Index IRVIX ( t,1y,1y + 3M ) the perod from July 30, 2004 to January 30, durng /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/2008 FIGURE 6. Frst ln-dfferences n the Impled Volatlty Index durng the perod from July 30, 2004 to January 30, IRVIX ( t,1y + 3M,1Y + 6M ) /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/

28 FIGURE 7. Frst ln-dfferences n the Impled Volatlty Index durng the perod from July 30, 2004 to January 30, IRVIX ( t,1y + 6M,1Y + 9M ) /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/2008 FIGURE 8. Frst ln-dfferences n the Impled Volatlty Index the perod from July 30, 2004 to January 30, IRVIX ( t,1y + 9M,2Y ) durng /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /12/

29 FIGURE 9. Hstograms of Frst ln-dfferences n the Impled Volatlty Indexes from July 30, 2004 to January 30, The contnuous curves overlayng the hstograms correspond to normal dstrbutons wth the same mean and standard devaton as the seres. 29

30 TABLE 1.- Summary statstcs of Frst dfferences n the Impled Volatlty Indexes. [1Y,1Y+3M] [1Y+3M,1Y+6M] [1Y+6M,1Y+9M] [1Y+9M,2Y] Panel A: Summary statstcs from the whole sample: July 30, 2004 to January 30, 2009 Mean Standard Dev Skewness Kurtoss ρ * 0.08 * * ADF ** ** ** ** Panel B: Summary statstcs from the frst subsample: July 30, 2004 to August 31, 2006 Mean Standard Dev Skewness Kurtoss Panel C: Summary statstcs from the second subsample: September 01, 2006 to January 30, 2009 Mean Standard Dev Skewness Kurtoss Entres report the summary statstcs of the four mpled volatlty ndexes n the frst daly dfferences for the whole sample (Panel A) and for the frst and second subsamples (Panel B and C, respectvely). The frst order autocorrelaton ρ 1 and the Augmented Dckey Fuller (ADF) test values are also reported for the entre sample. One astersk denotes statstcal sgnfcance at a 5% confdence level. Two astersks denote statstcal sgnfcance at a 1% confdence level. 30

31 TABLE 2.- Summary statstcs of Frst ln-dfferences n the Impled Volatlty Indexes. [1Y,1Y+3M] [1Y+3M,1Y+6M] [1Y+6M,1Y+9M] [1Y+9M,2Y] Panel A: Summary statstcs from the whole sample: July 30, 2004 to January 30, 2009 Mean Standard Dev Skewness Kurtoss ρ * * * ADF ** ** ** ** Panel B: Summary statstcs from the frst subsample: July 30, 2004 to August 31, 2006 Mean Standard Dev Skewness Kurtoss Panel C: Summary statstcs from the second subsample: September 01, 2006 to January 30, 2009 Mean Standard Dev Skewness Kurtoss Entres report the summary statstcs of the four mpled volatlty ndexes n the frst daly lndfferences for the whole sample (Panel A) and for the frst and second subsamples (Panel B and C, respectvely). The frst order autocorrelaton ρ 1 and the Augmented Dckey Fuller (ADF) test values are also reported for the entre sample. One astersk denotes statstcal sgnfcance at a 5% confdence level. Two astersks denote statstcal sgnfcance at a 1% confdence level. 31