Interest rate and stock return volatility indices for the Eurozone. Investors gauges of fear during the recent financial crisis *

Transcription

1 Interest rate and stock return volatlty ndces for the Eurozone. Investors gauges of fear durng the recent fnancal crss * Raquel López a, Elseo Navarro b Abstract We suggest a methodology for the constructon of a set of nterest rate volatlty ndces for the Eurozone (s) based on the mpled volatlty quotes of caps (floors), one of the most lqud nterest rate dervatves. These ndces reflect the market s aggregate expectaton of volatlty of forward rates over both short- and long-term horzons (from one to ten years ahead). Volatlty ndces n equty markets are referred to as nvestors gauges of fear because they usually spke n perods of market turmol. In ths paper, we extend the emprcal evdence by analyzng the effect of the recent fnancal crss on short- and long-term s. We fnd that the level of short-term s (70%) as of Aprl 2012 s stll far from returnng to the average pre-crss value (17%) and that the crss has also affected nvestors long-term expectatons of volatlty. In addton, usng two stock return volatlty ndces for the Eurozone, we fnd that the crss has had a deeper mpact on nvestors uncertanty about the evoluton of nterest rates than on stock market returns. Keywords: caps and floors, crss, nterest rates, nvestors gauge of fear, volatlty ndces EFM Classfcaton Codes: 410, 450, 630 a Raquel López s n the Department of Economc Analyss and Fnance, Unversty of Castlla-La Mancha (Span). Address: Plaza de la Unversdad 1, Albacete (Span). Telephone: Fax: E-mal: raquel.lopez@uclm.es. b Elseo Navarro s n the Department of Busness Scence, Unversty of Alcalá (Span). Address: Facultad de Cencas Económcas y Empresarales, Plaza de la Vctora 2, Alcalá de Henares (Span). Telephone: (+34) E-mal: elseo.navarro@uah.es. * Raquel and Elseo acknowledge the fnancal support provded by Mnstero de Educacón y Cenca grant ECO Any remanng errors are our responsblty alone.

2 1. Introducton We suggest a methodology for the constructon of a set of nterest rate volatlty ndces for the Eurozone (s), whch reflect the market estmate of the volatlty of threeand sx-month tenor forward rates over dfferent fxed horzons one, two, three, four, fve, seven and ten years. To ths end, we use data on caps (floors), one of the most lqud over-the-counter (OTC) nterest rate dervatve contracts (L and Zhao, 2009). Bank for Internatonal Settlements statstcs, as of December 2011, ndcate that nterest rate optons (caps, floors, collars and corrdors) were the second most-traded OTC nterest rate dervatves worldwde. Moreover, the notonal amount of OTC nterest rate optons exceeded that of exchange-traded optons by nearly $20 trllon. Sorted by currency, Euro nterest rate optons accounted for approxmately 46% of the total amount outstandng of OTC nterest rate optons traded n the world. 1 The number of mpled volatlty ndces has sgnfcantly ncreased over the last decade n the equty markets (e.g., VIX, VDAX, VCAC and VSTOXX among others). These ndces capture the market s expectaton of volatlty of stock ndces returns over the next 30 calendar days. 2 To the best of our knowledge, there are no volatlty ndces calculated by exchanges or other nsttutons for the Eurozone fxed-ncome market. Thus, ths paper contrbutes to the lterature on volatlty ndces by coverng ths gap. Moreover, ths s the frst tme that caps (floors) data are used for ths specfc purpose. Among other applcatons, caps data have been used for the mplementaton of a fxed In addton to VIX, the Chcago Board of Optons Exchange (CBOE) also dstrbutes a constant threemonth maturty volatlty ndex based on S&P 500 optons, VXV. In Europe, Deutsche Börse, SIX Swss Exchange and STOXX Ltd calculate three man volatlty ndces wth a fxed 30-day maturty for the German (VDAX), Swss (VSMI) and Eurozone (VSTOXX) equty markets, respectvely, as well as eght volatlty sub-ndces on the bass of eght expry months rangng from one month to two years, whch allow to construct volatlty ndces wth dfferent fxed days to expraton through lnear nterpolaton of the two nearest sub-ndces. 1

3 ncome volatlty arbtrage strategy (Duarte et al. 2007) and to estmate rsk premums n long-term nterest rates (Almeda et al. 2011). VIX s usually called the nvestors gauge of fear for the US stock market (Whaley, 2000, 2009). The name s approprate because t reflects the consensus market vew of the expected volatlty of the S&P 500, and t spkes durng perods of market turmol. In partcular, the recent fnancal crss has the second-largest burst of volatlty after the market crash n October 1987, although t seems that stock volatlty returned to more normal levels farly quckly after the burst of the crss (Schwert, 2011). Usng daly data from January 2004 to Aprl 2012, ths paper ams to provde novel emprcal evdence about the effect of the crss on the market s short- and long-term expectatons of nterest rate volatlty n the Eurozone. Concernng short-term s, we observe large spkes along upward and downward slopes snce the summer of As of Aprl 2012, volatlty levels of approxmately 70% are stll far from returnng to the average pre-crss value (17%). Thus, the recent crss has had a deep and lastng effect on nvestors short-term expectatons of volatlty n the fxed-ncome market. More nterestngly, we also fnd that as the crss deepened, t also eventually affected expectatons of volatlty fve and ten years ahead: the ndces ntate an upward trend n In addton, we also analyze whether the fnancal turmol has had a deeper mpact on nterest rate or stock return volatlty ndces by usng two VSTOXX volatlty subndces constructed from Dow Jones EURO STOXX 50 optons exprng n one and two years. We observe that VSTOXX ndces exhbt a lower rse than one- and two-year s along the crss perod and that the sze of the spkes s also smaller. Ths fndng suggests that s have played a greater role as nvestors gauge of fear durng the recent fnancal crss than VSTOXX ndces. In addton, we prove that there 2

4 s a statstcally sgnfcant correlaton between changes n s and VSTOXX ndces. The rest of the paper s organzed as follows. The next secton examnes caps (floors) valuaton accordng to the Lbor Market Model (LMM), whch s consstent wth the market standard approach for prcng these contracts usng the Black prcng formula. In Secton Three, we present the methodology for the calculaton of s. Secton Four descrbes the database. In Secton Fve, we analyze the behavor and statstcal propertes of s, and compare the effect of the fnancal turmol durng the recent crss on the market estmates of future volatlty of nterest rates and stock returns. Fnally, Secton Sx ncludes the conclusons of the study. 2. Caps and floors valuaton. The LMM and the Black formula Caps and floors are portfolos of optons on nterest rates, caplets and floorlets. Thus, the desgn and valuaton of caps (floors) can best be understood by frst descrbng the optons that comprse them. Caplets (floorlets) are European-style call (put) optons where the underlyng asset s a forward rate agreement (FRA). An FRA s an agreement between two partes to exchange an amount of money proportonal to the dfference between the fxed strke rate K (set at t) and the floatng nterest rate (reset at tme T ) whch prevals over the perod [T,T +τ], L(T,T +τ), ( t T < T + τ ). The payoff of an FRA at T +τ s: NP [ L T, T + τ K ] τ (, (1) ) where NP s the notonal prncpal of the contract and τ s the tenor nterval. Caplets (floorlets) are exercsed only f L(T,T +τ) s greater (smaller) than the strke K. The payoff of a caplet at T +τ s: 3

5 { L( T, T + τ ) K, } τ NP Max 0, (2) and the payoff of a floorlet s: { K L T, T +τ ), } τ NP Max 0. (3) ( The LMM assumes that the forward nterest rate f(t,t,t +τ) follows a lognormal stochastc process (see Brgo and Mercuro, 2006 for an extensve revew of LMM.). Takng nto account that the lmtng value of the forward rate when t approaches T s equal to the floatng nterest rate L(T,T +τ), and assumng there are no arbtrage opportuntes, the well-known Black (1976) prcng formulas for valung caplets (floorlets) are derved (see, e.g., Díaz et al., 2009): [ f ( t, T, T + τ ) N( h ) K N( h )] P( t, T + τ τ K Caplet( t, T, τ, K, σ, Black ) = 1 2 ) Floorlet t, T, τ, K, σ [ K N( h ) f ( t, T, T + τ ) N( h )] P( t, T + τ τ K (, Black 2 1 ) = ),, (4) (5) where h h 1 2 ln = ln = [ f ( t, T, T + τ ) / K ] σ K, Black [ f ( t, T, T + τ ) / K ] σ K, Black 1 K + ( σ, 2 ( T t) 1 K ( σ, 2 ( T t) Black Black ) ) 2 2 ( T ( T t), (6) t). (7) K K Caplet t, T, τ, K, σ ) and Floorlet t, T, τ, K, σ ) are the prces at t of a caplet (, Black (, Black and a floorlet, respectvely, T s the exercse date of the opton (and the maturty date of the underlyng forward rate), τ s the tenor of the underlyng forward rate (and T +τ s the maturty date of the opton), P(t,T +τ) s the prce at t of a unt-zero coupon bond wth maturty at T +τ, N( ) s the cumulatve normal dstrbuton, and s the socalled Black mpled volatlty of an opton wth exercse date T and strke K. 4

6 Black mpled volatlty can be understood, wthn the LMM, as an average of the nstantaneous volatlty of the log of the forward rate f(t,t,t +τ) over the perod [t,t ]: T 2 σ ( u, T ) du K 2 t ( σ, Black ) =, (8) ( T t) where s the nstantaneous volatlty at t of the lognormal process followed by the forward rate f(t,t,t +τ). Caps (floors) are portfolos of caplets (floorlets) wth the same strke and tenor but wth consecutve maturtes so that the maturty date of each caplet (floorlet) concdes wth the exercse date of the followng one. In the Eurozone, caps (floors) wth tme to expraton up to two years have a three-month tenor, whereas the tenor for caps (floors) wth maturtes beyond two years s sx months. Thus, a two-year cap (floor) conssts of a chan of seven caplets (floorlets) wth exercse dates n three, sx, nne, 12, 15, 18 and 21 months, whereas a three-year cap (floor) comprses fve caplets (floorlets) wth exercse dates n sx, 12, 18, 24 and 30 months. Please note that, unlke equty optons, caplets (floorlets) and caps (floors) have a constant lfe perod. The payoffs generated by a cap (floor) can be descrbed as follows. On the exercse date of the frst caplet (floorlet), the floatng rate s observed and compared to the strke. If the floatng rate s greater (smaller) than the strke, then on the second reset date the seller of the cap (floor) pays the holder the dfference between the floatng rate (strke) and the strke (floatng rate) multpled by the notonal prncpal and the tenor. If the floatng rate s less (more) than the strke, there s no payoff from the cap (floor). Thus, through the lfe of a cap (floor), payments are due at the end of each tenor nterval, 5

7 although the amount s known at the reset date (at the begnnng of the tenor nterval) when the floatng nterest rate s observed. 3 Then, the prce at tme t of an n-year cap wth strke K can be obtaned as the sum of the values of the caplets that comprse t. That s, n k 1 = 1 K Cap( t, Tn k, K) = caplet( t, T, τ, K, σ, Black ), (9) where k equals 4 (2) when the tenor nterval s three (sx) months, and T 1, T 2,...,T n k-1 are the reset dates of the cap that concde wth the exercse dates of the caplets that compose the cap and T n k = T n k-1 +τ,.e., the date that the last cash flow wll be due f L(T n k-1,t n k-1 +τ) > K. 4 However, quotatons n the cap market are computed assumng that the volatlty of all the caplets that compose a partcular cap s the same. In fact, an n-year cap wth strke K s quoted by the market through the so-called flat volatlty, whch s the constant value that equals the sum of the values of all the caplets that compose the cap accordng to the Black formula to ts market prce,.e., the value such that n k 1 = 1 K Cap( t, Tn k, K) = caplet( t, T, τ, K, σ n, Black ). 5 (10) Therefore, flat volatltes cannot be consdered to be a pure measure of the future evoluton of volatlty of a forward rate; rather, they are a mxture of the average future volatltes of a set of forward rates wth consecutve terms to maturty. 6 Thus, for 3 Caps (floors) are usually defned so that the ntal floatng rate, even f t s greater (smaller) than the strke, does not lead to a payoff on the frst reset date (Hull, 2009). 4 An analogous formula can be set up for the prce of a floor. 5 Actually, the market quotes flat volatltes 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 ther expectatons. 6 The dfference between and s smlar to the dfference between zero-coupon rates and the yelds to maturty of coupon-bearng bonds. 6

8 nstance, the flat volatlty of a two-year cap s a mxture of the average future volatlty of three-month tenor forward rates wth maturtes n three, sx, nne, 12, 15, 18 and 21 months. Fnally, note that accordng to the LMM and Equaton (8), the mpled volatlty of caplets should be the same for all caplets wth the same term to maturty, ndependent of the strke K. However, n practce, the mpled volatlty of caplets and caps (wth everythng else equal) vares wth the strke rate, gvng rse to volatlty surfaces (see Jarrow et al., 2007). 3. Methodology We develop a set of nterest rate volatlty ndces that capture the market s expected volatlty of a partcular forward rate over dfferent fxed horzons usng the mpled volatlty quotes of caps (floors). 7,8 However, the use of data from ths market poses the problem of havng to address a contract where the underlyng rate s not a sngle forward rate but a set of forward rates wth consecutve maturtes. Therefore, the constructon of s nvolves recoverng the mpled volatltes of the ndvdual caplets that compose caps usng a strppng procedure (see, e.g., Hernández, 2005). Ths process conssts of obtanng the prce at tme t of a caplet wth a strke K and reset date T, caplet(t,t,τ,k, ), by subtractng the prces of two consecutve caps wth the same strke K: 7 Stock return volatlty ndces are calculated usng the market prces of (exchange-traded) optons, rather than ther respectve mpled volatltes, based on the concept of the far delvery value of future realzed varance suggested by Demeterf et al. (1999). However, note that the quoted opton prce n the OTC market s actually mpled volatlty tself (.e., mpled volatlty does not need to be nferred from opton prces). Thus, to provde an mpled volatlty quote n the cap (floor) market means to gve the opton prce, smlar to how the yeld to maturty of a bond s an alternatve way of provdng the prce of the bond. 8 Impled volatltes of specfc forward rates could be drectly obtaned from caplet (floorlet) quotatons, however, these contracts are qute llqud; thus, obtanng a complete enough range of caplets (floorlets) wth dfferent maturtes can be complcated. Thus, the constructon of s from caps (floors) data can gve a much more accurate ndcaton of the actual uncertanty regardng the future behavor of nterest rates for a wde range of maturtes, wthout the ntruson of the nose caused by the lack of lqudty. 7

9 K Caplet t, T, τ, K, σ ) Cap( t, T, K) Cap( t, T, ), (11) (, Black = + 1 K where Cap( ) are defned as n Equaton (10). Once the prce of the caplet s obtaned, the Black prcng formula s used to derve the correspondng mpled volatlty. 9 Note that when mplementng the strppng procedure, the Black s model s used only to translate volatlty quotes nto opton prces and vce versa. Thus, we are not makng use of any of the assumptons of the Black s model. It s merely used as a tool to provde a one-to-one mappng between opton prces and mpled volatltes. As stressed n the prevous secton, t must be noted that we obtan dfferent mpled Black volatltes for the same caplet, dependng upon the strke rate K. Thus, a decson on the strkes of the caps used for the mplementaton of s needs to be made. Poon and Granger (2003) suggest usng at-the-money (ATM) optons because they are more lqud and less prone to measurement errors. In the cap market, an n-year cap s sad to be ATM f the strke of ths nstrument equals the fxed rate of a swap that has the same payment days as the cap (see, e.g., Hull, 2009). However, we cannot use ATM caps n the strppng process because two consecutve caps would have dfferent strkes to the extent that swaps wth dfferent maturtes usually have dfferent fxed rates. Therefore, we must address the problem of determnng the strke of an ATM caplet. Accordng to the Black formula, a caplet s sad to be ATM when the value of the underlyng forward rate equals the strke rate. Thus, we propose usng the avalable caps wth strkes closest to the outstandng forward rate f(t,t,t +τ) defned as P( t, T ) 1 f ( t, T, + τ ) = 1 (, τ ) T, (12) P t T + τ 9 Note that the same mpled volatlty s obtaned when the prcng formulas for floors (floorlets) are used nstead. 8

10 where P(t,T ) and P(t,T +τ) are the prces at t of unt zero-coupon bonds wth maturtes at T and T +τ, respectvely. In partcular, we wll use caps wth strkes mmedately above and below f(t,t, T +τ), and we wll refer to them as K A and K B, respectvely, wth K B < f(t,t,t +τ) < K A. Then, usng Equaton (11), we obtan the prces of caplets wth strkes K A (frst out-ofthe-money caplet) and K B (frst n-the-money caplet), and we derve ther mpled volatltes usng the Black formula. We denote these two mpled volatltes by and, respectvely. Fnally, we use lnear nterpolaton to obtan : t T A B K K f ( t, T, T + τ ) A K f ( t, T, T + τ K σ + ) B, Black A B σ, (13) Black A K K K K (, ) =, B where (t,t ) s the annualzed mpled volatlty of a theoretcal ATM caplet wth a constant tme to maturty, from t to T. Accordng to Carr and Lee (2003, 2009a), the volatlty swap rate wth expry at tme T s well approxmated by the ATM mpled volatlty maturng at the same tme. 10,11 Thus, (t,t ) approxmates the condtonal rsk-neutral expectaton of the realzed volatlty of the underlyng forward nterest rate f(t,t,t +τ) over the perod [t, T ]. In addton, because s based on the market quotes of very lqud optons, t represents a consensus market vew of the expected volatlty of the underlyng forward rate. 10 A volatlty swap s a contract traded OTC that pays at maturty the dfference between the realzed volatlty of the underlyng asset over the lfe of the contract and a fxed volatlty rate (the volatlty swap rate). Snce the contract has zero value at the tme of entry, by no arbtrage, the volatlty swap rate equals the condtonal rsk-neutral expected value of the realzed volatlty over the lfe of the contract. 11 Moreover, Carr and Lee (2009b) show that the payoff on a volatlty swap can be perfectly replcated by dynamcally tradng European optons and futures. 9

11 Usng the constructon method just descrbed, we create a daly set of nterest rate volatlty ndces for three- (sx-) month tenor forward rates exprng n one and two (three, four, fve, seven and ten) years. Accordng to Duarte et al. (2005), these are the most lqud cap maturtes. Accordng to Equaton (8), each provdes the average future volatlty of a forward nterest rate up to ts maturty. For nstance, (t,1y) measures the market s assessment at any tme t of the uncertanty regardng the evoluton of the forward rate f(t,t+1y,t+1y+3m) over the next year; and (t,10y) would ndcate the average volatlty of the forward rate f(t,t+10y,t+10y+6m) over the next ten years. Thus, unlke flat volatltes, s measure the volatlty of specfc forward rates. For nstance, σ K 1, flat would be some sort of average of the future volatltes of the forward rates f(t,t+3m,t+6m), f(t,t+6m,t+9m) and f(t,t+9m,t+1y) up to ther respectve maturtes. 4. Data For the constructon of s we use two sets of daly data from the Eurozone fxedncome market. The frst set conssts of closng md flat volatlty quotes of caps (floors) for a fxed set of maturtes and strkes retreved from Bloomberg. The data suppler for these quotes s the large OTC nterdealer broker ICAP. The second set conssts of zerocoupon curves provded by Reuters based on the most lqud rate nstruments avalable, a combnaton of deposts, lqud futures and nterest rate swaps. The sample extends from January 02, 2004 to Aprl 30, Flat volatltes correspond to caps (floors) wth maturtes of one to ten plus 12, 15 and 20 years and wth the followng range of strke rates: 0.01, 0.02, , 0.025, 0.03, 0.04, 0.05, 0.06 and These strkes cover the range of values of the forward rates 10

12 durng the sample perod to ensure that there wll be always a strke above and below the outstandng forward rates. 12 Note also that the two strkes closest to the forward rate f(t,t,t +τ) can dffer by only 25, 50 or 100 bass ponts. Thus, the assumpton that the volatlty smle s well approxmated by a lne that we use when the mpled volatltes of near-the-money optons are lnearly nterpolated for the constructon of s consdered reasonable due to the small range of strkes over whch the nterpolaton s made (Flemng et al., 1995). Note also that the strppng procedure nvolves usng the prces of caps wth maturtes n one (two) years and three months, and three (four, fve, seven and ten) years and sx months, whereas markets only provde caps wth annual terms to maturty (.e., wth an nteger number of years to maturty). Therefore, nterpolaton and extrapolaton technques must be used to obtan flat volatltes of caps wth a maturty dfferent from those quoted. Interpolaton and extrapolaton technques are appled between caps wth the same tenor nterval. Thus, when the requred maturty cap s below three years, we apply lnear nterpolaton/extrapolaton by usng the one- and two-year maturtes. For the rest of maturtes, we apply cubc splne nterpolaton (see Hernández, 2005) based on the flat volatltes of caps wth maturtes of three to ten years plus 12, 15 and 20 years. We use lnear nterpolaton/extrapolaton only when the number of avalable flat volatltes s less than sx. Note that these nterpolaton/extrapolaton technques must produce unquely determned values of unobservable flat volatltes wth any term to maturty up to ten years and sx months (the maturty date of the caplet wth an exercse date n 12 The only excepton occurs for the forward rate maturng n one year snce values below one percent are observed snce md Thus, n ths partcular stuaton, s just the mpled volatlty of the caplet (floorlet) wth a strke rate of

13 ten years). See the Appendx for a detaled descrpton of the nterpolaton/extrapolaton procedure. In regards to measurng the market s expectatons of volatlty n the equty market, we use two volatlty ndces dstrbuted by STOXX Ltd, VSTOXX 12M and VSTOXX 24M. Actually, they belong to the set of sub-ndces that are calculated n addton to the man ndex, VSTOXX. In partcular VSTOXX 12M and VSTOXX 24M are constructed based on the prces of Dow Jones EURO STOXX 50 optons exprng n 12 and 24 months, respectvely. Thus, they capture the market s expected volatlty of the Dow Jones EURO STOXX 50 returns over the next 12 and 24 months. Sub-ndces based on optons wth longer terms to expraton are not currently avalable Emprcal analyss In ths secton we analyze the behavor and statstcal propertes of the set of s. Then, we use two Eurozone equty market volatlty ndces to compare the effect of the crss on nvestors uncertanty one and two years ahead of the evoluton of nterest rates and stock market returns 5.1. Propertes of s The daly evoluton of s wth tmes to maturty of one, two, fve and ten years from January 02, 2004 to Aprl 30, 2012 s shown n Fgure 1. We observe a decreasng pattern n s wth the closest forecast horzons from the begnnng of the sample up to approxmately md-july Then, the ndces ntate an upward trend whch leaves market estmates of nterest rate volatlty over the next one and two years at approxmately 70% n May Thus, short-term s seem to reflect the fnancal turmol snce the begnnng of the crss. By May 2010, the 13 Addtonal nformaton on the ndces can be found on 12

14 upward trend turns a downward trend untl approxmately Aprl 2011, when the level of s s close to 30%. Then, s exhbt a new outstandng rse that drves market expectatons of nterest rate volatlty to a maxmum of approxmately 90% n fve months. Fnally, large (up and down) spkes are observed along a downward slope untl Aprl 2012, when s level (approxmately 70%) s stll far from returnng to the average pre-crss value. Concernng nvestors expectatons of volatlty over the next fve and ten years, they exhbt a rse durng the frst half of 2010 and approxmately double by the end of the sample. Thus, two man conclusons can be drawn from the graphcs. On the one hand, the fact that the ndces (especally the short-term ones) spke and sharply ncrease durng the recent fnancal crss supports the nterpretaton of as a gauge of fear for fxedncome markets smlar to the wdely held vew of VIX for equty markets (Whaley 2000, 2009). On the other hand, the fact that long-term s also respond to the fnancal turmol seems to suggest that nvestors foresee long perods of turbulence n nterest rate markets. Recall that provdes the average level of future volatlty untl the maturty of the underlyng forward rate (see Equaton (8)), and hence, only a lastng shft n the market estmates of future volatlty would make long-term s rase. The summary statstcs of the set of nterest rate volatlty ndces for the full sample as well as before and after the begnnng of the subprme crss are ncluded n Table 1 (Panels A, B and C, respectvely). The frst subsample spans the perod from January 13

15 02, 2004 to July 31, 2007 (885 observatons), and the second spans the perod from August 01, 2007 to Aprl 30, 2012 (1177 observatons). 14 The average value of all s ncreases durng the crss perod. In addton, we fnd that the mean of all maturty s s qute smlar before the crss, whereas t progressvely decays as the forecast horzon ncreases for the second subsample. Shortterm s also show greater varablty (standard devaton) than long-term ones before and durng the crss. The skewness and kurtoss measures suggest that the ndces are closer to a normal dstrbuton n the splt sample than when the whole sample s consdered. In any case, the Jarque-Bera test does not accept the null hypothess of a normal dstrbuton for any of the ndces n any of the two subperods. To nvestgate whether the seres are statonary, the augmented Dckey-Fuller (ADF) unt root test for ts most general specfcaton (.e., wth ntercept and lnear trend) s performed on the logarthm of the volatlty ndces. The null hypothess of a unt root s not rejected n any case. 15 Summary statstcs for the daly log-dfferences of s are also shown n Table 2. On the one hand, the excess kurtoss found n the seres s also reported by Dotss et al. (2007) for several equty market volatlty ndces n ther frst dfferences, where the non-normalty may be attrbuted to the presence of jumps n mpled volatlty. On the other hand, the sgnfcant negatve frst-order autocorrelaton supports the modelng of mpled volatlty ndces as mean-revertng processes. All the seres are statonary after dfferencng. 14 August 2007 s usually referred to as the onset date of the subprme crss, and a change n the values of the ndces s also especally perceptble around ths date. 15 Gven the lkely exstence of a structural break n the seres durng the crss perod, the modfed verson of the ADF test developed by Zvot and Andrews (1992) to allow for a structural break n the data s conducted. The null hypothess s that the seres follows a unt root process; the alternatve hypothess mples that the seres s a trend-statonary process wth a one-tme break n the trend functon occurrng at an unknown pont n tme. We obtan that the null hypothess contnues not beng rejected n all the cases, except for (t,4y) at the 5% sgnfcance level. Moreover, we fnd that the structural break dates dentfed by the test for short-term s belong to July

16 To formally nvestgate whether there are statstcally sgnfcant dfferences n the dstrbuton of the ndces before and after the begnnng of the crss we apply two nonparametrc tests. Panel A n Table 3 shows the results of the Wlcoxon/Mann-Whtney test for the equalty of medans and the Brown-Forsythe test for the equalty of varances for the seres n levels. The results show evdence of sgnfcant dfferences n both the medan and the varance between the frst and second subsamples at the 1% sgnfcance level for all the ndces. For the seres n frst log-dfferences (Panel B n Table 3), statstcally sgnfcant dfferences n the medans between the frst and second subsamples are unproven for all forecast horzons; however, the null hypothess of equalty of varances s rejected for s wth tme to expraton from one to fve years The role of nterest rate and stock return volatlty ndces as nvestors gauges of fear durng the fnancal crss Next, we use (t,1y) and (t,2y) along wth VSTOXX 12M and VSTOXX 24M to track how nvestors uncertanty about the future behavor of nterest rates and stock returns one and two years ahead changes n response to fnancal nstablty durng the recent fnancal crss. Fgure 2 plots the four mentoned ndces across the sample. Smlar to s, we can also see an ncrease n the VSTOXX ndces by the summer of However, the sze of the spkes observed n the VSTOXX seres along the crss perod s notably smaller than n the case of s. The standard devaton of VSTOXX 12M s 6% (Table 4), whereas t s 17% for (t,1y). The hghest volatlty level (49.73%) s reached by VSTOXX 12M on November 21, Also note that for approxmately one year before and after the burst of the crss, market estmates of future volatlty are hgher n the equty market than n the fxed-ncome market. However, by March 2009, 15

17 VSTOXX 12M and VSTOXX 24M decrease and from that moment untl the end of the sample the stock return volatlty ndces reman below s. Moreover, by Aprl 2012, VSTOXX ndces are approxmately 30%, whereas the average pre-crss level of the ndces s approxmately 20%. Thus, we fnd that the fnancal turmol has had a deeper mpact on nvestors uncertanty about the evoluton of nterest rates than on stock market returns. Put dfferently, s have played a greater role as nvestors gauge of fear durng the recent fnancal crss than VSTOXX ndces. It s nterestng to note that both fxed-ncome and equty market volatlty ndces depct a qute smlar pattern over most of the sample except from March 2009 to Aprl 2010, when VSTOXX ndces start to fall whle s keep an upward trend. Smlartes n the behavor of s and VSTOXX ndces seem to be partcularly notceable over two perods. On the one hand, both nterest rate and stock return volatlty ndces exhbt a remarkable rse n September 2008, when the falure of Lehman Brothers - the fourth-largest US nvestment bank took place. On the other hand, the reducton observed n VSTOXX ndces and s around May 2010 may be attrbuted to the resoluton of the European Central Bank to conduct nterventons n publc and prvate debt securtes markets of the Euro area to ensure depth and lqudty n certan market segments. Table 5 shows the cross-correlatons between weekly log-changes n the four ndces for the entre sample (Panel A) as well as for the pre-crss and crss perods (Panels B and C, respectvely). We fnd that there s a statstcally sgnfcant postve correlaton between changes n nterest rate and stock return volatlty ndces over the whole sample, although ths s stronger durng the crss perod - the hghest correlaton s obtaned for (t,1y) and VSTOXX 12M (31%). Ths outcome extends the 16

18 emprcal evdence n Äjö (2008), where the authors fnd a sgnfcant contemporaneous correlaton between the volatlty term structures of three European stock return volatlty ndces: VSTOXX, VDAX-NEW and VSMI. 6. Summary and conclusons We suggest for the frst tme a methodology for the constructon of a set of nterest rate volatlty ndces for the Eurozone (s) based on the mpled volatlty quotes of one of the most lqud fxed-ncome dervatves: caps (floors). These ndces reflect the market estmate of the volatlty of three- and sx-month tenor forward rates over dfferent fxed horzons one, two, three, four, fve, seven and ten years. s are constructed through a two-step process. Frst, we apply a strppng procedure consstng of recoverng the mpled volatltes of the ndvdual caplets (floorlets) that compose caps (floors), as these are the contracts that do have an underlyng specfc forward rate. Second, mpled volatltes of near-the-money caplets (floorlets) are lnearly nterpolated. Thus, each (t,t ) reflects the mpled volatlty of a theoretcal ATM caplet (floorlet) wth a constant tme to maturty, from t to T. The ATM mpled volatlty wth expry at tme T has a specfc theoretcal nterpretaton: t approxmates the volatlty swap rate (.e., the condtonal rsk-neutral expectaton of the future realzed volatlty of the underlyng asset over the perod [t, T]). Volatlty ndces n the equty markets are referred to as nvestors gauges of fear because they usually spke n perods of fnancal turmol. In ths paper, we extend the emprcal evdence by analyzng the effect of the recent fnancal crss on short- and long-term s. We fnd that the crss has had a deep and lastng effect on nvestors short-term expectatons of volatlty n the fxed-ncome market by Aprl 2012, volatlty levels are more than three-fold the average pre-crss value. More 17

19 nterestngly, we also fnd that as the crss deepened, t also eventually affected expectatons of volatlty fve- and ten-years ahead the ndces ntate an upward trend n The frst fndng seems to support the nterpretaton of as nvestors gauge of fear for the fxed-ncome market, whereas the second one mght be nterpreted as a sgnal that nvestors foresee long perods of turbulence n nterest rate markets. In addton, we compare the effect of the fnancal turmol on nvestors expectatons of volatlty of nterest rates and stock returns over the next one and two years by usng two equty market volatlty ndces, VSTOXX 12M and VSTOXX 24M. We observe that VSTOXX ndces exhbt a lower rse than one- and two-year s durng the crss perod and that the sze of the spkes s also smaller. Moreover, by Aprl 2012, VSTOXX ndces are approxmately 30%, whereas the average pre-crss level of the ndces s approxmately 20%. Ths fndng suggests that s have played a greater role as nvestors gauge of fear durng the recent fnancal crss than VSTOXX ndces. Fnally, we show that changes n s and VSTOXX ndces are postvely correlated, especally durng the crss perod. We observe that they both react to some partcular events such as the Lehman Brothers falure n September 2008 and the resoluton of the European Central Bank to conduct nterventons n publc and prvate debt securtes markets of the Euro area n May Appendx The strppng procedure nvolves usng nterpolaton/extrapolaton technques to obtan the flat volatltes of caps wth a maturty dfferent from those quoted (.e., non-annual maturty and mssng annual maturty quotes) for a partcular strke. One can expect to 18

20 obtan a smoother fttng of the term structure of flat volatltes by usng cubc splne nterpolaton nstead of lnear nterpolaton. However, we need to address the fact that the tenor of the underlyng forward rates of the caplets that compose caps s not the same for all maturty caps (.e., all maturty caps do not have the same structure); hence, we cannot apply cubc splne nterpolaton along the whole term structure. In partcular, one- and two-year caps have a shorter tenor (three months) than the rest of the caps. In ths case, we extrapolate the flat volatlty of the two- (one-) year cap when the requred maturtes are one year or two years and three months (two years). For the one year and three months maturty, we lnearly nterpolate between the one- and twoyear maturtes or extrapolate when there s one mssng quote. Then, cubc splne nterpolaton s appled over volatlty quotes of caps wth maturtes of three to ten years plus 12, 15 and 20 years (.e., the maxmum number of flat volatlty quotes avalable for a partcular strke s 11). When the number of flat volatlty quotes s less than sx, we propose smple lnear nterpolaton/extrapolaton. When applyng cubc splne nterpolaton, we dstngush two possbltes. If the number of observatons s greater than nne, we use two ntermedate knots; otherwse, we use a sngle knot. Knots are postoned n such a way that the observatons are unformly dstrbuted between knots. In partcular, the poston of the knots s set as follows. Let N denote the number of avalable flat volatltes for a partcular strke and t 1 and t 2 denote the postons of the knots. Then, we have the followng: If N = 11, t 1 = 5.5 and t 2 = 9.5. If N = 10, t 1 s postoned at the mdpont between the thrd and fourth observatons and t 2 at the mdpont between the seventh and eghth observatons. 19

21 If N = 9 or N = 8, the unque knot s settled at the mdpont between the fourth and ffth observatons. If N = 7 or N = 6, the knot s postoned at the mdpont between the thrd and fourth observatons. When N s less than 6, we apply lnear nterpolaton. For those caps wth maturtes out of the range of avalable maturtes, we proceed to extrapolate. Let us denote the flat volatltes of the caps wth the shortest and greatest terms to maturty by and, respectvely. Then, for caps maturng before the frst avalable cap, we assume that flat volatltes are equal to. For caps wth maturty greater than the last avalable cap, we assume that flat volatltes are equal to. To gve a hnt of the completeness of the sample for maturty caps rangng from three to ten years plus 12, 15 and 20 years, Table 6 shows the proporton of flat volatltes avalable correspondng to a sngle day and a gven strke durng the whole sample. As t s consdered desrable n 97% of cases, the sample s complete (eleven observatons). 20

22 References Äjö J Impled term structure lnkages between VDAX, VSMI and VSTOXX volatlty ndces. Global Fnance Journal 18: Almeda C., Gravelne J.J., Josln S Do nterest rate optons contan nformaton about excess returns? Journal of Econometrcs 164: Black F The prcng of commodty contracts. Journal of Fnancal Economcs 3: Brgo D., Mercuro F Interest Rate Models Theory and Practce: Wth Smle, Inflaton and Credt. Sprnger-Verlag: Berln. Carr P., Lee R At-the-Money Impled as a Robust Approxmaton of the Volatlty Swap Rate. Workng paper, New York Unversty. Carr P., Lee R. 2009a. Volatlty dervatves. Annual Revew of Fnancal Economcs 1: DOI: /annurev.fnancal Carr P., Lee R. 2009b. Robust replcaton of volatlty dervatves. Workng paper, New York Unversty. Demeterf K., Derman E., Kamal M. and Zou J More than you ever wanted to know about volatlty swaps. Goldman Sachs Quanttatve Strateges Research Notes. Díaz A., Meneu V., Navarro E Internatonal evdence on alternatve models of the term structure of volatltes. The Journal of Future markets 29: Dotss G., Psychoyos D., Skadopoulos G An emprcal comparson of contnuous-tme models of mpled volatlty ndces. Journal of Bankng and Fnance 31:

23 Duarte J., Longstaff F.A., Yu F Rsk and Return n Fxed-ncome Arbtrage: Nckels n front of a streamroller? UC Los Angeles: Anderson Graduate School of Management (avalable at Duarte J., Longstaff F.A., Yu F Rsk and Return n Fxed-ncome Arbtrage: Nckels n front of a streamroller? Revew of Fnancal Studes 20: Flemng J., Ostdek B., Whaley R.E Predctng stock market volatlty: A new measure. The Journal of Futures Markets 15: Hernández, L.G., Prcng of Game Optons n a market wth stochastc nterest rates. Thess (avalable at Hull J.C Optons, futures and other dervatves. Prentce Hall: Upper Saddle Rver. Jarrow R., L H., Zhao F Interest Rate Caps Smle Too! But Can the LIBOR Market Models Capture the Smle? The Journal of Fnance 62: L H., Zhao F Nonparametrc Estmaton of State-Prce Denstes Implct n Interest Rate Cap Prces. The Revew of Fnancal Studes 22: Poon S-H., Granger C.W.J Forecastng Volatlty n Fnancal Markets: A Revew. Journal of Economc Lterature 41: Schwert G.W Stock volatlty durng the recent fnancal crss. European Fnancal Management 17: Whaley R.E The Investor Fear Gauge. The Journal of Portfolo Management 26: Whaley R.E Understandng the VIX. Journal of Portfolo Management 35:

24 Zvot, E., Andrews, D.W.K., Further evdence on the Great Crash, the Ol-Prce Shock, and the Unt-Root Hypothess. Journal of Busness and Economc Statstcs 10:

25 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% IRVIX(t,1Y) IRVIX(t,2Y) IRVIX(t,5Y) IRVIX(t,10Y) FIGURE 1. Daly levels of (t,1y), (t,2y), (t,5y) and (t,10y) over the perod from January 02, 2004 to Aprl 30, % 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% IRVIX(t,1Y) IRVIX(t,2Y) VSTOXX 12M VSTOXX 24M FIGURE 2. Daly levels of (t,1y), (t,2y), VSTOXX 12M and VSTOXX 24M over the perod from January 02, 2004 to Aprl 30,

26 TABLE 1. Summary statstcs of s across the entre sample (Panel A) and for two subsamples: from January 02, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) (t,1y) (t,2y) (t,3y) (t,4y) (t,5y) (t,7y) (t,10y) Panel A: January 02, 2004 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera ρ ** 0.99 ** 0.99 ** 0.99 ** 0.99 ** 0.98 ** 0.97 ** ADF Panel B: January 02, 2004 to July 31, 2007 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Panel C: August 01, 2007 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Notes: a p-values of the Jarque-Bera test are nsde parenthess b ρ 1 denotes the frst-order autocorrelaton coeffcent. The sgnfcance of autocorrelatons s tested wth the Ljung-Box Q-statstc. c The ADF test s performed on the logarthm of the ndces. The null hypothess s that the seres contans a unt root. The optmal lag length s determned accordng to the Schwarz nformaton crteron. d One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. 25

27 TABLE 2. Summary statstcs of frst log-dfferences of s across the entre sample (Panel A) and for two subsamples: from January 02, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) (t,1y) (t,2y) (t,3y) (t,4y) (t,5y) (t,7y) (t,10y) Panel A: January 02, 2004 to June 30, 2011 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera ρ ** ** ** ** ** ** ** ADF ** ** ** ** ** ** ** Panel B: January 02, 2004 to July 31, 2007 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Panel C: August 01, 2007 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Notes: a p-values of the Jarque-Bera test are nsde parenthess b ρ 1 denotes the frst-order autocorrelaton coeffcent. The sgnfcance of autocorrelatons s tested wth the Ljung-Box Q-statstc. c The ADF test s performed on the logarthm of the ndces. The null hypothess s that the seres contans a unt root. The optmal lag length s determned accordng to the Schwarz nformaton crteron. d One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. 26

28 TABLE 3. Tests of equalty of medans and varances between the frst and second subsamples for the ndces n levels (Panel A) and n frst log-dfferences (Panel B) (t,1y) (t,2y) (t,3y) (t,4y) (t,5y) Panel A: Tests of equalty of medans and varances for s n levels (t,7y) (t,10y) Wlcoxon/Mann- Whtney test ** ** ** ** ** ** ** Brown-Forsythe test ** ** ** ** ** ** ** Panel B: Tests of equalty of medans and varances for s n frst log-dfferences Wlcoxon/Mann- Whtney test Brown-Forsythe test ** ** ** ** ** Notes: The null hypothess of the Wlcoxon/Mann-Whtney test s that the medans are equal. The null hypothess of the Brown-Forsythe test s that the varances are equal. One and two astersks denote rejecton of the null hypothess at the 5% and 1% sgnfcance level, respectvely. 27

29 TABLE 4. Summary statstcs of VSTOXX 12M and VSTOXX 24M n levels and frst log-dfferences across the entre sample (Panel A) and for two subsamples: from January 01, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) Levels Frst log-dfferences VSTOXX 12M VSTOXX 24M VSTOXX 12M VSTOXX 24M Panel A: January 01, 2004 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera ρ ** 0.99 ** ** ** ADF ** ** Panel B: January 01, 2004 to July 31, 2007 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Panel C: August 01, 2007 to Aprl 30, 2012 Observatons Mean Medan Maxmum Mnmum Std. Devaton Skewness Kurtoss Jarque-Bera Notes: a p-values of the Jarque-Bera test are nsde parenthess. b ρ 1 denotes the frst-order autocorrelaton coeffcent. The sgnfcance of autocorrelatons s tested wth the Ljung-Box Q-statstc. c The ADF test s performed on the logarthm of the ndces. The null hypothess s that the seres contans a unt root. The optmal lag length s determned accordng to the Schwarz nformaton crteron. d One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. 28

30 TABLE 5. Cross-correlatons between weekly log-changes n (t,1y), (t,2y), VSTOXX 12M and VSTOXX 24M across the entre sample (Panel A) and for two subsamples: from January 02, 2004 to July 31, 2007 (Panel B) and from August 01, 2007 to Aprl 30, 2012 (Panel C) Panel A: January 02, 2004 to Aprl 30, 2012 (t,1y) (t,2y) VSTOXX 12M VSTOXX 24M (t,1y) ** 0.25 ** 0.19 ** (t,2y) ** 0.15 ** VSTOXX 12M ** VSTOXX 24M 1 Panel B: January 02, 2004 to July 31, 2007 (t,1y) (t,2y) VSTOXX 12M VSTOXX 24M (t,1y) ** 0.06 * 0.06 * (t,2y) ** 0.07 * VSTOXX 12M ** VSTOXX 24M 1 Panel C: August 01, 2007 to Aprl 30, 2012 (t,1y) (t,2y) VSTOXX 12M VSTOXX 24M (t,1y) ** 0.31 ** 0.23 ** (t,2y) ** 0.18 ** VSTOXX 12M ** VSTOXX 24M 1 Note: One and two astersks denote statstcal sgnfcance at the 5% and 1% sgnfcance level, respectvely. 29

31 TABLE 6. Frequency of avalable flat volatlty quotes for maturty caps rangng from three to ten years plus 12, 15 and 20 years along the sample Number of avalable Frequency flat volatltes N = N = N = N = N = N = N = N = 4 0 N = 3 0 N = 2 0 N =