An Interest Rate Swap Volatility Index and Contract

Anonio Mele QUASaR Yoshiki Obayashi Applied Academics LLC Firs draf: November 10, 2009. This version: June 26, 2012. ABSTRACT Ineres rae volailiy and equiy volailiy evolve heerogeneously over ime, comoving disproporionaely during periods of global imbalances and each reacing o evens of di eren naure. While he Chicago Board Opions Exchange R (CBOE R ) VIX R re ecs he fair value of equiy volailiy, no ineres rae counerpars exis o he CBOE VIX R ; we ll his gap for swap rae volailiy. We use daa on ineres rae swapions and bonds o consruc wo indexes of ineres rae swap volailiy expeced o prevail in a risk-neural marke wihin any given invesmen horizon. The wo indexes mach marke pracices of quoing swapion-implied volailiies boh in erms of basis poin and percenage volailiy. The indexes are consruced such ha hey re ec he model-free fair value of variance swap conracs for forward swap raes ha we design o accoun for he uncerainy in he annuiy facor of he xed leg of forward swaps. The indexes are model-free in he sense ha hey do no rely on any paricular opion pricing model. While he end-resul can be hough of as he swap rae counerpar o he CBOE VIX R for equiy volailiy, he mahemaical formulaions, conrac designs, replicaion argumens, and derivaives underlying he index are maerially di eren. The di erences arise as we mus accoun no only for changes in forward swap raes bu also in he yield curve poins impacing forward swap values in a framework where ineres raes are no assumed o be consan as hey are in he case of he VIX R mehodology. We also consider a framework in which swap raes may experience jumps and nd ha our basis poin volailiy index can be expressed in a model-free forma even in he presence of disconinuiies. c by Anonio Mele and Yoshiki Obayashi 1

Conens 1. Inroducion.......................................................................... 3 2. Ineres rae ransacion risks......................................................... 6 3. Opion-based volailiy sraegies..................................................... 7 4. Ineres rae swap variance conracs................................................ 10 4.1 Conracs........................................................................10 4.2 Pricing.......................................................................... 12 4.3 Hedging......................................................................... 12 5. Basis poin volailiy................................................................ 14 5.1 Conracs and evaluaion........................................................ 14 5.2 Hedging......................................................................... 16 5.3 Links o consan mauriy swaps................................................ 17 6. Marking o marke and rading sraegies............................................18 6.1 Marks o marke of IRV swaps...................................................18 6.2 Trading sraegies............................................................... 18 7. Swap volailiy indexes.............................................................. 22 7.1 Percenage...................................................................... 22 7.2 Basis poin...................................................................... 23 8. Raes versus equiy variance conracs and indexes...................................26 9. Resilience o jumps..................................................................27 10. Index implemenaion.............................................................. 29 10.1 A numerical example........................................................... 29 10.2 Hisorical performance......................................................... 31 Technical Appendix.....................................................................40 A. Noaion, assumpions and preliminary facs.....................................40 B. P&L of opion-based volailiy rading........................................... 41 C. Spanning he conracs.......................................................... 43 D. Direcional volailiy rades......................................................46 E. Local volailiy surfaces..........................................................47 F. The conracs and index in he Vasicek marke...................................48 G. Spanning and hedging Gaussian conracs....................................... 52 H. Jumps...........................................................................58 References.............................................................................. 60 2 cby Anonio Mele and Yoshiki Obayashi

1. Inroducion Ineres rae volailiy and equiy volailiy evolve heerogeneously over ime, comoving disproporionaely during periods of global imbalances and each reacing o evens of di eren naure. While he Chicago Board Opions Exchange R (CBOE R ) VIX R re ecs he fair value of equiy volailiy, no ineres rae counerpars exis o he CBOE VIX R ; we ll his gap for swap rae volailiy. Figure 1 depics he 10 year spo swap rae over nearly wo decades and compares is 20 day realized volailiy wih ha of he S&P 500 R Index. While hese wo volailiies share some common rends and spikes, one is hardly a proxy for he oher. Their average correlaion over he sample size is a mere 51%. Moreover, Figure 2 shows ha he correlaion beween he wo volailiies experiences large swings over ime. For insance, before he nancial crisis saring in 2007, he correlaion was negaive. In he rs years of his crisis, he same correlaion was high, mos likely due o global concerns abou disorderly ail evens. However, even during he crisis, we see periods of marked divergences beween he wo volailiies. Afer 2010, for example, he correlaion beween he wo volailiies plummeed, reaching negaive values in 2011. How can one hedge agains or formulae views on uncerainies around changes in swap raes? In swap markes, which are he focus of his paper, volailiy is ypically raded hrough swapion-based sraegies such as sraddles: for example, going long boh a payer and a receiver swapion may lead o pro s should ineres raes experience a period of high volailiy. However, dela-hedged or unhedged swapion sraddles do no necessarily lead o pro s consisen wih direcional volailiy views. The reason for his discrepancy is, a leas parially, similar o ha explaining he failure of equiy opion sraddles o deliver P&L consisen wih direcional volailiy views. Sraddles su er from price dependency, he endency o generae P&L a eced by he direcion of price movemens raher han heir absolue movemens i.e. volailiy. These, and relaed reasons, have led o variance swap conracs in equiy markes designed o beer align volailiy views and payo s, and a new VIX R index mainained by he Chicago Board Opions Exchange since 2003. Despie he fac ha ransacion volumes in he swap and swapion markes are orders of magniude greaer han in equiy markes, no such index or variance conracs exis. Moreover, here are only weak linkages beween ineres rae swap volailiy and equiy volailiy, as illusraed by Figures 1 and 2, which furher moivaes he creaion of a VIX-like index for swap raes. A he same ime, he complexiy of ineres rae ransacions is such ha heir volailiy is more di cul o price. Inuiively, one prices asses by discouning heir fuure payo s, bu when i comes o xed income securiies, discouning raes are obviously random and, in urn, ineres rae swap volailiy depends on his randomness. An addiional criical issue peraining o swap rae volailiy is ha uncerainy in swap markes is driven by wo sources of randomness: rs, a direc source relaing o changes in swap raes; and second, an indirec one peraining o changes in poins of he yield curve ha a ec he value of swap conracs. This paper inroduces securiy designs o rade uncerainy relaed o developmens of ineres rae swap volailiy while addressing he above issues. The pricing of hese producs is modelfree, as i only relies on he price of raded asses such as (i) European-syle swapions, serving as 3 cby Anonio Mele and Yoshiki Obayashi

10 10Y spo (%) 5 0 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 100 50 S&P 500 realized volailiy (%) 10Y realized volailiy (%) 0 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Figure 1. Top panel: The 10 year, annualized, percen. Boom panel: esimaes of 20 day realized volailiies of (i) he S&P 500 R daily reurns (solid line), and (ii) he daily logarihmic changes in he 10 year, annualized, q percen, 100 12 P 21 =1 ln2 X i+1 X i, where X denoes eiher he S&P 500 Index R or he 10 year. The sample includes daily daa from January 29, 1993 o May 22, 2012, for a oal of 4865 observaions. 100 80 Corr. beween S&P 500 & SWAP Vol (%) 60 40 20 0 20 40 60 80 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 Figure 2. Moving average esimaes of he correlaion beween he 20 day realized volailiies of he S&P 500 R and he 10 year logarihmic changes. Each correlaion esimae is calculaed over he previous one year of daa, and he sample includes daily daa from January 29, 1993 o May 22, 2012, for a oal of 4865 observaions. 4 cby Anonio Mele and Yoshiki Obayashi

counerpars o he European-syle equiy opions underlying equiy volailiy (e.g., Demeerfi, Derman, Kamal and Zou, 1999), bu, also, (ii) he price of zero coupon bonds wih mauriies corresponding o he fixed paymen days of forward swaps underlying swapions. As such, our conrac designs lead o new indexes ha reflec marke expecaions abou swap marke volailiy, adjused for risk. The mos basic example of conracs we consider is srucured as follows. A ime, wo counerparies A and B agree ha a ime, A shall pay B he realized variance of he forward swap rae from o, scaled by he price value of a basis poin of he fixed leg of he forward swap a he fixed paymen daes 1 for he paymen periods 1 1 (see Figure 3). lengh of he volailiy conrac lengh of he swap conrac (enor) T T1 T 2 Tn Curren ime. Conrac characerisics are chosen: (i) lengh of he period over which he volailiy of he forward swap rae will be accouned, T-; (ii) enor of he paymens underlying he ime forward swap rae : T 1,, T n. Daes T 1,,T n. Daes of execuion of he ineres paymens relaing o he ime forward swap rae. Time T. Resoluion of uncerainy abou he volailiy of he forward swap rae occurred from ime o ime T. Paymens relaed o he volailiy conrac are execued, and he conrac expires. Figure 3. Timing of he ineres rae variance conrac and he execuion of swap paymens. As he pricing of his and oher variance swap conracs inroduced below rely on bonds and swapions, he nex secion provides a concise summary of risks inheren in relevan ineres rae ransacions. Secion 3 highlighs pifalls arising from opion-based rading sraegies aimed o express views abou developmens in swap rae volailiy. These pifalls moivae he inroducion of new ineres rae variance conracs. Secions 4 and 5 develop he design, pricing, hedging for hese new conracs: Secion 4 deals wih conracs relying on percenage volailiy, i.e. he volailiy of he logarihmic changes in he forward swap rae; Secion 5 inroduces conracs on basis poin or Gaussian volailiy, relaed o absolue changes in he forward swap rae. The conracs in Secion 4 and 5 are priced in a model-free forma, hrough spanning argumens such as hose in Bakshi and Madan (2000) and Carr and Madan (2001), alhough, as noed already, he payoffs and conrac designs we consider here are quie disinc since hey relae o fixed income marke volailiy and pose new mehodological issues. Our conracs and indexes aim o mach he marke pracice of quoing swapion implied volailiy boh in erms of percenage and basis poin implied volailiy. Secion 6 suggess poenial rading sraegies for he variance conracs inroduced in Secions 4 and 5. Secion 7 develops indexes of swap rae volailiy, which reflec he fair value of he variance conracs of Secions 4 of 5. Secion 8 explains in deail he main differences beween rae variance conracs and indexes and hose already in place in he equiy case. Secion 9 conains exensions where swap markes can experience jumps, and show he remarkable propery ha even in hese markes, 5 c by Anonio Mele and Yoshiki Obayashi

our basis poin index is expressed in a model-free forma. Secion 10 uses sample marke daa o illusrae a numerical implemenaion of one of he ineres rae swap volailiy indexes. An appendix conains echnical deails. 2. Ineres rae ransacions risks Consider a forward saring swap, i.e. a conrac agreed a ime, saring a ime T, and having n rese daes T 0 ; ; T n 1 and paymen periods T 1 T 0 ; ; T n T n 1, wih T T 0, summing up o a enor period equal o T n T 0. Sandard marke pracice ses T = T 0, as in Figure 3, alhough his pracice does no play any role for all he pricing and conrac designs pu forward in his paper. Le R (T 1 ; ; T n ) be he forward swap rae prevailing a ime, i.e. he xed rae such ha he value of he forward saring swap is zero a. In a swapion conrac agreed a ime, one counerpary has he righ, bu no he obligaion, o ener a swap conrac a T 0 < T 1, eiher as a xed rae payer or as a xed rae receiver, wih he agreed xed rae equal o some srike K. If he swapion is exercized a T, he value of he underlying swap is, from he perspecive of he xed rae payer: SWAP T (K; T 1 ; ; T n ) = PVBP T (T 1 ; ; T n ) [R T (T 1 ; ; T n ) K] ; (1) where PVBP T (T 1 ; ; T n ) is he presen value of receiving one dollar a each xed paymen dae a he swapion expiry, also known as he swap s price value of he basis poin (see, e.g., Brigo and Mercurio, 2006), i.e. he presen value impac of one basis poin move in he swap rae a T, R T (T 1 ; ; T n ). Naurally, R (T 1 ; ; T n ) is boh a forward and a spo swap rae when = T. Appendix A conains a succinc summary of he evaluaion framework underlying swapion conracs, as well as he mahemaical de niions of boh he forward swap rae, R (T 1 ; ; T n ), and he price value of he basis poin, PVBP (T 1 ; ; T n ), (see Eqs. (A.1) and (A.2)). Swapions can be priced hrough he Black (1976) formula once we assume he forward swap rae has deerminisic volailiy. The sandard marke pracice is o use his formula o conver premiums o implied Black volailiies and vice-versa. There are wo marke convenions for quoing implied volailiies: one in erms of percenage implied volailiies, similar o equiy opions; and anoher in erms of basis poin implied volailiies, which are percenage volailiies muliplied by he curren level of he forward swap rae. This paper develops variance conracs and volailiy indexes ha mach boh srands of marke pracice. An imporan feaure of swapions is ha he underlying swap value is unknown unil he swapion expiraion no only because forward swap raes ucuae bu also because he PVBP of he forward swap s xed leg also ucuaes. This PVBP is he presen value a ime T of each dollar arising from paying he xed srike rae K raher han he realized swap rae R T (T 1 ; ; T n ). I can be inerpreed as a porfolio of zero coupon bonds mauring a he xed paymen daes of he swap, weighed by he lenghs of he rese inervals and is deermined by he yield curve prevailing a ime T. Opions on equiies relae o a single source of risk, he sock price. Swaps, insead, are ied o wo sources of risk: (i) he forward swap rae, and (ii) he swap s PVBP realized a ime T. This adds complexiy o de ning and pricing swap marke volailiy. 6 cby Anonio Mele and Yoshiki Obayashi

3. Opion-based volailiy sraegies An Ineres Rae Swap Volailiy Index and Conrac The sandard insrumens for rading ineres rae swap volailiy are swapion payers and receivers. Consider he following sraegies: (a) A dela-hedged swapion posiion, i.e. a long posiion in a swapion, be i payer or receiver, hedged hrough Black s (1976) dela. (b) A sraddle, i.e. a long posiion in a payer and a receiver wih he same srike, mauriy, and enor. Boh sraegies are analyzed in Appendix. In Appendix B, Eq. (B.9), we show he delahedged swapion posiion leads, approximaely, o he following daily P&L a he swapion s expiry: P&L T 1 2 TX =1 $ 2 IV 2 0 PVBPT + TX =1 Track Vol (PVBP) g R R ; (2) where $ is he Dollar Gamma, i.e. he swapion s Gamma imes he square of he forward swap rae, is he insananeous volailiy of he forward swap rae a day, IV 0 is he swapion implied percenage volailiy a he ime he sraegy is rs implemened, Track is he racking error of he hedging sraegy, Vol (PVBP) is he volailiy of he PVBP rae of growh a and, nally, gr R denoes he series of shocks a ecing he forward swap rae. In Appendix B, Eq. (B.11), we show ha a sraddle sraegy leads o a P&L similar o ha in Eq. (2), wih he rs erm being wice as ha in Eq. (2), and wih he sraddle value replacing he racking error in he second erm: P&L sraddle T TX =1 $ 2 IV 2 0 PVBPT + TX =1 STRADDLE Vol (PVBP) g R R ; (3) where STRADDLE is he value of he sraddle as of ime. The advanage of his sraegy over he rs, is ha i does no rely on hedging and, hence, i is no expensive in his respec. However, i is sriking o see how similar he P&L in Eqs. (2) and (3) are. We emphasize ha he P&L in Eqs. (2) and (3) are only approximaions o he exac expressions given in Appendix B. For example, he exac version of he P&L in Eq. (3) has an addiional erm, arising because rading sraddles implies a dela ha is only approximaely zero. I may well occur ha in episodes of pronounced ineres rae swap volailiy, his dela drifs signi canly away from zero. Naurally, no including his erm in he approximaions of Eqs. (2) and (3) favorably biases he assessmen of opion-based based sraegies o rade ineres rae swap volailiy. This erm would only add uncerainy o oucomes, on op of he uncerainies singled ou below. Eqs. (2) and (3) show ha he wo opion-based volailiy sraegies lead o quie unpredicable pro s ha fail o isolae changes in ineres rae swap volailiy. There are wo main issues: (i) For boh sraegies, he rs componen of he P&L is proporional o he sum of he daily P&L, given by he volailiy view 2 IV 2 0, weighed wih he Dollar Gamma, $, and he 7 cby Anonio Mele and Yoshiki Obayashi

PVBP a he swapion expiry. This is similar, bu disinc (due o he PVBP a T ), from wha we already know from equiy opion rading (see Bossu, Srasser and Guichard, 2005, and Chaper 10 in Mele, 2011). Indeed, one reason equiy variance conracs are aracive compared o opions-based sraegies is ha hey overcome he price dependency he laer generaes. This dependency shows up in our conex as well: even when 2 IV 2 0 > 0 for mos of he ime, i may happen oo ofen ha bad realizaions of volailiy, 2 IV 2 0 < 0, occur precisely when he Dollar Gamma $ is high. In oher words, he rs erms in Eq. (2) and Eq. (3) can lead o losses even if 2 is higher han IV 2 0 for mos of he ime. (ii) The second erms in Eqs. (2) and (3) show ha opion-based P&L depend on he shocks driving he forward swap rae, gr R. In fac, even if fuure volailiy is always higher han he curren implied volailiy, 2 > IV 2 0, he second erms in Eq. (2) and Eq. (3) may acually dwarf he rs in he presence of adverse realizaions of gr R. These second erms in he P&L do no appear in opion-based P&L for equiy volailiy. They are speci c o wha swapions are: insrumens ha have as underlying a swap conrac, whose underlying swap value s PVBP is unknown unil he mauriy of he swapion, as Eq. (1) shows. Uncerainy relaed o he PVBP, Vol (PVBP) > 0, canno be hedged away hrough he previous wo opion-based sraegies. These wo issues lead o compelling heoreical reasons why opions migh no be appropriae insrumens o rade ineres rae swap volailiy. How reliable are opion-based sraegies for ha purpose in pracice? Figure 4 plos he P&L of wo direcional volailiy rading sraegies agains he realized variance risk-premium, de ned below, and calculaed using daily daa from January 1998 o December 2009. The wo sraegies are long posiions in: (i) an a-he-money sraddle; (ii) one of he ineres rae variance conracs inroduced in he nex secion. Boh posiions relae o conracs expiring in one and hree monhs and enors of ve years. The realized variance risk-premium is de ned as he di erence beween he realized variance of he forward swap rae during he relevan holding period (i.e. one or hree monhs), and he squared implied a-he-money volailiy of he swapion (a one or hree monhs) prevailing a he beginning of he holding period. Appendix D provides all echnical deails regarding hese calculaions. The op panel of Figure 4 depics he wo P&Ls relaing o one-monh rades (147 rades), and he boom panel displays he wo P&Ls relaing o hree-monh rades (49 rades). Do hese sraegies deliver resuls consisen wih direcional views? In general, he P&Ls of boh he sraddle-based sraegy and he ineres rae variance conrac have he same sign as he realized variance risk-premium. However, he sraddle P&Ls are dispersed, in ha hey do no always preserve he same sign of he realized variance risk-premium, a propery displayed by he ineres rae variance conrac. Wih sraddles, he P&L has he same sign as he variance risk-premium approximaely 62% of he ime for one-monh rades, and for approximaely 65% of he ime for hree-monh rades. The correlaion beween he sraddle P&L and he variance risk-premium is abou 33% for he one-monh rade, and abou 28% for he hree-monh rade. Furher simulaion sudies performed by Jiang (2011) under a number of alernaive holding period assumpions sugges ha if dynamically dela-hedged afer incepion, hese rades lead o higher correlaions wih an average correlaion over all he simulaion experimens equal o 63%. Insead, noe he performance of he ineres rae variance conrac in our experimen, which is a pure play in volailiy, wih is P&L lining up o a sraigh line and a correlaion wih he variance risk-premium indisinguishable from 100%. 8 cby Anonio Mele and Yoshiki Obayashi

P&Ls of Volailiy Conrac and Sraddle P&Ls of Volailiy Conrac and Sraddle An Ineres Rae Swap Volailiy Index and Conrac 0.06 One monh rade and conrac 0.04 0.02 0 0.02 0.04 0.06 Volailiy Conrac Sraddle 0.08 0.1 0.25 0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2 Change in Variance (1 monh ino 5 years) Three monh rade and conrac 0.15 0.1 0.05 0 0.05 0.1 Volailiy Conrac Sraddle 0.1 0.05 0 0.05 0.1 0.15 0.2 Change in Variance (3 monhs ino 5 y ears) Figure 4. Empirical performance of direcional volailiy rades. 9 cby Anonio Mele and Yoshiki Obayashi

The reason sraddles lead o more dispersed pro s in he hree-monh rading period relaes o price dependency. For example, even if volailiy is up for mos of he ime in a one monh period, i migh be down during a given week, which in uences he P&L. However, sraddles do no weigh hese ups and downs in he same way. Raher, hey migh aribue a large weigh (given by he Dollar Gamma) o he paricular week when volailiy is down, as Eq. (3) makes clear. The chances hese unforunae weighings happen increase wih he duraion of he rade. While hese facs are well-known in he case of equiy volailiy, he complexiy of ineres rae ransacions makes hese facs even more severe when i comes o rading ineres rae swap volailiy hrough opions. 4. Ineres rae swap variance conracs We assume ha he forward swap rae R (T 1 ; ; T n ) is a di usion process wih sochasic volailiy, and consider exensions o a jump-di usion case in Secion 9. I is well-known ha he forward swap rae is a maringale under he so-called swap probabiliy, such ha in a di usive environmen, d ln R (T 1 ; ; T n ) = 1 2 k (T 1 ; ; T n )k 2 d + (T 1 ; ; T n ) dw ; 2 [; T ] ; (4) where W is a mulidimensional Wiener process under he swap probabiliy, and (T 1 ; ; T n ) is adaped o W. Accordingly, de ne V n (; T ), as he realized variance of he forward swap rae logarihmic changes in he ime inerval [; T ], for enor lengh T n T : V n (; T ) Z T k (T 1 ; ; T n )k 2 d: (5) Appendix A summarizes addiional noaion and basic assumpions and facs abou forward saring swaps, swap raes, he swap probabiliy, and he pricing of swapions. This secion designs conracs o rade he forward swap rae volailiy, V n (; T ), ha overcome he issues inheren in opion rading as observed in Secion 3. The conracs are priced in a model-free fashion, and lead o model-free indexes of expeced volailiy over a reference period [; T ] inroduced in Secion 7. This secion considers conracs referencing percenage, or logarihmic variance, and Secion 5 develops heir basis poin counerpars. 4.1. Conracs We consider hree conracs: a forward agreemen, which requires an iniial paymen, and wo variance swaps, which are seled a he expiraion. These conracs are all equally useful for he purpose of implemening views abou swap marke volailiy developmens, as explained in Secion 6. We begin wih he forward agreemen. Our rs remark is ha uncerainy abou swap markes relaes o he volailiy of he forward swap rae, rescaled by he PVBP, as Eq. (1) reveals, wih he complicaion ha he PVBP is unknown unil T. Consider he de niion of he realized variance of he forward swap rae, V n (; T ) in Eq. (5), and he following forward agreemen: 10 cby Anonio Mele and Yoshiki Obayashi

De niion I (Ineres Rae Variance Agreemen). A ime, counerpary A promises o pay counerpary B he produc of forward swap rae variance realized over he inerval [; T ] imes he swap s PVBP prevailing a ime T, i.e. he value V n (; T ) PVBP T (T 1 ; ; T n ). The price counerpary B shall pay counerpary A for his agreemen a ime is called he Ineres Rae Variance (IRV) rae, and is denoed as F var;n (; T ). The PVBP is he price impac of a one basis poin move in he swap rae over he enor a ime T, which is unknown before ime T, as furher discussed afer De niion III. Rescaling by he forward PVBP is mahemaically unavoidable when he objecive is o price volailiy in a model-free fashion. Our goal is o simulaneously price ineres rae swap volailiy, swaps, swapions and pure discoun bonds in such a way ha he price of volailiy conveys all he informaion carried by all he raded swapions and bonds. Eq. (1) suggess ha he uncerainy relaed o he ineres rae swaps underlying he swapions is ied o boh he realized variance, V n (; T ), and he forward PVBP. To insulae and price volailiy, he payo of he variance conrac needs o be rescaled by PVBP T (T 1 ; ; T n ), jus as he value of he swap does in Eq. (1). Noe ha he erm PVBP T (T 1 ; ; T n ) also appears in he P&L of he swapionbased volailiy sraegies, as shown by Eqs. (2) and (3); he imporan di erence is ha he variance conrac underlying De niion I isolaes volailiy from Dollar Gamma. 1 Finally, from a pracical perspecive, such a rescaling does no a ec he abiliy of he conrac o convey views abou fuure developmens of V n (; T ): as noed, he correlaion beween he variance risk-premium and he P&L of IR-variance conracs is nearly 100% in he empirical experimen in Figure 4. In fac, he P&L in Figure 4 refer o a variance conrac cas in a swap forma, whereby counerpary A agrees o pay counerpary B he following di erence a ime T : Var-Swap n (; T ) V n (; T ) PVBP T (T 1 ; ; T n ) P var;n (; T ) ; (6) where P var;n (; T ) is a xed variance swap rae, deermined a ime, chosen in a way o lead o he following zero value condiion a ime : De niion II (IRV ). The IRV swap rae is he xed variance swap rae P var;n (; T ), which makes he curren value of Var-Swap n (; T ) in Eq. (6) equal o zero. Noe ha he variance conrac arising from De niion II is a forward, no really a swap. However, we uilize a erminology similar o ha already in place for equiy variance conracs, and simply refer o he previous conrac as a swap. Finally, we consider a second de niion of he IRV swap rae, which will lead o a de niion of he index of ineres rae swap volailiy in Secion 7. Consider he following payo : Var-Swap n (; T ) V n (; T ) P var;n (; T ) PVBP T (T 1 ; ; T n ) ; (7) where P var;n (; T ), a xed variance swap rae deermined a, is se so as o lead o a zero value condiion a ime : 1 If he risk-neural expecaion of he shor-erm rae is insensiive o changes in he shor-erm swap rae volailiy, he PBVT a ime T is increasing in he shor-erm swap rae volailiy (see Mele, 2003). In his case, he IRV forward agreemen is a device o lock-in boh volailiy of he forward swap rae from o T, and shor-erm swap rae volailiy a T. 11 cby Anonio Mele and Yoshiki Obayashi

De niion III (Sandardized IRV ). The Sandardized IRV swap rae is he xed variance swap rae P var;n (; T ) ha makes he curren value of Var-Swap n (; T ) in Eq. (7) equal o zero. The payo in Eq. (7) carries an inuiive meaning. I is he produc of wo erms: (i) he difference beween he realized variance and a fair srike and (ii) he PVBP a ime T. This payo is similar o ha of sandard equiy variance conracs: he holder of he conrac would receive PVBP T (T 1 ; ; T n ) dollars for every poin by which he realized variance exceeds he variance srike price. The added complicaion of he IRV conrac design is ha PVBP T (T 1 ; ; T n ) is unknown a ime. 4.2. Pricing In he absence of arbirage and marke fricions, he price of he IRV forward agreemen o be paid a ime, F var;n (; T ) in De niion I, is expressed in a model-free forma as a combinaion of he prices of a-he-money and ou-of-he-money swapions. One approximaion o F var;n (; T ) based on a nie number of swapions, is, keeping he same noaion: " X R SWPN (K i ; T ; T n ) F var;n (; T ) = 2 K K 2 i + X # P SWPN (K i ; T ; T n ) K i:k i <R i K 2 i i:k i R i where SWPN R (K i ; T ; T n ) (resp., SWPN P (K i ; T ; T n )) is he price of a swapion receiver (resp., payer), sruck a K i, expiring a T and wih enor exending up o ime T n, and K i = 1 (K 2 i+1 K i 1 ) for 1 i < M, K 0 = (K 1 K 0 ), K M = (K M K M 1 ), where K 0 and K M are he lowes and he highes available srikes raded in he marke, and M + 1 is he oal number of raded swapions expiring a ime T and wih enor exending up o ime T n. Appendix C provides he heoreical framework underlying Eq. (8). Appendix E provides an expression for he local volailiy surface, which we can use o inerpolae he missing swapions from hose ha are raded, so as o ll in Eq. (8). In Appendix C, we show ha he swap rae in De niion II is given by: (8) P var;n (; T ) = F var;n (; T ) P (T ) (9) where P (T ) is he price of a zero coupon bond mauring a T and ime-o-mauriy T. Finally, he sandardized swap rae in De niion III is: P var;n (; T ) = F var;n (; T ) PVBP (T 1 ; ; T n ) (10) 4.3. Hedging The ineres rae variance conracs can be hedged using swapions underlying he evaluaion framework of Secion 4.2. An IRV marke maker will be concerned wih he replicaion of he payo s needed o hedge he conracs. In he exposiion below, we ake he perspecive of a provider of insurance agains volailiy who sells any of he IRV conracs underlying De niions I hrough III of Secion 4.1. 12 cby Anonio Mele and Yoshiki Obayashi

The IRV forward conrac in De niion I can be hedged hrough he posiions in rows (i) and (ii) of Tables I and II, which we now discuss. Nex, consider he conracs relaing o he IRV swap rae of De niion II, P var;n (; T ). We aim o replicae he payo in Eq. (6). As shown in Appendix C, we hedge agains his payo hrough wo, idenical, zero cos porfolios, consruced wih he following posiions: (i) A dynamic posiion in a self- nanced porfolio of zero coupon bonds aiming o replicae he cumulaive incremens of he forward swap rae over [; T ], which urns ou o equal dr s(t 1 ; ;T n) R s(t 1 ; ;T n) = 1V 2 n (; T ) + ln R T (T 1 ; ;T n) R (T 1 ; ;T n), rescaled by he PVBP a ime T. R T (ii) Saic posiions in swaps saring a ime T and OTM swapions expiring a ime T, aiming o replicae he payo of he ineres rae counerpar o he log-conrac (Neuberger, 1994) for equiies, rescaled by he PVBP a T. The posiions are as follows: (ii.1) Shor 1=R (T 1 ; ; T n ) unis of a forward saring (a T ) xed ineres payer swap sruck a he curren forward swap rae, R (T 1 ; ; T n ). (ii.2) Long ou-of-he-money swapions, each of hem carrying a weigh equal o K i, Ki 2 where K i and K i are as in Eq. (8). (iii) A saic borrowing posiion aimed o nance he ou-of-he-money swapion posiions in (ii.2). The deails for he self- nanced porfolio in (i) are given in Appendix C. Table I summarizes he coss of each of hese rades a, as well as he payo s a T, which sum up o be he same as he payo in Eq. (6), as required o hedge he conrac. Table I Replicaion of he conrac relying on he IRV swap rae of De niion II. ZCB and OTM sand for zero coupon bonds and ou-of-he-money, respecively, and R R (T 1 ; ; T n ), PVBP T PVBP T (T 1 ; ; T n ). Porfolio Value a Value a T (i) long self- nanced porfolio of ZCB 0 V n (; T ) + 2 ln R T R PVBP T (ii) shor swaps and long OTM swapions F var;n (; T ) 2 ln R T R PVBP T F (iii) borrow F var;n (; T ) +F var;n (; T ) var;n(;t ) P (T ) = P var;n (; T ) Ne cash ows 0 V n (; T ) PVBP T P var;n (; T ) Finally, o hedge agains he Sandardized conrac of De niion III, we need o replicae he payo in Eq. (7). We creae a porfolio, which is he same as ha in Table I, excep now ha row (iii) refers o a borrowing posiion in a baske of zero coupon bonds wih value equal o PVBP (T 1 ; ; T n ) and noional of P var;n (; T ), as in Table II. By Eq. (10), he borrowing posiion needed o nance he ou-of-he-money swapions in row (ii), is jus F var;n (; T ) = P var;n (; T ) PVBP (T 1 ; ; T n ). Come ime T, i will be closed for a value of P var;n (; T ) PVBP T (T 1 ; ; T n ). The ne cash ows are precisely hose of he conrac wih he Sandardized IRV swap rae. 13 cby Anonio Mele and Yoshiki Obayashi

Table II Replicaion of he conrac relying on he Sandardized IRV swap rae of De niion III. ZCB and OTM sand for zero coupon bonds and ou-of-he-money, respecively, and R R (T 1 ; ; T n ), PVBP PVBP (T 1 ; ; T n ). Porfolio Value a Value a T (i) long self- nanced porfolio of ZCB 0 V n (; T ) + 2 ln R T R PVBP T (ii) shor swaps and long OTM swapions F var;n (; T ) 2 ln R T R PVBP T (iii) borrow baske of ZCB for P var;n (; T ) PVBP +F var;n (; T ) P var;n (; T ) PVBP T Ne cash ows 0 Vn (; T ) P var;n (; T ) PVBP T The replicaion argumens in his secion are alernaive means o deermine he no-arbirage value of he IRV forward agreemen of De niion I, and he IRV swap raes in De niions II and III, P var;n (; T ) and P var;n (; T ). Suppose, for example, ha he marke value of he Sandardized IRV swap rae, P var;n (; T ) $ say, is higher han he no-arbirage value P var;n (; T ) in Eq. (10). Then, one could shor he sandardized conrac underlying De niion III and, a he same ime, synhesize i hrough he porfolio in Table II. This posiioning coss zero a ime and yields a sure posiive pro a ime T equal o P var;n (; T ) $ P var;n (; T ) PVBP T. To rule ou arbirage, hen, we need o have P var;n (; T ) $ = P var;n (; T ). 5. Basis poin volailiy 5.1. Conracs and evaluaion We can price and hedge volailiy based on arihmeic, or basis poin (BP henceforh), changes of he forward swap rae in Eq. (4), dr (T 1 ; ; T n ) = R (T 1 ; ; T n ) (T 1 ; ; T n ) dw ; 2 [; T ] : (11) Accordingly, denoe he BP realized variance wih V BP n (; T ), Z T Vn BP (; T ) R 2 (T 1 ; ; T n ) k (T 1 ; ; T n )k 2 d (12) We can hen resae he ineres rae variance conracs in Secion 4.1 in erms of BP variance and, for reasons explained below, we refer o he ensuing conracs as Gaussian conracs, de ned as follows: De niion IV (Gaussian Conracs). Consider he following conracs and swap raes replacing hose in De niions I, II, and III: (a) (BP-IRV Agreemen). A ime, counerpary A promises o pay counerpary B he produc of he forward swap rae BP variance realized over he inerval [; T ] imes he swap s PVBP ha will prevail a ime T, i.e. he value Vn BP (; T ) PVBP T (T 1 ; ; T n ). The price counerpary B shall pay counerpary A for his agreemen a ime is called BP-IRV rae, and is denoed as F BP var;n (; T ). 14 cby Anonio Mele and Yoshiki Obayashi

(b) (BP-IRV ). The BP-IRV swap rae is he xed variance swap rae P BP var;n (; T ), which makes he curren value of Vn BP (; T ) PVBP T (T 1 ; ; T n ) P BP var;n (; T ) equal o zero. (c) (Sandardized BP-IRV ). The Sandardized BP-IRV swap rae is he xed variance swap rae P BP var;n (; T ), which makes he curren value of Vn BP (; T ) P BP var;n (; T ) PVBP T (T 1 ; ; T n ) equal o zero. The reason we refer o hese conracs as Gaussian relaes o he fac ha a benchmark case is one where he realized BP variance of he forward swap rae in Eq. (11) is consan and equal o N, dr (T 1 ; ; T n ) = N dw ; 2 [; T ] ; (13) such ha he forward swap rae has a Gaussian disribuion. We shall reurn o his absrac assumpion of a consan BP variance below o develop inuiion abou he pricing resuls relaing o he general sochasic BP variance case as de ned in Eq. (12). In Appendix G, we show ha in he general case of Eqs. (11) and (12), he price of he BP-IRV forward conrac is approximaed by: F BP var;n (; T ) 2 " X i:k i <R SWPN R (K i ; T ; T n ) K i + X i:k i R SWPN P (K i ; T ; T n ) K i where he srike di erences, K i, are as in Eq. (8). Given he BP-IRV forward, he BP-IRV swap raes in De niion IV-(a) and De niion IV-(b) are: # (14) var;n (; T ) = FBP var;n (; T ) P (T ) P BP and P BP var;n (; T ) = F BP var;n (; T ) PVBP (T 1 ; ; T n ) (15) The inuiion behind he forward price in Eq. (14) is ha he insananeous BP-variance is simply he insananeous variance of he logarihmic changes of he forward swap rae rescaled by he forward swap rae, as in Eq. (11), and squared. This propery ranslaes ino an analogous propery for he price of he forward variance agreemen: comparing Eq. (8) wih Eq. (14) reveals ha for he BP agreemen, each swapion price i carries he same weigh as ha for he agreemen in De niion I, K i, rescaled by he squared srike K 2 Ki 2 i, i.e. K i K 2 Ki 2 i. In heir derivaion of he fair value of equiy variance swaps, Demeer, Derman, Kamal and Zou (1999) develop an inuiive approach relying on he Black and Scholes (1973) marke, and explain ha a porfolio of opions has a vega ha is insensiive o changes in he sock price only when he opions are weighed inversely proporional o he square of he srike, which also relaes o he fair value of our forward variance agreemen in Eq. (8). We develop a similar approach o gain inuiion abou he reasons underlying he uniform swapions weighings in Eq. (14). Assume a Gaussian marke, i.e. one where he forward swap rae is as in Eq. (13). This assumpion is he obvious counerpar o ha of a consan percenage volailiy underlying he sandard Black-Scholes marke for equiy opions. Denoe wih O (R; K; T; N ; T n ) he price of a swapion (be i a receiver or a payer) a ime when he forward swap rae is R. We creae a porfolio wih a coninuum of swapions having he same mauriy and enor, and 15 cby Anonio Mele and Yoshiki Obayashi

denoe wih! (K) he porfolio weighings, assumed o be independen of R, and such ha he value of he porfolio is: Z (R (T 1 ; ; T n ) ; N )! (K) O (R (T 1 ; ; T n ) ; K; T; N ; T n ) dk: We require ha he vega of he porfolio, de ned as (R; ) @(R;), be insensiive o @ changes in he forward swap rae, @ (R; ) = 0: (16) @R In Appendix G, we show ha he vega of he porfolio does no respond o R if and only if he weighings,! (K), are independen of K, Eq. (16) holds rue ()! (K) = cons: (17) Tha is, a porfolio of swapions aiming o rack BP variance, hereby being immuned o changes in he underlying forward swap rae, is one where he swapions are equally weighed. 5.2. Hedging Replicaing BP-denominaed conracs requires posiioning in a way quie di eren from replicaing percenage volailiy as explained in Secion 4.3. As noed in he previous secion, he hedging argumens summarized in Tables I and II rely on he replicaion of log-conracs, i.e. conracs delivering ln( R T R ), rescaled by he PVBP. Insead, o hedge BP conracs, we need o rely on quadraic conracs, i.e. hose delivering RT 2 R 2, rescaled by he PVBP, as we now explain. Consider, rs, he payo of he BP-IRV forward conrac in De niion IV-(a), which can be replicaed hrough he porfolios in rows (i) and (ii) of Tables III and IV, as furher discussed below. To replicae he conrac for he BP-IRV swap rae of De niion IV-(b), we creae porfolios comprising he following posiions: (i) A dynamic, shor posiion in wo idenical, self- nanced porfolios of zero coupon bonds ha replicae he incremens of he forward swap rae over [; T ], weighed by he forward R T swap rae, which urns ou o equal R s (T 1 ; ; T n ) dr s (T 1 ; ; T n ) = V BP n (; T ) (RT 2 (T 1; ; T n ) R 2 (T 1 ; ; T n )), rescaled by he PVBP a ime T. 1 2 (ii) Saic posiions in swaps saring a ime T and OTM swapions expiring a ime T, aiming o replicae he payo of a quadraic conrac, rescaled by he PVBP a T. The posiions are as follows: (ii.1) Long 2R (T 1 ; ; T n ) unis of a forward saring (a T ) xed ineres payer swap sruck a he curren forward swap rae, R (T 1 ; ; T n ). (ii.2) Long ou-of-he-money swapions, each of hem carrying a weigh equal o 2K i, where K i and K i are as in Eq. (8). (iii) A saic borrowing posiion aimed o nance he ou-of-he-money swapion posiions in (ii.2). Coss and payo s of hese porfolios are summarized in Table III. The resul is a perfec replicaion of he payo relaing o he conrac of De niion IV-(b). 16 cby Anonio Mele and Yoshiki Obayashi

Table III Replicaion of he payo relaing o he BP-IRV conrac of De niion IV-(b). ZCB and OTM sand for zero coupon bonds and ou-of-he-money, respecively, and R R (T 1 ; ; T n ), PVBP T PVBP T (T 1 ; ; T n ). Porfolio Value a Value a T (i) shor self- nanced porfolio of ZCB 0 (ii) long swaps and long OTM swapions F BP var;n (; T ) RT 2 R 2 PVBPT (iii) borrow F BP var;n (; T ) +F BP F var;n (; T ) BP var;n (;T ) P (T ) = P BP var;n (; T ) Ne cash ows 0 Vn BP (; T ) PVBP T P BP var;n (; T ) V BP n (; T ) R2 T R 2 PVBPT Finally, Table IV is obained by modifying he argumens leading o Table III, in he same way as we did o obain Table II in Secion 4.3. Table IV Replicaion of he payo relaing o he Sandardized BP-IRV conrac of De niion IV- (c). ZCB and OTM sand for zero coupon bonds and ou-of-he-money, respecively, and R R (T 1 ; ; T n ), PVBP PVBP (T 1 ; ; T n ). Porfolio Value a Value a T (i) shor self- nanced porfolio of ZCB 0 (ii) long swaps and long OTM swapions F BP var;n (; T ) RT 2 R 2 PVBPT (iii) borrow baske of ZCB for P BP var;n (; T ) PVBP +F BP var;n (; T ) P BP var;n (; T ) PVBP T Ne cash ows 0 V BP n (; T ) R2 T R 2 PVBPT V BP n (; T ) P BP var;n (; T ) PVBP T 5.3. Links o consan mauriy swaps We show ha a consan mauriy swap (CMS) can be represened as a baske of he BP-IRV forwards of De niion IV-(a), suggesing ineres rae variance swaps can be used o hedge agains CMS. The echnical reason IRV forwards link o CMS is due o a classical convexiy adjusmen of he CMS, arising because he price of a CMS relaes o he wrong expecaion of fuure payo s, one under which he forward swap rae is no a maringale. In a sandard consan mauriy swap, a pary pays a counerpary he spo swap rae wih a xed enor over a sequence of daes, wih legs se in advance, i.e. R T0 +j (T 1 + j; ; T n + j) a imes T 0 + (j + 1), j = 0; ; N 1, where is a xed consan, for example 1, and N is he number of paymen daes. In Appendix G, we show ha 2 he price of a CMS is approximaely equal o: CMS N () + NX 1 j=0 NX 1 j=0 P ( j + k) R (T 1 + j; ; T n + j) G 0 (R (T 1 + j; ; T n + j)) F BP var;n (; T 0 + j) ; (18) where F BP var;n (; T ) is he price of he BP-IRV forward conrac in De niion IV-(a), as given by Eq. (14), and G () is a funcion given in he Appendix (see Eq. (G.8)). 17 cby Anonio Mele and Yoshiki Obayashi

Tha he price of a CMS relaes o he enire swapion skew is known a leas since Hagan (2003) and Mercurio and Pallavicini (2006), as furher discussed in Appendix G. As is also clear from Eq. (18), CMS can be hedged hrough a baske of IRV forwards in BP expiring a he poins preceding he CMS paymens. 6. Marking o marke and rading sraegies This secion provides expressions for marking o marke updaes of he IRV conracs of Secions 4 and 5 (see Secion 6.1). These expressions also help assess he value of rading sraegies based on hese conracs, as explained in Secion 6.2. 6.1. Marks o marke of IRV swaps In Appendix C, we show ha he value a any ime 2 (; T ), of he variance conrac sruck a ime a he IRV swap rae P var;n (; T ) in Eq. (9) is: M-Var n (; ; T ) V n (; ) PVBP (T 1 ; ; T n ) P (T ) [P var;n (; T ) P var;n (; T )] ; (19) and he value of he conrac sruck a he Sandardized IRV swap rae P var;n (; T ) in Eq. (10) is: M-Var n (; ; T ) PVBP (T 1 ; ; T n ) V n (; ) P var;n (; T ) P var;n (; T ) : (20) I is immediae o see ha he marking o marke updaes for Basis Poin conracs are he same as hose in Eqs. (19) and (20) di ering only by a mere change in noaion. In he nex secion, we uilize hese marking o marke expressions o calculae he value of rading sraegies. 6.2. Trading sraegies We consider examples of rading sraegies relying on he IRV conracs of Secions 4 and 5. We analyze sraegies relying on views abou developmens of: (i) spo IRV swap raes relaive o curren (in Secion 6.2.1), and (ii) he value of spo IRV swap raes relaive o forward IRV swap raes i.e. hose implied by he curren erm-srucure of spo IRV swap raes (in Secion 6.2.2). The sraegies we consider apply o boh percenage and BP conracs, alhough o simplify he presenaion, we only illusrae he case applying o percenage conracs. 6.2.1. Spo rading 6.2.1.1. Trading hrough IRV swap raes Firs, we sugges how o express views on IRV conracs ha have di eren mauriies and enors. Le m = 1, and consider, for example, Figure 5, which illusraes he iming of wo 12 conracs: (i) a conrac expiring in 3 monhs, relaing o he volailiy of 3m ino ^T 5 3m forward swap raes; and (ii) a conrac expiring in 9 monhs, relaing o he volailiy of 9m ino T 5 9m forward swap raes. Time unis are expressed in years, and we se = 0. We assume he wo conracs have he same enor, ^T 5 3m = T 5 9m. 18 cby Anonio Mele and Yoshiki Obayashi

Tˆ 1 T ˆ Tˆ 2 5 =0 T=3m T=9m T1 T2 5 Figure 5. Rolling enor rade. T Le P var ˆ ( )andp var ( ) be he IRV swap raes of hese wo conracs, and suppose wehaveheviewhaasomepoin 3, P var ˆ ( 3 ) P var ˆ (0 3 ) and P var ( 9 ) P var (0 9 ) (21) A rading sraegy consisen wih hese views migh be as follows: (i) go long he 3 conrac, (ii) shor unis of he 9 conrac, (iii) shor he IRV forward conrac in Definiion I for he 3 mauriy, wih conrac lengh equal o, (iv)golong unis of he IRV forward conrac for he 9 mauriy, wih conrac lengh equal o. The value of his sraegy a ime is, by marking-o-marke according o Eq. (19): M-Varˆ (0 3 ) M-Var (0 9 ) ˆ (0 ) PVBP ( ˆ 1 ˆ )+ (0 ) PVBP ( 1 ) = (3 )[P var ˆ ( 3 ) P var ˆ (0 3 )] + (9 )[P var (0 9 ) P var ( 9 )] 0 Is cos a =0is F var ˆ (0 3 )+ F var (0 9 ). We choose, = F var ˆ (0 3 ) F var (0 9 ) (22) o make he porfolio worhless a =0. The echnical reason o ener ino he IRV forward conrac is ha he ime value of he variance conracs (i) and (ii) enails, by Eq. (19), cumulaive realized variance exposures equal o ˆ (0 ) PVBP ( ˆ 1 ˆ )and (0 ) PVBP ( 1 ), which canno offse, as hey refer o wo disinc raes: he 3 ino ˆ 5 3 forward swap rae and he 9 ino 5 9 forwardswaprae.thedeviceoshor unis of he 9 conrac serves he purpose of offseing hese wo exposures a ime, and he expression for in Eq. (22) allows us o creae a zero-cos porfolio a =0. Nex, we develop an example of a rading sraegy relying on conracs wih differen mauriies bu he same forward swap raes. Assume ha a ime 0, he erm srucure of he IRV swap rae P var (0 )isincreasingin for some fixed enor, bu ha a view is held ha his erm srucure is abou o flaen soon. For example, we migh expec ha a some poin in ime, P var ( 3 ) P var (0 3 ) and P var ( 9 ) P var (0 9 ) (23) where 3. To express his view, one could go long a 3 monh IRV swap sruck a P var (0 3 ) and shors a 9 monh IRV swap sruck a P var (0 9 ). Noe ha his rading sraegy relies on 19 c by Anonio Mele and Yoshiki Obayashi

IRV swaps for forward swap raes ha are 9 monhs ino 9 years, a ime =0,as Figure 6 illusraes. =0 T=3m T 0 =9m T 1 T 2 T n Figure 6. Fixed enor rade. The payoffs o hese wo IRV swaps are as follows: (i) The payoff of he 3 monh IRV swap a =3 is (0 3 ) PVBP 3 ( 1 ), where he PVBP 3 ( 1 ) relaes o swaps paymens saring a ime 1. (ii) The payoff of he 9 monh IRV swap a =9 is (0 9 ) PVBP 9 ( 1 ), and PVBP 9 ( 1 ) relaes o swaps paymens saring a ime 1. The price of he 9 monh conrac relies on he price of ou-of-he-money swapions ha are 9monhsino 9 years, as of ime = 0. The price of he 3 monh conrac is insead calculaed hrough he price of ou-of-he-money swapions wih expiry =3 and underlying swap saring a 0 =9. Typically, swapion markes have = 0, alhough heoreically, he pricing framework in his secion is sill valid even when 0 as i is for he 3 monh conrac in Figure 6. Noe ha markes may no exis or, a bes, be illiquid, for he swapions needed o price volailiy producs raded for an expiry rade occurring prior o he beginning of he enor period of he underlying forward swap rae, 0. In his case, he pricing of hese producs could no be model-free. The simples soluion o price hese illiquid asses migh rely on he Black s (1976) formula, where one uses as an inpu he implied volailiies for mauriy 0 and replaces, everywhere in he formula, ime o mauriy 0 wih ime o mauriy. Noe ha we could simply no use swapions wih ime o mauriy o price hese producs, if he swaps and, hence, he forward swap raes underlying hese swapions sar exis a ime,noaime 0 as required in Figure 6. The sraegy of Figure 6 eliminaes he risk relaed o he flucuaions of realized ineres rae swap volailiy, as we now explain. Come ime, isvalueis: M-Var (0 3 ) M-Var (0 9 ) = (0 ) PVBP ( 1 )+ (3 )[P var ( 3 ) P var (0 3 )] ( (0 ) PVBP ( 1 )+ (9 )[P var (0 9 ) P var ( 9 )]) 0 where we have made use of Eq. (19), and he las inequaliy follows by he view summarized by Eqs.(23). Noe ha he sraegy eliminaes he cumulaive realized variance exposure (0 ) PVBP ( 1 ), as shown by he previous calculaions, precisely because we are rading he wo IRV swaps in Figure 6, which refer o he volailiy of he same forward swap rae a = 0: he forward swap rae of 9 monhs ino 9 years. As a resul, a fixed enor rade does no need addiional posiions in he IRV forward conrac, as in he case of he rolling enor rades in Figure 5. 20 c by Anonio Mele and Yoshiki Obayashi

6.2.1.2. Trading hrough Sandardized IRV swap raes While he rading sraegies of he previous secion rely on he IRV swap rae of De niion II, rading he Sandardized IRV swap rae of De niion III leads o comparable oucomes. For example, suppose ha, in analogy wih Eq. (21), he view is held ha, a some fuure poin 3m, P var;^n (; 3m) > P var;^n (0; 3m) and P var;n (; 9m) < P var;n (0; 9m) ; where he dynamics of he conracs are, now, as in Figure 5. Consider he following sraegy: (i) go long he 3m conrac, (ii) shor unis of he 9m conrac, (iii) shor he IRV forward conrac of De niion I for he 3m mauriy, wih conrac lengh equal o, (iv) go long unis of he IRV forward conrac for he 9m mauriy, wih conrac lengh equal o. Using he marking o marke updaes in Eq. (20), we nd ha he value of his sraegy a ime is: M-Var ^n (0; ; 3m) M-Var n (0; ; 9m) V^n (0; ) PVBP ( ^T 1 ; ; ^T n ) + V n (0; ) PVBP (T 1 ; ; T n ) = PVBP ( ^T 1 ; ; ^T n ) P var;^n (; 3m) P var;^n (0; 3m) + PVBP (T 1 ; ; T n ) P var;n (0; 9m) P var;n (; 9m) > 0; where is as in Eq. (22), which ensures he porfolio is worhless a = 0. Likewise, suppose ha, similarly as in Eq. (23), he view is ha, a some fuure poin 3m, P var;n (; 3m) > P var;n (0; 3m) and P var;n (; 9m) < P var;n (0; 9m) ; where he dynamics of he conracs are exacly as in Figure 6. We go long he 3m and shor he 9m Sandardized conracs. By Eq. (20), he value of his sraegy a ime is: M-Var n (0; ; 3m) M-Var n (0; ; 9m) = PVBP (T 1 ; ; T n ) P var;n (; 3m) P var;n (0; 3m) + P var;n (0; 9m) P var;n (; 9m) > 0: 6.2.2 rading Finally, we consider examples of rading sraegies o implemen views abou he implici pricing ha we can read hrough he curren IRV swap raes. Suppose, for example, ha a ime = 0, we hold he view ha in one year ime, he IRV swap rae for a conrac expiring in a furher year will be greaer han he implied forward IRV swap rae, i.e. he rae implied by he curren erm srucure of he IRV conracs, viz P var;n (1; 2) > P var;n (0; 2) P var;n (0; 1) : (24) The hree IRV swap raes refer o conracs wih enors saring in wo year ime from = 0, and exhausing a he same ime T n, similarly as for he conracs underlying he xed enor rades in Figure 6. We synhesize he cheap, implied IRV swap rae, by going long he following porfolio a 0: (i) long a wo year IRV swap, sruck a P var;n (0; 2) [P.1] (ii) shor a one year IRV swap, sruck a P var;n (0; 1) 21 cby Anonio Mele and Yoshiki Obayashi

This porfolio is cosless a ime zero, and if Eq. (24) holds rue in one year, hen in one year s ime we close (i) and access he payo in (ii), securing a oal payo equal o: M-Var n (0; 1; 2) + [P var;n (0; 1) V n (0; 1) PVBP 1 (T 1 ; ; T n )] : The rs erm is he marke value of he long posiion in (i) in one year s ime, wih M- Var n (0; 1; 2) as in Eq. (19). The second erm is he payo arising from (ii). Using Eq. (19), we obain, = [P var;n (1; 2) P var;n (0; 2)] P 1 (2) + P var;n (0; 1) [P var;n (1; 2) P var;n (0; 2) + P var;n (0; 1)] P 1 (2) > 0; where he rs inequaliy follows by P 1 (2) < 1, and he second holds by Eq. (24). Likewise, we can implemen rading sraegies consisen wih views abou implied Sandardized IRV swap raes. Suppose we anicipae ha: P var;n (1; 2) > P var;n (0; 2) P var;n (0; 1) ; where he dynamics of he conracs are again as in Figure 6. Consider he same porfolio as in [P.1] wih Sandardized IRV conracs replacing IRV conracs. In one year s ime, he value of his porfolio is: M-Var n (0; 1; 2) + P var;n (0; 1) V n (0; 1) PVBP 1 (T 1 ; ; T n ) = P var;n (1; 2) + P var;n (0; 1) P var;n (0; 2) PVBP 1 (T 1 ; ; T n ) > 0; where we have used he marking o marke updae in Eq. (20). 7. Swap volailiy indexes 7.1. Percenage The price of he IRV forward agreemen of De niion I can be normalized by he lengh of he conrac, T, and quoed in erms of he curren swap s PVBP, so as o lead o he following index of expeced volailiy: r 1 IRS-VI n (; T ) = T P var;n (; T ) (25) where P var;n (; T ) is he Sandardized IRV swap rae in Eq. (10). We label he index in Eq. (25) IRS-VI. In Appendix C.1, we show ha he expression in Eq. (25) can be simpli ed and fed wih only he Black s (1976) volailiies on he ou-of-he money swapion skew (see Eq. (C.5)). In he Appendix, we also show ha if V n (; T ) and he pah of he shor-erm rae in he ime inerval [; T ] were uncorrelaed, he IRS-VI in Eq. (25) would be he risk-neural expecaion of he fuure realized variance de ned as: r 1 E-Vol n (; T ) T E Q [V n (; T )]; (26) 22 cby Anonio Mele and Yoshiki Obayashi

where E Q denoes expecaion in a risk-neural marke. In pracice, V n (; T ) and ineres raes are likely o be correlaed. For example, if he shorerm rae is generaed by he Vasicek (1977) model, he correlaion beween V n (; T ) and PVBP T (T 1 ; ; T n ) is posiive a nding we esablished as a by-produc of experimens repored in Appendix F. Despie his heoreical dependence, Eq. (25) is, for all inens and purposes, he same as he risk-neural expecaion of fuure realized volailiy, Eq. (26). Figure 7 plos he relaion beween volailiy and he forward rae arising wihin he Vasicek marke. The lef panels plo he relaion beween he forward swap rae and: (i) he IRS-VI n (; T ) index in Eq. (25), for a mauriy T = 1 monh and enor n = 5 years (op panels), a mauriy T = 1 year and enor n = 10 years (boom panels), and regular quarerly rese daes; and (ii) he risk-neural expecaion of fuure realized volailiy of he forward swap rae, compued hrough Eq. (26). The righ panels plo he relaion beween he forward swap rae and he insananeous volailiy of he forward swap rae, as de ned in Appendix F, Eq. (F.4). Appendix F provides deails regarding hese calculaions, as well as addiional numerical resuls relaing o a number of alernaive combinaions of mauriies and enors (see Table A.1). In hese numerical examples, he IRS-VI index provides an upper bound o he expeced volailiy in a risk-neural marke, and i highly correlaes wih i. The IRS-VI and expeced volailiy respond precisely in he same way o changes in marke condiions, as summarized by movemens in he forward swap rae, and deviae quie insigni canly from each oher. Ineresingly, he Vasicek marke is one where we observe he ineres rae counerpar of he leverage e ec observed in equiy markes: low ineres raes are associaed wih high ineres rae volailiy. 7.2. Basis poin The BP counerpar o he IRS-VI index in Eq. (25) is: IRS-VI BP n (; T ) = r 1 T PBP var;n (; T ); P BP var;n (; T ) = F BP var;n (; T ) PVBP (T 1 ; ; T n ) (27) where F BP var;n (; T ) is he price of he IRV forward agreemen in Eq. (14). Similar o he percenage index in Eq. (25), he Basis Poin index in Eq. (27) can be calculaed by feeding Black s (1976) formula wih he swapion skew (see Eq. (G.5) in Appendix G.1). Ineresingly, our model-free BP volailiy index squared and rescaled by 1 coincides wih T he condiional second momen of he forward swap rae, as i urns ou by comparing our formula in Eq. (27) wih expressions in Trolle and Schwarz (2011). I is an ineresing saisical propery, which complemens he asse pricing foundaions lied down in Secions 4 and 5, peraining o securiy designs, hedging, replicaion, and hose relaing o marking o marke and rading sraegies in Secion 6. In Secion 9, we shall show ha our Basis Poin index enjoys he addiional ineresing propery of being resilien o jumps: is value remains he same, and hence model-free, even in he presence of jumps. Finally, Figure 8 repors experimens ha are he basis poin counerpars o hose in Figure 7. In hese experimens, we compare BP volailiy as referenced by our index for ve and en year enors, wih ha arising in a risk-neural Vasicek marke, as well as he insananeous BP volailiy of he forward rae, as prediced by he Vasicek model. Once again, he index and expeced volailiy are pracically he same, a propery arising in a number of addiional experimens repored in Appendix G.4. 23 cby Anonio Mele and Yoshiki Obayashi

1Y 10Y IRS VI and Expeced Volailiy (in %) Swap Insananeous Volailiy (in %) 1m 5Y IRS VI and Expeced Volailiy (in %) Swap Insananeous Volailiy (in %) An Ineres Rae Swap Volailiy Index and Conrac 36 34 IRS VI (in %) Expeced Volailiy (in %) 36 34 32 30 28 26 24 22 20 18 4 5 6 7 8 9 1m 5Y (in %) 32 30 28 26 24 22 20 18 4 5 6 7 8 9 1m 5Y (in %) 14.5 14 IRS VI (in %) Expeced Volailiy (in %) 12 11.5 13.5 11 13 10.5 12.5 10 12 9.5 11.5 5.5 6 6.5 7 7.5 8 1Y 10Y (in %) 9 5.5 6 6.5 7 7.5 8 1Y 10Y (in %) Figure 7. Lef panels: The IRS-VI index and expeced volailiy (boh in percenage) in a risk-neural Vasicek marke, as a funcion of he level of he forward swap rae. Righ panels: he insananeous forward swap rae volailiy (in percenage) as a funcion of he forward swap rae. Top panels relae o a lengh of he variance conrac equal o 1 monh and enor of 5 years. Boom panels relae o a lengh of he variance conrac equal o 1 year and enor of 10 years. 24 cby Anonio Mele and Yoshiki Obayashi

1Y 10Y IRS VI and Expeced Volailiy (in BP) Swap Insananeous Volailiy (in BP) 1m 5Y IRS VI and Expeced Volailiy (in BP) Swap Insananeous Volailiy (in BP) An Ineres Rae Swap Volailiy Index and Conrac 160 158 158 156 156 154 154 152 152 150 150 IRS VI (in BP) Expeced Volailiy (in BP) 148 4 5 6 7 8 9 1m 5Y (in %) 148 146 4 5 6 7 8 9 1m 5Y (in %) 89 73 88 72 87 86 71 70 85 84 83 69 68 82 IRS VI (in BP) Expeced Volailiy (in BP) 81 5.5 6 6.5 7 7.5 8 1Y 10Y (in %) 67 66 5.5 6 6.5 7 7.5 8 1Y 10Y (in %) Figure 8. Lef panels: The IRS-VI index and expeced volailiy (boh in Basis Poins) in a risk-neural Vasicek marke, as a funcion of he level of he forward swap rae. Righ panels: he insananeous forward swap rae volailiy (in Basis Poins) as a funcion of he forward swap rae. Top panels relae o a lengh of he variance conrac equal o 1 monh and enor of 5 years. Boom panels relae o a lengh of he variance conrac equal o 1 year and enor of 10 years. 25 cby Anonio Mele and Yoshiki Obayashi

8. Raes versus equiy variance conracs and indexes An Ineres Rae Swap Volailiy Index and Conrac The pricing framework underlying IRV conracs and indexes relies on spanning argumens, such as hose in Bakshi and Madan (2000) and Carr and Madan (2001), as is also he case wih equiy variance conracs and indexes. However, IRV conracs and indexes quie di er from he equiy case for a number of reasons. Firs, he payo s of IRV conracs inroduced in his paper have wo sources of uncerainy: one relaed o he realized variance of he forward swap rae, and anoher relaed o he forward PVBP. These feaures of he conrac design and he assumpion ha swap raes are sochasic disinguish our mehodology from he work ha has been done on pricing equiy volailiy. In he equiy case, he volailiy index is: v u 2 3 u VIX (; T ) = 1 T e 2 r(t ) 4 X i:k i <F (T ) Pu (K i ; T ) K Ki 2 i + X i:k i F (T ) Call (K i ; T ) K 2 i K i 5; (28) where r is he insananeous ineres rae, assumed o be consan, and Pu (K; T ) and Call (K; T ) are he marke prices as of ime of ou-of-he money European pu and call equiy opions wih srike prices equal o K and ime o mauriy T, F (T ) is he forward price of equiy, and K i are as in Eq. (8). Noe ha due o consan discouning, he fair srike for an equiy variance conrac is he same as he index, up o ime-rescaling and squaring, and is given by: P equiy (; T ) (T ) VIX 2 (; T ) : (29) As for IRV conracs, noe ha he fair value of he Sandardized IRV swap rae, P var;n (; T ) in Eq. (7), is sill he index, up o ime-rescaling and squaring and, by Eq. (25), is given by: IRS-VI n (; T ) v " u = 1 2 X R SWPN (K i ; T ; T n ) K T PVBP (T 1 ; ; T n ) K 2 i + X P SWPN (K i ; T ; T n ) K i:k i <R i K 2 i #: i:k i R i While he wo indexes in Eqs. (28) and (30) aggregae prices of ou-of-he money derivaives using he same weighs, hey di er for hree reasons. (i) he VIX in Eq. (28) is consruced by rescaling a weighed average of ou-of-he money opion prices hrough he inverse of he price of a zero wih he same expiry dae as ha of he variance conrac in a marke wih consan ineres raes, e r(t ). Insead, he IRS-VI index in Eq. (30) rescales a weighed average of ou-of-he money swapion prices wih he inverse of he price of a baske of bonds, PVBP (T 1 ; ; T n ), each of hem expiring a he end-poins of he swap s xed paymen daes; (ii) he di eren naure of he derivaives involved in he de niion of he wo indexes: European opions for he VIX (; T ) index, and swapions for he IRS-VI n (; T ) index; (iii) he exra dimension in swap rae volailiy inroduced by he lengh of he enor period underlying he forward swap rae, T n T 0. There are furher disincions o be made. Consider he curren value of a forward for delivery of he rae variance a ime T in Secion 4.1 (De niion I), which by Eq. (8) is: (30) F var;n (; T ) = PVBP (T 1 ; ; T n ) P var;n (; T ) : (31) 26 cby Anonio Mele and Yoshiki Obayashi

As for equiy, he price o be paid a, for delivery of equiy volailiy a T, is, insead: F equiy (; T ) P (T ) P equiy (; T ) : (32) The wo prices, F var;n and F equiy, scale up o he ime fair values of heir respecive noionals a ime T : (i) he fair value PVBP (T 1 ; ; T n ) of he random noional PVBP T (T 1 ; ; T n ), in Eq. (31); and (ii) he fair value P (T ) of he deerminisic noional of one dollar, in Eq. (32). This remarkable propery arises in spie of he di eren assumpions underlying he wo conracs: (i) random ineres raes, for he rae variance conrac, and (ii) consan ineres raes, for he equiy variance conrac. Finally, consider he Basis Poin IRV conrac in Secion 5. By using he expression in Eq. (14) for he fair value of he Basis Poin Sandardized IRV swap rae, he BP index in Eq. (27) can be wrien as: IRS-VI BP n (; T ) v " # u = 1 2 X X SWPN R (K i ; T ; T n ) K i + SWPN P (K i ; T ; T n ) K i : T PVBP (T 1 ; ; T n ) i:k i <R i:k i R For he index of he Basis Poin volailiy meric in Eq. (33), he weighs o be given o he ou-of-he money swapion prices are no inversely proporional o he square of he srike, as in he case for he equiy volailiy index in Eq. (28). In he heoreical case where we were given a coninuum of srikes, our BP index would acually be an equal-weighed average of ouof-he money swapion prices, as revealed by he exac expression of he BP-Sandardized IRV swap rae in Appendix G (see Eq. (G.4)), and consisenly wih he consan Gaussian vega explanaions of Secion 5.1 (see Eq. (17)) alhough due o discreizaion and runcaion, he weighs in Eq. (33), K i, migh no be exacly he same. Therefore, he Basis Poin ineres rae volailiy index and conracs di er from equiy, no only due o he aspecs poined ou for he percenage volailiy, bu also for he paricular weighing each swapion price eners ino he de niion of he index and conracs. Noe ha hese di eren weighings re ec di eren hedging sraegies. As explained in Secions 4 and 5, replicaing a variance conrac agreed over a percenage volailiy meric requires relying on log-conracs, whereas replicaing a variance conrac agreed over he basis poin realized variance, requires relying on quadraic conracs. (33) 9. Resilience o jumps This secion examines he pricing and indexing of expeced ineres rae swap volailiy in markes where he forward swap rae is a jump-di usion process wih sochasic volailiy: dr (T 1 ; ; T n ) R (T 1 ; ; T n ) = E Q swap; e jn() 1 () d + (T 1 ; ; T n ) dw () + e jn() 1 dn () ; 2 [; T ] ; (34) where W () is a mulidimensional Wiener process de ned under he swap probabiliy Q swap, (T 1 ; ; T n ) is a di usion componen, adaped o W (), N () is a Cox process under 27 cby Anonio Mele and Yoshiki Obayashi

Q swap wih inensiy equal o (), and j n () is he logarihmic jump size. (See, e.g., Jacod and Shiryaev (1987, p. 142-146), for a succinc discussion of jump-di usion processes.) By applying Iô s lemma for jump-di usion processes o Eq. (34), we have ha d ln R (T 1 ; ; T n ) = ( ) d + (T 1 ; ; T n ) dw () + j n () dn () ; such ha he realized variance of he logarihmic changes of he forward swap rae over a ime inerval [; T ], or percenage variance, is now: V J n (; T ) Z T k (T 1 ; ; T n )k 2 d + Z T j 2 n () dn () : (35) Insead, he realized variance of he arihmeic changes of he forward swap rae over [; T ], or basis poin variance, is: Vn J;BP (; T ) Z T R 2 (T 1 ; ; T n ) k (T 1 ; ; T n )k 2 d+ Z T R 2 (T 1 ; ; T n ) e jn() 1 2 dn () : (36) These de niions generalize hose in Eq. (5) and Eq. (12). We now derive he fair value of he IRV Sandardized conracs in Secions 4 and 5 in his new seing. In Appendix H, we show ha he fair value of he percenage IRV conrac payo Var-Swap n (; T ) in Eq. (7), P J; var;n (; T ) say, generalizes ha in Eq. (10) because of he presence of one addiional erm, capuring jumps, as follows: Z T P J; var;n (; T ) = P var;n (; T ) 2E Qswap; e jn() 1 1 j n () 2 j2 n () dn () ; (37) where P var;n (; T ) is he fair value of he Sandardized IRV conrac in Eq. (10). Suppose, for example, ha he disribuion of jumps is skewed owards negaive values. The fair value P J; var;n (; T ) should hen be higher han i would be in he absence of jumps. Assume, in paricular, ha he disribuion of jumps collapses o a single poin, j < 0 say, and ha he jump inensiy equals some posiive consan, in which case he fair value in Eq. (37) collapses o, P J; var;n (; T ) = P var;n (; T ) + 2 (T ) J ; (38) where we have de ned he posiive consan J (e j 1 j 1 2 j 2 ) > 0. The index of percenage swap volailiy is now, r IRS-VI J n (; T ) IRS-VI 2 2 R n (; T ) T E T Q swap; e jn() 1 1 j n () 2 j2 n () dn () (39) where IRS-VI n (; T ) is as in Eq. (25). In he special case of he parameric example underlying he price of he IRV conrac of Eq. (38), he previous formula reduces o, q IRS-VI J n (; T ) = IRS-VI 2 n (; T ) + 2J : In a noable conras, for he case of basis poin IRV, we prove in Appendix H ha he fair value of he basis poin IRV conrac payo in De niion IV-(c), denoed P J;BP var;n (; T ), is resilien o jumps, meaning P J;BP var;n (; T ) = P BP var;n (; T ) ; (40) 28 cby Anonio Mele and Yoshiki Obayashi

where P BP var;n (; T ) is as in Eq. (15). Accordingly, he basis poin index of swap volailiy, IRS- (; T ) say, is he same as ha we derived in he absence of jumps, VI BP;J n IRS-VI BP;J n where IRS-VI BP n (; T ) is as in Eq. (27). (; T ) = IRS-VI BP n (; T ) (41) 10. Index implemenaion This secion provides examples of a numerical implemenaion of he index. Secion 10.1 develops a sep-by-sep illusraion of he index calculaions, and Secion 10.2 conains examples of he index behavior around paricular daes. 10.1. A numerical example The following example is a non-limiing illusraion of he main seps involved ino he consrucion of he IRS-VI indexes IRS-VI BP n (; T ) and IRS-VI n (; T ) in Eq. (25) and Eq. (27). We uilize hypoheical daa for implied volailiies expressed in percenage erms for swapions mauring in one monh and enor equal o ve years. The rs wo columns of Table V repors srike raes, K say, and he skew, i.e. he percenage implied volailiies for each srike rae, denoed as IV (K) ( Percenage Implied Vol ). For reference, he hird column provides basis poin implied volailiies for each srike rae K, denoed as IV BP (K), and compued as: IV BP (K) = IV (K) R; (42) where R denoes he curren forward swap rae ( Basis Poin Implied Vol ). For example, from Table V, R = 2:7352% and IV (K)j K=R = 35:80% and, hen, IV BP (K) K=R = 2:7352% 35:80% = 0:979202%, which is 97:9202 basis poins volailiy. Basis poin volailiies are no needed o compue he indexes of his paper. The wo IRS-VI indexes in Eq. (25) and Eq. (27) are implemened hrough Black s (1976) formula, as explained in Appendix C.1, Eq. (C.5), and Appendix G.1, Eq. (G.5). Firs, we plug he skew IV (K) ino he Black s (1976) formulae, ^Z R; T; K i ; (T ) IV 2 (K i ) = Z R; T; K i ; (T ) IV 2 (K i ) + K R; (43) Z (R; T; K; V ) = R (d) K(d p V ); d = ln R K + 1 2 V p V ; (44) where denoes he cumulaive sandard normal disribuion, and ^Z () and Z () are he Black s prices, i.e. swapion prices (receivers, for ^Z (); and payers, for Z ()), divided by he PVBP. Second, we use Eqs. (43)-(44), and calculae wo indexes in Eq. (25) and Eq. (27): and: IRS-VI n (; T ) v 2 u = 100 2 4 X T i:k i <R ^Z R; T; K i ; (T K 2 i ) IV 2 (K i ) K i + X i:k i R Z R; T; K i ; (T K 2 i ) IV 2 (K i ) 3 K i 5; (45) IRS-VI BP n (; T ) 29 cby Anonio Mele and Yoshiki Obayashi

v 2 3 u = 100 100 2 4 X ^Z R; T; Ki ; (T ) IV 2 (K i ) Ki + X Z R; T; K i ; (T ) IV 2 (K i ) Ki 5: (46) T i:k i <R i:k i R The basis poin index in Eq. (46), is rescaled by 100 2, o mimic he marke pracice o express basis poin implied volailiy as he produc of raes imes log-volailiy, where boh raes and log-volailiy are muliplied by 100. The fourh column of Table V provides he values of ^Z and Z ( Black s prices ), for each srike rae. Table V Black s prices Srike Percenage Basis Poin Receiver Payer Rae (%) Implied Vol Implied Vol Swapion ( ^Z) Swapion Z 1.7352 36.1900 98.9869 0 10.000010 3 1.9852 36.1900 98.9869 0.000710 3 7.500710 3 2.2352 36.1200 98.7954 0.025910 3 5.025910 3 2.4352 35.9900 98.4398 0.177310 3 3.177310 3 2.5352 35.9300 98.2757 0.369210 3 2.369210 3 2.6352 35.8600 98.0843 0.679310 3 1.679310 3 2.6852 35.8300 98.0022 0.885510 3 1.385510 3 2.7352 35.8000 97.9202 1.127210 3 1.127210 3 2.7852 35.7600 97.8108 1.403710 3 0.903710 3 2.8352 35.7300 97.7287 1.714210 3 0.714210 3 2.9352 35.6700 97.5646 2.427010 3 0.427010 3 3.0352 35.6000 97.3731 3.240610 3 0.240610 3 3.2352 35.4700 97.0175 5.064410 3 0.064410 3 3.4852 35.3100 96.5799 7.509210 3 0.009210 3 3.7352 35.1400 96.1149 10.001010 3 0.001010 3 Table VI provides deails regarding he compuaion of he indexes IRS-VI BP n (; T ) and IRS- VI n (; T ) hrough Eqs. (46) and (45): he second column displays he ype of ou-of-he money swapion enering ino he calculaion; he hird column has he corresponding Black s prices; he fourh and fh columns repor he weighs each price bears owards he nal compuaion of 2 he index before he nal rescaling of ; nally, he sixh and sevenh columns repor each ouof-he money swapion price muliplied by he appropriae weigh (i.e. hird column muliplied T by he fourh column for he Basis Poin Conribuion, and hird column muliplied by he fh column for he Percenage Conribuion. Table VI Weighs Conribuions o Srikes Srike Rae (%) Swapion Type Price Basis Poin K i Percenage K i =Ki 2 Basis Poin Conribuion Percenage Conribuion 1.7352 Receiver 0 0.0025 8.3031 0 0 1.9852 Receiver 0.000710 3 0.0025 6.3435 0.001810 6 0.004610 3 2.2352 Receiver 0.025910 3 0.0022 4.5035 0.058310 6 0.116710 3 2.4352 Receiver 0.177310 3 0.0015 2.5294 0.266010 6 0.448510 3 2.5352 Receiver 0.369210 3 0.0010 1.5559 0.369210 6 0.574410 3 2.6352 Receiver 0.679310 3 0.0008 1.0800 0.509510 6 0.733710 3 2.6852 Receiver 0.885510 3 0.0005 0.6935 0.442810 6 0.614110 3 2.7352 ATM 1.127210 3 0.0005 0.6683 0.563610 6 0.753310 3 2.7852 Payer 0.903710 3 0.0005 0.6446 0.451810 6 0.582510 3 2.8352 Payer 0.714210 3 0.0007 0.9330 0.535710 6 0.666410 3 2.9352 Payer 0.427010 3 0.0010 1.1607 0.427010 6 0.495610 3 3.0352 Payer 0.240610 3 0.0015 1.6282 0.360910 6 0.391710 3 3.2352 Payer 0.064410 3 0.0023 2.1497 0.144810 6 0.138410 3 3.4852 Payer 0.009210 3 0.0025 2.0582 0.022910 6 0.018810 3 3.7352 Payer 0.001010 3 0.0025 1.7919 0.002410 6 0.001710 3 SUMS 4.156710 6 5.540510 3 The wo indexes are compued as IRS-VI n (; T ) = 100 r 2 12 1 5:5405 10 3 = 36:4653; 30 cby Anonio Mele and Yoshiki Obayashi

IRS-VI BP n (; T ) = 100 100 An Ineres Rae Swap Volailiy Index and Conrac r 2 12 1 4:1567 10 6 = 99:8803: In comparison, ATM implied volailiies are IV (R) = 35:8000 and IV BP (R) = 97:9202. 10.2. Hisorical performance This secion documens he performance of he IRS-VI index over seleced days by relying on daa provided by a major inerdelear broker. Firs, we consider he index behavior a four seleced daes: (i) February 16, 2007, (ii) February 15, 2008, (iii) February 13, 2009, and (iv) February 12, 2010. Second, we documen he index behavior over he Lehman s collapse occurred in Sepember 2008. We calculae he index hrough Eq. (25), using ATM swapions and ou-of-he-money (OTM, henceforh) swapions, wih moneyness equal o 200, 100, 50, and, nally, 25 basis poins away from he curren forward swap rae. We repor he index and implied volailiies in percenages. Basis poin volailiies are percenage implied volailiies muliplied by he curren forward swap rae. The Basis Poin IRS-VI index of Secion 5, IRS-VI BP n (; T ), de ned in Eq. (27), aims o aggregae informaion abou hese volailiies, in ha i racks expeced volailiy cas in basis poin erms. We only repor experimens relaing o he percenage index, as de ned in Eq. (25), IRS-VI n (; T ). Furhermore, noe ha he index is calculaed using rolling enors, as for he rading sraegy in Figure 9: for each ime-o-mauriy T, IRS-VI n (; T ) is he volailiy index for enor period equal o T n T, for all considered n. Figures 9 hrough 12 compare he IRS-VI index wih he implied volailiy for a-he-money swapions wih ve year enors (ATM volailiy, henceforh), as a funcion of he mauriy of he variance conrac (one, hree, six, nine and welve monhs), over he seleced daes. Figures 13 hrough 16 plo he IRS-VI surface for he same daes, ha is, he plo of he IRS-VI index agains (i) he mauriy of he variance conrac, and (ii) he enor of he forward rae. Figures 9-12 show ha quaniaively, he IRS-VI index behaves dissimilarly from ATM volailiies. I may be larger or smaller han ATM volailiies, depending on he speci c dae a which i is measured or he horizon of he variance conrac. For example, on February 15, 2008, he IRS-VI index is higher han ATM volailiies for low mauriies of he variance conrac, and is lower oherwise. The reason for hese di erences is ha he IRS-VI aggregaes informaion relaing o boh ATM and OTM volailiies, in such a way o isolae expeced volailiy from oher facors, such as changes in he underlying forward raes. ATM volailiies, insead, and by defaul, only rely on he curren forward swap raes. On February 12, 2010, o cie a second example, OTM volailiies were in general higher (resp. lower) han ATM volailiies, in correspondence of smaller (larger) mauriies. This paern is capured by he IRS-VI, which is higher han ATM volailiies for mauriies up o six monhs, and lower han ATM volailiies for mauriies of nine and welve monhs. Qualiaively, he shape of he IRS-VI index displays more pronounced characerisics han ATM volailiies, when assessed agains he lengh of he variance conrac. In wo days occurring during bad imes (February 15, 2008 and February 13, 2009), he IRS-VI slopes downwards more sharply han ATM volailiies do, hereby vividly signaling he marke expecaion ha bad imes will be followed by periods where resoluion of uncerainy abou ineres 31 cby Anonio Mele and Yoshiki Obayashi

raes will occur. In good imes (February 16, 2007), he IRS-VI index and ATM volailiies boh slope up, alhough moderaely, re ecing he marke concern hese good imes, while persisen, will be slighly more uncerain in he fuure; even in his simple case, he IRS-VI index exhibis a richer behavior han ATM volailiies, as i aens ou for mauriies higher han nine monhs. Finally, he hump shape of he IRS-VI index and ATM volailiies occurring on February 12, 2010 re ecs he marke expecaion ha he uncerainy abou ineres raes will be rising in he immediae, alhough hen i will subside in he following monhs. Naurally, adding o he fac he IRS-VI index and ATM volailiies are simply no he same hing, boh quaniaively and qualiaively, variance conracs such as hose in Secion 4 canno be priced using solely ATM volailiies. Raher, hese conracs need o rely upon he IRS-VI index, as explained in he previous secions. Figures 13 hrough 16 show how he IRS-VI index can be used o express views abou developmens in ineres rae swap volailiy. In fac, one of he added feaures of he IRS-VI index and conrac, compared o he VIX R index for equiies, is he possibiliy for marke paricipans o choose he enor of he forward swap rae, on op of he period lengh abou which o convey volailiy views. This added dimension arises quie naurally, as a resul of an increased complexiy of xed income derivaives, as opposed o equiy. As an example, he rading sraegies of Figure 5 can be implemened on a paricular enor period T n T, and rading can ake place over one of he enors in Figure 13-16, where he mispricing is hough o be occurring. Figures 17 hrough 21 depic he behavior of he index around he Lehman s collapse, occurred on Sepember 15, 2008. A few days before he collapse, he index was downward sloping wih respec o mauriy, re ecing marke expecaions of possibly imminen negaive ail evens. These expecaions would reinforce he day of he Lehman s collapse, and only parially weaken over he week of his even. 32 cby Anonio Mele and Yoshiki Obayashi

15 14.5 14 13.5 13 12.5 IRS VI for 5 year enor (in %) ATM Swapion Implied Volailiy (in %) 12 1 monh 3 monhs 6 monhs 9 monhs 12 monhs Figure 9. Term srucure of he Ineres Rae Swap Volailiy Index (IRS-VI) February 16, 2007. 45 40 35 30 IRS VI for 5 year enor (in %) ATM Swapion Implied Volailiy (in %) 25 1 monh 3 monhs 6 monhs 9 monhs 12 monhs Figure 10. Term srucure of he Ineres Rae Swap Volailiy Index (IRS-VI) February 15, 2008. 33 cby Anonio Mele and Yoshiki Obayashi

65 60 55 IRS VI for 5 year enor (in %) ATM Swapion Implied Volailiy (in %) 50 45 40 35 1 monh 3 monhs 6 monhs 9 monhs 12 monhs Figure 11. Term srucure of he Ineres Rae Swap Volailiy Index (IRS-VI) February 13, 2009. 40 39 38 37 36 35 34 33 32 IRS VI f or 5 y ear enor (in %) ATM Swapion Implied Volailiy (in %) 31 30 1 monh 3 monhs 6 monhs 9 monhs 12 monhs Figure 12. Term srucure of he Ineres Rae Swap Volailiy Index (IRS-VI) February 12, 2010. 34 cby Anonio Mele and Yoshiki Obayashi

IRS VI (in %) IRS VI (in %) An Ineres Rae Swap Volailiy Index and Conrac 15.5 15 14.5 14 13.5 13 12.5 2y 3y 4y 5y 6y 7y 8y 9y 10y 1m 3m 6m 9m 12m Underly ing swap enor, in y ears Lengh of he v olailiy conrac, in monhs Figure 13. Ineres Rae Swap Volailiy Index (IRS-VI) February 16, 2007. 55 50 45 40 35 30 25 12m 20 2y 3y 4y 5y 6y 7y Underly ing swap enor, in y ears 8y 9y 10y 1m 3m 6m 9m Lengh of he v olailiy conrac, in monhs Figure 14. Ineres Rae Swap Volailiy Index (IRS-VI) February 15, 2008. 35 cby Anonio Mele and Yoshiki Obayashi

IRS VI (in %) IRS VI (in %) An Ineres Rae Swap Volailiy Index and Conrac 80 70 60 50 40 12m 30 2y 3y 4y 5y 6y 7y Underly ing swap enor, in y ears 8y 9y 10y 1m 3m 6m 9m Lengh of he v olailiy conrac, in monhs Figure 15. Ineres Rae Swap Volailiy Index (IRS-VI) February 13, 2009. 70 60 50 40 30 20 2y 3y 4y 5y 6y 7y 8y 9y 10y 1m 3m 6m 9m 12m Underly ing swap enor, in y ears Lengh of he v olailiy conrac, in monhs Figure 16. Ineres Rae Swap Volailiy Index (IRS-VI) February 12, 2010. 36 cby Anonio Mele and Yoshiki Obayashi

IRS VI (in %) IRS VI (in %) An Ineres Rae Swap Volailiy Index and Conrac Sepember 10, 2008 (Wednesday ) 50 45 40 35 30 25 20 2y 3y 4y 5y 6y 7y 8y 9y10y 1m 3m 6m 9m 12m Underly ing swap enor, in y ears Lengh of he v olailiy conrac, in monhs Figure 17. Ineres Rae Swap Volailiy Index (IRS-VI) around he Lehman s collapse days: I. Sepember 12, 2008 (Friday ) 50 45 40 35 30 25 20 2y 3y 4y 5y Underly ing swap enor, in y ears 6y 7y 8y 9y 10y 1m 3m 6m 9m 12m Lengh of he v olailiy conrac, in monhs Figure 18. Ineres Rae Swap Volailiy Index (IRS-VI) around he Lehman s collapse days: II. 37 cby Anonio Mele and Yoshiki Obayashi

IRS VI (in %) IRS VI (in %) An Ineres Rae Swap Volailiy Index and Conrac Sepember 15, 2008 (Monday, Lehman's collapse) 90 80 70 60 50 40 30 20 2y 3y 4y 5y Underly ing swap enor, in y ears 6y 7y 8y 9y 10y 1m 3m 6m 9m 12m Lengh of he volailiy conrac, in monhs Figure 19. Ineres Rae Swap Volailiy Index (IRS-VI) around he Lehman s collapse days: III. Sepember 16, 2008 (Tuesday ) 70 60 50 40 30 20 2y 3y 4y 5y Underly ing swap enor, in y ears 6y 7y 8y 9y 10y 1m 3m 6m 9m 12m Lengh of he v olailiy conrac, in monhs Figure 20. Ineres Rae Swap Volailiy Index (IRS-VI) around he Lehman s collapse days: IV. 38 cby Anonio Mele and Yoshiki Obayashi

IRS VI (in %) An Ineres Rae Swap Volailiy Index and Conrac Sepember 18, 2008 (Thursday ) 65 60 55 50 45 40 35 30 25 2y 3y 4y 5y 6y 7y 6m 8y 9y 10y 1m 3m Underly ing swap enor, in y ears 9m 12m Lengh of he v olailiy conrac, in monhs Figure 21. Ineres Rae Swap Volailiy Index (IRS-VI) around he Lehman s collapse days: V. 39 cby Anonio Mele and Yoshiki Obayashi

Technical Appendix A. Noaion, assumpions and preliminary facs Le P (T ) be he price as of ime of a zero coupon bond expiring a ime T >. The forward swap rae as of ime is he xed ineres rae ha makes he value of a forward saring swap equal o zero. For a forward sar swap conrac wih fuure rese daes T 0 ; ; T n 1 and paymen periods T 1 T 0 ; ; T n T n 1, i is de ned as: R (T 1 ; ; T n ) = P (T 0 ) P (T n ) PVBP (T 1 ; ; T n ) ; (A.1) where PVBP (T 1 ; ; T n ) is he price value of he basis poin of he swap, PVBP (T 1 ; ; T n ) = nx i 1 P (T i ) ; (A.2) i=1 and i = T i+1 T i are he lenghs of he rese inervals. Le r s be he insananeous ineres rae as of ime s, and de ne he value of he money marke accoun as of ime (MMA, for shor) as M = e R rsds. Le Q be he risk-neural probabiliy, under which all raded asses discouned by he MMA are maringales. De ne he Radon-Nikodym derivaive of Q swap wih respec o Q as: dq swap dq FT = e R T rsds PVBP T (T 1 ; ; T n ) PVBP (T 1 ; ; T n ) ; (A.3) where F T is he informaion se as of ime T, E Q [] denoes he expecaion aken under he risk-neural probabiliy, and E Qswap [] is he expecaion aken under he Q swap probabiliy. The forward swap rae is soluion o (see, e.g., Mele, 2011, Chaper 12): dr s (T 1 ; ; T n ) R s (T 1 ; ; T n ) = s (T 1 ; ; T n ) dw s ; s 2 [; T ] ; (A.4) where W is a mulidimensional Wiener process under he Q swap probabiliy, and (T 1 ; ; T n ) is adaped o W. The payo of he IRV forward agreemen is: V n (; T ) PVBP T (T 1 ; ; T n ) ; (A.5) where V n (; T ) is he realized variance of he forward swap rae logarihmic changes in he inerval [; T ], V n (; T ) = where we use he simpli ed noaion, Z T 2 s (T 1 ; ; T n ) ds; (A.6) 2 (T 1 ; ; T n ) k (T 1 ; ; T n )k 2 : 40 cby Anonio Mele and Yoshiki Obayashi

In he absence of arbirage, he value of a forward saring swap payer wih xed ineres K is (see, e.g., Mele, 2011, Chaper 12), SWAP (K; T 1 ; ; T n ) = PVBP (T 1 ; ; T n ) [R (T 1 ; ; T n ) K] ; (A.7) and he prices of a payer and receiver swapions expiring a T are, respecively, and SWPN P (K; T ; T 1 ; ; T n ) = PVBP (T 1 ; ; T n ) E Qswap [R T (T 1 ; ; T n ) K] + ; (A.8) SWPN R (K; T ; T 1 ; ; T n ) = PVBP (T 1 ; ; T n ) E Qswap [K R T (T 1 ; ; T n )] + : (A.9) The pariy for payer and receiver swapions is: SWPN P (K; T ; T 1 ; ; T n ) = SWAP (K; T 1 ; ; T n ) + SWPN R (K; T ; T 1 ; ; T n ) : (A.10) B. P&L of opion-based volailiy rading B.1. Definiions The price of a payer swapion in Eq. (A.8) is known in closed-form, once we assume ha he volailiy in Eq. (A.4) and, hence, he inegraed variance V n (; T ) in Eq. (A.6), are deerminisic. I is given by Black s (1976) formula: SWPN_Bl P R n ; PVBP ; K; T ; V = PVBP Z R n ; T; K; V ; (B.1) where: Z (R ; T; K; V ) = R (d ) K(d p V ); d = ln R K + 1 2 V p V ; denoes he cumulaive sandard normal disribuion, and o alleviae noaion, R n R (T 1 ; ; T n ), PVBP PVBP (T 1 ; ; T n ), and V is he consan value of V n (; T ) in Eq. (A.6). By he de niion of he forward swap rae in Eq. (A.1), we have: SWPN_Bl P R n ; PVBP ; K; T ; V (P (T ) P (T n )) (d ) PVBP K(d p V ): (B.2) Eq. (B.2) shows he swapion can be hedged hrough porfolios of zero coupon bonds: (i) long (d ) unis of a porfolio which is long one zero expiring a T and shor one zero expiring a T n, and (ii) shor K (d s) unis of he bonds baske PVBP. Alernaively, noe ha: SWPN_Bl P R n ; PVBP ; K; T ; V p = SWAP (K; T 1 ; ; T n ) (d )+PVBP K (d ) (d V ) : (B.3) Eq. (B.3) shows he swapion p can equally be hedged as follows: (i) long (d ) unis of a swap, and (ii) long K( (d ) (d V )) unis of he bonds baske PVBP. Nex, and agains he assumpion underlying Eq. (B.2) and Eq. (B.3), we assume he forward swap rae has sochasic volailiy, say, as in Eq. (A.4), which i does, for example, in he models of Appendix F. We wish o have a view ha volailiy will raise, say, compared o he curren implied volailiy, de ned as he value, s V IV = T ; (B.4) such ha, once re-normalized again by T marke price. and, hen, insered ino Eq. (B.2), i delivers he swapion 41 cby Anonio Mele and Yoshiki Obayashi

B.2. Dela-hedged swapions sraegies Consider, rs, he sraegy o purchase he swapion, and hedge i using one of he porfolios underlying eiher Eq. (B.2) or Eq. (B.3). We derive he P&L of his sraegy assuming he hedging porfolio is ha relying on Eq. (B.2). The value of he hedging sraegy, say, sais es: where, denoing for simpliciy R R n, = a (P (T ) P (T n )) + b PVBP ; 2 [; T ] ; a = ln R = SWPN_Bl P R ; PVBP ; K; T; (T ) IV 2 ; K + 1 2 (T )! IV2 R 1 ln p K 2 and b = K (T )! IV2 p : T IV T IV (B.5) Because he hedging sraegy is self- nanced, d = a d [P (T ) P (T n )] + b dpvbp, and, hence: h i h i d = b + P b a (P (T ) P (T n )) d+ b + P b a (P (T ) P (T n )) dw ; (B.6) where W is a Wiener process under he physical probabiliy and, accordingly, b and b denoe he drif and insananeous volailiy of dpvbp PVBP, and P and P denoe he drif and insananeous d(p (T ) P (Tn)) volailiy of P (T ) P (T n). On he oher hand, consider he swapion price in Eq. (B.1) wih V replaced by (T ) IV 2, as in Eq. (B.4), i.e. SWPN_Bl P SWPN_BlP R ; PVBP ; K; T; (T ) IV 2. By Iô s lemma, he de niion of he forward swap rae in Eq. (A.1), and he parial di erenial equaion sais ed by he pricing funcion Z R; T; K; V, his price changes as follows: dswpn_bl P = PVBP dz + Z dpvbp + dz dpvbp! @Z = PVBP @ + 1 @ 2 Z 1 2 @R 2 R2 IV 2 @ 2 Z d + PVBP {z } 2 @R 2 R2 2 IV 2 @Z + @R R R d =0 + b SWPN_Bl P + @Z @R b (P (T ) P (T n )) d @Z + @R (P (T ) P (T n )) + b SWPN_Bl P dw ; (B.7) where R is he drif of dr R under he physical probabiliy, and equals R = p b b ; = p b ; (B.8) where he second relaion follows by Iô s lemma. By using again he de niion of he forward swap rae in Eq. (A.1), and hen Eq. (B.8) and he relaion a = @Z @R, we can inegrae he di erence beween dswpn_bl P in Eq. (B.7), and d in Eq. (B.6), so as o obain ha he P&L a he swapion mauriy is: SWPN_Bl P T T = [R T K] + T = 1 Z T @ 2 Z Z 2 @R 2 R2 2 IV 2 T (;T PVBP ) d + ;T b SWPN_Bl P dw ; (B.9) 42 cby Anonio Mele and Yoshiki Obayashi

R T where we have used he rs relaion in Eqs. (B.5), SWPN_Bl P =, and de ned ;T = e b s ds. The approximaion in Eq. (2) relies on: (i) PVBP T ;T PVBP ; (ii) gr R dw ; and (iii) disregarding he erm ;T inside he sochasic inegral in Eq. (B.9), which we merely made o simplify he presenaion. B.3. Sraddles sraegies Finally, we consider he P&L relaing o rading he sraddle. The value of he sraddle is, STRADDLE = SWPN P + SWPN R. By Black s (1976) formula, he price of he receiver swapion is: ^Z SWPN_Bl R R n ; PVBP ; K; T ; V = PVBP ^Z R n ; T; K; V ; R ; T; K; V p = K 1 (d V ) R (1 (d )) ; d = ln R K + 1 2 V p V : (B.10) Therefore, he dynamics of SWPN_Bl R are he same as SWPN_Bl P in Eq. (B.7), bu wih ^Z replacing Z. Moreover, we have, by Eq. (A.10) and Eq. (A.7), @STRADDLE @R = PVBP 1 + 2 @ ^Z @R! ; @ ^Z @R = @Z @R where Z is as in Eq. (B.1). Assuming he sraddle dela is su cienly small, which by he previous equaion i is when 2 @Z @R 1, he value of he sraddle is, by Eq. (B.7), and he previous argumens abou he dynamics of SWPN_Bl R : 1; STRADDLE T STRADDLE Z T @ 2 Z Z = @R 2 R2 2 IV 2 T (;T PVBP ) d + ;T b STRADDLE dw : (B.11) The approximaion in Eq. (3) relies on he same argumens leading o Eq. (2). Noe ha he approximaion 2 @Z @R 1 is quie inadequae as he forward swap raes drifs away from ATM a very well-known feaure discussed, e.g. by Mele (2011, Chaper 10) in he case of equiy sraddles. C. Spanning he conracs C.1. Pricing We rely on spanning argumens similar o hose uilized by Bakshi and Madan (2000) and Carr and Madan (2001) for equiies. To alleviae noaion, we se R R n = R (T 1 ; ; T n ). By he usual Taylor s expansion wih remainder, ln R T R = 1 R (R T R ) Z R 0 (K R T ) + 1 K 2 dk Z 1 R (R T K) + 1 K 2 dk: (C.1) Muliplying boh sides of he previous equaion by PVBP (T 1 ; ; T n ), and aking expecaions under Q swap, de ned hrough Eq. (A.3): PVBP (T 1 ; ; T n ) E Qswap ln R T = R Z R 0 SWPN R (K; T ; T n ) K 2 dk Z 1 SWPN P (K; T ; T n ) R K 2 (C.2) where we have used (i) he fac ha by Eq. (A.4), he forward swap rae is a maringale under Q swap, and (ii) he pricing Equaions (A.8) and (A.9). 43 cby Anonio Mele and Yoshiki Obayashi dk;

By Eq. (A.4), and a change of measure obained hrough Eq. (A.3), 2E Qswap ln R Z T T = E Qswap 2 s (T 1 ; ; T n ) ds R = 1 PVBP (T 1 ; ; T n ) E Q Combining his relaion wih Eq. (C.2) leaves: F var;n (; T ) = E Q e e R T Z T PVBP rsds T (T 1 ; ; T n ) 2 s (T 1 ; ; T n ) ds Z R SWPN R (K; T ; T 1 ; ; T n ) = 2 0 K 2 dk + R T Z T PVBP rsds T (T 1 ; ; T n ) 2 s (T 1 ; ; T n ) ds : (C.3) Z 1 SWPN P (K; T ; T 1 ; ; T n ) R K 2 dk : (C.4) By Eqs. (A.5) and (A.6), he lef hand side of he previous equaion is he price of he IR forward variance agreemen, and Eq. (8) is is approximaion. The claim in he main ex ha if V n (; T ) and he pah of he shor-erm rae in he ime inerval [; T ] were independen, he index IRS-VI n (; T ) in Eq. (25) would be he risk-neural expecaion of V n (; T ), follows by Eq. (C.3) and he de niion of he Radon-Nikodym derivaive of Q swap wih respec o Q in Eq. (A.3). To derive he IRV swap rae P var;n (; T ) in Eq. (9), noe ha by Eq. (6), i solves: R T 0 = E Q e rsds (V n (; T ) PVBP T (T 1 ; ; T n ) P var;n (; T )) ; which, afer rearranging erms, yields Eq. (9). As for he Sandardized IRV swap rae P var;n (; T ) in Eq. (10), noe ha by Eq. (7), i sais es: R T 0 = E Q e rsds V n (; T ) P var;n (; T ) PVBP T (T 1 ; ; T n ) ; which by he de niion of he Radon-Nikodym derivaive in Eq. (A.3), yields Eq. (10), afer rearranging erms. Finally, as claimed in he main ex, Eq. (25) can be simpli ed, using he Black s (1976) formula. By replacing Eq. (8) ino Eq. (25), and using he Black s formulae in Eqs. (B.1) and (B.10), IRS-VI n (; T ) v 2 u = 2 4 X T i:k i <R ^Z R n ; T; K i ; (T ) IV 2 i; K i + K 2 i X i:k i R Z Rn ; T; K i ; (T K 2 i ) IV 2 3 i; K i 5; (C.5) where he expressions for Z and ^Z are given in Eqs. (B.1) and (B.10), R n is he curren forward swap rae for mauriy T and enor lengh T n T, R n R (T 1 ; ; T n ), and, nally, IV i; denoes he ime implied percenage volailiy for swapions wih srike equal o K i. C.2. Marking o marke We rs derive Eq. (19). For a given 2 (; T ), we need o derive he following condiional expecaion of he payo Var-Swap n (; T ) in Eq. (6): R T e rudu Var-Swap n (; T ) E Q 44 cby Anonio Mele and Yoshiki Obayashi

= E Q e An Ineres Rae Swap Volailiy Index and Conrac R T rudu (V n (; ) + V n (; T )) PVBP T (T 1 ; ; T n ) P (T ) P var;n (; T ) = V n (; ) PVBP (T 1 ; ; T n ) + F var;n (; T ) P (T ) P (T ) F var;n (; T ) ; (C.6) where E Q denoes expecaion under Q, condiional upon all informaion up o ime, and we have used he de niion of he Radon-Nikodym derivaive of Q swap in Eq. (A.3) and expression for F var;n (; T ) in Eq. (C.4). Eq. (19) follows afer plugging he expression for P var;n (; T ) in Eq. (9) ino Eq. (C.6). Nex, we derive Eq. (20). We have, uilizing he expression of Var-Swap n (; T ) in Eq. (7), R T e rudu Var-Swap n (; T ) E Q = E Q e R T rudu V n (; ) + V n (; T ) P var;n (; T ) PVBP T (T 1 ; ; T n ) = PVBP (T 1 ; ; T n ) V n (; ) + P var;n (; T ) P var;n (; T ) ; where he second equaliy follows by he expression for F var;n (; T ) in Eq. (C.4), and by Eq. (10). C.3. Hedging We provide he deails leading o Table I, as hose for Table II are nearly idenical. By Iô s lemma: Z T PVBP T (T 1 ; ; T n ) 2 s (T 1 ; ; T n ) ds Z T dr s (T 1 ; ; T n ) = 2PVBP T (T 1 ; ; T n ) R s (T 1 ; ; T n ) 2PVBP T (T 1 ; ; T n ) ln R T (T 1 ; ; T n ) R (T 1 ; ; T n ) ; (C.7) where, by Eq. (C.1), he expression for he swap value in Eq. (1) and ha for he swapion premium in Eqs. (A.9)-(A.10), he second erm on he righ hand side is he payo a T of a porfolio se up a, which is wo imes: (a) shor 1=R (T 1 ; ; T n ) unis a xed ineres payer swap sruck a R (T 1 ; ; T n ), and (b) long a coninuum of ou-of-he-money swapions wih weighs K 2 dk. This porfolio is he saic posiion (ii) in Table I of he main ex. By Eqs. (C.1), (C.2) and (C.4), is cos is F var;n (; T ). We borrow F var;n (; T ) a ime, and repay i back a ime T, as in row (iii) of Table I of he main ex. Nex, we derive he self- nancing porfolio of zero coupon bonds (i) in Table I, by designing i o be worhless a ime, and o replicae he rs erm on he righ hand side of Eq. (C.7). Firs, noe ha by Eq. (A.1), he forward swap rae as of ime s can be replicaed hrough a porfolio ha is long zeros mauring a T 0 and shor zeros mauring a T n, wih equal weighs 1=PVBP s (T 1 ; ; T n ). Consider, hen, a self- nanced sraegy invesing in (a) his porfolio, which we call he forward swap rae for simpliciy, and (b) a MMA, as de ned in Appendix A, wih oal value equal o: = R (T 1 ; ; T n ) + M ; where are he unis in he forward swap rae and are he unis in he MMA. Consider he following porfolio: Z ^ R (T 1 ; ; T n ) = PVBP (T 1 ; ; T n ) ; ^ dr s (T 1 ; ; T n ) M = PVBP (T 1 ; ; T n ) 1 : R s (T 1 ; ; T n ) (C.8) We have ha ^ = ^ R + ^ M sais es: Z dr s (T 1 ; ; T n ) ^ = PVBP (T 1 ; ; T n ) R s (T 1 ; ; T n ) ; (C.9) 45 cby Anonio Mele and Yoshiki Obayashi

such ha Z T ^ = 0; and ^ T = PVBP T (T 1 ; ; T n ) dr s (T 1 ; ; T n ) R s (T 1 ; ; T n ) : Therefore, by going long wo porfolios (^ ; ^ ), he rs erm on he righ hand side and, hen, by he previous resuls, he whole of he righ hand side of Eq. (C.7), can be replicaed and, so, hedged, provided (^ ; ^ ) is self- nanced. We are only lef o show (^ ; ^ ) is self- nanced. We have: d^ = ^ R (T 1 ; ; T n ) dr (T 1 ; ; T n ) R (T 1 ; ; T n ) + ^ dm M M dr (T 1 ; ; T n ) = PVBP (T 1 ; ; T n ) r d R (T 1 ; ; T n ) dr (T 1 ; ; T n ) = PVBP (T 1 ; ; T n ) r d R (T 1 ; ; T n ) = ^ R dr (T 1 ; ; T n ) R (T 1 ; ; T n ) r d Z + PVBP (T 1 ; ; T n ) + r ^ d dr s (T 1 ; ; T n ) r d R s (T 1 ; ; T n ) + r ^ d; (C.10) where he second line follows by he porfolio expressions for (^ ; ^ ) in Eq. (C.8), he hird line holds by Eq. (C.9), and he fourh follows, again, by he expression for ^ R in Eq. (C.8). I is easy o check ha he dynamics of ^ in Eq. (C.10) are hose of a self- nanced sraegy. D. Direcional volailiy rades The P&L displayed in Figure 4 of Secion 3 are calculaed using daily daa from January 1998 o December 2009, and comprise yield curve daa as well as implied volailiies for swapions. Yield curve daa are needed o compue forward swap raes for a xed 5 year enor and he relaed realized volailiies, and implied volailiies are needed o compue P&Ls, as explained nex. As for he sraddle, he sraegy is o go long an a-he-money swapion sraddle. Le denoe he beginning of he holding period (one monh or hree monhs). The erminal P&L of he sraddle, ha of as of ime T say, is: PVBP T (T 1 ; ; T n ) [R T (T 1 ; ; T n ) K] + + [K R T (T 1 ; ; T n )] + Sraddle (T 1 ; ; T n ) ; where Sraddle (T 1 ; ; T n ) is he cos of he sraddle a, and T is eiher one monh (as in he op panel of Figure 4) or hree monhs (as in he boom panel of Figure 4). Insead, he erminal P&L of he volailiy swap conrac is calculaed consisenly wih De niion II and Eq. (9), as: 1 F var;n (; T ) PVBP T (T 1 ; ; T n ) V n (; T ) ; T P (T ) where F var;n (; T ) approximaes he unnormalized srike of he conrac o be enered a. Is exac value, F var;n (; T ), is ha in Eq. (8). By Eq. (25), which we approximae hrough: 1 T F var;n (; T ) = PVBP (T 1 ; ; T n ) IRS-VI n (; T ) 2 ; 1 T F var;n (; T ) PVBP (T 1 ; ; T n ) ATM n (; T ) 2 ; where ATM n (; T ) is he a-he-money implied volailiy. 46 cby Anonio Mele and Yoshiki Obayashi

Finally, he realized variance rescaled by T, 1 T RV m = 21 m 1 RV m ; where RV m = 252 V n (; T ), is calculaed as: i= X 21m ln R i (T 1 ; ; T n ) 2 ; R i 1 (T 1 ; ; T n ) wih m 2 f1; 3g, T 1 = 1 4 and T n = 5. The variance risk-premium is de ned as RV m ATM 2 n ( 21 m; T ), where T is eiher 1 3 12 (for one monh) or 12 (for hree monhs). Finally, o make he P&Ls of he volailiy swap conrac and he sraddle line up o he same order of magniude, he picures in Figure 4 are obained by re-muliplying he P&Ls of he volailiy swap conracs by 1 3 12 (op panel) and 12 (boom panel). E. Local volailiy surfaces We develop argumens ha hinge upon hose Dumas (1995) and Brien-Jones and Neuberger (2000) pu forh in he equiy case. Consider he price of he payer swapion in Eq. (A.8), SWPN P (K; T ; T n ). For simpliciy, le R R (T 1 ; ; T n ). We have: @SWPN P @T = @ ln PVBP @T SWPN P + PVBP de Q swap (R T K) + : (E.1) dt Since he forward swap rae is a maringale under Q swap, wih volailiy as in Eq. (A.4), we have: de Qswap (R T K) + dt = 1 2 E Q swap (RT K) R 2 T 2 T ; (E.2) where () is he Dirac s dela. We can elaborae he righ hand side of Eq. (E.2), obaining: ZZ E Qswap (RT K) RT 2 2 T = (R T K) RT 2 2 T c T ( T j R T ) m T (R T ) dr {z } T d T join densiy of ( T ;R T ) = K 2 m T (K) E Qswap 2 T R T = K @2SWPNP = K 2 @K 2 E Qswap 2 PVBP T R T = K ; where c T ( T j R T ) denoes he condiional densiy of T given R T under Q swap, m T (R T ) denoes he marginal densiy of R T under Q swap, and he hird line follows by a well-known propery of opion-like prices. By replacing his resul ino Eq. (E.2) and hen ino Eq. (E.1), we obain: @SWPN P @T = @ ln PVBP @T Rearranging his equaion, we obain: SWPN P + 1 2 K2 @2 SWPN P @K 2 E Qswap 2 T RT = K : 2 loc (K; T ) E Q swap 2 T R T = K @SWPN P @T = 2 @ ln PVBP @T SWPN P Nex, le he volailiy of he forward swap rae in Eq. (A.4) be given by: where v s is anoher random process. De ne: s = (R s ; s) v s ; K 2 @2 SWPN P @K 2 : (E.3) (R; s) = q loc (R; s) ; (E.4) E Qswap [vsj 2 R s = R] 47 cby Anonio Mele and Yoshiki Obayashi

where loc (R; s) is as in Eq. (E.3). Consider he following model of he forward swap rae, under he Q swap probabiliy, dr s = (R s ; s) v s dws ; s 2 [; T ] ; R s where (R s ; s) is as in Eq. (E.4). This model is, heoreically, capable o mach he cross-secion of swapions wihou errors, and can be used o price all of he non-raded swapions in Eq. (8), hrough Monecarlo inegraion. F. The conrac and index in he Vasicek marke We consider wo alernaive models. In he rs model, Vasicek s (1977), he shor-erm rae follows a Gaussian process: dr = (r r ) d + v d ^W ; (F.1) where ^W is a Wiener process under he risk-neural probabiliy, and, r and v are hree posiive consans. The price of a zero prediced by his model is P (r ; T ) = A (T ) e B(T )r for wo funcions A and B, wih obvious noaion. The forward swap rae prediced by his model is: R (r ; T 1 ; ; T n ) = P (r ; T 0 ) P (r ; T n ) PVBP (r ; T 1 ; ; T n ) ; (F.3) he price value of he basis poin is, PVBP (r ; T 1 ; ; T n ) = nx i 1 P (r ; T i ) ; i=1 he insananeous volailiy of bond reurns is, and, nally, he forward swap rae volailiy is, Vol-P (r ; T ) B (T ) v ; (F.3) (r ; T 1 ; ; T n ) = Vol-P (r ; T ) P (r ; T ) Vol-P (r ; T n ) P (r ; T n ) P (r ; T ) P (r ; T n ) P n i=1 i 1P (r ; T i ) Vol-P (r ; T i ) : (F.4) PVBP (r ; T 1 ; ; T n ) We simulae Eq. (F.1) using a Milsein approximaion mehod, for iniial values of he shor-erm rae in he inerval [0:01; 0:10]. We use parameer values se equal o = 0:3807, r = 0:072 and v = p 0:02 0:2341, aken from Veronesi (2010, Chaper 15), and generae simulaed values of (r ; T 1 ; ; T n ), by plugging in he simulaed values of r. We use he same parameer values for he Vasicek model in Eq. (F.1), wih v = p 0:02 r. The wo expecaions, E Q e and R T Z T PVBP rsds T (r T ; T 1 ; ; T n ) E Q Z T 2 s (r s ; T 1 ; ; T n ) ds 2 s (r s ; T 1 ; ; T n ) ds ; (F.5) are esimaed hrough Monecarlo inegraion. The forward raes in Figure 7 and Table A.1 are obained by plugging he iniial values of he shor-erm rae ino Eq. (F.3). 48 cby Anonio Mele and Yoshiki Obayashi

Table A.1 q The IRS-VIn (; T ) index in Eq. (25) (labeled IRS-VI), and fuure expeced volailiy in a risk-neural marke, 1 T E Q [Vn (; T )], where Vn (; T ) is as in Eq. (A.6) (labeled E-Vol), for (i) ime-o mauriy T equal o 1, 6 and 9 monhs, and 1, 2 and 3 years, and (ii) enor lengh n equal o 1, 2, 3 and 5 years, and for ime-o mauriy T equal o 1, 6 and 12 monhs, and (ii) enor lengh n equal o 10, 20 and 30 years. The index and expeced volailiies are compued under he assumpion he shor-erm ineres rae is as in he Vasicek model of Eq. (F.1), wih parameer values = 0:3807, r = 0:072 and v = 0:0331. Mauriy of he variance conrac Tenor = 1 year 1 monh 6 monhs 9 monhs Shor Term Rae 1.00 2.18 124.83 124.82 2.48 106.53 106.53 3.27 73.50 73.52 3.00 3.80 71.97 71.97 4.01 66.39 66.38 4.53 53.39 53.40 5.00 5.44 50.62 50.62 5.54 48.25 48.25 5.80 41.93 41.94 7.00 7.08 39.08 39.08 7.08 37.93 37.93 7.08 34.54 34.55 9.00 8.73 31.85 31.86 8.63 31.27 31.27 8.36 29.38 29.38 10.00 9.56 29.17 29.17 9.41 28.78 28.75 9.00 27.34 27.35 Tenor = 2 years 1 monh 6 monhs 9 monhs Shor Term Rae 1.00 2.93 78.12 78.12 3.53 60.11 60.12 3.84 52.74 52.76 3.00 4.31 53.66 53.66 4.70 45.49 45.50 4.91 41.59 41.61 5.00 5.68 40.93 40.93 5.88 36.62 36.62 5.98 34.36 34.37 7.00 7.07 33.12 33.12 7.07 30.66 30.67 7.06 29.29 29.30 9.00 8.47 27.84 27.84 8.26 26.39 26.40 8.15 25.53 25.55 10.00 9.18 25.80 25.80 8.86 24.68 24.69 8.69 24.01 24.02 Tenor = 3 years 1 monh 6 monhs 9 monhs Shor Term Rae 1.00 3.51 55.76 55.76 4.02 45.18 45.20 4.28 40.52 40.54 3.00 4.68 42.23 42.23 5.02 36.50 36.51 5.20 33.69 33.71 5.00 5.87 34.04 34.04 6.04 30.64 30.65 6.12 28.85 28.87 7.00 7.06 28.55 28.55 7.06 26.44 26.45 7.05 25.25 25.27 9.00 8.27 24.62 24.62 8.08 23.26 23.27 7.98 22.47 22.49 10.00 8.87 23.04 23.04 8.60 21.95 21.96 8.45 21.30 21.32 Tenor = 5 years 1 monh 6 monhs 9 monhs Shor Term Rae 1.00 4.31 34.63 34.62 4.69 29.56 29.58 4.89 27.16 27.18 3.00 5.20 29.11 29.11 5.46 25.75 25.77 5.59 24.05 24.07 5.00 6.11 25.17 25.17 6.23 22.84 22.86 6.30 21.61 21.63 7.00 7.04 22.20 22.20 7.02 20.55 20.56 7.01 19.64 19.66 9.00 7.97 19.89 19.89 7.82 18.70 18.71 7.74 18.01 18.03 10.00 8.44 18.92 18.92 8.22 17.90 17.91 8.11 17.30 17.32 49 cby Anonio Mele and Yoshiki Obayashi

Table A.1 coninued Mauriy of he variance conrac Tenor = 1 year 1 year 2 years 3 years Shor Term Rae 1.00 3.61 64.25 64.26 4.67 41.87 41.92 5.38 31.66 31.72 3.00 4.76 49.10 49.10 5.45 36.06 36.10 5.92 28.90 28.96 5.00 5.91 39.72 39.72 6.24 31.66 31.69 6.46 26.58 26.63 7.00 7.07 33.35 33.35 7.04 28.22 28.25 7.00 24.61 24.65 9.00 8.23 28.75 28.75 7.83 25.46 25.48 7.54 22.91 22.95 10.00 8.82 26.90 26.90 8.23 24.27 24.29 7.81 22.15 22.19 Tenor = 2 years 1 year 2 years 3 years Shor Term Rae 1.00 4.13 47.30 47.32 5.01 32.95 33.01 5.61 25.69 25.76 3.00 5.10 38.61 38.62 5.68 29.27 29.32 6.07 23.86 23.92 5.00 6.07 32.62 32.62 6.35 26.34 26.38 6.52 22.27 22.33 7.00 7.05 28.25 28.25 7.02 23.94 23.97 6.98 20.89 20.94 9.00 8.04 24.92 24.92 7.69 21.94 21.97 7.44 19.66 19.71 10.00 8.53 23.53 23.54 8.03 21.07 21.10 7.67 19.11 19.15 Tenor = 3 years 1 year 2 years 3 years Shor Term Rae 1.00 4.52 36.92 36.94 5.28 26.87 26.93 5.79 21.41 21.48 3.00 5.36 31.47 31.49 5.85 24.41 24.46 6.18 20.14 20.21 5.00 6.19 27.44 27.45 6.42 22.37 22.41 6.57 19.02 19.08 7.00 7.04 24.33 24.34 7.00 20.64 20.69 6.97 18.02 18.08 9.00 7.89 21.87 21.88 7.58 19.17 19.21 7.36 17.12 17.17 10.00 8.31 20.82 20.83 7.87 18.51 18.55 7.56 16.70 16.76 Tenor = 5 years 1 year 2 years 3 years Shor Term Rae 1.00 5.07 25.19 25.22 5.65 19.35 19.41 6.04 15.86 15.93 3.00 5.71 22.66 22.68 6.09 18.10 18.16 6.34 15.19 15.26 5.00 6.36 20.60 20.61 6.53 17.01 17.06 6.64 14.58 14.64 7.00 7.01 18.90 18.91 6.98 16.05 16.10 6.95 14.02 14.08 9.00 7.67 17.47 17.48 7.43 15.20 15.25 7.26 13.50 13.56 10.00 8.00 16.83 16.84 7.65 14.81 14.86 7.41 13.26 13.31 50 cby Anonio Mele and Yoshiki Obayashi

Table A.1 coninued Mauriy of he variance conrac Tenor = 10 years 1 monh 6 monhs 12 monhs Shor Term Rae 1.00 5.25 17.80 17.80 5.48 15.92 15.93 5.72 14.12 14.14 3.00 5.81 16.51 16.51 5.97 14.96 14.97 6.13 13.43 13.45 5.00 6.39 15.42 15.41 6.47 14.13 14.14 6.54 12.82 12.84 7.00 6.98 14.48 14.48 6.97 13.40 13.41 6.96 12.27 12.29 9.00 7.59 13.67 13.67 7.49 12.75 12.77 7.39 11.78 11.80 10.00 7.91 13.31 13.31 7.76 12.46 12.47 7.61 11.55 11.57 Mauriy of he variance conrac Tenor = 20 years 1 monh 6 monhs 12 monhs Shor Term Rae 1.00 5.79 10.55 10.55 5.9561 9.65 9.66 6.11 8.74 8.76 3.00 6.17 10.27 10.27 6.2765 9.44 9.45 6.38 8.59 8.60 5.00 6.55 10.03 10.03 6.6067 9.24 9.26 6.65 8.44 8.46 7.00 6.95 9.79 9.79 6.9470 9.06 9.07 6.93 8.30 8.32 9.00 7.36 9.57 9.57 7.2978 8.88 8.90 7.22 8.17 8.18 10.00 7.57 9.47 9.47 7.4771 8.81 8.82 7.37 8.10 8.12 Mauriy of he variance conrac Tenor = 30 years 1 monh 6 monhs 12 monhs Shor Term Rae 1.00 5.96 8.70 8.70 6.0928 8.01 8.02 6.22 7.30 7.31 3.00 6.27 8.59 8.59 6.3651 7.93 7.93 6.45 7.23 7.25 5.00 6.60 8.49 8.49 6.6466 7.84 7.85 6.69 7.17 7.19 7.00 6.94 8.39 8.39 6.9374 7.76 7.77 6.93 7.11 7.13 9.00 7.29 8.29 8.29 7.2378 7.68 7.69 7.17 7.05 7.07 10.00 7.47 8.24 8.24 7.3917 7.65 7.66 7.30 7.02 7.04 51 cby Anonio Mele and Yoshiki Obayashi

The lef panels of Figure 7 depic he approximaions o he square roo of he wo expecaions in Eq. (F.5): he rs, normalized by PVBP (r ; T 1 ; ; T n ) (T ), as in Eq. (25); and he second, normalized by (T ), as in Eq. (26). The righ panels of Figure 7 depic he insananeous volailiy of he forward swap rae, as de ned in Eq. (F.4). Finally, Eqs. (F.3), (F.4) and (F.5) are evaluaed assuming rese daes are quarerly and, accordingly, seing i consan and equal o 1 4. G. Spanning and hedging Gaussian conracs G.1. Pricing We price conracs where he variance payo in Eq. (A.5) is replaced by: V BP n (; T ) PVBP T (T 1 ; ; T n ) ; where V BP n (; T ) denoes he realized variance of he forward swap changes in he inerval [; T ], Z T Vn BP (; T ) = Rs 2 (T 1 ; ; T n ) 2 s (T 1 ; ; T n ) ds; (G.1) and 2 s (T 1 ; ; T n ) is as in Eq. (A.6). By a Taylor s expansion wih remainder, we have, denoing, as usual, for simpliciy, R R (T 1 ; ; T n ), Z R Z 1 RT 2 = R 2 + 2R (R T R ) + 2 (K R T ) + dk + (R T K) + dk : (G.2) 0 R Muliplying boh sides of his equaion by PVBP (T 1 ; ; T n ), and aking expecaions under he Q swap probabiliy, leaves: PVBP (T 1 ; ; T n ) E Qswap RT 2 R 2 = 2 Z R 0 SWPN R (K; T ; T 1 ; ; T n ) dk + Z 1 Moreover, by Iô s lemma, and Eqs. (A.4) and (G.1), By replacing his expression ino Eq. (G.3) yields: E Qswap RT 2 R 2 = EQswap V BP n (; T ) : Z R 2 SWPN R (K; T ; T 1 ; ; T n ) dk + 0 = PVBP (T 1 ; ; T n ) E Qswap RT 2 R 2 = PVBP (T 1 ; ; T n ) E Qswap V BP (; T ) = E Q e R T rsds n R Z 1 R PVBP T (T 1 ; ; T n ) Vn BP (; T ) SWPN P (K; T ; T 1 ; ; T n ) dk : (G.3) SWPN P (K; T ; T 1 ; ; T n ) dk F BP var;n (; T ) ; (G.4) where he las line follows by he Radon-Nikodym derivaive de ned in Eq. (A.3). The basis poin volailiy in Eq. (27) can also be simpli ed using he Black s formulae in Eqs. (B.1) and (B.10) o compue F BP var;n (; T ) in Eq. (G.4), as follows: IRS-VI BP n (; T ) 52 cby Anonio Mele and Yoshiki Obayashi

v u = 2 T An Ineres Rae Swap Volailiy Index and Conrac 2 4 X ^Z R n i; (T ) IV 2 i; Ki + X Z i:k i <R i:k i R R n; T; K i; (T 3 ) IV 2 i; Ki 5; (G.5) where Z and ^Z are as in Eqs. (B.1) and (B.10), R n is he curren forward swap rae for mauriy T and enor lengh T n T, R n R (T 1 ; ; T n ), and, nally, IV i; denoes he ime implied percenage volailiy for swapions wih srike equal o K i. G.2. Hedging We provide he deails leading o Table III only, as hose for Table IV are nearly idenical. By Iô s lemma: PVBP T (T 1 ; ; T n ) Vn BP (; T ) = 2PVBP T (T 1 ; ; T n ) Z T R s (T 1 ; ; T n ) dr s (T 1 ; ; T n ) + PVBP T (T 1 ; ; T n ) R 2 T (T 1 ; ; T n ) R 2 (T 1 ; ; T n ) : (G.6) By Eq. (G.2), he second erm on he righ hand side is he payo a T of a porfolio se up a, which is: (a) long 2R (T 1 ; ; T n ) unis a xed ineres payer swap sruck a R (T 1 ; ; T n ), and (b) long a coninuum of ou-of-he-money swapions wih weighs 2dK. I is he saic posiion (ii) in Table III of he main ex. By Eqs. (G.2), (G.3) and (G.4), and he de niion of he Radon-Nikodym derivaive in Eq. (A.3), is cos is F BP var;n (; T ), which we borrow a, o repay i back a T, as in row (iii) of Table III. The self- nanced porfolio o be shored, as indicaed by row (i) of Table III, is obained similarly as he porfolio in row (i) of Table I (see Appendix C.3), bu wih he porfolio, Z ^ M = PVBP (T 1 ; ; T n ) 2R s (T 1 ; ; T n ) dr s (T 1 ; ; T n ) 1 ; replacing ha in Eq. (C.8). G.3. Consan Mauriy Swaps Consider, iniially, he fair price of he paymen of a Consan Mauriy Swap (CMS) occurring a ime T 0 +, and se, for simpliciy, S (T 0 ) R T0 (T 1 ; ; T n ). The curren value of S (T 0 ) o be paid a ime T 0 + is he same as he curren value of S (T 0 ) P T0 (T 0 + ) o be paid a T 0, such ha: cms (; T 0 + ) = E Q e R T0 r udu S (T 0 ) P T0 (T 0 + ) = P (T 0 + ) E Qswap S (T 0 ) G T 0 G ; (G.7) P where G (T 0 +) PVBP (T 1 ; ;T. We calculae G n), by discouning hrough he rese imes, T i T 0, using he a rae formula, P T0 (T i ) = (1 + S (T 0 )) i, where is he year fracion beween he rese imes, such ha, G P (T 0 + ) PVBP (T 1 ; ; T n ) (1 + S ()) (T0+ ) P n i=1 (1 + S ()) (T i ) = S () (1 + S ()) 1 1 (1+S()) n G (S ()) : (G.8) The previous derivaions closely follow hose in Hagan (2003), and are provided for compleeness. We now make he connecion beween he price of he CMS in Eq. (G.7) and he price of he IRV forward conrac in BP of De niion IV-(a) in he main ex. Replacing he approximaion in Eq. (G.8) ino Eq. (G.7) leaves: cms (; T 0 + ) 53 cby Anonio Mele and Yoshiki Obayashi

= P (T 0 + ) E swap S (T 0 ) G (S (T 0)) G (R ) P (T 0 + ) E swap S (T 0 ) 1 + G0 (R ) G (R ) (S (T 0) R ) An Ineres Rae Swap Volailiy Index and Conrac = P (T 0 + ) E swap (S (T 0 )) + PVBP (T 1 ; ; T n ) G 0 (R ) E Qswap [S (T 0 ) (S (T 0 ) R )] = P (T 0 + ) R + G 0 (R ) PVBP (T 1 ; ; T n ) E Qswap S 2 (T 0 ) R 2 = P (T 0 + ) R + G 0 (R ) F BP var;n (; T 0 ) ; where he second line follows by a rs order Taylor approximaion of he funcion G abou R, he hird from he de niion of G (R ), he fourh from he maringale propery of he forward swap rae under Q swap, E swap (S (T 0 )) = R and, nally, he fh equaliy from Eq. (G.3) and Eq. (G.4). Eq. (18) of he main ex follows by calculaing and summing every single CMS paymen, cms (; T 0 + j), for j = 1; ; N. As menioned in Secion 5.3 of he main ex, i is well-known since a leas Hagan (2003) and Mercurio and Pallavicini (2006) ha CMS link o he enire skew. However, our represenaion of he price of a CMS in Eq. (18) quie di ers from previous ones. Mercurio and Pallavicini (2006), for example, uilize spanning argumens di eren from ours. We explain he di erences. Consider a Taylor s expansion wih remainder of he funcion f (R T ) R 2 T abou some poin R o, Z Ro Z 1 RT 2 = Ro 2 + 2R o (R T R o ) + 2 (K R T ) + dk + (R T K) + dk : (G.9) 0 R o Mercurio and Pallavicini (2006) consider he poin R o = 0, such ha, Eq. (G.9) collapses o such ha E Qswap R 2 T = 2 Z 1 RT 2 2 = PVBP (T 1 ; ; T n ) 0 (R T K) + dk; Z 1 0 SWPN P (K; T ; T n ) dk: Our approach di ers as we ake R o = R in Eq. (G.9), leading o Eq. (G.2) and, hen, o, Z E Qswap RT 2 R 2 R Z 1 = 2EQswap (K R T ) + dk + (R T K) + dk 0 R Z 2 R = SWPN R (K; T ; T n ) dk + PVBP (T 1 ; ; T n ) G.4. Numerical experimens 0 Z 1 R SWPN P (K; T ; T n ) dk : Table A.2 repors experimens relaing o he calculaion of BP volailiy as referenced by he index IRS-VI BP n (; T ) in Eq. (33), as well as fuure expeced volailiy in a risk-neural marke, as prediced by he Vasicek model, using he same parameer values as hose in Appendix F. 54 cby Anonio Mele and Yoshiki Obayashi

Table A.2 The IRS-VI BP n (; T ) index in Eq. (33) (labeled IRS-VI BP ), and fuure expeced volailiy in a risk-neural marke, q 1 T E Q[V BP n (; T )], where V BP n (; T ) is as in Eq. (G.1) (labeled E-Vol BP ), for (i) ime-o mauriy T equal o 1, 6 and 9 monhs, and 1, 2 and 3 years, and (ii) enor lengh n equal o 1, 2, 3 and 5 years, and for ime-o mauriy T equal o 1, 6 and 12 monhs, and (ii) enor lengh n equal o 10, 20 and 30 years. The index and expeced volailiies are compued under he assumpion he shor-erm ineres rae is as in he Vasicek model of Eq. (F.1), wih parameer values = 0:3807, r = 0:072 and v = 0:0331. Mauriy of he variance conrac Tenor = 1 year 1 monh 6 monhs 9 monhs Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 2.18 272.43 272.43 2.90 252.92 252.97 3.27 242.33 242.41 3.00 3.80 273.79 273.79 4.28 253.99 254.05 4.53 243.26 243.36 5.00 5.44 275.15 275.16 5.68 255.07 255.13 5.80 244.19 244.30 7.00 7.08 276.52 276.54 7.08 256.15 256.22 7.08 245.13 245.26 9.00 8.73 277.90 277.92 8.49 257.23 257.32 8.36 246.07 246.21 10.00 9.56 278.59 278.62 9.19 257.78 257.87 9.00 246.54 246.69 Tenor = 2 years 1 monh 6 monhs 9 monhs Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 2.93 229.60 229.60 3.53 213.30 213.37 3.84 204.45 204.55 3.00 4.31 231.15 231.15 4.70 214.53 214.60 4.91 205.51 205.63 5.00 5.68 232.71 232.72 5.88 215.76 215.84 5.98 206.57 206.72 7.00 7.07 234.29 234.30 7.07 217.00 217.09 7.06 207.65 207.81 9.00 8.47 235.88 235.89 8.26 218.25 218.35 8.15 208.73 208.90 10.00 9.18 236.68 236.69 8.86 218.87 218.99 8.69 209.28 209.46 Tenor = 3 years 1 monh 6 monhs 9 monhs Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 3.51 196.13 196.14 4.02 182.41 182.48 4.28 174.93 175.05 3.00 4.68 197.97 197.97 5.02 183.85 183.93 5.20 176.18 176.32 5.00 5.87 199.82 199.82 6.03 185.31 185.40 6.12 177.45 177.60 7.00 7.06 201.68 201.69 7.05 186.78 186.88 7.05 178.72 178.90 9.00 8.27 203.57 203.58 8.08 188.26 188.38 7.98 180.01 180.20 10.00 8.87 204.52 204.53 8.60 189.01 189.13 8.45 180.66 180.85 Tenor = 5 years 1 monh 6 monhs 9 monhs Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 4.31 149.31 149.31 4.69 139.19 139.27 4.89 133.66 133.79 3.00 5.20 151.57 151.58 5.46 140.98 141.07 5.59 135.22 135.36 5.00 6.11 153.87 153.87 6.23 142.79 142.90 6.30 136.80 136.96 7.00 7.03 156.20 156.21 7.02 144.63 144.75 7.01 138.39 138.57 9.00 7.97 158.56 158.57 7.82 146.49 146.61 7.74 140.01 140.20 10.00 8.44 159.76 159.76 8.22 147.43 147.56 8.11 140.82 141.02 55 cby Anonio Mele and Yoshiki Obayashi

Table A.2 coninued Mauriy of he variance conrac Tenor = 1 year 1 year 2 years 3 years Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 3.61 232.52 232.62 4.67 199.65 199.91 5.38 174.98 175.33 3.00 4.76 233.32 233.44 5.45 200.12 200.39 5.92 175.25 175.62 5.00 5.91 234.13 234.27 6.24 200.58 200.88 6.46 175.53 175.91 7.00 7.07 234.95 235.10 7.04 201.05 201.36 7.00 175.81 176.21 9.00 8.23 235.77 235.93 7.83 201.53 201.85 7.54 176.09 176.50 10.00 8.82 236.18 236.35 8.23 201.76 202.10 7.81 176.23 176.64 Mauriy of he variance conrac Tenor = 2 years 1 year 2 years 3 years Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 4.13 196.23 196.36 5.01 168.67 168.95 5.61 147.95 148.30 3.00 5.10 197.16 197.30 5.68 169.21 169.51 6.07 148.26 148.64 5.00 6.07 198.09 198.25 6.35 169.74 170.07 6.52 148.58 148.97 7.00 7.05 199.03 199.20 7.02 170.29 170.62 6.98 148.91 149.30 9.00 8.04 199.98 200.16 7.69 170.83 171.19 7.44 149.23 149.64 10.00 8.53 200.45 200.64 8.03 171.11 171.47 7.67 149.39 149.81 Mauriy of he variance conrac Tenor = 3 years 1 year 2 years 3 years Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 4.52 167.99 168.13 5.28 144.62 144.91 5.79 127.00 127.35 3.00 5.36 169.09 169.24 5.85 145.26 145.57 6.18 127.38 127.75 5.00 6.19 170.19 170.36 6.42 145.90 146.23 6.57 127.76 128.14 7.00 7.04 171.31 171.49 7.01 146.55 146.89 6.97 128.14 128.54 9.00 7.89 172.43 172.63 7.58 147.20 147.56 7.36 128.53 128.93 10.00 8.31 172.99 173.20 7.87 147.53 147.89 7.56 128.73 129.13 Mauriy of he variance conrac Tenor = 5 years 1 year 2 years 3 years Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 5.07 128.51 128.66 5.65 111.04 111.33 6.04 97.75 98.09 3.00 5.71 129.88 130.04 6.09 111.84 112.14 6.34 98.23 98.58 5.00 6.36 131.25 131.43 6.53 112.64 112.96 6.64 98.71 99.07 7.00 7.01 132.65 132.84 6.98 113.45 113.79 6.95 99.20 99.56 9.00 7.67 134.06 134.26 7.43 114.27 114.62 7.26 99.69 100.06 10.00 8.00 134.77 134.97 7.65 114.68 115.04 7.41 99.93 100.31 56 cby Anonio Mele and Yoshiki Obayashi

Table A.2 coninued Mauriy of he variance conrac Tenor = 10 years 1 monh 6 monhs 12 monhs Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 5.25 93.49 93.50 5.48 87.58 87.65 5.7299 81.26 81.39 3.00 5.81 96.00 96.00 5.97 89.57 89.65 6.1350 82.78 82.93 5.00 6.39 98.56 98.56 6.47 91.60 91.69 6.5477 84.33 84.49 7.00 6.98 101.18 101.19 6.97 93.67 93.77 6.9682 85.91 86.07 9.00 7.59 103.86 103.87 7.49 95.78 95.89 7.3965 87.51 87.68 10.00 7.90 105.23 105.23 7.76 96.85 96.96 7.6137 88.32 88.49 Mauriy of he variance conrac Tenor = 20 years 1 monh 6 monhs 12 monhs Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 5.7998 61.21 61.21 5.95 57.56 57.62 6.1146 53.62 53.72 3.00 6.1705 63.43 63.43 6.27 59.33 59.40 6.3827 54.99 55.10 5.00 6.5546 65.72 65.72 6.60 61.15 61.23 6.6577 56.38 56.50 7.00 6.9526 68.09 68.09 6.94 63.02 63.10 6.9396 57.80 57.93 9.00 7.3649 70.53 70.53 7.29 64.94 65.03 7.2287 59.25 59.39 10.00 7.5766 71.77 71.78 7.47 65.92 66.01 7.3759 59.99 60.13 Mauriy of he variance conrac Tenor = 30 years 1 monh 6 monhs 12 monhs Shor Term Rae IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P IRS-VI B P E-Vol B P 1.00 5.96 51.91 51.91 6.09 48.88 48.93 6.22 45.59 45.68 3.00 6.27 53.95 53.95 6.36 50.51 50.57 6.45 46.85 46.94 5.00 6.60 56.06 56.07 6.64 52.19 52.25 6.69 48.13 48.24 7.00 6.94 58.25 58.25 6.93 53.91 53.98 6.93 49.45 49.56 9.00 7.29 60.51 60.51 7.23 55.69 55.77 7.17 50.79 50.91 10.00 7.47 61.66 61.66 7.39 56.59 56.68 7.30 51.47 51.60 57 cby Anonio Mele and Yoshiki Obayashi

G.5. Consan vega in Gaussian markes We show ha he saemen in (17) is rue consan vega requires swapions o be equally weighed in a Gaussian marke. If he forward swap rae is soluion o Eq. (13), he price of a swapion payer is: where O P (R (T 1 ; ; T n ) ; K; T; N ; T n ) PVBP (T 1 ; ; T n ) Z P (R (T 1 ; ; T n ) ; K; T; N ; T n ) ; Z P (R; K; T; ; T n ) = (R R K) p T K + p R T p T and denoes he sandard normal densiy. The vega of a payer is he same as he vega of a receiver, and equals: O (R; ) @OP (R; K; T; ; T n ) @ K = PVBP (T 1 ; ; T n ) p R T p T ; K ; such ha he vega of a porfolio of swapions (be hey payers and/or receivers) is: (R; ) @ (R; ) @ = PVBP (T 1 ; ; T n ) p Z T R! (K) p T K dk: (G.10) As for he if par in (17), le! (K) = cons:, such ha by Eq. (G.10), and he obvious fac ha he Gaussian densiy inegraes o one, we have indeed ha he vega is independen of R, (R; ) = PVBP (T 1 ; ; T n ) p T cons:: As for he only if par, le us di ereniae (R; ) in Eq. (G.10) wih respec o R, @ (R; ) @R = PVBP (T 1 ; ; T n ) 2p T Z R! (K) p T K (R K) dk: The only funcion! which is independen of R, and such ha @(R;) @R is idenically zero, is he consan weighing. H. Jumps We derive he fair value of he Sandardized IRV swap rae in De niion III, and he Sandardized BP-IRV swap rae of De niion IV-(c), under he assumpion ha he forward swap rae follows he jump-di usion process in Eq. (34). These derivaions lead o he wo indexes in Eq. (39) (percenage) and Eq. (41) (basis poin). Secion H.1 develops he argumens applying o he percenage conrac and index, and Secion H.2 conains derivaions relaing o he basis poin. H.1. Percenage We apply Iô s lemma o Eq. (34), obaining, d ln R (T 1 ; ; T n ) = E Qswap; e jn() 1 () d 1 2 k (T 1 ; ; T n )k 2 d + (T 1 ; ; T n ) dw () + j n () dn () = 1 k (T 1 ; ; T n )k 2 d + jn 2 () dn () + (T 1 ; ; T n ) dw () 2 E Qswap; e jn() 1 () d + j n () dn () + 1 2 j2 n () dn () : 58 cby Anonio Mele and Yoshiki Obayashi

We have, E Qswap; e jn() 1 dn () = E Qswap; e jn() 1 () d; (H.1) such ha by he de niion of V J n (; T ) in Eq. (35), and Eq. (H.1), we obain: 2E Qswap; ln R T (T 1 ; ; T n ) R (T 1 ; ; T n ) = E Qswap; [V n (; T )] = 1 PVBP (T 1 ; ; T n ) E F var;n (; T ) = PVBP (T 1 ; ; T n ) = P J; var;n (; T ) ; e 2E Qswap; Z T e jn() 1 j n () R T r s ds PVBPT (T 1 ; ; T n ) Vn J (; T ) 1 2 j2 n () dn () (H.2) where he hird equaliy follows by he de niion of he IRV forward agreemen in De niion I, and he fourh by a sraighforward generalizaion of Eq. (10). Comparing Eq. (H.2) wih Eq. (10) and Eq. (C.2) produces Eq. (37). H.2. Basis poin Apply Iô s lemma for jump-di usion processes o Eq. (34), obaining, dr 2 (T 1 ; ; T n ) R 2 (T 1 ; ; T n ) = 2 E Qswap; e jn() 1 () d + 2 (T 1 ; ; T n ) dw () + k (T 1 ; ; T n )k 2 d + e 2jn() 1 = 2 E Qswap; e jn() 1 () d + 2 + k (T 1 ; ; T n )k 2 d + dn () e jn() 1 dn () + 2 (T 1 ; ; T n ) dw () 2 e 1 jn() dn () : (H.3) By inegraing, aking expecaions under Q swap, and using he de niion of basis poin variance, Vn J;BP (; T ) in Eq. (36), leaves: E Qswap; RT 2 (T 1 ; ; T n ) R 2 (T 1 ; ; T n ) Z T Z T = 2 E Qswap; e jn() 1 () d + 2E Qswap; e jn() 1 dn () {z } Z T + 2E Qswap; (T 1 ; ; T n ) dw () {z } =0 + E Qswap; [Vn J;BP (; T )]; where he rs erm is zero by Eq. (H.1). By Eq. (H.3), and Eq. (G.3), his cancellaion leads o Eq. (40) and, hence, Eq. (41). =0 59 cby Anonio Mele and Yoshiki Obayashi

References Bakshi, Gurdip and Dilip Madan, 2000. Spanning and Derivaive Securiy Evaluaion. Journal of Financial Economics 55, 205-238. Black, Fisher and Myron Scholes, 1973. The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy 81, 637-659. Black, Fisher, 1976. The Pricing of Commodiy Conracs. Journal of Financial Economics 3, 167-179. Bossu, Sebasien, Eva Srasser and Regis Guichard, 2005. Jus Wha You Need To Know Abou Variance Swaps. JPMorgan Equiy Derivaives. Brigo, Damiano and Fabio Mercurio, 2006. Ineres Rae Models Theory and Pracice, wih Smile, In aion and Credi. Springer Verlag (2nd Ediion). Brien-Jones, Mark and Anhony Neuberger, 2000. Opion Prices, Implied Price Processes and Sochasic Volailiy. Journal of Finance 55, 839-866. Carr, Peer and Dilip Madan, 2001. Opimal Posiioning in Derivaive Securiies. Quaniaive Finance 1, 19-37. Demeer, Kresimir, Emanuel Derman, Michael Kamal and Joseph Zou, 1999. More Than You Ever Waned To Know Abou Volailiy Swaps. Goldman Sachs Quaniaive Sraegies Research Noes. Dumas, Bernard, 1995. The Meaning of he Implici Volailiy Funcion in Case of Sochasic Volailiy. Available from: hp://www.insead.edu/faculyresearch/faculy/personal/bdumas/research/index.cfm. Hagan, Parick S., 2003. Convexiy Conundrums: Pricing CMS Swaps, Caps, and Floors. Wilmo Magazine, March 38-44. Jacod, Jean and Alber N. Shiryaev, 1987. Limi Theorems for Sochasic Processes. Springer Verlag, Berlin. Jiang, Rosanne, 2011. An Analysis of Gamma Trading Sraegy in Swapion Marke. Working Paper London School of Economics. Mele, Anonio, 2003. Fundamenal Properies of Bond Prices in Models of he Shor-Term Rae. Review of Financial Sudies 16, 679-716. Mele, Anonio, 2011. Lecures on Financial Economics. Book manuscrip. Available from: hp://www.anoniomele.org. Mercurio, Fabio and Andrea Pallavicini, 2006. Swapion Skews and Convexiy Adjusmens. Banca IMI. Neuberger, Anhony, 1994. Hedging Volailiy: he Case for a New Conrac. Journal of Porfolio Managemen 20, 74-80. Trolle, Anders B. and Eduardo S. Schwarz, 2011. The Swapion Cube. Working paper EPFL and Swiss Finance Insiue. 60 cby Anonio Mele and Yoshiki Obayashi

Vasicek, Oldrich A., 1977. An Equilibrium Characerizaion of he Term Srucure. Journal of Financial Economics 5, 177-188. Veronesi, Piero, 2010. Fixed Income Securiies. John Wiley & Sons. No saemen wihin his paper should be consrued as a recommendaion o buy or sell a nancial produc or o provide invesmen advice. Pas performance is no indicaive of fuure resuls. Opions involve risk and are no suiable for all invesors. Prior o buying or selling an opion, a person mus receive a copy of Characerisics and Risks of Sandardized Opions. Copies are available from your broker, by calling 1-888-OPTIONS, or from The Opions Clearing Corporaion a www.heocc.com. CBOE R, Chicago Board Opions Exchange R and VIX R are regisered rademarks of Chicago Board Opions Exchange, Incorporaed (CBOE). S&P R and S&P 500 R are rademarks of Sandard & Poor s Financial Services, LLC. 61 cby Anonio Mele and Yoshiki Obayashi