Convexity Adjustments in Inflation linked Derivatives using a multi-factor version of the Jarrow and Yildirim (2003) Model

Transcription

1 Imperal College of Scence, echnology and edcne Deparmen of ahemacs Convexy Adjusmens n Inflaon lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Hongyun L Sepember 007 Submed o Imperal College London n fulflmen of he requremens for he Degree of aser of Scence n ahemacs and Fnance

2 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Absrac In hs paper, we use a Gaussan HJ-ype (Heah e al 99 model for he valuaon of nflaon-lnked dervaves. he model s essenally ha of Jarrow and Yldrm (003, whch n urn s essenally analogous o a cross-currency model (modellng he spo foregn exchange rae, domesc currency neres-raes and foregn currency neres-raes. In he cross-currency F analogy of Jarrow and Yldrm (003, nomnal zero coupon bonds are analogous o zero coupon bonds n he domesc currency, real zero coupon bonds are analogous o zero coupon bonds n he foregn currency and he spo consumer prce ndex (CPI s analogous o he spo foregn exchange rae. We exend he Jarrow and Yldrm (003 model by modellng neres-rae yeld curves wh a mul-facor (raher han one facor Gaussan HJ (Heah e al 99 model. Our paper s organzed as follows: Frsly, we nroduce he model and our noaon. hen, we explan he valuaon of sandard zero coupon nflaon swaps. We hen examne popular and acvely-raded nflaon producs such as zero coupon nflaon swaps wh delayed paymen, perod-on-perod nflaon swaps wh boh no delayed paymens and wh delayed paymens, usng he Gaussan model (we explan wha we mean by delayed paymens n secon.4. oreover, we gve he analycal prces of hese nflaon-lnked dervaves, conssen wh no-arbrage. Specfcally, we focus on he convexy adjusmens nvolved n prcng hese producs. We provde an applcaon of our convexy adjusmen formulae o he valuaon of lmed prce ndexaon (LPI swaps. Fnally, o specfy he model, we use he same mehod as n Jarrow and Yldrm (003 o esmae he model parameers whch are needed o evaluae he convexy adjusmens.

3 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Acknowledgemens Sncere hanks o ark Davs for recommendng me o Lloyds SB o ake he nernshp. Specal hanks o John Crosby, my ndusry supervsor, for hs dedcaed gudance, suppor and paence hroughou he enre projec. eanwhle, hanks o hs dsclosng o me he secres of he nflaon dervaves marke and for he smulaon provded. Sncere hanks o Dorje C. Brody, my projec supervsor, for hs help and gudance. hanks o Lloyds SB whch provde sponsorshp o me. hanks o erry organ of Lloyds SB for provdng he hsorcal daa used n chaper 6. any hanks o all my Imperal College lecurers, from whom I ganed nvaluable knowledge and whou whch hs hess would no have been possble. Very specal hanks o my moher and faher for her encouragemen and suppor. 3

4 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel ABLE OF CONENS Inroducon Background o nflaon-lnked dervaves Oulne of he hess... 7 Foregn Exchange Analogy and odelng Inflaon Inroducon o foregn exchange mehodology Noaon A basc valuaon formula n he foregn exchange analogy odelng Inflaon Noaon An Imporan Observaon for nflaon dervaves Zero Coupon Inflaon Swaps....4 Indexaon lag and delayed paymens... 3 Framework of he model Sochasc evoluon of nomnal bond prces Sochasc evoluon of real bond prces Sochasc evoluon of he spo CPI ndex Exoc nflaon dervaves Perod-On-Perod Inflaon Swaps Zero coupon nflaon swaps wh delayed paymen Perod-on-perod nflaon swaps wh delayed paymens An applcaon of our convexy adjusmen formulae o prcng LPI swaps LPI swaps Prcng LPI swaps

5 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel 6 Calbraon o marke daa Ge hsorcal daa Ge zero coupon nflaon swaps and CPI Ge nomnal and real dscoun facors Ge model parameers usng Jarrow and Yldrm (003 mehod One facor model wo facor model Gve he values of convexy adjusmens of exoc dervaves Conclusons Appendx Appendx Appendx 3 Proof of Proposon Appendx 4 Proof of Proposon... 5 Appendx Appendx 6 Parameer Esmaes References

6 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Inroducon. Background o nflaon-lnked dervaves In recen years, he marke for nflaon-lnked dervaves has grown very rapdly. hey are used by marke parcpans o manage he rsks of changng nflaon and changng nflaon expecaons n an effcen way. I s far o say ha nflaon s now regarded as an ndependen asse class. here are, broadly speakng, wo ypes of parcpans n he nflaon dervaves markes: hose who wsh o receve and hose who wsh o pay nflaon-lnked cash flows. Acvely-raded nflaon dervaves nclude sandard zero coupon nflaon swaps, and more complcaed producs such as perod-on-perod nflaon swaps (ercuro (005, nflaon caps, nflaon swapons, and fuures conracs wren on nflaon (Crosby 007b. Inflaon s descrbed n erms of an nflaon ndex. In pracce, here are a number of acvely referenced nflaon ndces. he man ndces are he HICPx ndex (whch measures nflaon n he Euro zone and s publshed by Euro sa, he RPI (Real Prce ndex (whch measures nflaon n he U and s publshed by Naonal Sascs, and he US-CPI (consumer prce ndex (whch measures nflaon n he US and s publshed by BLS. hroughou hs paper, we wll, for he sake of brevy, refer o he nflaon ndex as he CPI ndex or he spo CPI ndex (even hough n he U, would be probably he RPI and n he Eurozone, would probably be he HICPx ndex. All of hese ndces are a measure of real or consumer prce nflaon. hey are calculaed by collecng and comparng he prces of a se baske of goods and servces, as bough by a ypcal consumer, a regular nervals over me. (Reuers 6

7 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel. Oulne of he hess he remander of hs paper s srucured as follows: In chaper wo, he foregn exchange analogy s explaned brefly. hen, we wll provde noaon and dscuss he smples ype of nflaon dervave, namely sandard (.e. wh no delayed paymen zero coupon nflaon swaps, and show how hey can be valued n a model-ndependen fashon. oreover, we wll explan n deal wha s mean by ndexaon lag and delayed paymens. In chaper hree, workng whn a mul-facor verson of he Jarrow and Yldrm (003 model, we wll nroduce he dynamcs of nomnal zero coupon bond prces, real zero coupon bond prces and he spo CPI ndex. In chaper four, we compue he convexy adjusmens requred o value perod-on-perod nflaon swaps wh no delayed paymens, zero coupon nflaon swaps wh delayed paymen and perod-on-perod nflaon swaps wh delayed paymens n deal. o our bes knowledge, some of hese resuls, a leas n he conex of a mul-facor Jarrow and Yldrm (003 model, have no appeared n he leraure before. In chaper fve, we provde an applcaon of he convexy adjusmens we compued n chaper four, o he valuaon of lmed prce ndexaon (LPI swaps, n whch we use he quas-analyc mehodology of Ryen (007. In chaper sx, we use he mehods of Jarrow and Yldrm (003 o esmae he model parameers from hsorcal daa. We wll also llusrae our model wh some examples and comparsons. In chaper seven, we wll gve he conclusons of hs paper. In he appendces, we gve dealed dervaons of some of he formulae ha we use. 7

8 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Foregn Exchange Analogy and odelng Inflaon We are concerned, n hs paper, wh Gaussan models for nflaon whch are arbrage-free and conssen wh any nal erm srucure of neres-raes (boh nomnal and real. In hs and all subsequen secons, we wll always make he assumpons ha he marke s frconless, complee and arbrage-free. hese assumpons guaranee (Harrson and Plska (98 he exsence of a unque equvalen marngale measure whch s denoed by Q. We use he noaon E [ ] o denoe expecaons a me, wh respec o hs equvalen marngale measure. All sochasc processes are defned on a common flered probably space ( Ω, FQ,, where he flraon F s assumed o be he naural flraon generaed by he Brownan moons, whch we shall shorly nroduce, drvng he nomnal and real neres-rae yeld curves and he spo CPI ndex. We denoe calendar me by. We defne oday (he nal me o be me 0.. Inroducon o foregn exchange mehodology. Noaon he foregn exchange (F analogy relaes o he valuaon of foregn exchange opons wren on a spo foregn exchange rae. We denoe he prce, n domesc currency, of a (cred rsk free zero coupon bond denomnaed n domesc currency, a me, maurng a me by P(,, and he correspondng domesc shor rae by r (. We denoe he prce, n foregn currency, of a (cred rsk free zero coupon bond denomnaed n foregn currency, a me, maurng a me by P (, f, and he correspondng foregn shor rae 8

9 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel by rf (. Le ( denoe he spo foregn exchange (F rae, a me, quoed as he number of uns of domesc currency per un of foregn currency.. A basc valuaon formula n he foregn exchange analogy he basc dervaves valuaon formula s (Harrson and Plska (98: he prce, H 0, of a dervave, a me 0, s: H = E exp 0 r s ds H 0, where H 0 s he random payou a me. here s a key observaon for modellng cross-currency dervaves whch s: E exp r( s ds ( = Pf (, (. whch we wll refer o laer n secon.. Remark: he above equaon s rue and model-ndependen. o see ha s rue, observe ha, Pf (, s he prce of a zero coupon bond n foregn currency, a me maurng a me, and ( s he prce, n domesc currency, of one un of foregn currency pad a me.herefore he RHS of he equaon represens he prce, a me, n domesc currency, of one un of foregn currency pad a me. In erms of he LHS, snce ( denoes he prce, n domesc currency, of one un of foregn currency pad a me, hen he condonal expecaon of, dscouned o me, represens he prce, a me, n domesc currency, of one un of foregn money pad a me. Hence, he equaon mus be rue.. odelng Inflaon he key o modelng nflaon and o prcng nflaon-lnked dervaves s o noce ha here s a oal and complee analogy beween nflaon-lnked dervaves and cross-currency dervaves. he analogy s ha nomnal neres raes are he equvalen of domesc neres raes, real neres raes are he equvalen of foregn neres raes and he spo CPI nflaon ndex s he equvalen of he spo foregn exchange rae. he F analogy gves an nuve way o hnk abou nflaon. See he fgure below 9

10 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel able he analogy. Noaon F rae ( CPI ndex ( Domesc neres rae r ( Nomnal neres rae r ( Foregn neres rae rf ( Real neres rae r ( Le us explan some noaon. We wll use a subscrp r o ndcae real neres raes and real zero coupon bond prces. Le ( r and r r denoe he (connuously compounded rsk-free nomnal and real shor raes, a me, respecvely. Le P(, and (, P denoe he prce of a (cred rsk free nomnal and real zero coupon bond, a me, maurng a me, respecvely. hroughou hs paper, we wll ofen use he words zero coupon bond and dscoun facor almos nerchangeably, wh he provso ha dscoun facors are known oday, me. 0 r Le ( denoe he spo CPI ndex, a me,.e. s he value, n uns of nomnal currency, of a ypcal baske of goods and servces.. An Imporan Observaon for nflaon dervaves he key observaon for prcng nflaon dervaves s ha, for any mes and, wh, we have: E exp r( s ds ( = Pr(, (. Remark: he above equaon s model ndependen. I s he analogous equaon o (. for modelng nflaon dervaves. o see ha s rue, noe ha Pr (, s he prce of a real zero coupon bond, a me, maurng a me, and ( s he prce, n nomnal currency, a me, of one un of real currency pad a me. herefore he RHS of he equaon represens he prce, a me, n nomnal currency, of one un of real currency pad a me. In erms of he LHS, snce ( denoes he prce, n nomnal currency, of one un of real currency pad a me, hen he condonal expecaon of dscouned o me, represens he prce, a me, n nomnal currency, of one un of real currency pad a me. Hence, nuvely, equaon (. holds. 0

11 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Remark: In Appendx, we gve a more mahemacal proof of equaon (.. We can use hs key observaon o help prce several dfferen ypes of nflaon-lnked dervaves, ncludng zero coupon nflaon swaps and perod-on-perod nflaon swaps, whch we wll explan n deal laer..3 Zero Coupon Inflaon Swaps Suppose ha oday, me 0, we ener no a year sandard zero coupon nflaon swap. As wh a sandard neres-rae swap, here s no up-fron cos o enerng no a zero coupon nflaon swap. So he value of he swap oday, me 0, mus be zero. he exchange of cash flows beween he wo pares only occurs a he maury of he swap. We wsh o value he swap, a me, where 0. By defnon, he payoff of he zero coupon nflaon swap a me s: ( N N + 0 (.3 where s he fxed rae on he swap and N s he noonal amoun. We can call ( 0 he frs erm N n he expresson (.3 he floang ( (nflaon-lnked sde and he second erm N ( + he fxed sde. In he absence of arbrage, he value of he swap, a me, s: 0 ( E exp r( s ds N N( ( + 0 ( = E Nexp r( s ds ( + 0 N = E exp r s ds NP, + ( 0 N = ( P r(, NP (, ( + where n he las lne, we have used he key observaon, equaon (..

12 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel So he value of he swap, a me, s: N ( P (, (, ( r NP + (.3 0 whch gves us a valuaon formula for he value of he swap a any me. In parcular, we know ha he value of he swap oday, me 0, mus be zero. So seng = 0 n equaon (.3 and equang he value o zero mples: N 0 = ( 0 P ( 0, ( 0, ( r NP +. 0 herefore, P (, P(, ( r 0 0 = + (.33 Remark: Zero coupon nflaon swaps are acvely raded n he marke and one can ge prces n he brokers. hey are quoed by he fxed rae for varous maures. Hence, we can use he las equaon o oban he real neres-rae yeld-curve.e. oban a se of real neres-rae dscoun facors (gven a se of nomnal neres-rae dscoun facors whch, of course, we can ge n he usual way from he sandard neres-rae swaps marke, whch we can hen use o prce more exoc srucures such as perod-on-perod swaps. Remark: Comparng he mehodology of Jarrow and Yldrm (003, n whch hey use a srppng mehod o ge nomnal and real zero coupon bond prces from he observed marke prces of coupon bearng bonds, s much easer and qucker o ge real dscoun facors from equaon (.33. Noe ha, n pracce, s usually a whole number of years. hs means we oban a se of real neres-rae dscoun facors o mes whch are a whole number of years from oday. When nerpolang beween hese mes, o esmae real dscoun facors o mes whch are fraconal numbers of years, one needs o be aware of he mpac of seasonaly. We wll no dscuss seasonaly furher here bu we refer he reader o Belgrade and Benhamou (004 and erkhof (005. Remark: Equaons (.3 and (.33 are model ndependen and are no based on specfc assumpons concernng he evoluon of neres rae yeld curves or he spo CPI ndex, bu, ndeed, smply follow from he absence of arbrage..4 Indexaon lag and delayed paymens he man purpose of nflaon-lnked dervaves s o proec he real (.e. afer allowng for nflaon value of fuure cash flows. In order o acheve a hgh degree of cerany n he real value of fuure cash flows, he nflaon-lnked cash flows should be as closely lnked as possble o conemporaneous nflaon. However, hs s no compleely possble owng o he exsence of ndexaon lag. hs s bes explaned as follows:

13 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel In pracce, here s a delay of a few weeks beween he dae on whch he CPI ndex s measured and he dae on whch he value of he CPI ndex s announced n he marke. hs me nerval s he me requred o collec and process he consumer prces requred by sascans o compue he CPI ndex. For example, n he Uned ngdom, he value of he CPI ndex (acually, one of he mos closely wached ndces s called he RPI bu we shall connue o call he CPI for brevy for a gven monh s publshed on abou he 5h of he followng monh. So for example, he CPI ndex for ay 007 was publshed on abou he 5h June 007. Furhermore, he marke for serlng denomnaed zero coupon nflaon swaps adops he convenon ha hroughou a calendar monh, he base ndex (he value of he ndex appearng n he denomnaor of he payoff s he ndex for wo monhs before. So, for example, hroughou July 007, all 5 year zero coupon nflaon swaps would have a fuure nflaon-lnked payoff (n July 03 equal o: he value of he CPI ndex for ay 03 (whch wll be announced n June 03 dvded by he value of he CPI ndex for ay 007 (whose value was known on approxmaely 5h June 007 mnus one. hs means ha an nvesor who receves he nflaon-lnked paymen on a 5 year zero coupon swap s no compensaed for nflaon over he perod ay o July 03 alhough he nvesor wll receve compensaon for nflaon over he perod ay o July 007 (before he swap commenced. When we wre ( as he value of he spo CPI ndex, wha we really mean s ha ( ε s he acual publshed value of he CPI ndex a a me ε earler. he value of ε can acually vary slghly (beween abou one monh and wo monhs. Snce he value of ε only changes slghly compared o he ypcally maury of nflaon swaps (whch s ofen greaer han 0 years, s he marke convenon o assume ha s effecvely consan. hs s he convenon we wll adop. here s very lle o be los by dong so snce he same convenon apples a he maury of he nflaon swap as appled a he sar of he nflaon swap and so 3

14 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel any msspecfcaon, a leas parally, cancels ou. One convenen benef of adopng hs convenon s ha we can connue o use equaon (.33 o oban real neres-rae dscoun facors and o do so n a model-ndependen fashon. here s one furher ssue wh nflaon swaps whch s he ssue of delayed paymens. hs s somemes called paymen lag alhough o avod confuson wh he concep of ndexaon lag, we wll refer o as delayed paymens. For sandard zero coupon nflaon swaps, he paymen me of he payoff concdes wh he argumen of he value of n he numeraor of he nflaon-lnked erm n he payoff. So, he paymen of he cash flow n equaon ( (.3, namely, N ( ( N +, occurs a me. Alhough, 0 hs s ndeed he mos common suaon, ofen, n pracce, he paymen s delayed unl some laer me N. hs delay s no jus he sandard day spo selemen lag bu can be a perod of a few weeks, a few monhs or even several years. We wll refer o such nflaon swaps as nflaon swaps wh delayed paymens. o see how such nflaon swaps have an mporan economc raonale, consder a commercal propery company. Suppose has deb n he form of fxed-rae loans. I receves rens from s enans whch wans o pay ou as he nflaon-lnked leg of an nflaon swap. I wll receve fxed paymens on he nflaon swap whch wll use o pay s fxed-rae deb. Ofen rens wll reman consan for a perod of 5 years before beng revewed. hey wll hen be revsed upwards o reflec nflaon over hose nervenng fve years. So for example, suppose, he commercal propery company waned o ener no an nflaon swap rade, n whch pad nflaon-lnked cash flows and receved fxed cash flows. he company wans o hedge he cash flows ha wll receve from s enans n years 6, 7, 8, 9 and 0. So a suable nflaon swap rade would be a srp of fve zero coupon nflaon swaps as follows. he payoff of he fve zero coupon swaps would be (we wre only he nflaon-lnked leg wh un noonal: 4

15 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel ( 5 0 A he end of year 6, he company pays. A he end of year 7, ( 5 0 agan pays ( 5 ( 0. A he end of year 8, agan pays ( 5 0 he end of year 9, agan pays ( ( 5 ( 0.. A. A he end of year 0, agan pays We can see ha hese are zero coupon nflaon swaps wh delayed paymen wh he delay on he fnal swap of he srp beng 5 years. Perod-on-perod swaps wh delayed paymens also rade n he markes. If nomnal neres-raes were deermnsc, hen valung hese nflaon swaps wh delayed paymens would be rval gven a prcng mehodology for valung he correspondng ype of nflaon swap wh no delayed paymens. However, snce we wll have sochasc neres-raes, valuaon s more dffcul and wll nvolve he evaluaon of addonal erms whch we wll loosely refer o as convexy adjusmens. 5

16 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel 3 Framework of he model In hs secon, we se up he dynamcs of nomnal zero coupon bond prces, real zero coupon bond prces and he spo CPI ndex. We work whn a mul-facor verson of he Jarrow and Yldrm (003 model. I s clear ha he Jarrow and Yldrm (003 model s a model whch s, frsly, arbrage-free and, secondly, conssen wh any nal erm srucure of nomnal and real neres raes, snce s a HJ (Heah e al (99 model. 3. Sochasc evoluon of nomnal bond prces We assume ha, under he equvalen marngale measure defned wh respec o he nomnal money marke accoun numerare, nomnal zero coupon bond prces are sochasc and follow a Gaussan HJ model (Heah e al. 99: (, (, dp P n ( σ (, ( = r d + dz. (3. k = kn kn where s he number of Brownan moons, n dzkn, for each k, k,..., n =, denoes sandard Brownan ncremens. Furhermore, he correlaon beween dz ( and dzkn ( s ρ jnkn, for each k and each j, j,..., n =, and σ (, kn jn, for each k, are volaly erms whch are purely deermnsc funcons of and, sasfyng ( σ, 0. kn 3. Sochasc evoluon of real bond prces We now descrbe he rsk-neural dynamcs of real zero coupon bond prces. We assume ha, under he equvalen marngale measure defned wh respec o he nomnal money marke accoun numerare, real zero coupon bond prces follow a Gaussan HJ model (Heah e al.(99: r (, (, dp r P = r kr kr d+ kr dzkr k= k= (3. r r ( ρ σ ( σ (, σ (, ( 6

17 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel where s he number of Brownan moons, r dzkr, for each k, k,..., r =, denoes sandard Brownan ncremens and where, for each k, k =,..., r, ρ kr s he correlaon beween he spo CPI and he respecve Brownan moon drvng real zero coupon bond prces. We denoe he correlaon beween dz jr ( and dzkr ( by ρ jrkr, for each k and each j, j =,..., r. Remark: Noe he quano drf adjusmen n equaon ( Sochasc evoluon of he spo CPI ndex he dynamcs of he spo CPI, under he equvalen marngale measure defned wh respec o he nomnal money marke accoun numerare, are gven by: d ( r( r ( d σ ( dz ( = + (3.3 where dz ( denoes sandard Brownan ncremens, he drf s he dfference beween he nomnal and real shor raes, and ( σ s he volaly whch we assume o be a purely deermnsc funcon of. Furhermore, we nroduce he noaon ha he correlaon beween dz and dz kn, for each k, k,..., n =, s ρ kn and he correlaon beween dz kn and dz jr, for each k, k,..., n =, and for each j, j =,..., r s ρ knjr. Remark: I s convenen o assume all he correlaons are consan (whch we do n he mplemenaon alhough all he equaons n hs paper would hold f hey are, a mos, deermnsc funcons of. We assume he correlaons form a symmerc posve-defne marx wh elemens uny down he leadng dagonal. If we defne he forward CPI a me o (.e. for he delvery a me by F (,, hen by no-arbrage argumens (see Appendx, we know ha, Furher, by Io s lemma, F (, r (, (, P =. (3.3 P 7

18 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel df F (, (, n n n = ρ σ σ ρ σ σ (, (, (, knjn jn kn kn kn k= j= k= r n k= j= ρ jnkrσ jn (, σkr (, d r n ( dz ( ( dz ( ( dz ( + σ + σ, σ, (3.33 kr kr kn kn k= k= hen he forward CPI ndex, F (, value F, 0, a me 0 (, = (, F F 0, as follows:, a me, can be expressed n erms of s exp σ + σ, σ, 0 r n ( s dz ( s ( s dz ( s ( s dz ( s kr kr kn kn k= k= n n r r exp ρjnknσ jn, σkn, ρkrjrσ jr, σkr, k j = = k= j= 0 ( s ( s ( s ( s r σ ( s ρkrσ ( s σkr( s, ds (3.34 k = Remark: Noce ha he drf and volaly erms n he sochasc dfferenal equaon for F (, are deermnsc and ha F (, s log-normally dsrbued. 8

19 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel 4 Exoc nflaon dervaves In chaper wo, we have shown ha, gven he raes on sandard (.e. wh no delayed paymen zero coupon nflaon swaps quoed n he marke (and gven nomnal dscoun facors, we can ge real dscoun facors. We were able o oban real dscoun facors by valung zero coupon nflaon swaps n a model-ndependen fashon. hs s analogous o obanng nomnal dscoun facors from LIBOR depos raes and by boosrappng swap raes, whch can also be done n a model-ndependen fashon. Jus as nomnal dscoun facors are he buldng blocks upon whch we could value more exoc neres-rae dervaves, so real dscoun facors are he buldng blocks upon whch we can value more exoc nflaon dervaves. hs s he am of hs secon. We wll see ha he prces of hese more exoc nflaon dervaves are model-dependen and herefore we wll am o value hem n he Jarrow and Yldrm (003 model we nroduced n he las secon. In hs secon, we wll value hree ypes of nflaon swap, namely, perod-on-perod nflaon swaps wh no delayed paymens, zero coupon nflaon swaps wh delayed paymen and perod-on-perod nflaon swaps wh delayed paymens. he key pon abou he las wo ypes of nflaon swap s ha hey have he same payoff as he correspondng nflaon swap wh no delayed paymens bu he payoff s pad a a laer me. When he delay n paymen s very small (for example, a few weeks, we would, nuvely, expec he dfference beween he values of he correspondng swaps wh no delayed paymens and wh delayed paymens o be small. Conversely, we shall see ha he dfference n values can be subsanal when he delay n paymens s, for example, a few years. As we noed n secon.4, nflaon swaps wh delayed paymens of fve years or more are que commonly raded n he markes. We now urn our aenon o prcng perod-on-perod nflaon swaps wh no delayed paymens. 9

20 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel 4. Perod-On-Perod Inflaon Swaps Suppose ha oday, me 0, we ener no a perod-on-perod nflaon swap. he swap s defned va a se of fxed daes 0 < < <... < <, where 0 0. hese daes are usually approxmaely equally spaced (for example, approxmaely one year apar bu hey need no be. As wh a sandard neres-rae swap, a perod-on-perod nflaon swap s made up of a seres of swaples. here are paymens of Nτ,nf ( ( agans a fxed rae Nτ, fxed a each me. herefore, he payoff of he h swaple, for =,,...,, a me s: ( Nτ,nf Nτ, fxed (4. where s he fxed rae on he swap, N s he noonal amoun, τ,nf s he day-coun adjused me from o for he floang (nflaon-lnked leg and τ, fxed s he day-coun adjused me from o for he fxed leg. In he absence of arbrage, he value of he swaple, a me, s : ( ( E exp r( s ds Nτ,nf Nτ, fxed = Nτ,nf E exp r( s ds P(, N( τ,nf + τ, fxed (4. o value he floang (nflaon-lnked sde, we have o consder separaely wo dfferen cases dependng upon wheher or <. Frs case:. In hs case, ( s known a me. herefore we can ake 0

21 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel ousde of he expecaon and hen use he key observaon, namely equaon (., and wre E exp becomes: (, ( P = r r s ds. Hence, equaon (4. (, ( P Nτ P N τ + τ (4.3 (, r,nf,nf, fxed Second case: 0 <. Usng he law of eraed expecaons n equaon (4., we can wre he value of he swaple, a me, as ( N τ,nf E exp r( s ds E exp r( s ds P, N,nf, fxed ( τ + τ Bu now he key observaon of equaon (., ells us ha ( ( Pr(, ( ( (, r E exp r( s ds = E exp r s ds = = P herefore, he value of he swaple, a me, s : Nτ,nf E exp r( s ds Pr (, P(, N( τ,nf + τ, fxed (4.4 We can show (usng he mehodology of Appendx 4 ha: (, (, Pr E exp r( s ds Pr(, = P(, exp A( s,, ds (4.5 P r { } n r = ρ σ σ σ where A s,, ds s, s, s, ds knjr kn jr jr k= j= r r k= j= r { } (, (, (, + ρ σ s σ s σ s ds krjr kr jr jr { } + ρ σ s σ s, σ s, ds (4.6 k= kr kr kr

22 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Remark: We call he expresson (4.6 he convexy adjusmen for a perod-on-perod nflaon swap. Noce when σ kr ( s, 0, for all k, he convexy adjusmen wll be dencally equal o zero. Bu, n hs specal case, real neres raes would be deermnsc. From equaon (4.4 and (4.5, when 0 <, he equaon (4. becomes: (, (, Pr,nf (,nf +, fxed Pr ( Nτ P, exp A s,, ds P, N τ τ (4.7 We can value a perod-on-perod nflaon swap by summng up he value of all he swaples, beng careful o use equaon (4.3 when, and equaon (4.7 when 0 <. We know ha he value of he swap oday, me 0, mus be zero. So we can se = 0 n he las formula and equae he value of he swap o zero, o relae he fxed rae o he erm srucure of real neres-rae dscoun facors and o he parameers of he sochasc processes for he neres-rae yeld curves and he spo CPI ndex. Perod-on-perod nflaon swaps are no as acvely raded n he marke as zero coupon nflaon swaps alhough s somemes possble o ge some prces. hey are quoed by he fxed rae for varous maures. As explaned earler, we can use zero coupon nflaon swaps o ge real neres-rae dscoun facors. In prncple, we could hen use perod-on-perod nflaon swaps (assumng we have enough of hem o calbrae he parameers (volales, mean reverson raes, CPI volaly, correlaons of he sochasc processes for he real neres-rae yeld curve and he spo CPI ndex. 4. Zero coupon nflaon swaps wh delayed paymen In secon.3, we valued sandard zero coupon nflaon swaps when he paymen of he payoff of he swap occurred a he same me as he argumen of he spo CPI

23 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel ndex appearng n he numeraor of he payoff. As we noed and explaned n secon.4, s now relavely common o rade zero coupon nflaon swaps where he paymen s delayed for some me, perhaps several years or more. We refer o hese nflaon swaps as zero coupon nflaon swaps wh delayed paymen. Our am, n hs secon, s o derve a valuaon formula for hem. Unlke wh a sandard (.e. wh no delayed paymen zero coupon nflaon swap, he valuaon of zero coupon nflaon swaps wh delayed paymen nvolves a convexy adjusmen whch s model-dependen. Frsly, we derve a formula whch, n a sense, exends he key observaon of equaon (. o he suaon of delayed paymen, alhough we should sress ha s less general han equaon (., n he sense ha s no longer model-ndependen. Proposon : For any mes and N, wh 0 N, he followng equaon holds: (, N (, N P E exp r( s ds ( = Pr(, exp C( s,, N ds (4. P n r { } where Cs (,, ds= ρ σ s, σ ( s, σ ( s, ds Proof: See Appendx 3. N knjr kn N kn jr k= j= n n + ρknjn { σkn σkn ( N } σ jn k= j= n + ρknσ ( s { σkn ( s N σkn s } ds k = (, s s, (, s ds, (, (4. Remark: When = N, s sraghforward o verfy ha Cs (,, N 0, n whch case, equaon (4. agrees wh equaon (.. Remark: hs formula wll be used below o prce zero coupon nflaon swaps wh delayed paymen. Suppose ha oday, me 0, we ener no a zero coupon nflaon swap wh delayed paymen. We denoe he paymen me of he payoff of he swap by N and 3

24 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel we denoe he maury of he swap by. We wsh o value he swap, a me, where 0 N. he payoff of he zero coupon nflaon swap wh delayed paymen s sll: ( N N + 0 where s he fxed rae on he swap and N s he noonal amoun. Bu he payoff s pad a me N whch s some me greaer han or equal o. he value, a me, of he zero coupon nflaon swap wh delayed paymen s : ( N E exp r( s ds N N( ( + 0 ( N = E Nexp r( s ds ( + 0 N N = E exp r s ds NP, N + ( 0 (, N (, N P = ( P r(, exp C ( s,, N ds NP, N + ( 0 P Remark: Noce ha n he las lne we have used proposon. So he value of he swap, a me, s: (, N (, N P ( P r(, exp C ( s,, N ds NP (, N( + ( 0 P (4.3 whch gves us a valuaon formula for he value of a zero coupon nflaon swap wh delayed paymen a any me. Remark: Comparng equaon (4.3 wh he equaon for he value of a sandard zero coupon nflaon swap wh no delayed paymen (equaon (.3, we can see ha here s an exra erm (, N (, P P exp Cs (,, N ds n he nflaon-lnked leg. 4

25 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel 4.3 Perod-on-perod nflaon swaps wh delayed paymens Our am n hs secon s o value, a any me, a perod-on-perod nflaon swap wh delayed paymens. he followng proposon wll be he key o hs because shows ha when here are delayed paymens, he valuaon of perod-on-perod nflaon swaps nvolves addonal convexy adjusmen erms. Equaon (4.3 of proposon exends equaons (4.5 and (4.6 whch we used n he valuaon of perod-on-perod nflaons swaps wh no delayed paymens. Proposon : When 0 < < N ( N E exp r( s ds (4.3 (, PN P (, r = P(, exp Cs (,, N ds+ { As (,, P (, Pr(, + B( s,,, N } ds where Cs (,, N s gven by (4., As (,, s gven by (4.6 and n n N = { } ρknjn σkn σkn σ jn N k= j= n n + ρknjn { σkn(, σkn(, } σ jn(, k= j= n r + ρknjr { σ jr σ jr } σkn N k= j= Bs (,,, ds (, s (, s (, s ds + Proof: See Appendx 4. n r k= j= ρ knjr s s s ds (, s (, s (, s ds { } σ ( s, σ ( s, σ ( s, ds (4.3 jr jr kn Remark: Noce ha when =, s sraghforward o confrm B(, s,, and N N Cs (,, N n equaon (4.3 becomes zero, whch confrms conssency wh equaon (4.5. 5

26 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Suppose ha oday, me 0, we ener no a perod-on-perod nflaon swap wh delayed paymens. he swap s defned va a se of fxed daes 0 < < <... < <, where 0 0. he perod-on-perod nflaon swap s made up of a seres of swaples. he key ssue s ha he value of he payoff of each swaple s he same as he payoff of he correspondng swaple of a perod-on-perod nflaon swap wh no delayed paymens bu now he payoff s pad a me N whch s some me greaer han or equal o. From equaon (4., he payoff of he h swaple, for =,,...,, a me N s: ( Nτ,nf Nτ, fxed where he noaon s he same as n equaon (4.. he value, a me, of he swaple wh delayed paymen,.e. ( N E exp r( s ds Nτ,nf Nτ, fxed N, s: ( N = Nτ,nf E exp r( s ds P(, N N( τ,nf, + τ fxed (4.33 o value he floang (nflaon-lnked sde, we have o consder separaely wo dfferen cases dependng upon wheher or <. Frs case:. In hs case, ( s known a me. herefore we can ake ousde of he expecaon and wre 6

27 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel ( ( N N E exp r( s ds = E exp r( s ds where we have used proposon n he las lne. Hence, equaon (4.33 becomes: r (, N (, N P = ( Pr(, exp Cs (,, N ds ( P(, P (, P Nτ,nf exp C( s,, N ds (, (,nf, (4.34 P N N τ + τ fxed P(, Second case: 0 <. Usng he law of eraed expecaons n equaon (4.33 and by proposon, we can wre he value of he swaple, a me, as : (, (, N Pr P Nτ,nf P(, exp C( s,, N ds+ { A( s,, (,,, } + B s N ds Pr, P(, (, N ( τ,nf τ, fxed P N + (4.35 herefore, we can value a perod-on-perod nflaon swap wh delayed paymens by summng up he value of all he swaples, beng careful o use equaon (4.34 when, and equaon (4.35 when 0 <. 7

28 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel 5 An applcaon of our convexy adjusmen formulae o prcng LPI swaps Lmed prce ndexaon (henceforh LPI swaps are a ype of exoc nflaon dervave and are very common n he Uned ngdom owng o he rules by whch U penson funds are governed. he rules ofen requre ha he fuure benefs of people payng no many penson funds have o rse by he year-on-year nflaon rae whenever he year-on-year nflaon rae, expressed as a percenage, s beween some gven levels f % and c %, where c f. If he year-on-year nflaon rae s less han f %, he fuure benefs have o ncrease by a leas f % and f he year-on-year nflaon rae s greaer han c %, he fuure benefs are ncreased by only c %. In pracce, f s ofen 0 % and c s ofen eher 3 % or 5 % bu varaons do occur. hese rules effecvely defne he fuure lables of U penson funds. Unsurprsngly, here has been subsanal demand for nflaon dervaves, from U penson funds, whch wll gve payoffs whch can hedge agans hose lables. hs has provded he economc raonale for LPI swaps. We wll see ha we can use he convexy adjusmen formulae, ha we derved n chaper 4, o help prce LPI swaps. 5. LPI swaps Suppose ha oday, me 0, we ener no an LPI swap. he LPI swap s defned va a se of fxed daes 0 < < < < <, where 0 0. In pracce, hese daes are usually approxmaely one year apar bu hey need no be. he paymen of he payoff of he swap occurs a me, where =. he payoff of he nflaon-lnked leg of he swap a me s: + mn max, F, C or equvalenly = ( mn max, + F,+ C = 8

29 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel where C and F are consans, wh C F. In pracce, F s ofen zero bu we wll assume n he followng ha C and F can ake on any values (posve, negave or zero provded C F. he perod-on-perod rae of nflaon beween and s gven by ( (. So he role of he consans C and F s o cap and floor he perod-on-perod nflaon rae over each perod. Remark: When =, LPI swaps could be prced by a varan of he Black (976 formula. When C = and F =, he produc elescopes and he LPI swap has he same payoff as a zero coupon nflaon swap. However, when C and F are fne and >, we would need o prce a swap whose payoff s pah-dependen. Because of he pah-dependency, hey are no, n general, rval o prce. When = or = 3 hey could be prced by numercal negraon echnques (.e. quadraure for he case = and cubaure for he case = 3. However, n pracce, LPI swaps ypcally have maures anywhere beween 5 years and 40 years mplyng ha s beween 5 and 40. When 4, he only feasble mehodology o precsely prce LPI swaps s one Carlo smulaon bu hs s CPU nensve. Hence, would be desrable o have a fas, even f approxmae, quas-analyc mehodology o prce hem. Such a mehodology s proposed n Ryen ( Prcng LPI swaps In hs secon, we wll use he mehodology of Ryen (007 o ge an approxmae prcng formula for he nflaon-lnked leg of LPI swaps. However, frsly, we nroduce some noaon: We denoe by Q he probably measure defned wh respec o he numerare whch s he zero coupon bond maurng a me denoe by [ ] E expecaons, a me, wh respec o Q.. We he mehodology of Ryen (007 s fully explaned n Ryen (007 so we wll jus oulne he approach here. I uses he dea of common facor represenaon. Suppose ha we have a year LPI swap wh perods. Le denoe (, for =,,,. We wll show n Appendx 5 ha: ( ln ( ln, for each, =,,,, s dsrbued as mul-varae ( normal n our model. 9

30 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel We can calculae he covarance marx cov( ln,ln j for each, j, =,,,, j =,,,. (5. ln In general, none of he elemens of hs covarance marx wll be zero because wll no be ndependen of ln j for any and j. hs lack of ndependence severely complcaes he problem of prcng an LPI swap. he key dea of Ryen (007 (see also Jackel (004 s o replace he covarance marx (5. for each, j by anoher marx, whch s close o he acual correlaon marx n some sense, bu n whch he off-dagonal elemens have a smple srucure. We use he same noaon as n Ryen (007. We wre n he form : = exp( az + b where z N(0, ; cov(ln,ln = cov( z, z a a ; E[ ] = exp( b + a. j j j he key dea of Ryen (007 s o replace by : = exp( b + a( aw+ dε, wh he followng addonal properes: he sysem w, ε, ε s a sysem of, ndependen N (0, varaes, and for each, a + d =. Ryen (007 shows how o calculae a and d, for each. he varaes,, are a represenaon of he varaes,, va one common facor w and addonal ndvdual dosyncrac random varables ε, =,,,. By changng measure o Q and usng Grsanov s heorem, he prce, a me 0, of he nflaon-lnked leg of he LPI swap s: E exp r( s ds mn max, + F,+ C 0 ( 0 = ( = P ( 0, E mn max, F, C = ( 30

31 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel P ( 0, E 0 mn max, + F,+ C = = P ( 0, E E mn max,, 0 F C w = = P ( 0, E E mn max,, 0 + F + C w 0 = (5. Remark: By assumpon, he ε are ndependen, and consequenly, condonal on a specfc value of w, he varaes are ndependen,.e. cov( w, j w 0 =, when j. herefore, we see ha he condonal expecaon of he produc n he las bu one lne of equaon (5. becomes a produc of condonal expecaons n he las lne. We have used (approxmaely equals n he hrd lne of equaon (5. because he varaes are, n general, only an approxmae represenaon of he varaes, =,,,. When, he represenaon s exac. When 3, he represenaon s only approxmae. I s rue ha E [ ] = E [ ], var [ln ] = var [ln ] for all, for any value of bu when 3, hen cov(, j s only an approxmaon o cov(,, when j. j We can use he mehodology of Ryen (007 o evaluae equaon (5. provded ha we can compue he expecaon and varance of n he probably measure Q. We do hs n Appendx 5. Snce s lognormal (see Appendx 5, hen denong by μln and σ ln he mean and varance of ln, hen E [ ] exp( 0 = μln + σln for =,,,. Hence, we can ge he expecaon of ln,.e. μln = ln ( E [ ] 0 σ. ln Now we can use he followng resul: If N( μ, σ, W N(0, and ρ W s he correlaon beween and W, hen W = w s normally dsrbued and, furhermore, 3

32 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel E W= w = μ + ρ σ w W, Var W = w = σ ( ρw In Appendx 5, we show ha he correlaon beween ln and w s a, for each, =,,,. Now, we recall ha E [ln ] = E [ln ] = μ, var [ln ] = var [ln ] = σ, 0 0 ln ln 0 0 hen, usng he resul above, we ge [ln ] μ 0 ln σ ln E w = + a w, = var w = σln a σ : [ln ] 0 F : = E [ w] = exp 0 μln + aσ ln w+ σ,,,, =. Fnally (see Ryen (007, equaon (5. becomes: ( + σ + + σ P( 0, E F Call( F, C, Pu( F, F, 0 (5.3 = where Call( F, + C, σ and Pu( F, + F, σ are he undscouned prces of a call opon, wh srke + C, and a pu opon, wh srke + F, n he Black (976 formula, when he forward prce s F and he negraed varance s σ. Remark: Noe ha each erm n he produc n equaon (5.3 depends on he common facor w, hrough F and σ, and w has a sandard normal N (0, dsrbuon. = We can hen evaluae equaon (5.3 by numercally negrang he produc of ( F Call( F, + C, σ + Pu( F, + F, σ and he sandard normally densy funcon. hs gves us he prce of he LPI swap a me 0 (noe ha when 3, s only an approxmaon. 3

33 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel 6 Calbraon o marke daa We wll now calbrae our model o marke daa n hs secon. 6. Ge hsorcal daa 6. Ge zero coupon nflaon swaps and CPI We obaned he fxed raes beng quoed n he marke for serlng (GBP denomnaed zero coupon nflaon swap raes, for every workng day beween 9h July 003 and 4h June 007, wh maures, equal o 5 years, 0 years, 5 years, 0 years, 5 years and 30 years. Usng equaon (.33, hs allowed us o ge daa for real zero coupon prces, for every workng day, for hese sx maures. We also obaned hsorcal daa on he CPI ndex (recall ha, hroughou hs paper, we call he ndex he CPI for brevy bu, n acual fac, we used he U RPI ndex. Because he CPI daa s only avalable monhly, we decded o use only monhly daa (even hough we had daly daa for nomnal and real zero coupon bond prces. We decded o use daa for he 8h (or he closes workng day of each monh. Remark: Snce CPI s announced n he mddle of a monh (.e. 0h-0h, he daa s somewha nosy so we decded no o use daa from hs perod. he choce s farly arbrary bu we decded o use daa from he 8h of each monh (or he closes workng day. 6. Ge nomnal and real dscoun facors We obaned nomnal dscoun facors, n serlng (GBP, for every workng day beween s July 003 and nd June 007, wh maures, agan, equal o 5 years, 0 years, 5 years, 0 years, 5 years and 30 years. hese were obaned, n he sandard fashon, from GBP LIBOR depos raes and by boosrappng GBP swap raes. Jarrow and Yldrm (003 consder he valuaon of nflaon-lnked nsrumens n he conex of he marke for reasury Inflaon Proeced Secures (henceforh IPS. IPS are US reasury bonds whose coupon and prncpal paymens are lnked o US CPI. Snce hese are coupon bearng bonds, Jarrow and Yldrm (003 had o 33

34 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel use a srppng mehodology o exrac he prces of real zero coupon bonds from he prces of coupon bearng bonds. hey hen used hsorcal daa of hese real zero coupon bond prces o esmae he parameers of her model. By conras, we are workng whn he conex of nflaon swaps. We know from equaon (.33 ha here s a smple relaonshp beween he fxed rae quoed on sandard zero nflaon swaps and real zero coupon bond prces. Hence, we do no need o employ he srppng algorhm of Jarrow and Yldrm (003 n order o ge real zero coupon bond prces. We smply used equaon (.33. Before we descrbe how we esmaed he model parameers, we need o be more precse abou he specfc form of he model ha we used. In chapers 3, 4, and 5, we worked wh a very general mul-facor verson of he Jarrow and Yldrm (003 model. However, we need o bear n mnd ha here s no oo much hsorcal daa for nflaon swaps and wha daa here s, may be somewha nosy. Hence, n order o make for a smpler esmaon of parameers, hroughou hs chaper, we assumed ha real zero coupon bond prces are drven by jus a sngle Brownan moon.e. we assumed r = n equaon (3. o smplfy parameer esmaon. In addon, we assume ha he volaly σ ( of he spo CPI ndex s consan,.e. we assume ha σ σ. We consdered wo possble specfcaons for nomnal zero coupon bond prces, namely ha here s eher one Brownan moon drvng nomnal zero coupon bond prces or ha here are wo.e. eher n = or n =. 6. Ge model parameers usng Jarrow and Yldrm (003 mehod In order o prce he exoc nflaon dervaves we dscussed n chapers 4 and 5, we need model parameers whch are dependen on he specfc model. herefore, frsly, we need o specfy he volaly funcons σ (, and (, kn σ. Poenally, here are dfferen forms of he volaly funcons, (, jr σ, kn 34

35 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel (, σ jr for each k, j, k =,..., n, j =,..., r, bu we wll consder only he exended Vascek form, where for each k, j we assume σ σ kn jr ( ( kn σ, exp (6. αnk σ, ( exp ( jr (6. α nk α ( rj α ( rj where, for each k, each j, σ nk, α nk, σ rj and α rj are posve consans. 6. One facor model In hs sub-secon, we esmae model parameers for he case where we have one Brownan moon drvng nomnal zero coupon bond prces.e. when n =. In addon as already saed, we assume r =. Usng he mehod of Jarrow and Yldrm (003, he varance of zero coupon bond prces over he me nerval [ +, ] sasfes he followng equaons: αr( Pr(, σ r( e var = Pr(, αr (6.3 αn( Pn(, σ n( e var = Pn(, αn (6.4 Usng Excel Solver, we ran a cross seconal non lnear regresson based on he equaons (6.3 and (6.4 acoss he sx dfferen maures o esmae he parameers ( σn, α n and ( σ r, α r. o be precse, n our calbraon, we solved for he parameers whch mnmzed he sums of squares of dfferences beween he hsorcal volales of zero coupon bond prces of he sx dfferen maures and he model volales n equaons (6. and (6.. he hsorcal volales were esmaed usng monhly daa.e. = /. he esmaes of hese parameers are σr = , αr = , σn = , 35

36 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel α n = as gven n able 3. hese parameers provde he volaly npus needed for he convexy adjusmens (see expresson (4.6, (4., (4.3 In hs sub-secon, we only use a one facor model when esmang model parameers, whch means we only need ρ kr, ρ jn, ρ krjn when k = and j =. From he mehod of Jarrow and Yldrm (003, hese parameers have he followng expresson: var ρ r cor Pr (,, r σ = = ( P (, ( ρ P (, ( P (, P (, = cor, ρ = cor, P (, ( P (, P (, n n r n nr n n r (6.5 Remark: We use he hsorcal daa of CPI, nomnal zero coupon bond prces and real zero coupon bond prces, as of he 8 h of each monh, o esmae hese parameers. he parameers obaned are n ables and wo facor model For he case where we have wo Brownan moons drvng nomnal zero coupon prces.e. when n =, we obaned he parameers σ n, α n, σ n, α n, ρ nn by calbrang a wo-facor Gaussan HJ model (Heah e al (99, Babbs (990, Hull and Whe (993 model o he marke prces of lqud European swapons. he resuls of he calbraon were ha we obaned model parameers as follows: σ = , α = , σ = , α = n n n n ogeher wh he correlaon beween hese wo facors ρ nn = as gven n ables 4 and 5. hese parameers were provded by John Crosby and Lloyds SB. We assumed ha σ, α, ρ were as above (see also able 3. We also need o n n n r r r r esmae ρ, ρ, ρ and ρ. We used he approach descrbed on page 45 of nr Brgo and ercuro (00. In essence, we se ρ ρ and we se ρ ρ. n n nr nr We assume ha ρ n and ρ n are equal o he correspondng values we obaned 36

37 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel for he one facor case above (see able. ha s, we assume ρ ρ = We would lke o do lkewse wh ρ and ρ. n n However, f we se ρ ρ = , we fnd ha he correlaon marx s no nr nr posve defne. herefore, we decded o se ρ ρ = 0.58 because 0.58 s he closes value whch makes he correlaon marx posve defne. Whls we concede ha hs s unlkely o yeld anyhng lke perfec esmaes of hese parameers, here s (as Brgo and ercuro (00 explan a leas a measure of mahemacal conssency abou and, n addon, gven he relavely scarce amoun of daa for nflaon, s a pragmac smplfcaon. 6.3 Gve he values of convexy adjusmens of exoc dervaves We have now obaned esmaes of he model parameers needed for he convexy adjusmens (see equaons (4.6, (4., (4.3. We wll, laer n hs secon, use hese parameers o es he analycal formulae we derved n secons 4., 4. and 4.3, and hen gve some numercal examples and comparsons of he convexy adjusmens, for he hree ypes of nflaon swaps we consdered n chaper 4, for dfferen swap enors and paymen mes. John Crosby (my ndusry supervsor also provded daa whch gves he values of he convexy adjusmen (ogeher wh sandard errors of hese esmaes usng a one Carlo mehodology whch was used o es and benchmark he analycal formulae (equaons (4.6, (4., (4.3. he one Carlo smulaon smulaed he CPI ndex level and he nomnal and real yeld curves by smulang underlyng Gaussan sae varables and herefore had no dscresaon error bas. For he sake of brevy, we om he full deals snce hey can be found n, for example, Crosby (005, Crosby (007a, Dempser and Huon (997 and Glasserman (004. he one Carlo values we repor were compued usng 30 mllon runs (65 mllon runs plus 65 mllon anhec runs whch ook several hours of CPU me. nr nr nr nr 37

38 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Here, hanks agan John Crosby for hs kndly help. In examples o 4, we use he model parameers for he case when here s one Brownan moon drvng nomnal neres-raes.e. when n =. We plo he convexy adjusmens, graphcally, n he dfferen examples below. Example : Comparson of he one Carlo resuls and he analycal formulae In hs example (fgures,, and 3, we consder he convexy adjusmens for zero coupon nflaon swaps wh delayed paymen, perod-on-perod swaples wh no delayed paymen and perod-on-perod swaples wh a 5 year paymen delay. Fgure Fgure 38

39 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Convexy Adjusmen for Perod-on-Perod Swaple wh 5 Years delayed paymen convexy adjusmen one Carlo Analycal (---( h sw aple me (n years Fgure 3 From fgure, and 3, we can see ha he dfference beween he resuls from he one Carlo and he analycal resuls of chaper 4 are very small - n fac, hey are almos zero and, whls we have no dsplayed he sandard errors n he graphs, we can confrm ha he analycal resuls are conssen wh he sandard errors of he one Carlo smulaon. We can conclude ha he formulae we derved n chaper 4 are correc and ha hey have been correcly mplemened. Example : Convexy adjusmens for zero coupon nflaon swaps In hs example (fgure 4, we compare he convexy adjusmens for zero coupon nflaon swaps, wh maures equal o 5, 0, 5, 0 and 5 years, when here s no delayed paymen, when here s a one year paymen delay and when here s a fve year paymen delay. From fgure 4, we can see ha, frsly, when here s no delayed paymen me, he convexy adjusmen always equals one, whch s wha we expec. However, when he paymen delay s ncreased, from zero o one year o 5 years delay, he convexy adjusmens ge furher away from one. In addon, as he maury ncreases from 5 years o 5 years, he convexy adjusmens also ge furher away from one. hs llusraes ha he convexy adjusmens become more sgnfcan for longer maures and longer paymen delays. 39

40 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel Convexy Adjusmen for Zero Coupon nflaon swap wh no delay, Year delay and 5 Years delayed paymen convexy adjusmen no delay year delay 5 years delay maury of he swap (n years Fgure 4 Example 3: Convexy adjusmens for perod-on-perod swaples In hs example (fgure 5, we perform a smlar analyss o example, bu hs me for perod-on-perod swaples. Comparson Convexy Adjusmen for Perod-on-Perod Swaple wh no delay, Year delay and 5Years delayed paymen.0005 convexy adjusmen no delay year delay 5 years delay (---( h swaple me (n years Fgure 5 In fgure 5, we agan see ha longer maures and longer paymen delays produce convexy adjusmens whch are furher away from one. Example 4: he effec of he convexy adjusmen on he fxed rae for zero coupon nflaon swaps. Fgure 6 shows he fxed rae on zero coupon nflaon swaps, wh a 40

41 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel paymen delay of 5 years, for swaps of dfferen enors from 5 years o 5 years. he fxed rae on he swaps when we evaluae he convexy adjusmen, usng equaon (4. and he parameers for he one facor case (see ables and 3, s always lower han he fxed rae we would oban on he swaps f we navely assumed ha no convexy adjusmen was necessary. Furhermore, he dfference ncreases wh ncreasng swap enor. A 5 years, he dfference s more han 0.035% whch s, from a rader s perspecve, sgnfcan as he bd-offer spread n he marke, for zero coupon nflaon swaps, s approxmaely 0.03%, or somemes even less. Example 5: Fgure 6 In example 5, we use he model parameers (see ables 4 and 5, for he case when here are wo Brownan moons drvng nomnal neres-raes.e. n =. We compare he esmaes of he convexy adjusmens, obaned by one Carlo smulaon (we also repor he sandard errors n he column marked s/e and hose obaned usng he analycal formulae, for perod-on-perod swaples when here s no delayed paymen, when here s a one year paymen delay and when here s a fve year paymen delay. he able shows agan ha he formulae we derved n chaper 4 are correc and 4

42 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel ha hey have been correcly mplemened. able 6 Now, n examples 6 and 7, we wll gve some examples of he prces of LPI swaps. We use he one facor model parameers (see ables and 3. For he purposes of hese llusraons, we assumed ha he neres-rae (boh nomnal and real yeld curves were nally fla and ha nomnal neres raes o all maures were 0.05 and real neres raes o all maures were 0.05.e. we assumed P ( and P ( r 0, exp( 0.05 for all. 0, exp( 0.05 Example 6:.6000% Fxed rae on LPI swaps.5000% fxed rae on LPI swaps.4000%.3000%.000%.000%.0000% (0%,3% one Carlo (0%,3% Analycal (%,4% one Carlo (%,4% Analycal (0%,5% one Carlo 3 (0%,5% Analycal perods (year, each perod year Fgure 7 4

43 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel In fgures 7, we consder hree dfferen combnaons of floors and caps (whch are commonly raded n he marke namely, (0%, 3%, (0%, 5% and (%, 4%. For all hree dfferen combnaons, we consdered LPI swaps where each perod was equal o one year, bu he number of perods vared from one perod, hrough 5, 0, 5, 5 o 30 perods and hence he maures of he LPI swaps vared from one year o 30 years. We can see ha he fxed raes obaned from he quas-analycal mehodology of Ryen (007 (see chaper 5 are very close o he resuls obaned from one Carlo smulaon for shorer maures alhough he dfferences do ncrease for LPI swaps wh longer maures. Example 7: In hs example, we consdered eleven dfferen combnaons of floors and caps as ndcaed n able 7. We consdered LPI swaps whose maures were one year, sx years, 0 years and 5 years. For all he swaps, excep hose wh sx year maures, each perod was a year and hence he number of perods equaled he number of years o maury. By conras, he LPI swaps wh sx year maures had only wo perods as each perod was equal o 3 years. 43

44 Convexy Adjusmens n Inflaon-lnked Dervaves usng a mul-facor verson of he Jarrow and Yldrm (003 odel able 7 We can see ha here s (o he probablsc errors mpled by he sandard errors perfec agreemen beween he prces of he LPI swaps obaned by one Carlo smulaon and hose obaned by he quas-analycal mehodology of Ryen (007 (see chaper 5, for he LPI swaps wh one year maury (one perod and hose wh sx years maury (wo perods of hree years each. hs s no surprsng snce we know ha he quas-analycal mehodology s exac for he cases when. However, we see for he LPI swaps wh 0 years maury and 5 years maury, he level of approxmaon nvolved n he quas-analycal mehodology. As a rough gude, he bd-offer spread n he marke for LPI swaps s approxmaely 0.06% (expressed as he fxed rae on he swap. For he LPI swaps wh 0 years maury, he maxmum (absolue dfference n he fxed rae, mpled by he one Carlo resuls and he quas-analycal mehodology, s less han 0.06% whch mples, f no perfec, ceranly very accurae prcng as s less han half he bd-offer spread. For he LPI swaps wh 5 years maury, he accuracy does deerorae somewha and s, n some cases, greaer han he bd-offer spread n he marke. 44