Convexity adjustments in inflation-linked derivatives with delayed payments

Transcription

1 Convexiy adjusmens in inflaion-linked derivaives wih delayed paymens Dorje C. Brody,JohnCrosby and Hongyun Li Deparmen of Mahemaics, Imperial College London, London SW7 2BZ, UK Deparmen of Economics, Universiy of Glasgow, Glasgow, G12 8RT Daed: July 29, 28 Several ypes of inflaion-linked derivaives are valued using a muli-facor version of he model of Hughson 1998 and Jarrow and Yildirim 23. Expressions for he prices of zero coupon inflaion swaps wih delayed paymen and period-on-period inflaion swaps wih delayed paymens are obained in closed forms by explicily compuing he relevan convexiy adjusmens. These laer resuls are hen applied o value limied price indexaion LPI swaps using he common facor represenaion mehodology of Ryen 27. I. INTRODUCTION In recen years, he marke for inflaion-linked derivaives has grown rapidly. I is fair o say ha inflaion is now regarded as an independen asse class. Acively-raded inflaion derivaives include sandard zero coupon inflaion swaps, as well as more complicaed producs such as period-on-period inflaion swaps Mercurio 25, inflaion caps Mercurio 25, inflaion swapions Kerkhof 25, and fuures conracs wrien on inflaion Crosby 27. Consider a sandard zero coupon inflaion swap wih mauriy T M,fixedraeK, and noional amoun N, which we ener ino a ime. Le X denoe he spo CPI a ime. The payoff a ime T M of he sandard zero coupon inflaion swap is N X TM /X 1 N 1 + K T M 1. Noice ha he ime T M a which he CPI is measured o specify he payou agrees wih he ime a which he paymen akes place. While his is he common siuaion, ofen in pracice he paymen is delayed unil some laer ime T N T M. This delay is no jus he sandard wo-day spo selemen lag bu can be a period of a few weeks, a few monhs, or even several years. We will refer o such inflaion swaps as inflaion swaps wih delayed paymens. To see how such inflaion swaps have an imporan economic raionale, consider a commercial propery company. Suppose i has deb in he form of fixed-rae loans. I receives rens from is enans which i wans o pay ou as he inflaion-linked leg of an inflaion swap. I will receive fixed paymens on he inflaion swap which is used o pay is fixed-rae deb. Ofen rens will remain consan for a period of 5 years before being reviewed. They will hen be revised upwards o reflec inflaion over hose inervening five years. So for example, suppose ha he commercial propery company waned o ener ino an inflaion swap rade, in which i paid inflaion-linked cash flows and i received fixed cash flows. The company wans o hedge he cash flows ha i will receive from is enans in years 6, 7, 8, 9 and 1. A suiable inflaion swap rade would be a srip of five zero coupon inflaion swaps, where he payoffs of he five zero coupon swaps are we wrie only he inflaion-linked leg wih uni noional as follows: A he end of year 6, he company pays X 5 /X 1. A he end of year 7, i again pays X 5 /X 1. Likewise, i pays X 5 /X 1 a he end of years 8,

2 2 9 and 1. We see ha hese are zero coupon inflaion swaps wih delayed paymen, wih he delay on he final srip being 5 years. Period-on-period swaps wih delayed paymens are also raded in he markes. We will provide formulae for boh hese ypes of inflaion swap by compuing he relevan convexiy adjusmens. Noe ha he issue of delayed paymens should no be confused wih he issue of indexaion lag. Indexaion lag refers o he fac ha he value of he CPI in he denominaor of he inflaion-linked erm in he payoff is, in fac, he CPI published ypically a few weeks earlier, which, in urn, was calculaed from consumer prices observed a few weeks before ha. This is a differen issue alhough i would be possible o relae he wo and we refer he reader o Kerkhof 25 and Li 27. Limied price indexaion henceforh LPI swaps are a ype of exoic inflaion derivaive and are very common in he Unied Kingdom owing o he rules by which UK pension funds are governed. We will see ha he convexiy adjusmens required o value inflaion swaps wih delayed paymens have a furher applicaion in he valuaion of LPI swaps. This aricle is srucured as follows: In Secion II we inroduce he dynamics of nominal and real zero coupon bond prices and he spo CPI. In Secion III we sae he convexiy adjusmens required o value zero coupon inflaion swaps wih delayed paymen and periodon-period inflaion swaps wih delayed paymens. To our bes knowledge, hese resuls, in he conex of a muli-facor Hughson and Jarrow-Yildirim model, have no appeared in he lieraure before, alhough some similar resuls in he conex of a wo-facor Hull- Whie ype model are in Dodgson and Kainh 26. These resuls are hen applied o he valuaion of limied price indexaion LPI swaps, aided by he quasi-analyic mehodology of Ryen 27. A number of examples and comparisons are given in Secion IV. We finish wih a brief concluding remark in Secion V. The appendix conains proofs of he convexiy adjusmen formulae as well as explici formulae for he valuaion of zero coupon inflaion swaps wih delayed paymen and period-on-period inflaion swaps wih delayed paymens. II. MODELS FOR BOND PRICES AND THE SPOT CPI We model he marke wih he specificaion of a probabiliy space Ω, F, Q wih filraion {F } < generaed by a muli-dimensional Brownian moion. The probabiliy measure Q denoes he risk-neural measure, and marke prices and oher informaion-providing processes are adaped o {F }. Throughou he paper we assume he absence of arbirage and he exisence of a pricing kernel hese condiions ensure he exisence of a unique pricing measure Q. WeleE [ ] denoe he expecaion in Q condiional on {F }. We denoe calendar ime by ; ime = will denoe he iniial ime. Le {r N} and {rr } denoe, respecively, he coninuously compounded risk-free nominal and real shor rae R } and {PT } denoe, respecively, he price process of a nominal and real zero coupon bond mauring a T.ThespoCPIaimeis denoed by X. A key observaion for pricing inflaion derivaives is ha, for any imes and T M, T M, we have Hughson 1998: processes. Le {P N T X P R T M = E [X TM exp TM ] rs N ds. 1 This follows from he fac ha he righ side of 1 is he price a ime of an index-linked bond, which pays he amoun X TM a ime T M. Dividing i by X we obain he value in

3 3 real erms of a bond ha pays one uni of goods and services a ime T M. Mercurio 25 uses his relaion o value sandard zero coupon inflaion swaps, and shows how, given he fixed raes quoed in he markes for hese swaps, he erm srucure of real discoun facors can be obained. Now, we inroduce he models for he dynamical equaions saisfied by nominal zero coupon bond prices, real zero coupon bond prices, and he spo CPI, wihin he muli-facor version of he Hughson and Jarrow-Yildirim model. These are given by: and dp R T P R T = r R dp N T P N T = r N d + σkt N dzk, N 2 ρ RX k σ X σkt R d + σkt R dzk, R 3 dx = r N r R d + σ X X dz X. 4 Here K N and K R are he number of Brownian moions driving nominal and real zero coupon bond prices respecively, {dzk N},...,K, {dzk R},...,K R, and {dz X } denoe sandard Q- Brownian incremens. Furhermore, {σkt N },...,K N and {σkt R },...,K R are volailiy erms, which are assumed o be deerminisic, saisfying σktt N =,and{σx } is he spo CPI volailiy which we also assume o be deerminisic. We denoe correlaions all assumed consan by ρ wih appropriae subscrips: Corrdzj N, dzn k =ρnn ρ RR jk d, CorrdzX, dzk N=ρNX k d, Corrdz X, dz R j =ρ RX j jk d, CorrdzR j, dzr k = d. d, and Corrdz N j, dz R k =ρnr jk III. LIMITED PRICE INDEXATION LPI SWAPS In his secion, we will provide a valuaion formula for LPI swaps. Before discussing LPI swaps, we sae wo preliminary proposiions, he proofs of which are in he Appendix A. We will use hem in his secion o value LPI swaps. However, as we show in he Appendix B, hey can also be used o value zero coupon inflaion swaps wih delayed paymen and periodon-period inflaion swaps wih delayed paymens. Proposiion 1 Given he assumpions of Secion II, for any imes and T N, T M T N, he following relaion holds: TN ] E [X TM exp rs N ds P N TM = X PT R T N M exp C PT N s T M,T N ds, 5 M where C s T M,T N = σ N kstn σkst N M ρ NR kj σr jst M ρ NN kj σjst N M + j=1 σ N kstn σ N kst M ρ NX k σ X s. 6 j=1

4 4 We remark ha when T M = T N i is sraighforward o verify ha C s T M,T N =,in which case equaion 5 agrees wih equaion 1. Proposiion 2 Given he assumpions of Secion II, we have, for <T i 1 <T i T Ni, [ TNi ] XTi E exp rs N X ds 7 Ti 1 = P N PT N Ni T i 1 PT N i P R T i P R T i 1 exp Ti T i 1 C s T i,t Ni ds + where C s T i,t Ni is given by 6 and where A s T i 1,T i = j=1 + σ RjsTi σ RjsTi 1 Ti 1 [A s T i 1,T i +B s T i 1,T i,t Ni ] ds, ρ NR kj σkst N i 1 ρ RR kj σ R kst i 1 σ RksTi 1 σ RksTi ρ RX k σ X s, 8 and B s T i 1,T i,t Ni = j=1 ρ NN kj + j=1 ρ NR kj σ NksTi 1 σ NksTi σ NjsTNi σ NjsTi σ RjsTi σ RjsTi 1 σ NksTNi σ NksTi. 9 We now proceed o he valuaion of LPI swaps. Suppose ha oday, a ime, we ener ino an LPI swap. The LPI swap is defined via a se of fixed daes T <T 1 <T 2 < <T M 1 <T M,whereT =. The paymen of he payoff of he swap occurs a ime T,whereT = T M. The payoff of he inflaion-linked leg of he swap a ime T is given by M i=1 XTi min max, 1+F, 1+C, X Ti 1 where C and F are consans wih C F. In pracice, F is ofen zero bu we will assume in he following ha C and F can ake on any values posiive, negaive or zero provided ha C F. We see ha he role of he consans C and F is o cap and floor he periodon-period inflaion rae over each period. We remark ha when C = and F = he produc elescopes and he LPI swap has he same payoff as a zero coupon inflaion swap. However, when C and F are finie and when M>1, we need o price a swap whose payoff is pah-dependen. For ypical values of M beween 5 and 4, say, he only feasible mehodology o price LPI swaps is by Mone Carlo simulaion, bu his is CPU inensive. Hence i would be desirable o have a fas, even if approximae, quasi-analyic mehodology o price hem. Such a mehodology, based on he idea of common facor represenaion, is proposed in Ryen 27. Noe, however, ha Ryen s model seup is raher differen from ours. We will apply Ryen s idea, in order

5 5 o value LPI swaps, wihin he seup of our muli-facor version of he model of Hughson 1998 and Jarrow and Yildirim 23. Le us begin by inroducing some addiional noaion. We le Q T denoe he probabiliy measure defined wih respec o he numeraire which is he zero coupon bond mauring a ime T. Similarly, we le E T [ ] denoe he expecaion wih respec o he measure Q T condiional on {F }. Suppose ha we have a T M year LPI swap wih M periods. Le X i denoe X Ti /X Ti 1 for i = 1, 2,...,M. In Li 27, i is shown ha ln X i for each i =1, 2,...,M is normally disribued in our model, and ha we can calculae he covariance marix covln X i, ln X j foreachi, j. In general, none of he elemens of his covariance marix vanish because ln X i is no independen of ln X j for any i, j. This lack of independence complicaes he problem of pricing an LPI swap. The idea of Ryen see also Jackel 24 is o replace he covariance marix covln X i, ln X j foreachi, j by anoher marix, which is close o he acual correlaion marix in some sense, bu in which he off-diagonal elemens have a simple srucure. This is achieved by generaing all he codependence beween ln X i and ln X j hrough a single common facor in fac, Ryen also considers he case of wo common facors bu we will, for he sake of breviy, only consider one. We remark ha i is easy o show Li 27 ha ln X i = lnx Ti / ln X Ti 1, for each i =1, 2,...,M, is disribued as muli-variae normal random variables in he measure Q T. Tha is o say, ln X i is Gaussian wih deerminisic drif and volailiy under Q T. Hence we can wrie X i in he form X i =expa i z i + b i, where z i N, 1; covln X i, ln X j = covz i,z j a i a j ;ande [X i ]=exp b i a2 i. The key idea of Ryen 27 is o replace X i by ˆX i defined via ] ˆX i exp [b i + a i â i w + 1 â 2 i ε i, where he sysem {w, ε 1,...,ε M } is a family of independen N, 1 variaes. The variaes ˆX 1,..., ˆX M represen he variaes X 1,...,X M via one common facor w and addiional individual idiosyncraic random variables {ε i } i=1,2,...,m. Noe ha he common facor w is an absrac facor and does no necessarily correspond o any marke-observable. From Ryen 27, which in urn references Jackel 24, we know ha when M 3 we can approximae â k by [ 1 M â k exp k k k ] i=1 i, M 2 2M 1 where k k = M i k ln [covln X i, ln X k ], k =1, 2,...,M. In he cases for which M =1or M = 2, we do no need an approximaion. Indeed, if M = 1 hen we have rivially â 1 = 1; likewise if M = 2, hen we have from Cholesky decomposiion â 1 = 1 and â 2 =CorrlnX 1, ln X 2. Noe ha he relaions E T [ ˆX i ]=E T [X i ] and var[ln ˆX i ] = var [ln X i ] are valid for all i =1, 2,...,M and for all value of M. However, if M 3, hen cov ˆX i, ˆX j isonlyan approximaion o covx i,x j wheni j. We now apply Ryen s idea in order o value LPI swaps. By changing he measure o Q T and using Girsanov s heorem, he price a ime T = of he inflaion-linked leg of he

6 6 LPI swap is: T E [exp rs N ds M i=1 [ M = PT N ET min P N T ET = P N T ET = P N T ET i=1 [ M [ i=1 E T [ M i=1 XTi min max max X Ti 1, 1+F XTi X Ti 1, 1+F ], 1+C ], 1+C ] min max ˆXi, 1+F, 1+C [ M E T i=1 min max ˆXi, 1+F, 1+C ]] w [ min ] max ˆXi, 1+F, 1+C ] w 1 By assumpion he random variables ε i are independen, and consequenly, condiional on w, hevariaes ˆX i are also independen, i.e. cov ˆXi, ˆX j w =,wheni j. Therefore, we see ha he condiional expecaion of he produc in he las bu one line of equaion 1 becomes a produc of condiional expecaions in he las line. We have used approximaely equals in he hird line of equaion 1 because he variaes ˆX i are, in general i.e. when M 3, only an approximae represenaion of he variaes X i for i =1, 2,...,M. In order o evaluae equaion 1 we need o compue he Q T -expecaion of X i and he covariance marix covln X i, ln X j. The laer is shown in Li 27 o be given by cov ln X i, ln X j = T i 1 K N cov σkst R i σkst R i 1 dzks R σpst N i σpst N i 1 dzps N, T i p=1 K N σkst R j σkst R j 1 dzks R σpst N j σpst N j 1 dzps N ds p=1 + covσs X dzs X + σkst R i dzks R σpst N i dzps, N T p=1 i 1 when j>i,whereaswhenj = i we have var ln X i σ 2 ln X i = + T i T i 1 T i 1 var KR K N σkst R j σkst R j 1 dzks R σpst N j σpst N j 1 dzps N ds p=1 σkst R i σkst R i 1 dzks R σpst N i σpst N i 1 dzps N ds p=1 var σs X dzs X + σkst R i dzks R σpst N i dzps N ds. p=1

7 7 The former can also be compued since i follows from he Girsanov heorem ha he Q T -expecaion of X i is [ ] E T XTi = 1 [ T ] E X Ti 1 PT N exp rs N ds XTi. 11 X Ti 1 The Q T -expecaion of X i can hen be evaluaed explicily by use of Proposiions 1 and 2. Specifically, when i = 1, we find, since T =, ha 11 implies whereas when i>1 we obain E T [X i] = P N T i 1 P N T i E T [X i]= P R T 1 P N T 1 P R T i P R T i 1 exp + Ti 1 T1 exp C s T 1,T ds, 12 Ti C s T i,t ds T i 1 [A s T i 1,T i +B s T i 1,T i,t ] ds. 13 Furhermore, since X i is lognormal, we can use he sandard resul ha if we denoe by μ ln Xi and σln 2 X i he mean and variance of ln X i,hene T [X i ]=exp μ ln Xi σ2 ln X i for i =1, 2,...,M. Hence we obain he expecaion of ln X i : μ ln Xi =ln E T [X i ] 1 2 σ2 ln X i. Now we can use he following well-known resul: If X N μ X,σX 2, W N, 1, and ρ XW is he correlaion beween X and W,henX W = w is normally disribued and, furhermore, E [X W = w] =μ X + ρ XW σ X w and var [X W = w] =σx 2 1 ρ2 XW. We can calculae he correlaion beween ln ˆX i and he common facor w. Indeed, since ln ˆX i is normally disribued wih variance a 2 i, and since cov ln ˆX i,w =cov a i â i w + 1 â 2 i ε i,w = a i â i, we deduce ha he correlaion beween ln ˆX i and w is â i for each i =1, 2,...,M.Nowwe recall ha E T [ln ˆX i ]=E T [ln X i ]=μ ln Xi and ha var[ln ˆX i ]=var[lnx i ]=σln 2 X i. Then using he resul above we ge [ ln ˆX ] i w = μ ln Xi +â i σ ln Xi w, E T σ i [ln 2 var ˆX ] i w = σln 2 X i 1 â 2 i, and F i E T [ ˆX i w] =exp μ ln Xi +â i σ ln Xi w σ2 i for i =1, 2,...,M. Finally equaion 1 becomes: [ M Fi Call ] Fi, 1+C, σ i 2 + Pu Fi, 1+F, σ i 2, 14 P N T ET i=1

8 8 where Call Fi, 1+C, σ i 2 and Pu Fi, 1+F, σ i 2 are, respecively, he undiscouned prices of a call opion wih srike 1 + C and a pu opion wih srike 1 + F, in he Black 1976 formula, when he forward price is F i and he inegraed variance is σ i 2. Noe ha each erm in he produc in equaion 14 depends on he common facor w hrough F i and σ i 2,and w has a sandard normal N, 1 disribuion. Hence he price of he inflaion-linked leg of he LPI swap a ime noe ha when M 3, i is only an approximaion is: P N T + 1 2π exp w2 2 M Fi Call Fi, 1+C, σ i 2 + Pu Fi, 1+F, σ i 2 dw. i=1 I follows ha we can value LPI swaps wih jus a single numerical inegraion. IV. NUMERICAL EXAMPLES We now examine some numerical examples. There are differen forms ha he volailiy funcions σkt N and σr jt can ake, bu here we will consider he exended Vasicek form in which we assume σ N kt = σn k α N k 1 e αn k T, σ R kt = σr k α R k 1 e αr k T, 15 where, for each k, σk N, σr k, αn k,andαr k are posiive consans. We will use he model parameers esimaed for GBP in Li 27. In order o simplify parameer esimaion, we assume ha real zero coupon bond prices are driven by a single Brownian moion so ha K R = 1 in equaion 3. In addiion, we assume ha he volailiy of he spo CPI is consan, i.e. σ X = σ X. We assume ha here are wo Brownian moions driving nominal zero coupon bond prices so ha K N = 2. This assumpion adds nohing o he complexiy of he calibraion since he associaed parameers can be and were obained by calibraing o he marke prices of GBP vanilla ineres-rae swapions see Li 27. The esimaed values of he parameers are: σ1 N = , α1 N = , σ2 N = , α2 N = , σr 1 =.69394, αr 1 = , σ X =.14, ρ NN 12 = , ρ RX 1 = , ρ NR 11 = ρ NR 21 =.5181, ρ NX 1 = ρ NX 2 = We will use hese parameers o give some numerical examples and comparisons for inflaion swaps wih differen swap enors and paymen imes. Example 1: The effec of he convexiy adjusmen on he fixed rae for zero coupon inflaion swaps. Figure 1 shows he fixed rae K on zero coupon inflaion swaps, wih a paymen delay of 5 years, for swaps of differen enors from 5 years o 25 years. The ineres-rae boh nominal and real yield curves were he GBP marke implied raes as of June 27 see Appendix C for he se of marke daa. The volailiy and correlaion parameers were as above. The fixed rae on he swaps when we evaluae he convexiy adjusmen, using Proposiion 1, is always lower han he fixed rae we would obain on he swaps if we naively assumed ha no convexiy adjusmen was necessary. Furhermore, he difference increases wih increasing swap enor. A 25 years, i.e. when T M =25andT N = 3,

9 9 The fixed rae K in % on zero coupon inflaion swaps, wih a paymen delay of 5 years, for swaps of differen mauriies 3.2% 3.15% 3.1% 3.5% 3.% 2.95% 2.9% Mauriy of zero coupon inflaion swap in years if assumed w ih no convexiy adjusmen w ih convexiy adjusmen FIG. 1: he difference is more han.65% which is, from a rader s perspecive, significan as he bid-offer spread in he marke, for zero coupon inflaion swaps, is approximaely.3%, or someimes even less. Some examples of period-on-period inflaion swaps are provided in Li 27 so here, in Examples 2 and 3, we will give some examples of he prices of LPI swaps, again using he volailiy and correlaion parameers above. For he purposes of hese illusraions, we assumed, for boh he examples below, ha he ineres-rae boh nominal and real yield curves were iniially fla and ha nominal ineres raes o all mauriies were.5 and real ineres raes o all mauriies were.25, i.e. we assumed PT N =exp.5t and PT R =exp.25t. We used Mone Carlo simulaion wih 13 million runs 65 million runs plus 65 million aniheic runs in order o es and benchmark he accuracy of our applicaion of he Ryen mehodology. Example 2: LPI swaps wih floors and caps a %, 3%, %, 5%, 1%, 4%. Here we consider hree differen combinaions of floors and caps which are commonly raded in he marke, namely, %, 3%, %, 5%, and 1%, 4%. For all hree differen combinaions, we consider LPI swaps where each period is equal o one year, and he number of periods varies from one period, hrough 2, 5, 1, 15, 2, 25 o 3 periods and hence he mauriies of he LPI swaps varied from one year o 3 years. We see from Figure 2 ha he fixed raes obained from he quasi-analyical mehodology of Ryen labelled QA are very close o hose obained from Mone Carlo labelled MC simulaion for shorer mauriies as explained above, he Ryen mehodology is, in fac, essenially exac for M 2. However, he differences do increase for LPI swaps wih more periods.

10 1 2.6% Fixed rae in % on LPI swaps 2.5% 2.4% 2.3% 2.2% 2.1% Mauriy of LPI sw ap in years QA,3 MC,3 QA 1,4 MC 1,4 QA,5 MC,5 FIG. 2: Example 3: LPI swaps wih mauriies of 1 years and 25 years.hereweconsider eleven differen combinaions of floors and caps as indicaed in Table 1. We consider LPI swaps whose mauriies were 1 years and 25 years. Again, each period is equal o one year. We know ha he Ryen mehodology is essenially exac when M 2. However, we see for he LPI swaps wih 1 years mauriy and 25 years mauriy he level of approximaion involved when M 3. As a rough guide, he bid-offer spread in he marke for LPI swaps is approximaely.6% expressed as he fixed rae on he swap. For he LPI swaps wih 1 years mauriy, he maximum difference Table 1, 8h column beween he fixed raes implied by he Mone Carlo resuls 6h column and by he Ryen mehodology 7h column, is less han.19%, which implies very accurae pricing as i is less han one hirieh he ypical bid-offer spread. For he LPI swaps wih 25 years mauriy, he accuracy does deeriorae somewha. The maximum difference in he fixed raes is approximaely.53%, which is close o he bid-offer spread. Having given some examples of he valuaion of LPI swaps, we can make one furher commen abou he accuracy of he quasi-analyical mehodology. In ables 1 and 2, we observe ha he accuracy deerioraes when he cap level is high and he floor level is low. This migh iniially seem surprising since in he limiing case ha C = and F = he LPI swaps become he same as sandard zero coupon swaps. However, he reason for he deerioraion in accuracy is ha he quasi-analyical mehodology approximaes he correlaion srucure. Alhough in he noaion of Secion III i is rue ha E T [ ˆX i ]= [ M E T [X i ] for all i, and i is also rue ha E T i=1 X i = E T [X TM /X ]=PT R M = PT R, he price of a sandard zero coupon swap, he approximaion of he correlaion srucure means ha E T [ M i=1 ˆX i ] does no equal E T ] [ M i=1 X i ], excep in he special cases for

11 11 1 year, 1 period LPI swap Price Price imp. rae imp. rae diff Cap floor san. error Mone Carlo Ryen QA MC % QA % raes %.3 7.8E E E E E E E E E E E Table 1 25 year, 25 period LPI swap Price Price imp. rae imp. rae diff cap floor san. error Mone Carlo Ryen QA MC % QA % raes % E E E E E E E E E E E Table 2 which M 2. For he sake of breviy, we only considered he Ryen mehodology for he case of condiioning on one common facor. Ryen 27 also considers he case of condiioning on wo common facors which means ha evaluaing he price of a LPI swap requires a double numerical inegraion and shows, in his model se-up which is differen o ours, ha unsur-

12 12 prisingly his gives a significan improvemen in accuracy. We would cerainly conjecure ha using wo common facors would also significanly improve he accuracy of he prices of he LPI swaps which we repored in Tables 1 and 2. However, we leave confirmaion of his conjecure for fuure research. V. CONCLUSION In recen years here has been a subsanial increase in he demand for more exoic inflaion derivaive producs. Working wihin a muli-facor version of he model of Hughson 1998 and Jarrow and Yildirim 23, we have provided he economic raionale for, and he valuaion formulae for, zero coupon inflaion swaps wih delayed paymen and periodon-period inflaion swaps wih delayed paymens. We have also valued LPI swaps, wih he aid of he quasi-analyic mehodology of Ryen 27. Dorje C. Brody dorje@imperial.ac.uk is Reader in Mahemaics a Imperial College London, John Crosby johnc225@yahoo.com is a Visiing Professor of Quaniaive Finance a Deparmen of Economics, Universiy of Glasgow, and Hongyun Li hongyun.li6@imperial.ac.uk is a Ph.D suden a Imperial College London. We hank Mark Davis for discussions and Terry Morgan for providing he marke daa used in he examples. Feedback from wo anonymous referees is graefully appreciaed. References. [1] Black F The pricing of commodiy conracs Journal of Financial Economics 3 p [2] Crosby J. 27 Valuing inflaion fuures conracs Risk magazine March p88-9. [3] Dodgson M. and D. Kainh 26 Inflaion-linked derivaives Risk raining course, 8h Sepember 26. Available a Royal Bank of Scoland Quaniaive Research Cenre. [4] Hughson L. P Inflaion derivaives Working paper, Merrill Lynch and King s College London wih added noe 24, available a [5] Jackel P. 24 Spliing he core Working paper available a [6] Jarrow R. and Y. Yildirim 23 Pricing reasury inflaion proeced securiies and relaed derivaives using an HJM Model Journal of Financial and Quaniaive Analysis 38 p [7] Kerkhof J. 25 Inflaion derivaives explained: Marke, producs, and pricing Lehman Brohers publicaion. [8] Li H. 27 Convexiy adjusmens in inflaion-linked derivaives using a muli-facor version of he Jarrow and Yildirim model M.Sc disseraion, Deparmen of Mahemaics, Imperial College London. Available online a: hp://

13 13 [9] Mercurio F. 25 Pricing inflaion-indexed derivaives Quaniaive Finance 5 p [1] Ryen M. 27 Pracical modelling for limied price index and relaed inflaion producs Presenaion given a he ICBI Global Derivaives conference in Paris on 22nd May 27.

14 14 APPENDIX A: PROOF OF PROPOSITIONS 1 & 2 The sochasic discouning erm exp T N r N s ds is log-normally disribued and can be wrien in he form TN TN exp rs N ds = PT N 1 N exp ρ NN kj σkst N 2 N σjst N N ds j=1 TN exp σkst N N dzks N. If we define he forward CPI a ime o ime T by F X T we have FT X = X P R 27. Since T /P T N, where F X T F X T M T M, hen by no-arbirage argumens, is log-normally disribued see, for example, Crosby PT R = X M T M TM = X PT N TM, M T M we find TN ] TN ] E [exp rs N ds X TM = E [exp rs N ds FT X M T M. This expecaion can be compued by noing ha i is he expecaion of a produc of wo log-normally disribued random variables, each of which has deerminisic mean and variance erms. Li 27 provides full deails. The proof of Proposiion 2 is very similar o ha for Proposiion 1 excep ha now we will compue an expecaion involving hree log-normally disribued random variables. APPENDIX B: INFLATION SWAPS WITH DELAYED PAYMENTS In his appendix, we will value wo ypes of inflaion swap, namely, zero coupon inflaion swaps wih delayed paymen and period-on-period inflaion swaps wih delayed paymens. The key poin abou hese ypes of inflaion swap is ha hey have he same payoff as he corresponding inflaion swap wih no delayed paymens bu he payoff is paid a a laer ime. When he delay in paymen is very small for example, a few weeks, we would, inuiively, expec he difference beween he values of he corresponding swaps wih no delayed paymens and wih delayed paymens o be small. Conversely, he difference in values can be subsanial when he delay in paymens is, for example, a few years. As we noed in Secion I, inflaion swaps wih delayed paymens of five years or more are quie commonly raded in he marke. 1. Zero coupon inflaion swaps wih delayed paymen As menioned above, i is now relaively common o rade zero coupon inflaion swaps where he paymen is delayed for some ime, perhaps several years or more, compared o he payoff of a sandard zero coupon inflaion swap. Unlike wih a sandard i.e. wih no delayed paymen zero coupon inflaion swap, he valuaion of zero coupon inflaion swaps

15 15 wih delayed paymen will involve a convexiy adjusmen which is model-dependen. We can explicily compue i wihin our model seup by using Proposiion 1. Suppose ha oday, a ime, we ener ino a zero coupon inflaion swap wih delayed paymen. We denoe he mauriy of he swap by T M and he paymen ime by T N T M. We wish o value he swap, a ime, where T M T N. The payoff of he zero coupon inflaion swap wih delayed paymen is sill N X TM /X 1 N 1 + K T M 1, where K is he fixed rae on he swap and N is he noional amoun, bu his is paid a ime T N T M.Thevalue,aime, of he zero coupon inflaion swap wih delayed paymen is: TN E [exp rs N N ds XTM 1 N 1 + K T M 1 ] X TN ] = E [N exp rs N ds XTM 1 + K T M X = N TN ] E [X TM exp rs N ds NPT N X N 1 + K T M = N P N TM X PT R T N X M exp C s T M,T N ds NPT N N 1 + K T M. B1 PT N M Noe ha in obaining he las line we have used Proposiion 1. Compared o he value of a sandard zero coupon inflaion swap wih no delayed paymen, we see ha here is an exra erm PT N N /PT N M e R T M C st M,T N ds in he inflaion-linked leg. 2. Period-on-period inflaion swaps wih delayed paymens Our aim now is o value, a any ime, a period-on-period inflaion swap wih delayed paymens. Proposiion 2 will be he key o his. Suppose ha oday, a ime, we ener ino a period-on-period inflaion swap wih delayed paymens. The swap is defined via a se of fixed daes T <T 1 <T 2 < < T M 1 <T M,whereT =. These daes are usually approximaely one year apar bu hey need no be. As wih a sandard ineres-rae swap, a period-on-period inflaion swap is made up of a series of swaples. The key issue is ha he value of he payoff of each swaple is he same as he payoff of he corresponding swaple of a period-on-period inflaion swap wih no delayed paymens bu now he paymen is made a ime T Ni which is some ime greaerhanorequalot i. The payoff of he ih swaple, for i =1, 2,...,M,aimeT Ni, is Nτi I XTi /X 1 Ti 1 Nτi F K,whereK ishefixedraeonheswap,n is he noional amoun, τi I is he day-coun adjused ime from T i 1 o T i for he floaing inflaion-linked leg, and τi F is he day-coun adjused ime from T i 1 o T i for he fixed leg. The value a ime of he swaple wih delayed paymen T Ni T i is TNi ] E [exp rs N XTi ds Nτi F K = Nτ I i E Nτi I 1 X Ti 1 [ TNi ] exp rs N ds XTi X Ti 1 NP N T Ni τ I i + τ F i K. To value he floaing inflaion-linked side, we have o consider separaely wo differen cases depending upon wheher T i 1 or <T i 1. B2

16 16 In he firs case for which T i 1 T i,hevalueofx Ti 1 is known a ime. Therefore, we can ake X Ti 1 ouside of he expecaion in B2 and obain, from Proposiion 1, TNi ] E [exp rs N ds X Ti = X P R PT N Ni T i PT N i Subsiuion of his expression in he righ side of B2 hen yields Nτ I i X X Ti 1 P R PT N Ni T i PT N i Ti exp C s T i,t Ni ds NPT N Ni τi I Ti exp C s T i,t Ni ds. B3 + τ F i K. In he second case for which <T i 1 we use he resul of Proposiion 2 o obain Nτ I i P N P R N P T i T Ni T i 1 PT R i 1 PT N i NP N T Ni τ I i Ti Ti 1 exp C s T i,t Ni ds + [A s T i 1,T i +B s T i 1,T i,t Ni ] ds T i 1 + τ F i K. Therefore, we can value a period-on-period inflaion swap wih delayed paymens by summing up he value of all he swaples, bearing in mind he wo disinc expressions arising from he case T i 1 T i and he case <T i 1. Noe ha when T i = T Ni, B s T i 1,T i,t Ni andc s T i,t Ni in 9 and 6 vanish. Hence, one can confirm, afer some algebra, ha he resuls we have jus given, in he case of exended Vasicek bond volailiies see equaion 15, reduce o he same as hose given in Mercurio 25 for he value of a period-on-period inflaion swap wih no delayed paymens. APPENDIX C: MARKET DATA FOR EXAMPLE 1 Tenor Nominal Discoun Facors Real Discoun Facors