Market Models for Inflation

Marke Models for Inflaion Ferhana Ahmad Lady Margare Hall Universiy of Oxford A hesis submied for he degree of Msc. in Mahemaical and Compuaional Finance Triniy 2008

This work is dedicaed o my family, for all heir love and care...

Acknowledgemens Praise is o he One, he Almighy, he merciful and he beneficen. Allah, who is he source of all knowledge and wisdom, augh us wha we knew no. I also send Darood o he Holy Prophe Muhammad (saw), who is he Allah s messenger o guide me in affairs of life by his examples and perceps. I express my hearies graiude o my course and hesis Supervisor, Dr. Ben Hambly for his guidance, suppor and moivaion. His valuable suggesions and encouragemen during my work made i possible. I would also like o menion my funding body Higher Educaion Commission, Pakisan for heir grea funding scheme ha provided so many scholars of Pakisan a chance of higher sudies in repuable universiies of he world. In he end I would like o pay hanks o my family and friends for heir coninuous encouragemen and moral suppor.

Conens 1 Inroducion 1 2 Inflaion 3 2.1 Inflaion.............................. 3 2.2 Nominal Raes, Real Raes and Expeced Inflaion...... 4 2.3 Inflaion-Indexed Securiies................... 5 2.4 Derivaives Traded........................ 6 3 Mahemaical Background 9 3.1 Fundamenals of Mahemaical Finance............. 9 3.2 Change of Numeraire....................... 11 3.3 Ineres Raes........................... 12 3.4 Heah-Jarrow-Moron Framework (HJM)............ 14 3.5 Exended Vasicek Model..................... 15 4 Models for Inflaion 16 4.1 Jarrow and Yildirim Model (2003)................ 16 4.1.1 The Model......................... 16 4.1.2 Exended Vasicek Model and Hedge Raios....... 20 4.1.3 Pricing Opions on he Inflaion Index.......... 22 4.2 Mercurio Marke Models..................... 25 4.3 Beldgrade-Benhamou-Koehlar Marke Model (2004)........................... 25 5 Marke Models of Inflaion 26 5.1 Inflaion-Indexed Swaps..................... 26 5.1.1 Pricing Zero-Coupon IIS................. 26 5.1.2 Pricing Year-on-Year IIS................. 28 5.2 Pricing wih he Jarrow Yildirim Model............. 29 5.3 Pricing wih a Firs Marke Model............... 31 5.4 Pricing wih a Second Marke Model.............. 35 i

6 Conclusion 38 Bibliography 39

Chaper 1 Inroducion Inflaion is currenly a big financial issue. No only i have an affec on a counry s economy bu i also changes he hinking of an average person. In paricular, people sar wih drawing heir money from bank accouns and ry o buy a sronger currency or wan o inves heir money in producs ha guaranee he value of heir money over he ime of inflaion. I affecs people on fixed income, such as pensioners and reirees, he mos because a healhy amoun of savings oday may no have he same purchasing power over ime. Simple quesions come ino one s mind such as how o proec agains inflaion? How o manain your sandard of living given uncerainiy abou fuure inflaion? The answer o hese quesions is inflaion indexed securiies. Producs in he inflaion indexed marke are ied o inflaion. The idea is o inves your money, for example in he bond marke, in such a way ha gives you a real reurn even hough he value of your principal amoun and he coupons may change. The Unied Kingdom (UK) inroduced he firs inflaion indexed bonds in 1981 afer he remarkable 1970 s high inflaion. The inflaion ouched double digis in he 1970 s in he UK and o proec again such spikes in he inflaion rae he governmen issued inflaion securiies. The inflaion-indexed marke has grown seadily over he pas en years. The governmen and differen corporaions are now esablishing a wide variey of inflaion-indexed securiies. The main objecive of his hesis is o sudy and discuss some of he models for inflaion ha have been developed. Inspie of he fac ha some work has already been done in his field, his sill remains an acive area of research hese days. The need o proec agains inflaion gives rise o inflaion-indexed derivaives and heir pricing in an arbirage free way. We 1

sar by looking a inflaion, is effecs and causes in chaper 2. This chaper includes some basic definiions such as nominal and real raes, provides some deails on he inflaion derivaives raded in financial markes and inflaionindexed securiies. The hird chaper consiss of he mahemaical background ha is needed in chapers 4 and 5. We sar by looking a maringale heory and presen some key ools of mahemaical finance. A small secion is devoed o reviewing he change of numeraire echnique. We hen provide some common and useful definiions ha we mus know for our work. We also include he HJM framework and exended Vasicek model o help he reader. Our main work is presened in chapers 4 and 5, in which we discuss some models of inflaion. Chaper 4 discusses he Jarrow and Yildirim model of inflaion in full deail. This model is considered as he firs model of inflaion and is based on an analogy wih exchange rae models. We also price call opions on he inflaion index using heir echnique. In he end, he model developed by Beldgrade, Benhamou and Koehlar in [5] is also included along wih he models of Mercurio in [3] for he sake of compleeness. Chaper 5 presens he marke models of Mercurio [3] in complee deail. We price he ZCIIS and YYIIS by using Mercurio s firs and second marke model and by he Jarrow and Yildirim model. One can find he exensions of hese models in [4,11,12]. 2

Chaper 2 Inflaion In his chaper, we presen some heories and definiions relaed o inflaion. Inflaion has is effecs on everyone from an average person o he economy of a counry and hence is a big area of oday s research. One migh hink ha everyone would be equally affeced by inflaion bu has no he case. Inflaion effecs you badly when you plan your reiremen period because your savings will buy less and less over ime and you don have any source of income. Of course you hen hink o proec yourself agains inflaion and hence here is a need for inflaion indexed securiies. These are he producs ha guarenee you he same purchasing power from your savings. This chaper sars wih he definiion of inflaion and is effecs. The difference beween he nominal and real raes is highlighed wih a small discussion on inflaion-indexed securiies and derivaives raded. 2.1 Inflaion Prices of differen goods and services do no never same. A candy bar coss more han i used o be 20 years ago. Your bills do no remain he same even if you purchase he same groceries. Inflaion is he erm o describe when he same goods and services cos more over ime. I means you can buy he same hing for he same price as you could a few years back. Inflaion occurs when prices coninue o creep upward. Inflaion is he rae of increase in he general level of prices for he same goods and services. In simple words, he rise in prices of goods when here is no compaible rise in wages cause inflaion. An increase in he money in circulaion may cause a sudden fall in is value and a sharp increase in prices of goods - hence inflaion. Inflaion is measured as a percenage in- 3

crease in some kind of inflaion index, which may no be unique as here are many ypes of inflaion indexes. A general fall in price level is called deflaion. Inflaion does no effec you much if i says for a small period of ime bu hisory shows ha he prices never fell back o heir previous values excep in 1930 s when here was a deflaion. Since he prices added up each year, a small rae as 4 % could be huge if i says for a long period of ime. In he pas fify years he developed counries have experienced a decrease in he level of inflaion and reduced his rae o abou 2.00-3.00 % Inflaion can be caused by many reasons. Economiss are rying o locae is causes and esablished many heories regarding his. One of hose is he demand and supply, in which if he demand increases more han he supply, he prices of he goods increase also and causes inflaion. Macroeconomiss also hink ha he increase in he wages of workers can cause inflaion because o make significan profis, he company s owner increases he prices of producs. Some hink ha inflaion is an indicaion of growing economy. There are many groups of hough which give differen heories bu hey all consider i a big issue in oday s world. 2.2 Nominal Raes, Real Raes and Expeced Inflaion We have wo choices o look a when valuing he cashflows, he nominal or real cashflows. When we alk abou raes or raes of reurn we refer o he amoun in erms of nominal raes. The price of he uni of money never remains he same raher i decays over ime due o inflaion. The nominal raes are he ones which indicae he raes in erms of money bu no in he real sense of is purchasing power. On he oher hand he real raes do consider he effec of inflaion also. In oher words he nominal raes ell you he growh of your money and no he purchasing power. The real raes ell you wha is he purchasing power of your money. I is no possible o ell he real raes a he ime of invesmen or borrowing because no one knows he inflaion rae for he coming years. From an economics perspecive i is imporan o differeniae beween he real and nominal raes because he real raes are he ones ha maer. The nominal raes ells you abou he growh of your money a presen pur- 4

chasing power ignoring he effec of inflaion. This makes you hink ha you will ge a big amoun of money in reurn bu no he purchasing power of he money a ha ime. The fac is ha he nominal raes are more fanciful and he real raes ell you he rue value of your money even hough i is a small number as compared o he nominal. We give an example below. We suppose ha we buy a 1 year bond wih an ineres rae of 8 percen ha means if we pay $ 100 a he sar of he year we will ge $ 108 a he end of he year. This rae of ineres is nominal as we didn consider he inflaion. Suppose now ha he inflaion rae is 4 % in he same year. Tha means if you can buy a baske of goods for $ 100 oday, he same baske will cos $ 104 a he end of he year. So if we buy a bond wih 8 % nominal ineres rae for $ 100, sell i afer an year for $ 108 and buy ha baske of goods we lef wih $ 4. Tha means your real ineres rae is 4 %. There is a relaionship beween he real and nominal raes and he expeced inflaion called Fisher s equaion. The Fisher hypohesis is ha he real rae of ineres remain consan and he nominal rae of ineres changes wih inflaion. Since one canno predic he inflaion and nominal inerses raes Fisher equaion roughly says ha Real ineres raes = Nominal ineres rae - Inflaion. If he inflaion is posiive, he real raes would be lower han he nominal raes and if he here is deflaion he nominal raes would be lower hen he real ones. Bu because we canno predic he inflaion for he coming year we can only deal wih eiher real or nominal raes. 2.3 Inflaion-Indexed Securiies In general invesors focus on he nominal rae of reurn on heir invesmens and do no consider he real reurns. The inflaion do effec he rae of nominal reurns on your invesmen especially when you inves for a long erm. Purchasing a normal bond ensures you a nominal rae of reurn on your invesmens a mauriy and you do no know he real reurn or in oher words he purchasing power of your money because you can no predic he rae of inflaion. For example, he purchasing power of $ 100 would no be he same 30 years from now. If he inflaion is a 1 % on average hen $ 100 would have a purchasing power of $ 74 afer 30 years. So one has o ake care of inflaion when invesing money. 5

Of course you would be ineresed if someone ells you abou a securiy ha guaranees a real rae of reurn over inflaion wih no credi risk. A securiy ha guaranees a reurn higher han he rae of inflaion if i is held o mauriy is called an inflaion-indexed securiy. These securiies guaranee ha he holder is going o have he same purchasing power plus a fixed reurn on his invesmen. The use of indexed securiies is no new. I was raced back o 18 h - cenury, when he sae of Masschuses inroduced bills linked o he price of silver. During he American Revoluion, soldiers were issued wih depreciaion noes in order o preserve he real value of heir wages. Experience showed ha inflaion based on a single good was no a healhy idea and more complex indexaion mehods have been developed on differen ses of baskes hroughou hisory. Several counries experienced a very high inflaion during he las half of he 20 h -cenury and began o issue he indexed bonds as means of mainaining he accepabiliy of long-erm conracs. Some recen issues include UK(1981), Ausralia(1985), Canada(1991), Sweden(1994), US(1997), France(1998), Greece and Ialy(2003), Japan(2004) and Germany(2006). Governmens mosly issue he inflaion-indexed securiies bu corporaions could also benefi from i. There are sill some counries which have decided no o issue hese securiies and some have sopped issuing hese securiies, for example he Ausralian governmen has sopped issuing he CIBs in 2003. Inflaion-indexed securiies are insrumens ha can proec heir buyers from changes in he general level of he economy. These securiies could be a means of measuring he markes expecaions of inflaion. The mos common cash flow srucures are ineres indexed bond, curren pay indexed, capial indexed bonds and indexed zero-coupon bonds. These securiies proec asses and fuure income agains inflaion and provide diversificaion wih oher asses. 2.4 Derivaives Traded Inflaion derivaives are no a new concep and have been raded for more han a decade. They are generally a form of conrac in which wo paries agree o paymens which are based on he value of an underlying asse or some oher daa a some paricular ime. The purpose of inflaion derivaives is o minimise he inflaion risk in an efficien way for one pary while 6

offering he poenial for a high reurn o anoher. There are a number of insrumens ha can be used as inflaion derivaives. We assume ha he mauriy of our conrac is M years wih paymen daes as T 1,...T M. A swap where Pary A pays he inflaion rae over a predefined period o Pary B, while Pary B pays a fixed rae o Pary A on each paymen dae is defined as inflaion-indexed swap. The inflaion rae will be measured as he percenage reurn of he Consumer Price Index (CPI) over he inerval i applies o and is denoed by I (). The benchmark is he Zero-Coupon Inflaion Indexed Swap (ZCIIS) and is defined as: Zero-Coupon Inflaion Indexed Swap: In his conrac, Pary B agrees o pay a fixed amoun ] N [(1 + K) M 1 (2.1) o Pary A a he ime of mauriy i.e. a T M = M, where K and N are he conrac fixed rae ( srike ) and nominal value, respecively. In exchange, Pary A pays he floaing amoun in erms of inflaion [ ] I = N (TM ) 1 (2.2) I 0 o Pary B a he ime of mauriy T M. There is no exchange beween paries unil he ime of mauriy in a ZCIIS. These are he mos flexible conracs and are he building blocks for oher srucures. There is anoher common srucure of inflaion derivaives in which here is an annual paymen in beween paries known as Year-on-Year Inflaion Indexed Swaps (YYIIS) and is defined as: Year-on-Year Inflaion Indexed Swap: Le ϕ i denoe he year fracion of he fixed leg for he ime inerval [T i 1,T i ] and ψ i denoe ha of he floaing leg wih T 0 = 0 ha is we sar he conrac oday. In his conrac, Pary B agrees o pay a fixed amoun N ϕ i K 7

o Pary B a each ime T i. In exchange, Pary A pays Pary B he floaing amoun [ ] I N (Ti ) ψ i I (T i 1 ) 1 (2.3) a each ime T i. There are oher ypes of srucures in inflaion derivaives known as Inflaion-Indexed Caples (IIC) and Inflaion-Indexed floorles (IIF). We will no deal wih hese ypes in deail bu for he sake of compleeness we presen he definiions here. Inflaion Indexed Caples / Floorles An inflaion-indexed cap is a sring of IICs and an IIC, in inflaion marke, can be regarded as a call opion on he inflaion rae ha is implied by he CPI index. An inflaion-indexed floor is a sring of IIF and an IIF is a pu opion on he same inflaion rae. The payoff a ime T i is given as ( )] + I N (Ti ) ψ i [ω I (T i 1 ) 1 K, (2.4) where K is he srike of IIC or IIF, ψ i is he conrac year fracion for he inerval [T i 1,T i ], as before, and ω = 1 for caple and ω = 1 for a floorle. A cap is used o proec is buyer from he inflaion erosion above some rae and a floor is used o proec he invesmens from downside risks due o inflaion. Boh he invesors and he issuers can hedge heir inflaion exposures wih he help of floors and caps. 8

Chaper 3 Mahemaical Background In his chaper, we will review some basic definiion and heories ha we need in our laer chapers. We sar wih some fundamenals of mahemaical finance ha we need in our laer work. In he nex secion we discuss he change of numeriare echnique ha we use in pricing a call opion in chaper 4. We hen move on o he Heah-Jarrow-Moron (HJM) framework and exended Vasicek model. We need hese in chaper 4 as he Jarrow and Yildirim inflaion model is based on HJM framework and discussed his model by considering he dynamics of an exended Vasicek model. 3.1 Fundamenals of Mahemaical Finance This secion deals wih basic mahemaical finance opics ha are needed in he nex wo chapers. We define our financial marke and some heoraical conceps and han presen wo fundamenal heorem of asse pricing. These heorems sae he condiions o have a complee and arbirage free marke for opion pricing. We also included he Girsanov s heorem. This heorem ells us how o conver a physical measure o a risk neural measure. One can find he deails on hese opics in [1,9,10]. We sar by considering a financial marke consising of n + 1 asses S = (S 0,S 1,...,S n ) on a ime inerval [0,T]. We assume ha S 0 is a risk free asse and he remaining ones are risky asses. We consider a filered probabiliy space (Ω, F, F,P), where he filraion F is generaed by a n- dimensional Brownian moion. A rading sraegy ( φ ) is a predicable process where ( φ ) = (φ 0,...,φ n ). The value of his rading sraegy is V (φ) = φ. S = Σ n i=0φ i S i. 9

A rading sraegy is self-financing if dv (φ) = φ.d S = Σ n i=0φ i ds i, ha is here is no injecion or removal of money. We call a saegy admissible if V (φ) α for some α > 0 and [0,T] almos surely. 3.1 Definiion A measure Q, on a filered probabiliy space is equivalen o he physical measure P, denoed by Q P, if hey have he same null ses. This is called a maringale measure if all discouned price processes are maringales over [0,T].. In our case Q is such ha S i S 0, i are maringales under Q. 3.2 Definiion A financial marke is complee if every coningen claim is aainable. A coningen claim X T in our financial marke is a payoff a ime T which depends on he asses S i over [0,T] and a coningen claim is aainable if here is an admissible self-financing sraegy such ha V T (φ) = X T almos surely Theorem 3.1 (The Firs Fundamenal Theorem of Asse Pricing). Our financial marke is arbirage free if and only if here exiss an equivalen maringale measure Q P. Theorem 3.2 (The Second Fundamenal Theorem of Asse Pricing). An arbirage free financial marke is complee if and only if he equivalen maringale measure Q is unique and hence he prices of he coningen claim are also unique. There are wo fundamenal heorems of sochasic calculus ha allow us o apply maringale heory o arbirage heory. The Girsanov Theorem shows ha a change of measure is equal o a change of drif wih respec o he underlying Brownian moion. The Maringale Represenaion Theorem is an exisence heorem ha shows he exisence of an adaped process such ha any maringale adaped o he Brownian filraion can be wrien as an inegral of his adaped process adaped o he Brownian fileraion. We do no sae his as we do no use he maringale represenaion heorem explicily in our work. 10

Theorem 3.3 (The Girsanov Theorem). Consider a probabiliy space (Ω, F,P) wih a n-dimensional Brownian moion W. Le θ = (θ 1,...,θ n ) be an adaped n-dimensional process and pu ( z = exp θ ) 0 u.dw u 1 θ 2 0 u 2 du, where θ u 2 = Σ n i=1 (θ i u) 2. Le W = W + θ 0 udu and assume ha ( ) E θ 0 u 2 zudu 2 < and pu P (A) = A z T (w)p (dw). Then W is a n-dimensional Brownian moion under P. Theorem 3.4 (Risk Neural Pricing). In a complee, arbirage free financial marke, here exiss a unique measure Q such ha he discouned asse prices are maringales and he price V (,S ) of any coningen claim wih payoff X T is ) ( XT V (,S ) = S 0 E Q S 0 T 3.2 Change of Numeraire A numeraire is any sricly posiive, non-dividend paying radeable asse. If we assume ha he risk free asse S 0 is sricly posiive hen S 0 is a numeriare. We will find ha he T-bonds are useful numeraire in fixed income seings. We presen a heorem on change of numeraire, oher deails can be find in Brigo-Mercurio [1]. Theorem 3.5. Le N 1 be a numeriare and Q N 1 be he measure equivalen o P, such ha he asse prices S/N 1 are maringales under Q N 1. For an arbirary numeraire U, here exiss an equivalen measure Q U such ha any coningen claim X T has price V (,S ) = U E Q U [X T /U T ] 11

and moreover dq U dq N 1 and S is a maringale under Q U U. F = U T N T 1 N 1 U 3.3 Ineres Raes In his secion, we firs presen some known definiions and hen presen some resuls ha we use in our laer work. A main assumpion in our work is ha we are dealing wih a bond marke of ZCBs. All our definiions and resuls are presened in erms of he ZCBs. We presen he definiion of zero-copon bond as: 3.3 Definiion (Zero-Coupon Bond) A zero-coupon bond is a promis o pay 1 uni a a fixed mauriy a ime T. We call i a T bond and wrie P (,T) for is price a ime. 3.4 Definiion (LIBOR Rae) The LIBOR rae is a simple ineres ha banks charge each oher for loans for he ime period [S,T], available a. Mahemaically i is defined as F (,S,T) = 1 T S [ 1 P (,S) P (,T) The spo LIBOR is F (S,T) = F (S,S,T) = 1 T S ] [ 1 1 P(S,T) ]. Before giving he definiion of forward raes, we define he coninuously compounded rae R (,S,T). A coninuously compounded rae can be defined in erms of ZCBs as: e R(,S,T)(T S) = P (,S) P (,T). Or, ( ) log (P (,T)) log (P (,S)) R (,S,T) =. T S A spo coninuously compounded rae is ( ) log (P (S,T)) R (S,T) =. T S 12

We now define forward raes in erms of a coninuous compounded rae 3.3 Definiion (Forward Rae) The insananeous forward rae is he rae a for borrowing over (T,T + δ) and can be defined from he prices of a T-bond by log P (,T) f (,T) = lim R (,S,T) =. S T T The shor rae is he rae for borrowing now for an infiniesimal ime and is defined as r = f (,). Before going o he Heah Jarrow Moron approach, we sae a resul ha we use in chaper 4 in proving Proposiion 1. We consider our forward rae o be modelled as: df (,T) = α (,T) d + σ (,T) dw f (0,T) = f (0,T) (3.1) where α (,T) and σ (,T) are adaped processes, he denoes he marke observed value and W is an n-dimensional Brownian moion under P. We assume ha T T α (,T) d <, σi 2 (,T) d < i P a.s. 0 0 We assume ha for a fixed mauriy T > 0, he insananeous forward raes evolve as (3.1) where W = (W 1,...,W n ) is an n-dimensional Brownian moion, σ (,T) = (σ 1 (,T),...,σ n (,T)) is a vecor of adaped processes. The produc σ (,T) dw () is a scalar produc of wo vecors. The iniial condiion is he observed forward rae curve. Under he risk neural measure Q, we have P (,T) = E Q (e R T and we also have from Definiion 3.3 ( P (,T) = exp T r(u)du ) ) f (,s) ds. As boh formulae mus be rued and for he consisency in our model, we have o be careful o ensure ha here is no arbirage in he marke. This gives us he following heorem: 13

Theorem 3.6. Under our forward rae model, he bond prices mus saisfy dp (,T) = P (,T) (r () + A (,T) + 12 ) S (,T) 2 d+p (,T) S (,T) dw under P, where For proof, see [13]. T A (,T) = S (,T) = T α (,s)ds, σ (,s)ds. 3.4 Heah-Jarrow-Moron Framework (HJM) In 1986, Ho and Lee developed an alernaive o shor-rae models and modelled he enire yield curve in a binomial ree seing, [7]. Heah, Jarrow and Moron (HJM), inspired by his, developed a general framework for he modelling of ineres-rae dynamics, [8]. They modelled he insananeous forward raes direcly and showed ha here is a relaionship beween he drif and volailiy parameers of he forward rae dynamics in an arbirage free marke. The forward rae dynamics given in (3.1) do no necessarily lead o arbirage free bond marke. Heah, Jarrow and Moron showed ha for a unique maringale measure o exis, he funcion α mus have a specified form. They showed ha his mus be equal o a quaniy depending on he vecor volailiy σ and on he drif raes in he dynamics of n seleced zero-coupon bond prices. HJM Drif Condiion The processes α and σ mus saisfy α (,T) = σ (,T) T σ (,s) ds, (3.2) under he maringale measure Q for every T. The forward raes can also be expressed as For proof, see [13]. f (,T) = f (0,T) + 0 α (s,t)ds + 14 0 σ (s,t)dw (s).

3.5 Exended Vasicek Model The Vasicek model for he shor rae of ineres has mean reversion. As we indicaed above, an HJM model can be described by he insananeous forward rae volailiy. The Vasicek ype volailiy is given as σ (,T) = σe λ(t ), (3.3) where σ and λ > 0 are consans. The shor rae process for he exended Vasicek model is dr () = [θ () a ()r ()]d + σ ()dw (). (3.4) If we assume a () and σ () are posiive consans, we recover he exended Vasicek model of Hull and Whie (1994). We ge he dynamics of he shor rae as dr () = (θ () ar ())d + σdw (). (3.5) The ime dependen θ () is chosen o exacly fi he erm srucure of ineres raes currenly observed in he marke. Le f (0,T) and P (0,T) denoe he marke insananeous forward raes and he marke value of T-bonds, respecively, a ime 0 for all mauriies T, hen One can find ha in his case θ () = f (0,T) T f (0,T) = log P (0,T). (3.6) T + af (0,T) + σ2 ( ) 1 e 2a. (3.7) 2a One can use he affine erm srucure o find he value of a T-bond a ime, ha is P (,T) = A (,T)e B(,T)r(), (3.8) where B (,T) = 1 [ ] 1 e a(t ), a A (,T) = P (0,T) {B P (0,) exp (,T)f (0,) a2 ( ) } 1 e 2a B (,T) 2. 4a The exended Vasicek model has an affine erm srucure, hus i has an explici soluion for bond prices and i is relaively simple o price oher insrumens in his model. 15

Chaper 4 Models for Inflaion In his chaper, we will consider some simple models for pricing inflaion index derivaives from he lieraure. The emphasis is going o be on Jarrow and Yildirim (2003) in which hey used he HJM model o develop a hree facor model for inflaion. We presen he full deails of his model wih all he mahemaical se up. For he res of he models we presen only a brief descripion. 4.1 Jarrow and Yildirim Model (2003) The main reference on pricing inflaion-indexed securiies and relaed derivaives using he HJM model is Jarrow and Yildirim (2003)[2]. They considered he Treasury Inflaion Proeced Securiies (TIPS) which is an inflaion indexed bond in which he principal is adjused consanly for inflaion and also modify he semi-annual ineres paymens accordingly. Under he noarbirage assumpion, hey considered a cross-currency economy where nominal dollars correspond o he domesic currency, real dollars corresponds o he foreign currency and he inflaion index corresponds o he spo exchange rae. The flucuaions of he real and nominal ineres raes and he inflaion rae are correlaed in heir se up. 4.1.1 The Model We define he noaion as used in [2]: r for real, n for nominal. P n (,T) : ime price of a nominal zero-coupon bond mauring a ime T in dollars. 16

I () : ime CPI-U inflaion indexed ha is CPI for all urban consumers. P r (,T) : ime price of a real zero-coupon bond mauring a ime T in CPI-U unis. f k (,T) : ime forward raes for dae T where k {r,n}, i.e. { T } P k (,T) = exp f k (,u) du. (4.1) r k () = f k (,) : he ime spo rae where k {r,n}. { } T B k () = exp r k (v)dv : ime money marke accoun value for k {r,n}. B k (0) : ime 0 value of a coupon bearing bond in dollars where C is he coupon paymen in dollars per period, T is he mauriy and F is he face value in dollars, i.e. B n (0) = T CP n (0,) + FP n (0,T). (4.2) =1 B T IPS (0) : ime 0 price of a TIPS coupon bearing bond in dollars issued a ime 0 0 wih a coupon paymen C unis of he CPI-U, he mauriy is ime T, and he face value is F unis of he CPI-U, { T } B T IPS (0) = CI (0)P n (0,) + FI (0)P n (0,T) /I ( 0 ). (4.3) =1 In [2], Jarrow and Yildirim defined he price in dollars of a real zero-coupon bond wihou an issue dae adjusmen ha is wihou muliplying by he raio I(0) I( 0 ) as P T IPS (,T) = I ()P r (,T). (4.4) The uncerainy in he economy is characerized by a probabiliy space (Ω,F,P) where Ω is a sae space, F is a σ-algebra and P is he probabiliy measure on (Ω,F). Le{F : [0,T]} be he filraion generaed by he hree Brownian moions (W n (),W r (),W I () : [0,T]). These are sandard Brownian moions iniialized a zero wih correlaions give by dw n ()dw r () = ρ nr d,dw n ()dw I () = ρ ni d,dw r ()dw I () = ρ ri d. 17

Given he iniial forward rae curve f n (0,T), we assume ha he nominal T mauriy forward rae evolves as: df n (,T) = α n (,T)d + σ n (,T)dW n () (4.5) where α n (,T) is random and σ n (,T) is a deerminisic funcion. We assume ha α n (,T) is F adaped and joinly measurable wih T 0 α n (v,t) dv < in P a.s. and σ n (,T) saisfies T 0 σ n (v,t) 2 dv <, P a.s. In he same way, given he iniial forward rae curve f r (0,T), we assume ha he real T mauriy forward rae evolves as: df r (,T) = α r (,T)d + σ r (,T) dw r () (4.6) where α r (,T) and σ r (,T) saisfies he same condiions as forward raes for nominal. The inflaion index s evoluion is given by di I = µ I ()d + σ I ()dw I () (4.7) where µ I () is random and σ I () is a deerminisic funcion of ime saisfying µ I ()is F adaped wih E [ τ µ 0 I () 2 d ] < and σ I () saisfies τ σ 0 I () 2 dv < P a.s. These evoluions are arbirage-free and he marke is complee if here exis a unique equivalen probabiliy measure Q such ha: P n(,t), I()Pr(,T) B n() B n(), and I()Br() B n() are Q maringales. By Girsanov s heorem, given ha (W n (),W r (),W I () : [0,T]) is a P Brownian moion and ha Q is a probabiliy measure equivalen o P, hen here exis marke prices of risk (λ n (),λ r (),λ I () : [0,T]) such ha W k () = W k () 0 λ k (s)ds for k {n,r,i} are Q Brownian moions. We presen a proposiion ha gives necessary and sufficien condiions for he arbirage free erm srucure. 18

Proposiion 1: P n(,t), I()Pr(,T) B n() B n(), and I()Br() B n() are Q maringales if and only if he following condiions hold: ( T ) α n (,T) = σ n (,T) σ n (,s)ds λ n () (4.8) ( T ) α r (,T) = σ r (,T) σ r (,s) ds σ I ()ρ ri λ r () (4.9) µ I () = r n () r n () σ I ()λ I (). (4.10) Proof: We prove he second of he above and he ohers follow in he same way. Since he marke is arbirage free, all radables mus be maringales under Q. Le ξ = IPr(,T) B n(). Then by Io s formula dξ = r n ()d+ 1 B n () {I ()dp r (,T) + P r (,T) di () + dp r (,T) di ()}. By using he dynamics of P r (,T) in an arbirage-free marke from chaper 3 and I (), we ge = ξ [A r (,T) + 1 ] 2 S r (,T) 2 +σ I ()S r (,T) ρ ri + µ I () d + ξ [S r (,T)dW r + σ I ()dw I ], where A r (,T) = T α r (,s)ds and S r (,T) = T σ r (,s) ds. Now for ξ o be a maringale, we need [ dξ = ξ S r (,T) d W r + σ I ()d W ] I where W k is as defined above. Tha is we need, σ I ()S r (,T) ρ ri + A r (,T) + 1 2 S r (,T) 2 +µ I () Differeniaing wih respec o T, we ge = λ r ()S r (,T) λ I ()σ I (). σ I ()σ r (,T) ρ ri ) α r (,T) σ r (,T) S r (,T) = λ r ()σ r (,T). So we have he resul, [ T α r (,T) = σ r (,T) ] σ r (,s) ds σ I ()ρ ri ) λ r ()σ r (,T). 19

The firs expression in he proposiion is he arbirage-free forward rae drif resricion as in he original HJM model. The second is he analogous arbirage-free forward rae drif resricion for he real forward rae. We can see he correlaion and volailiy of he inflaion in his expression. The las expression is he Fisher equaion relaing he nominal ineres rae o he real ineres rae and expeced inflaion rae. The above proposiion and Io s lemma give he following erm srucure evoluion under he maringale measure. Proposiion 2: Under he maringale measure Q, we have he following price processes: T df n (,T) = σ n (,T) σ n (,s)dsd + σ n (,T)d W n () [ T ] df r (,T) = σ r (,T) σ r (,s) ds ρ ri σ I () d + σ r (,T)d W r () di () I () dp n (,T) P n (,T) dp TIPS (,T) P TIPS (,T) dp r (,T) P r (,T) = [r n () r r ()] d + σ I ()d W I () = r n ()d T σ n (,s)dsd W n () T = r n ()d + σ I ()d W I () σ r (,s) dsd W r () T ] T = [r r () ρ ri σ I σ r (,s)ds d σ r (,s)dsd W r () From hese expressions, he real and nominal forward raes are normally disribued and he inflaion index follows a geomeric Brownian moion. 4.1.2 Exended Vasicek Model and Hedge Raios. In his secion we presen he exended Vasicek model for nominal and real raes and also presen he hedge raios as given in [2]. Assuming an exponenially declining volailiy of he form σ k (,T) = σ k e a k(t ) (4.11) 20

and ha σ k is consan for k {n,r}, one can easily obain he exended Vasicek model for he shor raes dr n () = [θ n () a n r n ()] d + σ n dwn (), (4.12) dr r () = [ θ r () ρ ri σ I σ ( r) a r r r () ] d + σ r dwr (), (4.13) di () = [r n () r r ()] + σ I dw I (), (4.14) I () where θ k () are deerminisic funcions o be used o fi he erm srucures for nominal and real raes and are given as: θ k () = f k (0,T) T + a k f k (0,) + σ2 k 2a k ( 1 e 2a k ),k {n,r}. (4.15) where fx T denoes he parial derivaive of f x wih respec o is second argumen. Assumming ha boh nominal and real raes are normally disribued under heir respecive risk-neural measures Jarrow and Yildirim showed ha r is an Ornsein-Uhlenbeck process and he inflaion index I (), a each ime, is lognormally disribued under Q n. So we can wrie, for each < T, { T I (T) = I () exp [r n (u) r r (u)] du 1 2 σ2 I (T ) + σ I (W I (T) W I ())}. Using he equaion (4.11), we can easily find he delas as in he proposiion saed below. 21

Proposiion 3: P r (,T) r r () [I ()P r (,T)] r n () [I ()P r (,T)] I () [I ()P r (,T)] r r () P n (,T) I () P n (,T) r r () = P r (,T) b r (,T) σ r, (4.16) = 0, (4.17) = P r (,T), (4.18) ( ) Pr (,T) = I () r r () = I ()P r (,T) b r (,T) /σ r, (4.19) = 0, (4.20) = 0. (4.21) This indicaes ha he TIPS zero-coupon bond prices do no depend direcly on he nominal spo ineres rae and hence he TIPS erm srucure has only wo facors. The las wo equaions i.e. (4.20) and (4.21) show ha he nominal zero-coupon bond prices do no depend on he real spo ineres rae or he inflaion index. Bu hese facors are correlaed across he wo erm srucures. 4.1.3 Pricing Opions on he Inflaion Index. In heir paper, Jarrow and Yildirim presened he closed form formulas for he valuaion of he European call opion on he inflaion index. They consider a European call opion on he inflaion index wih a srike price of K index unis and a mauriy dae T. Noing ha he index is no deermined in dollars, bu dollars per CPI-U uni, hey conver he opion payoff o dollars by assuming ha each uni of opion is wrien on one CPI-U uni. Thus, he payoff of he opion a mauriy in dollars is: C T = max [I T K, 0]. (4.22) Under he risk neural measure, we ge he value of he opion as: C = E Q (max [I (T) K, 0]e R ) T r n(s)ds, (4.23) where E Q is he expecaion based on he maringale measure Q. 22

This can be solved by using P n (,T) as numeraire and using (4.6) for he evoluion of he inflaion index and given he exended Vasicek model for boh real and nominal erm srucures of ineres raes. We proceed as follows: C = E Q ( = P n (,T) I (T) 1 {I(T) K>0} e R T ) r n(s)ds KE Q ( 1 {I(T) K>0} e R ) T r n(s)ds, [ ( ) ( ) ] E Qn I (T) 1{I(T) K>0} KE Q n 1{I(T) K>0}, (4.24) where Q n is an equivalen measure under which ξ = I()Pr(,T) P n(,t) is a maringale. Now using proposiion 2, we find he dynamics of ξ where we do no give he drif erm explicily. We ge, dξ = ξ [(...) d + σ I ()d W I () T ] + σ n (,s)dsd W n (), Then by Girsanov s heorem, = ξ [σ I ()d W I () T T σ r (,s)dsd W r () + By solving he above SDE, we have { T = ξ exp σ I () d W I () where, ξ T + T where k {n,r}, and T σ P n (u,t) d W n (u) 1 2 η2 }, σ P k (,T) = T T σ r (,s)dsd W r () T σ k (,u) du ] σ n (,s)dsd W n (), σ P r (u,t) d W r (u) η 2 = σi 2 (T ) + σr P (u,t) 2 du + σn P (u,t) 2 du T T + 2 ρ nr σn P (u,t) σr P (u,t) du + 2ρ ni σ I () σn P (u,t) du 2ρ ri σ I () T σ P r (u,t) du. 23 T

Now we solve he second facor in equaion (4.24) above i.e, consider E Qn ( 1{I(T) K>0} ) = Q n (I (T) K > 0), = Q n (ξ T > K), ( = Q n Z < log ( ξ ) ) K 1 2 η2, η where Z N (0, 1) leing N be he sandard cumulaive normal disribuion funcion we have, ( ) ( ) E Qn 1{I(T) K>0} = N log I()Pr(,T) P n(,t)k 1 2 η2. η The firs facor in equaion (4.24) follows similarly by noing ha he nominal price of a real zero-coupon bond equals he nominal price of he conrac paying off one uni of he CPI index a he bond mauriy. In formulas, ( E Q I (T)e R ) ( T r n(s)ds = I ()E Q I (T)e R ) T r r(s)ds Repeaing he above process o calculae Q n we ge, ( ) C = I ()P r (,T)N log I()Pr(,T) P n(,t)k η ( ) KP n (,T)N log I()Pr(,T) P n(,t)k η + 1 2 η2 1 2 η2 (4.25) The example of exended Vasicek model shows ha here are several shorcomings wih his model, he mos imporan is ha he parameers are no direcly observable in he marke. The second drawback is ha i does no allow a link beween insrumens ha are raded such as zerocoupon and year-on-year producs. This las poin should no be considered if we are ineresed in pricing insrumens in markes where here is no a liquid marke for hese producs. 24

4.2 Mercurio Marke Models In 2004 [3], Mercurio gave wo marke models for inflaion-indexed swaps and opions. He also priced hese producs wih he Jarrow and Yildirim model in is equivalen shor-rae formulaion. In his firs marke model he used a lognormal LIBOR model for boh nominal and real raes and a geomeric Brownian moion for he inflaion index. In his second mehod he used he fac ha he forward index is a maringale a some paricular ime. We discuss hese and his paper in deail in our nex chaper. In (2005), Mercurio and Moreni [4] exended he resuls of [3] by inroducing sochasic volailiy as in he Heson model and derived closed form formulae for Inflaion Indexed caps and floors under he marke model wih sochasic volailiy. 4.3 Beldgrade-Benhamou-Koehlar Marke Model (2004) To fill he gape in he Jarrow and Yildirim model, Belgrade, Benhamou and Koehler derived a model ha is based on raded and liquid marke insrumens in [5]. They also derived a no-arbirage relaionship beween zero coupon and year-on-year swaps using he concep of marke model. They focused on a model ha is simple enough o have only a few parameers and robus in he sense ha i can replicae he marke prices. They assumed a marke model in which he forward CPI reurn is modelled as a diffusion wih a deerminisic volailiy. They discussed he case of Black and Scholes volailiy and he Hull and Whie framework. In he Black-Scholes case, where he volailiy is deerminisic and homogeneous, hey found he formula for year-on-year volailiies as a funcion of he zero-coupon volailiies. They also performed a convexiy adjusmen of he inflaion swaps ha are derived from he difference of maringale measures beween he numeraor and denominaor. They also suggesed some boundary condiions o esimae he implici correlaions from he marke daa. 25

Chaper 5 Marke Models of Inflaion In his chaper we will consider some marke models. Our main reference is Mercurio [3] in which he proposed wo marke models of inflaion o price general Inflaion-Indexed Swaps (IIS) and opions. He also modelled II swaps and opions using he Jarrow and Yildirim mehod as in [2]. In inflaionindexed swaps he inflaion rae is eiher payed on an annual basis or a single paymen a he swap mauriy. We sar by defining he Zero-Coupon Inflaion Indexed Swaps (ZCIIS) and Year-on-Year Inflaion Indexed Swaps (YYIIS). We hen presen he models as proposed by Mercurio [3] for YYIIS. We keep he noaion inroduced in our las chaper. 5.1 Inflaion-Indexed Swaps We have already defined he ZCIIS and YYIIS in chaper 2. We now look a he pricing of hese conracs. 5.1.1 Pricing Zero-Coupon IIS We apply sandard no-arbirage pricing heory, ha we developed in chaper 3, o value he inflaion-indexed leg of he ZCIIS a ime, 0 T M. We ge, { ZCIIS (,T M,I 0,N ) = N E n e R T M r r(u)du [ ] } I (TM ) 1 F, (5.1) I 0 where F denoes he σ-algebra generaed by he relevan underlying processes up o ime and E n {. F } denoes he expecaion under he risk neural measure Q n as defined in chaper 4 when dealing wih he exended 26

Vasicek model. The nominal price of a real-zero coupon bond is I ()P r (,T) ha is by eliminaing he effec of inflaion from he real price of he bond. So he nominal price of real zero-coupon bond equals he nominal price of he conrac paying off one uni of he CPI index a he bond mauriy. Mahemaically, for each < T : { I ()P r (,T) = I ()E r e R } { T r r(u)du F = E n e R } T r n(u)du I (T) F. We can wrie quaion (5.1) as: ZCIIS (,T M,I 0,N ) = N I 0 E n { e R T M r r(u)du I (T M ) F } N E n { e R T M r r(u)du F }, (5.2) Now by using (5.2) and he definiion of nominal zero bond, we ge [ ] I () ZCIIS (,T M,I 0,N ) = N P r (,T M ) P n (,T M ). (5.3) I 0 A ime = 0, he above equaion simplifies o ZCIIS (0,T M,N ) = N [P r (0,T M ) P n (0,T M )]. (5.4) Mercurio in [3] noed ha he above wo equaions give model independen prices which are no based on any specific assumpions on he evoluion of he ineres rae marke bu follows from he absence of arbirage. This enables us o srip real zero-coupon bond prices from he quoed prices of zero-coupon inflaion-indexed swaps, as given below. If he marke quoes he values of K = K (T M ) for some given mauriies T M, hen we equae he presen nominal value of equaion (2.1), ha is [ ] N P n (,T M ) (1 + K) M 1 wih equaion (5.4). We ge he unknown P r (0,T M ) in he real economy for mauriy T M as P r (0,T M ) = P n (0,T M ) (1 + K (T M )) M. 27

5.1.2 Pricing Year-on-Year IIS Finding he value of YYIIS is no as direc as finding he value of ZCIIS. Using he definiion as in chaper 2, he value of he payoff (2.3) a ime T i is YYIIS (,T i 1,T i,ψ i,n ) = N ψ i E n { e R T i [ ] } r n(u)du I (Ti ) I (T i 1 ) 1 F. (5.5) If > T i 1 equaion (5.5) reduces o pricing he floaing leg of a ZCIIS as in equaion (5.4). When < T i 1, equaion (5.5) can be wrien in erms of ieraed expecaions as { [ = N ψ i E n e R T i 1 r n(u)du E n e R [ ] T i T r i 1 n(u)du I (Ti ) I (T i 1 ) 1 F Ti 1 ] F }. (5.6) The inner expecaion is he ZCIIS (T i 1,T i,i (T i 1 ),1), shown above, so we ge he resul { YYIIS (,T i 1,T i,ψ i,n ) = N ψ i E n e R T i 1 r n(u)du [P r (T i 1,T i ) P n (T i 1,T i )] F }, { = N ψ i E n e R } T i 1 r n(u)du P r (T i 1,T i ) F N ψ i P n (,T i ). (5.7) This expecaion is he nominal price of a payoff equal o he real zero-coupon bond price P r (T i 1,T i ) a ime T i 1 in nominal unis. If we assumed ha he real raes were deerminisic, he above equaion gives he presen value of he forward price of he real bond and we ge { E n e R } T i 1 r n(u)du P r (T i 1,T i ) F = P r (T i 1,T i )P n (,T i 1 ), = P r (,T i ) P r (,T i 1 ) P n (,T i 1 ). Since we have sochasic real raes in our model of chaper 4 in equaions (4.12) o (4.14), he forward price of a real bond mus be correced by a facor depending on boh he real and nominal ineres rae volailiies and heir respecive correlaions. We price YYIIS firs using he Jarrow-Yildirim model and hen he wo marke models as proposed by Mercurio [3]. 28

5.2 Pricing wih he Jarrow Yildirim Model Here onward, we denoe he nominal T forward measure for mauriy T by Q T n and he associaed expecaions by E T n. So we can wrie: YYIIS (,T i 1,T i,φ i,n ) = N φ i P n (,T i 1 ) E T i 1 n {P r (T i 1,T i ) F } N ψ i P n (,T i ). (5.8) We use he same formula for zero-coupon bond prices ha were given in (3.8), afer changing he noaion according o our model. Tha is, we consider P r (,T) = A r (,T) e Br(,T)rr(), B r (,T) = 1 [ ] 1 e a r(t ), a r A r (,T) = P { r M (0,T) (0,) exp B r (,T) fr M P M r (0,) σ2 ( ) } r 1 e 2a r B r (,T) 2. 4a r Now by using he change-of-numeraire echnique developed in secion (3.2), we ge he real insananeous rae evoluion under Q T i 1 n as follows: dr r () = [θ r () a r () ρ ri σ r σ I ρ nr σ n σ r B n (,T i 1 )] d σ r dw T i 1 r (), (5.9) where W T i 1 r is a Q T i 1 n -Brownian moion. Since he real raes are normally disribued under Q T i 1 n, our real bond price P r (T i 1,T i ) is lognormally disribued under he same measure. The real rae has he mean and variance, hese derivaions are similar o ha done in [1], p 73, given as follows: E [r r (T i 1 ) F ] = r r ()e ar(t i 1 ) + β (T i 1 ) β ()e ar(t i 1 ) ρ ri σ r σ I B r (,T i 1 ) ρ nrσ n σ r [B r (,T i 1 ) B n (,T i 1 ) a r + a n + a r B r (,T i 1 )B n (,T i 1 )]. V ar [r r (T i 1 ) F ] = σ2 r 2a r [ e 2a r(t i 1 ) 1 ], where, β () = f r (0,) + σ2 r 2a r ( e a r 1 ) 2. Now as he real bond price is lognormally disribued under Q T i 1 n, we ge he closed form formulae for YYIIS as in Mercurio [3] YYIIS (,T i 1,T i,ψ i,n ) = N ψ i P n (,T i 1 ) P r (,T i ) P r (,T i 1 ) ec(,t i 1,T i ) N ψ i P n (,T i ), (5.10) 29

where C (,T i 1,T i ) = σ r B r (,T i 1 ) + ρ nrσ n a n + a r (1 + a r B n (,T i 1 )) [ ( B r (,T i 1 ) ρ ri σ I 1 2 σ rb r (,T i 1 ) ) ρ nrσ n a n + a r B n (,T i 1 ) This shows ha in he Jarrow-Yildirim model he expecaion of a real zero-coupon bond is equal o he curren forward price of he zero-coupon bond muliplied by a correcion facor where he expecaion is aken under a nominal forward measure. The correcion facor depends on he volailiy of he nominal rae, real rae and he CPI as well as he correlaion beween he nominal and real raes and he real raes and he CPI, where hese volailiies and correlaions are insananeous. This correcion facor is due o he fac ha he real raes are sochasic in his model and i vanishes for σ r = 0. The value of he inflaion-indexed leg of he swap a ime is he sum of all floaing paymen values a ime. So we ge, [ ] YYIIS (, T, Ψ,N ) = N I () ψ ι() I ( ( ) ( ) )P r,tι() Pn,Tι() T ι() 1 ]. + N Σ M i=ι()+1ψ i [ P n (,T i 1 ) P r (,T i ) P r (,T i 1 ) ec(,t i 1,T i ) P n (,T i )], (5.11) where T = {T 1,T 2,...,T M }, Φ = {φ 1,φ 2,...,φ M } and ι () = min {i : T i > }. The value of YYIIS a ime = 0 is YYIIS (0, T, Ψ,N ) = N ψ 1 [P r (0,T 1 ) P n (0,T 1 )] + N Σ M i=2ψ i [P n (0,T i 1 ) ] P r (0,T i ) P r (0,T i 1 ) ec(0,t i 1,T i ) P n (0,T i ) (5.12) [ 1 + = N Σ M τi F n (0;T i 1,T i ) i=1ψ i P n (0,T i ) 1 + τ i F r (0;T i 1,T i ) ec(0,t i 1,T i ) 1] Modelling using he Hull-Whie-Vasicek model, which is a Gaussian model, helps us o find he formulae analyically bu he possibiliy of negaive raes and he difficuly in esimaing he real rae parameers pushed Mercurio o invesigae alernaive approaches. In he nex wo secion, we presen wo differen marke models for valuaion of a YYIIS and inflaion-indexed opions as proposed by Mercurio [3]. 30

5.3 Pricing wih a Firs Marke Model To price he above YYIIS, Mercurio noiced ha we can change he measure and can re-wrie he expecaion in (5.5) as follows { } P n (,T i 1 )E T i 1 n {P r (T i 1,T i ) F } = P n (,T i )E T Pr (T i i 1,T i ) n P n (T i 1,T i ) F { = P n (,T i )E T i 1 + τi F n (T i 1 ;T i 1,T i ) n 1 + τ i F r (T i 1 ;T i 1,T i ) F }, (5.13) where F r and F n are he LIBOR as defined in chaper 3. This expecaion can be calculaed when we specify he disribuion of forward raes under he nominal T i forward measure. Mercurio suggesed a lognormal LIBOR model and posulaed a model for he evoluion of he compounded forward raes ha appear in (5.13) In he nominal economy, price of an asse is given by I ()P r (,T i 1 ). This is a maringale under he risk neural measure Q T i 1 n when discouned by P n (,T i 1 ), by defininion, and is he forward CPI. Noing his fac, we le I i () := I () Pr(,T i) P n(,t i ). In [3], Mercurio assumed he lognormal dynamics of I i, wih σ ii, a posiive consan and Wi I a Brownian moion under Q T i 1 n ha is, di i () = σ Ii I i ()dw i I (). (5.14) Now we assume ha boh nominal and real forward raes follow a LI- BOR marke model, he analogy wih cross-currency derivaive pricing gives he dynamics of F n (.;T i 1,T i ) and F r (.;T i 1,T i ) under Q T i n as observed by Schlögl, 2002 [6] df n (;T i 1,T i ) = σ ni F n (,T i 1,T i )dw n i (), df r (;T i 1,T i ) = F r (;T i 1,T i ) [ ρ I,ri σ Ii σ ri d + σ ri dw r i ()], (5.15) where Wi n and Wi r are wo Brownian moions wih insananeous correlaion ρ i and σ ni and σ ri are posiive consans. ρ I,ri is he insananeous correlaion beween I i (.) and F r (.;T i 1,T i ), ha is dwi I ()dwi r = ρ I,ri d. 31

The expecaion in (5.16) can calculaed by noing ha he pair ( (X i,y i ) = log F n (T i 1 ;T i 1,T i ), log F ) r (T i 1 ;T i 1,T i ) F n (;T i 1,T i ) F r (;T i 1,T i ) (5.16) is disribued as a bivariae normal random variable under Q T i n. Using he dynamics of F n (;T i 1,T i ) and F r (;T i 1,T i ) as in (5.15 ), he mean vecor and variance-covariance marix of his bivariae random variable are given as where M Xi,Y i = [ µx,i () µ y,i () ] [, V Xi,Y i = σx,i 2 () ρ i σ x,i ()σ y,i () ] ρ i σ x,i ()σ y,i () σy,i 2, () (5.17) µ x,i () = 1 2 σ2 ni (T i 1 ), σ x,i () = σ ni Ti 1, µ y,i () = [ 12 ] σ2ri ρ I,ri σ Ii σ ri (T i 1 ), σ y,i () = σ ri Ti 1. We can decompose a densiy funcion f Xi,Y i (x,y) of (X i,y i ) as f Xi,Y i (x,y) = f Xi Y i (x,y) f Yi (y). (5.18) Where in our case, we have x µ x,i () y µ 1 f Xi Y i (x,y) = σ x,i () 2π σ exp x,i ρ y,i () () i σ y,i () 1 ρ 2 2 (1 ρ 2 i i ), [ 1 f Yi (y) = σ y,i () 2π exp 1 ( ) ] 2 y µy,i (). (5.19) 2 σ y,i () As x = log Fn(T i 1;T i 1,T i ) F n(;t i 1,T i ), we have 1 + τ i F n (T i 1 ;T i 1,T i ) = 1 + τ i F n (,T i 1,T i )e x Therefore using his and decomposiion in (5.18), we ge E T i n { } 1 + τi F n (T i 1 ;T i 1,T i ) 1 + τ i F r (T i 1 ;T i 1,T i ) F = + [ + f Yi (y)dy, 1 1 + τ i F r (;T i 1,T i )e y ] (1 + τ i F n (;T i 1,T i ) e x ) f Xi Y i (x,y) dx 32

which is hen equal o = + 1 + τ i F n (;T i 1,T i ) e µ y µ x,i()+ρ i σ x,i () y,i() + 1 σ y,i () 2 σ x,i() 2 (1 ρ 2 i) «2 y µy,i () 1 + τ i F r (;T i 1,T i )e y e 1 2 σ y,i () σ y,i () 2π dy Using he ransformaion z = y µ y,i() σ y,i (), = + 1 + τ i F n (;T i 1,T i )e ρ iσ x,i ()z 1 2 σ x,i() 2 ρ 2 i 1 + τ i F r (;T i 1,T i )e µ y,i()+σ y,i ()z 1 2π e 1 2 z2 dz, Therefore we ge, + YYIIS (,T i 1,T i,ψ i,n ) = N 1 ψ i P n (,T i ) e 1 2 z2 2π 1 + τ i F n (;T i 1,T i )e ρ iσ x,i ()z 1 2 σ x,i() 2 ρ 2 i 1 + τ i F r (;T i 1,T i )e µ y,i()+σ y,i ()z dz N ψ i P n (,T i ). (5.20) We canno value he whole of he inflaion-indexed leg of his swap by summing up values in (5.12), as in he Jarrow Yildirim model. This was noiced by Schlögl [6] ha he assumpions of volailiies σ Ii,σ ni and σ ri as posiive consans for all i is no righ. We can find he relaion beween wo consecuive forward CPI s and corresponding nominal and real forward raes as follows: By definiion of forward CPI I i () I i 1 () = P r (,T i ) P n (,T i 1 ) P r (,T i 1 ) P n (,T i ), and since P k (,T i 1 ) P k (,T i ) where k {r,n}. Tha is we have, = 1 + τ i F k (,T i 1,T i ). I i () I i 1 () = 1 + τ if n (;T i 1,T i ) 1 + τ i F r (;T i 1,T i ). (5.21) 33

So if we assume σ I,i,σ ni and σ ri are posiive consan for some i, hen is clear from he above equaion ha σ r,i 1 is no longer a consan. To deal wih his Mercurio [3] used he echnique of freezing he forward raes a heir ime 0 value in he diffusion coefficien of he righ hand side of equaion (5.21) and noiced ha we can sill ge approximaely consan CPI volailiy. Noe ha his gives an approximaed value for he model. He hen applied his Freezing procedure saring from σ I,M for each i < M or equivalenly for each i > 2 and saring from σ I,i are all consans and se o one of heir admissible values. Then he value of he inflaion-indexed leg of he swap a ime is given by [ ] YYIIS (, T, Ψ,N ) = N I () ψ ι() I ( ( ) ( ) )P r,tι() Pn,Tι() T ι() 1 In paricular for = 0, + N Σ M ι()+1ψ i P n (,T i ) [ + 1 2π e 1 2 z2 1 + τ i F n (;T i 1,T i )e ρ iσ x,i ()z 1 2 σ x,i() 2 ρ 2 i 1 + τ i F r (;T i 1,T i ) e µ y,i()+σ y,i dz ()z 1]. (5.22) YYIIS (0, T, Ψ,N ) = N ψ 1 [P r (0,T 1 ) P n (0,T 1 )] + N Σ M i=2ψ i P n (0,T i ) [ + 1 e 1 2 z2 2π 1 + τ i F n (0;T i 1,T i )e ρ iσ x,i (0)z 1 2 σ x,i(0) 2 ρ 2 i 1 + τ i F r (0;T i 1,T i )e µ y,i(0)+σ y,i dz 1 (0)z = N Σ M i=1ψ i P n (0,T i ) [ + 1 2π e 1 2 z2 1 + τ i F n (0;T i 1,T i )e ρ iσ x,i (0)z 1 2 σ x,i(0) 2 ρ 2 i 1 + τ i F r (0;T i 1,T i )e µ y,i(0)+σ y,i dz (0)z 1]. (5.23) Hence for his case YYIIS depends on he insananeous volailiies of nominal and real forward raes and heir correlaions, and he insananeous volailiies of forward inflaion indices and heir correlaion wih real forward raes, for each paymen ime T i, i = 2,...,M. ] 34

If we compare Mercurio s firs marke model wih (5.12), (5.22) looks more complicaed in erms of he calculaions and in erms of inpu parameers. The inpu parameers can be deermined more easily han hose coming from he previous shor-rae approach. This is a srong feaure of marke models due o absence of arbirage. In his respec (5.22) is preferable o (5.12). Esimaing he volailiies of real raes is a drawback of he Jarrow Yildirim model when valuing a YYIIS wih a LIBOR marke model. To overcome his problem Mercurio [3] proposed a second marke model approach, which we discuss in he nex secion. 5.4 Pricing wih a Second Marke Model In his second marke model, Mercurio [3] used he definiion of forward CPI and he fac ha I i is a maringale under Q T i n. Using his for < T i 1, { } YYIIS (,T i 1,T i,ψ i,n ) = N ψ i P n (,T i )E T i I (Ti ) n I (T i 1 ) 1 F { = N ψ i P n (,T i )E T Ii (T i i ) n I i 1 (T i 1 ) 1 F } (5.24) = N ψ i P n (,T i )E T i n { Ii (T i 1 ) I i 1 (T i 1 ) 1 F Since he purpose of his echnique is o overcome he shor comings of he Jarrow-Yildirim model in Secion 5.2, Mercurio considered he dynamics of I i under Q T i n as in (5.14) and a similar one for I i 1 under Q T i 1 n. Mercurio solved he quesion regarding he dynamics of I i 1 under Q T i n by he changeof-numeraire echnique, ha is τ i σ ni F n (;T i 1,T i ) di i 1 () = I i 1 ()σ I,i 1 1 + τ i F n (;T i 1,T i ) ρ I,nid+σ I,i 1 I i 1 ()dwi 1 I (), (5.25) where σ I,i 1 is a posiive consan, Wi 1 I is a Q T i n Brownian moion wih dwi 1 I ()dwi I () = ρ i,i d and ρ I,ni is he insananeous correlaion beween I i 1 (.) and F n (.;T i 1,T i ). }. The calculaions of (5.24) involve he evoluion of I i 1 under Q T i n depends on he nominal rae F n (.;T i 1,T i ). ha 35

This could cause complicaions like hose induced by higher dimensional inegraions. To avoid his, Mercurio again used he freezing echnique for he drif par of equaion (5.25) a is curren ime. This makes he disribuion of I i 1 (T i 1 ) condiional on F lognormal under Q T i E T i n { } Ii (T i 1 ) I i 1 (T i 1 ) F n and hence = I i () I i 1 () ed i(), where [ ] τi σ ni F n (;T i 1,T i ) D i () = σ I,i 1 1 + τ i F n (;T i 1,T i ) ρ I,ni ρ Ii σ Ii + σ I,i 1 (T i 1 ), so ha YYIIS (,T i 1,T i,n ) = N ψ i P n (,T i ) [ ] Pn (,T i 1 )P r (,T i ) P n (,T i )P r (,T i 1 ) ed i() 1. (5.26) Finally, he value a ime of he inflaion-indexed leg of he swap is given by [ ] YYIIS (, T, Ψ,N ) = N I () ψ ι() I ( ( ) ( ) )P r,tι() Pn,Tι() T ι() 1 In paricular a = 0, + N Σ M i=ι()+1ψ i [ P n (,T i 1 ) P r (,T i ) P r (,T i 1 ) ed i() P n (,T i )]. (5.27) YYIIS (0, T, Ψ,N ) = N ψ 1 [P r (0,T 1 ) P n (0,T i )] + N Σ M i=2ψ i [ P n (0,T i 1 ) P ] r (0,T i ) P r (0,T i 1 ) ed i(0) P n (0,T i ) (5.28) [ ] 1 + = N Σ M τi F n (0;T i 1,T i ) i=1ψ i P n (0,T i ) 1 + τ i F r (0;T i 1,T i ) ed i(0) 1. The above formula for YYIIS depends on insananeous volailiies of nominal forward raes; he insananeous volailiies of forward inflaion indices and heir correlaions; he insananeous correlaions beween forward inflaion indices and nominal forward raes. This expression combines he advanage of a fully-analyical formula wih he marke-model approach and may be preferred o (5.23) for his reason. 36

The correcion erm in his formula does no depend on he volailiy of real raes as in equaion (5.22). There is a drawback of his formula in ha he approximaion i is based on can be rough for long mauriies T i. In fac his formula is exac when he correlaions beween I i 1 (.) and F n (.;T i 1,T i ) ha is ρ I,ni are se o zero and he erms D i simplified accordingly. This is a major problem as many purchases of inflaion producs, such as a pension funds, wan very long daed insrumens. However, in general, such correlaions can have a non-negligible impac on he D i. In his paper [3], Mercurio hen inroduced he inflaion-indexed caples and floorles and priced hem using he mehods described above. His second marke model gives a formula ha has he same advanages and drawbacks as he swap price, in erms of inpu parameers, because he caple price depends on he insananeous volailiies of nominal forward raes and he insananeous correlaions beween he nominal forward raes and he forward inflaion index. The formula is analogous o he Black-Scholes formula and hence is more pracical suppor for he modelling of forward CPIs as geomeric Brownian moion. Mercurio and Moreni hen give an exension o he above models of pricing he caples and floorles. They noiced ha he Black-Scholes like formula is good for pricing II-caps wih differen mauriies bu he same srike. They assume ha he nominal forward raes are lognormally disribued wih consan velociies and ha forward CPI s have a Heson evoluion wih common volailiy bu sochasic his ime. As he payoff ha depends on he raio of wo differen asses a wo differen ime hey use Fourier ransforms. They go some parial differenial equaion ha does no seem o solve explicily and hen resric o some paricular cases. They found closed form formulae when assuming he correlaion beween forward raes and forward CPI s and beween he forward raes and volailiy are all zero. They noiced ha using freezing drif echniques, i is possible o derive an efficien approximaion o he price. They also found ha inroducing he sochasic volailiy gives beer resuls and is a much beer fi han in he deerminisic case. There is always some opening o find more efficien formulae and one can use a differen sochasic volailiy process for each forward CPI. 37

Chaper 6 Conclusion To undersand and price he inflaion indexed derivaives one needs o have an undersanding of ineres rae modelling as his is a building block for he heories on inflaion-indexed derivaives. The dynamics of real and nominal ineres raes as well as he inflaion index is an imporan assumpion in pricing hese derivaives. These procducs was iniiaed for penion holders bu now growing rapidly and is a need of oday s world. In our work we discussed he firs model of inflaion developed by Jarrow and Yildirim in full deails. Noing he weaknesses of his model, Mercurio developed wo marke models ha are given in chaper 5. We priced ZCIIS and YYIIS using hese models as developed by he auhors. We can also price oher derivaives inflaion indexed caps and floors using hese models. In our work we discussed he models developed up ill 2005 and did no look a he furher exensions. Hinnerich in 2006 exended he HJM approach proposed by Jarrow and Yildirim [2] o mulifacor HJM model and allowed he possibiliy of jumps in he economy [11]. He proposed he explici closed form soluions of he derivaives. In 2007, Zhu exended he sandard LIBOR marke models by inroducing mean reversion and by using all sochasic volailiies ha are correlaed o he LIBOR individually, [12]. There are many oher exensions o he models we have presened here as well as oher models ha one can look a for fuher consideraions. 38

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