Pricing Inflation-Indexed Derivatives Using the Extended Vasicek Model of Hull and White

Transcription

1 Pricing Inflaion-Indexed Derivaives Using he Exended Vasicek Model of Hull and Whie Alan Sewar Exeer College Universiy of Oxford A hesis submied in parial fulfillmen of he MSc in Mahemaical Finance April 19, 27

2 Conens 1 Inroducion Inflaion Sae of he marke Main users Characerisics of he marke Choice of Index Seasonaliy Indexaion Pricing inflaion-indexed derivaives in he Heah-Jarrow-Moron framework HJM no-arbirage dynamics in a single currency seing The exended Vasicek model of Hull and Whie in he HJM framework Inroducing he real economy Dynamics in he forward measure Derivaion of prices for inflaion-indexed derivaives Zero-coupon inflaion-indexed swaps Year-on-year inflaion-indexed swaps Year-on-year inflaion-indexed swap for Hull Whie model wih consan volailiy parameers Inflaion indexed Caps and Floors Calibraion Hull-Whie zero coupon bond dynamics Hull-Whie zero coupon bond opion Swapions Swapion marke quoes Hull-Whie swapion Caps and Floors

3 4.4.1 Cap/Floor marke quoes Hull-Whie caps and floors Implemenaion of he inflaion model calibraion A Inflaion Indexed Caple formula for consan Hull-Whie parameers 34 B Marke Daa 38 References 39 3

4 Chaper 1 Inroducion The purpose of his hesis is o review he framework for pricing inflaion-indexed derivaives using he wo currency Heah-Jarrow-Moron approach inroduced by Jarrow and Yildirim 11] and o derive prices for he mos commonly raded inflaion-indexed derivaives using he Hull Whie model. The firs chaper gives an overview of he inflaion markes and gives a brief descripion of he securiies ha are raded and heir liquidiy in he major markes. Some of he consideraions peculiar o he inflaion-indexed markes such as seasonaliy and indexaion are hen reviewed. The second chaper provides he mahemaical background for he model. Gaussian Markov shor-rae models are described in he HJM framework and he resricions on he HJM volailiy srucure ha allow he dynamics o be represened by a Markovian shor rae model are described. The dynamics of he zero-coupon bond in he single currency seing is hen derived in erms of he shor rae parameers. The real economy is hen inroduced in erms of he economy of he foreign currency and he maringale measure is consruced for he exended se of nominal radables. The hird chaper describes he mos popular inflaion-indexed derivaive securiies in more deail and in paricular derives prices for year-on-year inflaion-indexed swaps and inflaion-indexed caps and floors using he model. The fourh chaper describes calibraion consideraions. In paricular i reviews how he mos popular nominal derivaives such as swapions and caps/floors can be expressed in erms of opions on zero-coupon bonds and hence how hey can be priced using he dynamics of he zero-coupon bond ha were derived in he second chaper. A simple approach o calibraing he inflaion model is also described. 1.1 Inflaion Economiss generally define inflaion as a susained increase in he level of prices in an economy. The price level may be defined eiher in erms of he GDP deflaor or by he consumer price index. The GDP deflaor is a measure of he average price of goods produced by an economy whereas he consumer price index is concerned wih he price of goods consumed. The wo alernaive measures are in general no he same bu are no significanly differen. We are only concerned wih consumer 4

5 prices here as almos all price indexed securiies are linked o his index. The consumer price index is defined in erms of a baske of goods and services. We migh ask why economiss are concerned wih inflaion and why conrolling he rae of inflaion is such a significan elemen of cenral bank policy. If he inflaion rae was consan and all prices and wages rose a he same rae, inflaion would no really be a significan problem. The concern is wih uncerainy and he relaive changes in prices of differen iems and wages. Changes in prices relaive o wages has an obvious effec on he sandard of living and he lack of cerainy is also of concern o businesses making invesmen decisions and negoiaing wages. Over he pas housand years, periods of susained increases in prices have acually been in he minoriy and price sabiliy has been he norm. However, during he second half of he las cenury inflaion ook off and has only been conained again in recen years. Global CPI Growh 4% CPI 4-year average 3% 2% 1% % Sae of he marke Alhough inflaion-indexed securiies have been around in some form for hundreds of years, significan liquid markes only really sared developing in he early 198s. The firs producs were governmen bonds, originally issued by Canada and Ausralia and hen by he UK in France followed and hen in 1997 he US Federal Reserve issued is firs inflaion indexed reasury bonds, which are known as Treasury Inflaion Proeced Securiies (TIPS. The Inflaion-indexed derivaive marke didn really exis prior o 22, bu has developed rapidly since hen. In Europe here is now a 5

6 developed marke in vanilla inflaion indexed derivaives - primarily in zero coupon inflaion indexed swaps. From a governmen s perspecive issuing inflaion indexed bonds sends a srong saemen o he markes abou is inflaion fighing inenions. If marke inflaion expecaions are higher han hose of he governmen, i also provides a relaively cheap source of governmen funding as he marke is willing o pay more for he higher expeced nominal cash flows. For boh of hese reasons, he proporion of governmen deb issued in inflaion-linked form has been increasing in mos developed counries in recen years. Inflaion indexed derivaives have all of he normal benefis associaed wih derivaive producs. They can be raded over he couner and can herefore be ailored o mee specific needs, hey are off-balance shee and in heory have limiless supply. Inflaion swap curves are now well defined ou o abou 3 years in Europe and for he firs ime he swap marke is saring o drive he bond markes. However, zero-coupon inflaion-indexed swaps are he only really liquid inflaion-indexed derivaives. Year-on-year swaps and inflaion-indexed caps and floors are becoming increasingly common, bu are sill considered o be exoic producs in mos respecs. The pricing formulae for zero-coupon swaps, year-on-year swaps and inflaion-indexed caps and floors are derived using he seleced model in chaper Main users The economy conains some eniies ha are naural payers of inflaion and ohers ha are naural receivers. Driven o some exen by governmen accouning legislaion, here is currenly a srucural surplus of demand for inflaion proecion over supply in mos developed economies. The excess demand has caused real yield levels o fall and real yield curves o flaen driven by demand for long daed inflaion proecion and guaraneed real yields. In he Unied Saes and he Unied Kingdom, he real yield curves are currenly invered for longer mauriies. 1.4 Characerisics of he marke Choice of Index There are many alernaives o choose from when deciding which index o link a securiy o. In heory securiies could pay cashflows linked o he level of wages, core inflaion (excluding energy or he more common headline consumer prices. In he Unied Kingdom mos index linked securiies are linked o he RPI index, bu he governmen s official inflaion arge is now expressed in erms of he CPI index. The CPI index is more consisen wih he European harmonised measure of inflaion, bu someone hedging CPI exposure wih inflaion derivaives is exposed o basis risk as he wo measures are no perfecly correlaed. In a similar way, exposure o increases in wages 6

7 canno be hedged perfecly using RPI linked insumens. Because he indices are generally based on headline CPI (including food and energy, inflaion linked securiies may be sensiive o shor erm flucuaions caused by volaile facors such as he price of oil. Cenral banks generally arge core inflaion as hey are more concerned wih he longer erm rend han wih shor erm flucuaions, bu his means ha he wo measures can deviae significanly for shor periods and his inroduces hedging risk for hose hedging core inflaion exposure wih inflaion indexed derivaives Seasonaliy Mos inflaion indexed securiies are linked o unrevised indices ha are no seasonally adjused. Consumer prices generally exhibi seasonal behaviour and he calculaion of seasonal adjusmens is an imporan aspec of pricing inflaion indexed producs, paricularly for vanilla producs when cash flows occur a differen imes of year. Unil recenly seasonal adjusmens have been calculaed from hisorical index daa using saisical echniques such as hose reviewed by Belgrade and Benhamou 2]. They discuss a parameric leas squares echnique and a non-parameric (X11 approach. The following seasonal adjusmens were calculaed using he parameric approach fiing a log-linear model o hisorical index values. Seasonal adjusmens for he European HICP index Jan Feb Mar Apr May Jun Jul Aug Sep Oc Nov Dec In he European inflaion markes prices are now quoed for inflaion-indexed bonds in boh he cash and asse swap markes. The relaionship beween he wo prices is dependen on he seasonal adjusmens used and in general adjusmens derived from hisoric daa will resul in prices ha are inconsisen beween he wo markes. 7

8 I is possible o infer seasonal adjusmens from he bond prices, asse swap spreads and inflaion swap raes ha are quoed in he marke. Swap spreads for inflaion indexed bonds are mos commonly quoed in erms of he Z-spread, which is he consan spread o he inerbank curve ha would make he discouned fuure cash flows equal o he marke price of he bond. For he inflaion indexed bonds, he fuure cashflows are deermined using inflaion swap raes. If a bond has annual coupons, he Z-spread will conain informaion abou he relaive seasonaliy of he reference index for oday, and ha of he coupon paymen monh. Given a selecion of bonds wih coupon paymens in differen monhs, seasonal adjusmens may be fied o marke prices using a leas squares approach. Recen developmens in he asse swap marke has made i possible o infer seasonal adjusmens from marke inflaion swap raes and asse swap raes on a range of bonds wih cash flows a differen imes of year Indexaion In mos developed economies, inflaion indices are released monhly and in general he index level for each monh is released around he middle of he following monh. Bu inflaion indexed securiies are raded coninuously and in order o calculae he real cash flows i is necessary o define a reference index which can be applied a any poin in ime. The reference index mus be lagged so as o cope wih he delayed release in he publicaion of he index. There are wo alernaive mehods of defining he reference index for days during he monh. The original UK IL Gils and European inflaion swaps use a consan reference index during he monh which reses discreely a he end of each monh. For mos oher insrumens he reference index is calculaed by linear inerpolaion beween wo lagged monh-end index seings. Indexaion can have quie a disoring effec on changes in real yields observed in he markes as he reference index is highly seasonal and he real yield a any poin is defined in he conex of a reference index a ha ime. 8

9 Chaper 2 Pricing inflaion-indexed derivaives in he Heah-Jarrow-Moron framework This chaper inroduces he model for inflaion and derives he no-arbirage dynamics ha are used in he following chaper o derive pricing formulae for inflaion indexed derivaives. The model is based on he foreign currency analogy approach inroduced by Jarrow & Yildirim in 23 11]. In he wo currency model, he erm srucure of he domesic and foreign economies are defined as Heah Jarrow & Moron (HJM models and he spo FX rae beween he wo currencies is modelled as a lognormal process. The model suppors correlaions beween he domesic raes, foreign raes and he exchange rae beween he wo economies. This approach o modelling FX derivaives was inroduced by Amin & Jarrow in ]. The inflaion model considers he nominal erm srucure o be he domesic erm srucure, he real erm srucure o be he foreign erm srucure and he spo inflaion index o be he spo exchange rae. The wo currency model is an exension of he single currency model and so he firs secion of his chaper reviews he derivaion of he HJM risk neural dynamics in he single currency seing. Alhough he model is derived in he full generaliy of he HJM framework, in order o derive analyic price formulaions for he mos commonly raded inflaion producs i is necessary o resric he HJM volailiy srucure so as o resul in Gaussian forward raes and lognormal bond prices. The nex secion describes how he exended Vasicek model of Hull and Whie can be defined in he HJM framework and he resricions on he volailiy srucure ha allow an HJM model o be expressed as a Hull Whie Markov process. This secion is also discussed in he single currency seing. The nex secion inroduces he real erm srucure and inflaion index ino he model. The risk neural measure is defined by considering asses radable in he nominal economy and he dynamics of he zero coupon bonds and inflaion index are derived under his measure. The dynamics under he forward measures are also reviewed as hese are of use when deriving he price formulae in he following chaper. 9

10 2.1 HJM no-arbirage dynamics in a single currency seing The Heah, Jarrow & Moron approach models he insananeous forward ineres rae as f T = f T + α ut du + σ ut dw u (2.1 where f T is he insananeous forward rae observed a ime for borrowing a ime T and W is a muli-dimensional Brownian moion in he risk neural measure. This secion shows ha in a complee marke which is arbirage free, he drif α T of he forward rae is uniquely deermined by he volailiy srucure as T α T = σ T σ u du (2.2 Le P T denoe he price a ime of he zero-coupon bond mauring a ime T. P T can be expressed in erms of he forward rae as ( P T = exp = exp ( = P T P exp = P T P exp T T ( f u du ( f s + σ us dw u + ( T σ us ds dw u ( (Σ ut Σ u dw u α us du ds ( T α us ds du (A ut A u du where A T = T α s ds and Σ T = T σ u du. The money marke accoun B has he dynamics db = r B d where r is he spo exchange rae and can be wrien as r = f. B can herefore be wrien as B ( = exp ( = exp = = he discouned bond PT B 1 P exp 1 P exp f ss ds ( f s + s σ us dw u + s ( ( σ us ds dw u + ( u Σ u dw u + given by ( P T = P T exp B mus be a maringale and herefore implies A ut du = 1 2 Σ ut dw u α us du ds A u du ( u A ut du α us ds du (2.3 Σ ut Σ utdu (2.4 where denoes he adjoin. This implies ha A T = 1 2 Σ TΣ T for T and hence he no-arbirage condiion

11 2.2 The exended Vasicek model of Hull and Whie in he HJM framework Alhough he no-arbirage dynamics for he inflaion model will be derived in he general HJM framework, in order o derive explici prices for inflaion indexed derivaives i is necessary o resric he HJM volailiy so ha he dynamics can be represened by a Gaussian Markov process for he shor rae. The exended Vasicek model of Hull & Whie is he mos general formulaion of such a process. This secion reviews he exended Vasicek model and derives he resricions on he HJM volailiy ha allow i o be represened in his form. The sochasic differenial equaion for he exended Vasicek process is wrien as dr = (a( b(r d + σ(dw (2.5 where W is a Brownian moion in he maringale measure. The soluion o his equaion is given by r = e β( (r + where β( = b(udu and hence T r s ds = T e β(s (r + s e β(u a(udu + e β(u σ(udw u (2.6 ( T T e β(u a(udu ds + e β(s ds e β(u σ(udw u (2.7 u Because W is a Brownian moion in he maringale measure he process PT B mus be a maringale under his measure. P T can be wrien as P T = E exp( ] T r sds F and B can be wrien ( B = exp r sds and so P T B = E exp = P T exp ( ( T r s ds F ] (φ T φ u g u dw u 1 2 (φ T φ u 2 gudu 2 where φ = eβ(u du and g = e β( σ( aking logarihms and differeniaing wih respec o T his gives an expression for he insananeous forward rae in erms of he exended Vasicek parameers f T = T log P T = T log P T = f T + φ T T B g u dw u + φ T T (φ T φ u 2 g 2 u du and so under he maringale measure, he dynamics of he forward rae are given by φ T df T = g T dw g 2 φ T T (φ φ T 2 d (2.8 11

12 This shows ha he exended Vasicek model is an HJM model wih he volailiy of he insananeous forward rae given by σ T = g φ T T = σ(eβ(β(t. The derivaion of he prices of many ineres rae derivaive producs can be expressed in erms of he dynamics of he zero coupon bond price under an appropriae maringale measure. Using he above formulaion of he exended Vasicek parameers we now derive an expression for he dynamics of he zero coupon bond in erms of he exended Vasicek parameers, firsly under he maringale measure and hen under he T-forward measure when discouned by he zero coupon bond P T. P T = exp = exp ( ( = P T P exp T f s ds ( T φ s s ( (φ T φ u g u dw u ( g u dw u ( T φ s s (φ s φ u 2 ds g 2u du (φ T + φ 2φ u g 2 u du I urns ou ha i is easier o price mos of he producs of ineres in he T-forward measure insead of he risk neural measure. Under he T-forward measure P T he expression PS P T wih dynamics given by P S = P ( S exp (φ S φ T P T P T where W T g u dwu T 1 2 (φ S φ T 2 gu 2 du is a maringale (2.9 is a Brownian moion under he P T measure. 2.9 expresses he variance of he zero coupon bond in erms of he original Hull-Whie parameers and is useful when calibraing his model o marke prices. A his poin he noaion E is inroduced o represen he Doléans-Dade exponenial of a coninuous semi-maringale X which is Using his noaion 2.9 can be re-wrien as ( E(X = exp X 1 2 X, X P S P T = P S P T E ( (φ S φ T g u dw T u (2.1 ( Inroducing he real economy In his secion he model is exended o include he dynamics of he real erm srucure and ha of he inflaion index. Using he foreign currency analogy he nominal ineres rae is considered o be he ineres rae in he domesic economy, he real ineres rae is he ineres rae in he foreign economy and he inflaion index is he exchange rae beween he wo economies. The analysis presened so far applies equally o he nominal and real economies if hey are considered 12

13 in isolaion. There exiss a unique maringale measure P r in he real economy under which he real insananeous forward rae follows he arbirage free dynamics df r T = σ r TdW r σ r T(Σ r T d (2.12 where he superscrip r has been inroduced o denoe he real economy and from now on n will be used o denoe he nominal economy. W r is a Brownian moion under Pr, he risk neural measure in he real economy. The wo economies are relaed by he inflaion index I, which is he price of a uni of real currency in unis of nominal currency. Every asse in he real economy can be convered ino a radable asse in he nominal economy via he inflaion index. We wish o exend he nominal risk neural measure P n o include his addiional se of nominal radable asses. Every nominal radable asse mus be a maringale under he P n measure when discouned by he nominal money marke accoun. In paricular his is rue for he real money marke accoun B r, so here mus exis a pre-visible process σ I such ha I B r B n ( = I E σsd I W s I where W I is a Brownian moion in he nominal risk neural measure P n and B r = Bn (2.13 = 1. When normalised by I, his is he Radon-Nikodym densiy of he P r measure wih respec o he P n measure and is given by Z = I B r I B n ( = E σs I d W s I (2.14 (2.15 The dynamics of he real insananeous forward rae under he P r measure were given by 2.12 and using he Radon-Nikodym densiy 2.15 we can see ha f r T exended nominal measure P n ft r = fr T + σst r d W s r has he following dynamics under he σst r ((Σr st + ρ ri s σi s ds (2.16 where ρ ri is he insananeous correlaion beween he P n -Brownian moions W I and W r. From he definiion of he maringale measure P n, he produc I P r T he nominal money marke accoun B n wih dynamics given by I PT r ( B n = I PT r E σsd I W s I + Σ r std W s r or, in erms of he Hull-Whie parameerisaion inroduced earlier is a maringale when discouned by I P r T B n = I P r T E ( (φ r T φr u gr u d W u r + σu I d W u I 13

14 2.3.1 Dynamics in he forward measure I urns ou ha i is easier o derive prices for he mos popular inflaion indexed producs in he appropriae forward measure. In his secion he dynamics under he forward measure are reviewed and in paricular expressions for he forward inflaion index and he Radon-Nikodym densiy of he real forward measure P r T wih respec o he nominal forward measure Pn T are derived. These expressions will be useful in he derivaion of prices for year-on-year swaps and inflaion-indexed caps and floors ha are derived in he nex chaper. As in he single currency case, he raio P r S P r T where W r,t P r S P r T is a maringale in he real T-forward measure P r T. = P S r ( PT r E (Σ r ss Σ r stdws r,t (2.17 is a Brownian moion in he real T-forward measure P r T. Wih he appropriae choice of T and wih = T, he price a ime T of a bond paying 1 uni of real currency a ime S is herefore given by which is a maringale under P r T PTS r = P S r PT r E ( T (Σ r ss Σ r stdw r,t s wih expeced price equal o he curren forward price ] E r T PTS r F = P S r PT r (2.18 (2.19 Using he Hull-Whie exended Vasicek parameerisaion inroduced earlier, 2.17 and 2.18 are given by and P n T where P r S P r T = P S r ( PT r E (φ r S φ r T PTS r = P S r PT r E ( (φ r S φr T T gudw r u r,t gu r r,t dwu (2.2 (2.21 When discouned by PT n, he produc I PT r is a maringale under he nominal T-forward measure and has dynamics measure P n T I,T r,t W, W I P r S P n T we obain = I PS r ( PT n E σ I I,T sd W s + and W n,t are correlaed P n T Σ r r,t ssd W s Σ n n,t st dws (2.22 Brownian moions. Under he nominal T-forward E n T I T F ] = I P r T P n T (

15 and I T = I P r T P n T ( T E σsd I T T I,T W s + Σ r r,t std W s Σ n stdws n,t (2.24 The dynamics of he inflaion index are used in he derivaion of pricing formulae for inflaionindexed caps/floors which depend on he disribuion of he index a T. The Radon-Nikodym densiy of he real T-forward measure P r T wih respec o he nominal T-forward measure P n T is given by Z T ( = E σsd I W I,T s + Σ r std W r,t s Σ n stdws n,t (

16 Chaper 3 Derivaion of prices for inflaion-indexed derivaives This chaper describes he mos liquid insrumens in he inflaion-indexed derivaive marke and derives prices for hem using he Hull-Whie exended Vasicek model. The inflaion-indexed derivaive markes are relaively new and here are only a small number of producs ha are acively raded. Zero-coupon inflaion-indexed swaps are he mos liquid produc and have model independen prices. Year-on-year inflaion swaps are probably he nex mos common followed by inflaionindexed caps/floors and forward saring zero coupon swaps. 3.1 Zero-coupon inflaion-indexed swaps Zero coupon inflaion-indexed swaps are acively raded in he European, UK and US markes. They are also he mos simple inflaion-indexed insrumens and because hey have model independen prices, he erm srucure of real raes can be easily derived from he nominal erm-srucure and marke inflaion swap raes. A zero-coupon inflaion swap is defined by is mauriy T, is fixed rae r and is base index I which is he reference index when he swap is iniially raded. A ime T Pary A pays he floaing inflaion leg IT I 1 and Pary B pays he fixed amoun (1 + r k 1, where k is he number of years o mauriy when he swap is iniially raded and I T is he reference index a mauriy. The fixed rae r is chosen so as o make he value of he fixed leg equal o ha of he floaing leg when he swap is iniially raded and inflaion swaps are quoed in erms of he fixed rae. The value of he inflaion leg is denoed by ZCIIS(, T, I and is given by ZCIIS(, T, I P n T = E n T ] 1 I F where E n T denoes he expecaion wih respec o he nominal T-forward measure Pn T. Because I P r T P n T is a maringale under his measure IT (3.1 16

17 E n TI T F ] = I P r T P n T (3.2 under P n T he expeced index value a ime T is he curren forward value. The price of he inflaion leg a ime is hen given by ZCIIS(, T, I = I P r T I P n T (3.3 and a =, when he swap is iniially raded, I = I and he price of he floaing leg simplifies o ZCIIS(, T, I = P r T P n T (3.4 The value of he fixed side is simply he value of he fixed cash flow a ime T discouned by he nominal discoun facor and is given by P n T((1 + r k 1 (3.5 Equaing his o he value of he inflaion leg, i is clear ha he price of he real zero coupon bond P r T is given by PT r = PT(1 n + r k (3.6 This expression provides a simple, model independen, soluion for he real discoun facors given he nominal discoun facors and he marke fixed raes for zero coupon inflaion-indexed swaps and proves o be very useful in pricing he more complicaed producs, all of which depend on he iniial erm srucure of real raes 3.2 Year-on-year inflaion-indexed swaps A year-on-year inflaion-indexed swap consiss of a fixed and a floaing leg wih annual paymens a T 1,..., T N. A he end of each period Pary A pays he floaing inflaion leg which is based on he inflaion rae over he previous period. The floaing paymen on a swap rese a ime T i1 and paid a ime T i would be IT i I Ti1 1. I is similar o he floaing paymen on a zero coupon inflaion swap, and in fac he firs paymen is he same as a zero coupon swap because I Ti1 is known a he rade dae. However for subsequen paymens I Ti1 is no known unil he end of he previous period and because boh I Ti and I Ti1 canno boh be maringales under he same measure, he price of each paymen on he floaing leg of a year-on-year swap conains a convexiy correcion erm which accouns for he required measure change. As wih zero coupon swaps, year-on-year swaps are quoed in erms of he fixed rae ha would make he value of he floaing inflaion leg equal o ha of he fixed leg a rade dae =. A he end of each period, Pary B pays he fixed leg r and so he price a = of he fixed paymen paid a ime T i is P n T i r. 17

18 As discussed above, each floaing paymen on a year-on-year swap can be considered as he floaing paymen on a forward saring zero coupon swap. The value a ime of he floaing paymen rese a T 1 and paid a T 2 is herefore given by Y Y IIS(, T 1, T 2 P n T 1 = E n T 1 ZCIIS(T 1, T 2, I T1 ] (3.7 where E n T 1 denoes he expecaion wih respec o he nominal T 1 -forward measure. A T 1 his is a zero coupon swap raded a marke raes wih base index given by he reference index a ha ime and 3.4 can herefore be applied o give Y Y IIS(, T 1, T 2 P n T 1 = E n T 1 P r T 1T 2 P n T 1T 2 F ] (3.8 Noing ha P n T 1 E n T 1 P n T 1T 2 ] = P n T 2, he price of he floaing inflaion paymen a ime is given by Y Y IIS(, T 1, T 2 = P n T 1 E n T 1 P r T 1T 2 F ] P n T 2 (3.9 This expecaion is model dependen and is evaluaed mos easily by noing ha P r T 1T 2 is a maringale under he P r T 1 measure and hen using he Radon-Nikodym densiy for he P r T 1 measure wih respec o he P n T 1 measure ha was developed in he previous chaper o derive he expecaion for P r T 1T 2 in he nominal T 1 -forward measure. I is his measure change ha inroduces he convexiy adjusmen in he price of year-on-year swaps. As shown in he previous chaper P r T 2 P r T 1 is a maringale in he real T 1 -forward measure wih dynamics given by PT r 2 PT r = P r ( T 2 1 PT r E (Σ r st 2 Σ r st 1 dws r,t1 1 (3.1 where W r,t1 is a Brownian moion under he P r T 1 measure. Choosing = T 1, he price a T 1 of a real zero coupon bond mauring a T 2 is herefore given by P r T 1T 2 = P r T 2 P r T 1 E ( (Σ r st 2 Σ r st 1 dw r,t1 s and under his measure has an expeced price equal o he curren forward price. Wrien in erms of he hree correlaed P n T 1 Brownian moions r,t1 I,T W, W 1 (3.11 and W n,t1, he Radon-Nikodym densiy 2.25 of he real T 1 -forward measure P r T 1 wih respec o he nominal T 1 - forward measure P n T 1 is given by ( Z = E σsd I W I,T1 s + Σ r st 1 d W r,t1 s Σ n st 1 dw n,t1 s and so he required dynamics for P r T 1T 2 under he nominal T 1 -forward measure P n T 1 are given by (

19 I is herefore clear ha P r T 1T 2 = P r T 2 P r T 1 E exp ( (Σ r st 2 Σ r st 1 d ( W r,t1 s (Σ r st 2 Σ r st 1 (Σ n st 1 ρ nr s σ I sρ Ir s Σ r st 1 ds where E n T 1 P r T 1T 2 F ] = P r T 2 P r T 1 e C(,T1,T2 (3.13 C(, T 1, T 2 = and he floaing inflaion leg is given by (Σ r st 2 Σ r st 1 (Σ n st 1 ρ nr s σ s ρ Ir s Σ r st 1 ds (3.14 ( P r Y Y IIS(, T 1, T 2 = PT n T2 1 PT r e C(,T1,T2 P n T 2 1 PT n ( The convexiy adjusmen C is dependen on he correlaion beween he nominal and real raes and beween he inflaion index and real raes Year-on-year inflaion-indexed swap for Hull Whie model wih consan volailiy parameers When he volailiy parameers and correlaion coefficiens are consan i is possible o derive an explici formula for he year-on-year inflaion indexed swap in As shown in he previous chaper, consan volailiy parameers b and σ resul in a HJM forward volailiy of he form σ T = σe b(t (3.16 wih he zero-coupon bond volailiy Σ T given by T Σ T = σe b(u du = σ b eb(u ] T = σ b eb(t 1] 19

20 The convexiy correcion C(, T 1, T 2 can hen be wrien as C(, T 1, T 2 = σ r (e br(t2s e br(t1s ( σn ρ nr (e bn(t1s 1 σ I ρ Ir σ r = σ rσ n ρ nr σ rσ I ρ Ir σ2 r b 2 r (e br(t1s 1 e br(t2sbn(t1s e br(t2s e br(t1sbn(t1s + e br(t1s ds e br(t2s e br(t1s ds e br(t1+t22s e 2br(T1s e br(t2s + e br(t1s ds ds = σ rσ n ρ nr 1 e br(t2sbn(t1s 1 e br(t2s + 1 e br(t1sbn(t1s + 1 e br(t1s + σ rσ I ρ Ir 1 e br(t2s 1 ] T1 e br(t1s σ2 r b 2 r 1 2 e br(t1+t22s 1 2 e 2br(T1s 1 e br(t2s + 1 e br(t1s ] T1 ] T1 = σ rσ n ρ nr 1 e br(t2t1 1 e br(t2t e br(t2bn(t1 + 1 e br(t2 1 + e br(t1bn(t1 1 ] e br(t1 + + σ rσ I ρ Ir e br(t2t1 1 e br(t2 + e br(t1] σ2 r b 3 r b 2 r 1 2 ebr(t2t1 1 2 ebr(t2t ebr(t1+t e2br(t1 + e br(t2 e br(t1 ] and finally he convexiy adjusmen C is given by C(, T 1, T 2 = σ r B r T 1T 2 B r T 1 ρ nr σ n + (1 + B n T 1 + σ r B r T 1T 2 ρ nr σ n + B n T 1 (3.17 +σ r σ I ρ ri B r T 1T 2 B r T σ2 r(b r T 1 2 B r T 1T 2 where B T = 1 b (1 eb(t 2

21 3.3 Inflaion indexed Caps and Floors An inflaion indexed caple/floorle is a call/pu opion on he inflaion rae wih payoff a ime T 2 defined by ( ] + IT2 Nψ ω 1 κ (3.18 I T1 where κ is he srike, ψ is he year fracion for he inerval T 1, T 2 ], N is he noional and ω = 1 for a caple and ω = 1 for a floorle. Defining K = 1 + κ, he price of he opion a < T 1 is given by { ( ] + } IICF(, ψ, N, T 1, T 2, κ, ω PT n = NψE n IT2 T 2 ω K F ( I T1 where E n T 2 denoes he expecaion under he nominal T 2 -forward measure P n T 2. As shown in he previous chaper, under he nominal T 2 -forward measure P n T 2 I T2 = I P r T 2 P n T 2 E ( T2 Under he nominal T 1 -forward measure I T1 = I P r T 1 P n T 1 E ( σsd I and under he nominal T 2 -forward measure T2 T2 σs I I,T2 d W s + Σ r r,t2 st 2 d W s Σ n st 2 dws n,t2 T1 I,T1 W s + Σ r r,t1 st 1 d W s Σ n st 1 dws n,t1 (3.2 (3.21 I T1 = I P r T 1 P n T 1 E ( σsd I T1 I,T2 W s + Σ r r,t2 st 1 d W s Σ n st 1 dws n,t2 e D(,T1,T2 (3.22 where e D(,T1,T2 is he change in drif due o he measure change. The inflaion rae IT 2 I T1 is herefore lognormally disribued under he nominal T 2 -forward measure wih dynamics given by I T2 = P T r 2 P n ( T2 T 1 I T1 PT n 2 PT r E σsd I 1 Σ r st 1 d T 1 W r,t2 s + T2 I,T2 W s + Σ r st 2 d Σ n st 1 dw n,t2 s W r,t2 s e D(,T1,T2 T2 Σ n st 2 dws n,t2 (3.23 because he inflaion rae is lognormally disribued he opion prices can be derived from he following propery of he lognormal disribuion. If ln(x is normally disribued wih EX] = m and he variance of ln(x = v 2 hen Eω(X K] + = ωmφ (ω ln m K v2 ωkφ (ω ln m K 1 2 v2 v v (3.24 The expeced value of IT 2 I T1 can be derived by considering he analysis in he previous secion. The YYIIS price was given boh by 21

22 ] Y Y IIS(, T 1, T 2 PT n = E n IT2 T 2 1 F 2 I T1 or alernaively in erms of a forward saring zero coupon inflaion swap (3.25 Y Y IIS(, T 1, T 2 P n T 1 = E n T 1 P r T1T 2 P n T 1T 2 F ] (3.26 So he required expecaion can be given in erms of T 1 -forward expecaions as ] E n IT2 T 2 1 F = P T n 1 I T1 PT n E n T 1 PT r 1T 2 PT n 1T 2 F ] ( and so, following he working from he year-on-year swap ] E n IT2 T 2 1 F = P T n 1 PT r 2 I T1 PT n 2 PT r e C(,T1,T2 ( wih C(, T 1, T 2 given by The value of he opion is herefore given by IICF(, ψ, N, T 1, T 2, κ, ω = P n ωnψpt n T1 PT r 2 2 PT n 2 P,T r 1 ln P n T 1 P r T 2 e C(,T1,T2 KPT n P + C(, T r 1, T Φ ω 2 T 2 V 2 (, T 1, T 2 1 V (, T 1, T 2 ln P T n 1 PT r 2 KPT n P + C(, T r 1, T 2 1 KΦ ω 2 T 2 V 2 ] (, T 1, T 2 1 V (, T 1, T 2 where V 2 (, T 1, T 2 is he variance of IT 2 I T1 and can be derived from 3.23 V 2 (, T 1, T 2 = T2 + + T 1 (σ I s 2 ds + +2ρ Ir T2 +ρ nr ( 2 T2 Σ r st 1 (Σ r st 1 ds + Σ n st 1 (Σ n st 1 ds 2 Σ r st 2 (Σ r st 2 ds (3.29 T2 T 1 σ I s Σr st 2 ds 2ρ In T2 Σ n st 2 (Σ n st 2 ds T2 Σ r st 2 (Σ n st 2 ds Σ r st 1 (Σ n st 2 ds 2 Σ r st 2 (Σ r st 1 ds 2 T 1 σ I s Σn st 2 ds Σ r st 2 (Σ n st 1 ds Σ r st 1 (Σ n st 1 ds Σ n st 1 (Σ n st 2 ds For consan parameers his expands o 22

23 V 2 (, T 1, T 2 = σ2 n 2b 3 (1 e bn(t2t1 2 1 e 2bn(T1 ] + σi(t 2 2 T 1 n + σ2 r 2b 3 (1 e br(t2t1 2 1 e 2br(T1 σ n σ r ] 2ρ nr r ( + (1 e bn(t2t1 (1 e br(t2t1 1 e (bn+br(t1 ] + σ2 n b 2 n + σ2 r b 2 r T 2 T e bn(t2t1 1 e 2bn(T2T1 3 ] 2 2 T 2 T e br(t2t1 1 e 2br(T2T1 3 ] 2 2 T 2 T 1 1 ebn(t2t1 2ρ nr σ n σ r 1 ebr(t2t1 + 1 ] e(bn+br(t2t1 + σ n σ I +2ρ ni T 2 T 1 1 ] ebn(t2t1 2ρ ri σ r σ I The working for his is given in Appendix A. T 2 T 1 1 ] ebr(t2t1 23

24 Chaper 4 Calibraion The objecive of calibraion is o choose he model parameers in such a way ha he model prices are consisen wih he marke prices of simple insrumens. In he case of nominal ineres raes, he mos common producs in he marke are swapions and caps/floors. Saring wih he Hull-Whie exended Vasicek model for he spo rae, we derive an expression for he price of a zero coupon bond a a fuure ime in erms of he Hull-Whie parameers. The bond price is lognormally disribued, and we are herefore able o use he properies of he lognormal disribuion, combined wih he firs fundamenal heorem of finance o derive he price of an opion on a zero-coupon bond. We hen show ha boh swapions and caps/floors can be expressed in erms of opions on zero-coupon bonds and consequenly we are able o price hese insrumens in erms of he Hull-Whie volailiy parameers. The calibraion process is hen a maer of choosing a paricular form of he volailiy parameers and fiing hem so as o mach he prices of seleced marke insrumens. In wha follows we will use he following resul which is a propery of he lognormal disribuion and allows us o price European opions when he asse price is lognormally disribued under he maringale measure. If V is lognormally disribued wih he variance of ln(v given by w, hen E(V K + ] = E(V N(d 1 KN(d 2 (4.1 E(K V + ] = KN(d 2 E(V N(d 1 (4.2 where d 1 = lne(v /K] + w2 /2 w d 2 = lne(v /K] w2 /2 w 24

25 4.1 Hull-Whie zero coupon bond dynamics As shown previously, when a( is chosen so as o fi he iniial erm srucure, he risk-neural Hull-Whie spo rae dynamics are given by dr = (a( b(r d + σ(dw (4.3 and are equivalen o a one facor Heah-Jarrow-Moron model wih forward volailiy srucure given by σ T = σ(e β(β(t (4.4 where β( = b(udu. The dynamics of he zero coupon bond P S in he T-forward measure is given by 2.11 where φ = eβ(u du and g = e β( σ(. P S = P ( S E (φ S φ T P T P T g u dw T u ( Hull-Whie zero coupon bond opion We now derive he price a ime of a call opion V (, K, T, S expiring a ime T and sruck a K on a zero-coupon bond wih uni value a ime S where S > T. Using he firs fundamenal heorem of finance, we can see ha under he T-forward measure V (, K, T, S P(, T = E T (P(T, S K + F ] (4.6 In secion 4.1 we showed ha under he T-forward measure he zero-coupon bond is lognormally disribued wih expeced price equal o is forward price and wih he variance of he logarihm given by (φ S φ T 2 G. We can herefore use resul 4.2 o derive he opion price in erms of he Hull-Whie volailiy parameers as follows where V (, K, T, S = P(, T(FN(d 1 KN(d 2 (4.7 d 1 = lnf/k] + w2 /2 w d 2 = lnf/k] w2 /2 w and F is he expeced price of he bond in he T-forward measure i.e. he forward price P(, S/P(, T and w is he variance of he logarihm of he bond, which is given above as 25

26 w 2 = (φ S φ T 2 G (4.8 In he Hull-Whie case wih consan volailiy parameers, his expression reduces o he form w 2 = σ ( 1 e a (1 ea(st 2 2aT 2a ( Swapions A swapion is an opion on a swap wih he fixed rae on he swap given by he srike. A receiver swapion gives he holder he righ o ener ino a swap receiving he fixed srike rae and paying he floaing rae and a payer swapion gives he holder he righ o pay he fixed srike rae and receive he floaing rae Swapion marke quoes Marke prices of swapions are quoed as lognormal volailiies where he price of he swapion is implied using Black s swapion formula. Swapion volailiies are generally quoed as a marix of volailiies for a combinaion of opion expiries and underlying swap erms. We now give a brief derivaion of Black s swapion formula. If we define he forward annuiy A(, T 1, T N as he price a ime of an annuiy ha pays regular cash flows a imes T 1,..., T N, we can wrie he forward swap rae saring a T and mauring a T N as S(, T, T N = P(, T P(, T N A(, T 1, T N I is clear ha he forward swap rae is a maringale in he forward annuiy measure, and so (4.1 E A S(T, T, T N F ] = S(, T, T N (4.11 Black s formula assumes a lognormal disribuion for he forward swap rae and he opion price is calculaed in he forward annuiy measure. Defining rec(s k, S,, T, T 1, T N as he price of a receiver swapion sruck a S k and expiring a T on a swap saring a T wih paymen daes T 1,..., T N and wriing S for he forward swap rae a ime, i.e. S(, T, T N, we have he relaionship: rec(s k, S,, T, T 1, T N A(, T 1, T N = E A ( (S T S k A(T, T 1, T N A(T, T 1, T N + (4.12 which simplifies o rec(s k, S,, T, T 1, T N = A(, T 1, T N E A (S T S k + ] (

27 Using resul 4.2 we have Black s swapion formula for a receiver swapion ( rec(s k, S,, T, T 1, T N = A(, T 1, T N S N(d 1 S k N(d 2 (4.14 where d 1 = d 2 = ln( S S k σ2 T σ T ln( S S k 1 2 σ2 T σ T wih he corresponding payer swapion formula given by ( pay(s k, S,, T, T 1, T N = A(, T 1, T N S k N(d 2 S N(d 1 ( Hull-Whie swapion The payoff a ime T of a receiver swapion which expires a ime T wih a fixed rae srike of S k on a swap wih cashflows a imes T 1,..., T N is given by ( N + S k P(T, T i 1 + P(T, T N (4.16 i=1 This is idenical o he payoff of a call opion sruck a 1 on a bond paying a coupon S k a imes T 1...T N. To price an opion on a coupon paying bond we use a echnique proposed by Jamshidian. The echnique is based on he observaion ha in a one-facor model, he price of each cash flow decreases monoonically wih he spo ineres rae. I is herefore possible o price an opion on a coupon bond as a porfolio of opions on he individual cash flows, wih each opion sruck a he respecive zero coupon bond rae a opion expiry when he spo rae is he criical rae r. Here r is he spo rae ha makes he price of he coupon bond a opion expiry equal o he srike on he coupon bond opion. The approach is herefore as follows: 1. Using he Hull-Whie zero-coupon bond formula P(T, T i = A(T, T i exp(rb(t, T i solve for r, he value of r ha makes he price a ime T of a bond paying a coupon S k a imes T 1,..., T N equal o The price of he swapion is hen he price of a porfolio of zero coupon bond opions. Each opion expires a T, wih he underlying zero coupon bond paying 1 a T i where 1 i N. The srike of each opion is he price a ime T of a zero coupon bond mauring a ime T i if he spo rae were r and he noional of he opion is size of he cash flow. Each opion can be priced using

28 4.4 Caps and Floors A cap is a porfolio of call opions on a Libor rae. Each opion is known as a caple and has he payoff δ(l i K + a ime T i where L i is he Libor rae rese a ime T i1 and paid a ime T i and δ is he accrual fracion for he period. In he same way, a floor is a porfolio of floorles wih payoff δ(k L i +. In wha follows we will only discuss caps, bu he logic applies equally o floors and boh are conneced by he pu/call pariy relaion: Value of cap = value of floor + value of swap Cap/Floor marke quoes Caps and floors are quoed in he marke as a-he-money Black lognormal fla volaiies. Tha is o say each caple is sruck a he swap rae wih he same enor and cashflow frequency as he cap and he price of he cap is he sum of he prices of he individual caples implied via he quoed lognormal volailiy using Black s caple formula. Black s caple formula is based on he Libor rae L(, T i1, T i which defines he value a ime of he ineres rae ha reses a ime T i1 and pays a ime T i. I can be wrien in erms of zero coupon bonds as follows which can be wrien as L(, T i1, T i = 1 δ ( P(, Ti1 P(, T i 1 (4.17 L(, T i1, T i = 1 ( P(, Ti1 P(, T i δ P(, T i Clearly his is a maringale under he T i -forward measure so (4.18 E Ti L(T i1, T i1, T i F ] = L(, T i1, T i (4.19 By he firs fundamenal heorem of finance we have he value of a caple wih srike K in he T i forward measure as and using he resul 4.2 we ge Black s caple model caple P(, T i = δe T i (L(T i1, T i1, T i K + ] (4.2 where δp(, T i L N(d 1 KN(d 2 ] (

29 d 1 = ln(l /K σ2 T i1 σ T i1 d 2 = ln(l /K 1 2 σ2 T i1 σ T i1 Here L is defined as he forward Libor given by L(, T i1, T i Hull-Whie caps and floors Here we show ha each caple can be considered as an opion on a zero coupon bond and can herefore be priced using he Hull-Whie zero coupon bond opion formula discussed in secion 4.2 The payoff from a caple a ime T i sruck a K is δ (L i K + (4.22 Where L i is he rae rese a ime T i1 and paid a ime T i. This is equivalen o he paymen a ime T i1 and can be simplified o give δ 1 + L i δ (L i K + (4.23 he expression ( Kδ + ( L i δ 1 + Kδ 1 + L i δ is he value a ime T i1 of a zero-coupon bond ha pays off 1 + Kδ a ime T i and he caple is herefore a pu opion wih mauriy T i1 on a zero-coupon bond wih mauriy T i wih a srike of 1 and he nominal value of he bond equal o 1 + Kδ. The caple and floorle can hen be priced using he bond opion formula 4.7 floorle = (1 + KδP Ti N(d 1 P Ti1 N(d 2 caple = P Ti1 N(d 2 (1 + KδP Ti N(d 1 (1+KδPTi ln( P Ti1 + w2 2 d 1 = w (1+KδPTi ln( P Ti1 w2 2 d 2 = w where w 2 is he variance a ime T i1 of he logarihm of he zero-coupon bond mauring a ime T i. 29

30 4.5 Implemenaion of he inflaion model calibraion In his secion we show how he Hull Whie inflaion model wih consan volailiy parameers can be calibraed o marke daa. The iniial erm srucures are derived from he nominal zero coupon curve and marke inflaion swap raes. The nominal volailiy parameers are hen fied o a-he-money caps and he oher parameers are hen fied o he inflaion insrumens. The mos liquid insrumens conaining informaion on he inflaion volailiy and correlaion are year-on-year inflaion swaps. Alhough inflaion-indexed caps and floors are becoming increasing popular, hey are no ye sufficienly liquid o permi saisfacory calibraion and so here we concenrae on fiing he model o year-on-year inflaion swaps. The firs sage in he calibraion process is o derive he iniial erm srucures. The nominal erm srucure is derived from raded insrumens in he cash, fuures and swap markes. We do no discuss he deails of yield curve inerpolaion here, bu assume a coninuous se of nominal discoun facors which are derived from he raded marke insrumens using a proprieary model. The real discoun facors are derived from he nominal discoun facors and he zero coupon inflaion swap raes using he model independen expression 3.6 PT r = P T n (1 + rk There are liquid zero-coupon inflaion swap markes for European CPI, French domesic CPI, US CPI and UK RPI. We choose o calibrae o European CPI because of he more liquid year-on-year swap markes. In he European marke, zero coupon swaps are liquid ou o a mauriy of abou hiry years. The inflaion swap erm srucure on he European HICPXT index on 1 April is shown here. The erm srucures for he oher markes menioned above are shown in appendix A. 2.35% EUR HICP Zero Coupon Swap Raes (1-Apr % Swap Rae 2.25% 2.2% 2.15% 2.1% Years o mauriy The one year inflaion swap rae is no very liquid and is highly dependen on he reference index and economiss forecass and so we do no include i in his analysis. On 1 April, he nominal and 3

31 real zero coupon curves for he European HICPXT index were as follows. (The real curves for he oher indices are also shown in appendix A European nominal and real zero coupon raes (1-Apr % 2.52% 4.62% Nominal (lhs Real (rhs Nominal Rae 4.41% 4.2% 2.31% 2.1% Real Rae 3.99% Mauriy Dae 1.89% The year-on-year inflaion swap pricing formula developed in chaper 3 conains he seven parameers:, σ n,, σ r, σ I, ρ nr and ρ Ir. We firs fi he nominal volailiy parameers and σ n o a-he-money nominal caps and hen fi he remaining five parameers o year-on-year inflaion swaps. The calibraion o nominal caps is done by choosing and σ n so as o minimise he sum of he square difference beween marke and model cap prices using he goodness-of-fi measure n (MarkeCap i ModelCap i 2 i=1 where here are n calibraing insrumens. The Hull Whie cap price was inroduced earlier in he chaper and is dependen on he variance of he logarihm of he zero coupon bond for each opion. The model implied volailiy (he implied Black volailiy backed ou from he calibraed model cap prices is shown here wih he marke Black implied volailiy srucure Calibraion o nominal ATM caps 14% Marke Implied Model Implied Implied Volailiy 12% 1% 8% Years o Mauriy 31

32 Alhough he fi is no bad for longer mauriies, he model does no suppor he humped srucure ypically observed in he cap marke. The behaviour is o be expeced from he consan parameer model wih he forward volailiy decaying exponenially wih mauriy. This is he volailiy srucure proposed by Jarrow and Yildirim bu, as described in chaper 2, he model can be exended o include deerminisic ime dependen parameers and hus provide a much closer fi o he iniial marke volailiy srucure. In his case eiher σ or b or boh parameers may be made ime dependen, bu his exension is ouside he scope of his hesis. Year-on-year inflaion swap raes rade below he corresponding zero-coupon swap raes due o he convexiy adjusmen. The following char shows marke zero-coupon and year-on-year swap raes on 1 April 27 Marke Inflaion Swap Term Srucure (1-April 2.35% Zero Coupon Year-on-year 2.3% 2.25% 2.2% 2.15% 2.1% Mauriy Year The model was also fied o marke year-on-year swaps using a leas-squares approach, wih he remaining five parameers chosen so as o minimise he square of he difference beween he marke and model year-on-year swap raes. The calibraed model raes and marke raes for year-on-year swaps are shown here Marke and model Year-on-year inflaion swap raes (1-Apr % Marke Model 2.25% 2.2% 2.15% 2.1%

33 The fi seems quie saisfacory, paricularly given ha bid/offer spreads are ypically of he order of five basis poins. However, given he raher poor fi o he nominal volailiy srucure, i seems unlikely ha he consan parameer model should be used o price more exoic producs. In his case exending he model o include deerminisic ime dependen volailiy parameers would allow a much beer iniial fi and resul in more reliable prices for more complex inflaion-linked producs. 33

34 Appendix A Inflaion Indexed Caple formula for consan Hull-Whie parameers In his appendix he variance V 2 (, T 1, T 2 of he logarihm of he inflaion rae IT 2 I T1 is derived for he exended Vasicek model of Hull and Whie wih consan volailiy parameers. As shown in 3.29 he variance V 2 (, T 1, T 2 is given by V 2 (, T 1, T 2 = T2 + T 1 (σ I s 2 ds + +2ρ Ir T2 T2 Σ n st 1 (Σ n st 1 ds 2 T 1 σ I sσ r st 2 ds T2 2ρ In σs I Σn st 2 ds T 1 +ρ nr ( 2 T2 +2 Σ r st 1 (Σ n st 2 ds 2 Σ r st 2 (Σ r st 2 ds + Σ r st 2 (Σ n st 2 ds + 2 Σ r st 2 (Σ r st 1 ds 2 T2 Σ r st 1 (Σ r st 1 ds + Σ r st 2 (Σ n st 1 ds Σ r st 1 (Σ n st 1 ds Σ n st 1 (Σ n st 2 ds Σ n st 2 (Σ n st 2 ds The volailiy of he zero coupon bond P T under his model is given by Σ T = σ b eb(t 1] (A.1 The ρ Ir coefficien comes from 34

35 and he ρ In coefficien comes from The ρ nr coefficiens come from T2 2σ I ρ Ir σ r (e br(t2s 1ds T 1 = 2σ Iρ Ir σ r e (T 2s ] T2 s = 2σ Iρ Ir σ r 2σ Iρ In σ n = 2σ Iρ In σ n = 2σ Iρ In σ n 1 e (T 2T 1 T2 T 1 + T 1 T 2 (e bn(t2s 1ds T 1 e (T 2s ] T2 s 1 e (T 2T 1 T 1 ] T 2 + T 1 ] 2ρ nrσ r σ n + 2ρ nrσ r σ n + 2ρ nrσ r σ n which would expand o give 2ρ nrσ r σ n T2 (e br(t2s 1(e bn(t2s 1ds (e br(t2s 1(e bn(t1s 1ds (e br(t1s 1(e bn(t2s 1ds (e br(t1s 1(e bn(t1s 1ds 2ρ nr σ r σ n and in urn his gives + + T2 (e br(t2sbn(t2s e br(t2s e bn(t2s + 1ds (e br(t2sbn(t1s e br(t2s e bn(t1s + 1ds (e br(t1sbn(t2s e br(t1s e bn(t2s + 1ds ] (e br(t1sbn(t1s e br(t1s e bn(t1s + 1ds 35

36 = 2ρ ( nrσ r σ n + + Finally for he ρ nr coeffs we ge e (T 2s(T 2s + e (T 2s(T 1s + e (T 1s(T 2s + e (T 1s(T 1s + ebr(t2s ebr(t2s ebr(t1s ebr(t1s ebn(t2s ebn(t1s ebn(t2s ebn(t1s ] T2 + s + s + s + s ] T1 ] T1 ] T1 2ρ nrσ n σ r (T 2 T 1 1 ebn(t2t1 1 ebr(t2t1 + 1 e(bn+br(t2t1 + 2ρ nr σ n σ r ( + (1 ebn(t2t1 (1 e br(t2t1 (1 e (bn+br(t1 Now we move on o hose erms wih no correlaion coefficens = σ 2 I(T 2 T 1 + σ2 r + σ2 n b 2 n 2σ2 r b 2 r 2σ2 n b 2 n T2 and doing he inegraions we ge b 2 r T2 (e bn(t2s 1 2 ds + σ2 n b 2 n (e br(t2s 1 2 ds + σ2 r (e br(t2s 1(e br(t1s 1ds (e bn(t2s 1(e bn(t1s 1ds b 2 r (e bn(t1s 1 2 ds (e br(t1s 1 2 ds = σi(t 2 2 T 1 + σ2 r e 2(T 2s b 2 r 2 + σ2 r e 2(T 1s b 2 r 2 2σ2 r e (T 2+T 22s b 2 r 2 2σ2 n e (T 2+T 12s b 2 n 2 2ebr(T2s 2ebr(T1s ] T2 + s + s ebr(t2s ebn(t2s + σ2 n b 2 n ] T1 + σ2 n b 2 n ebr(t1s ebn(t1s e 2(T 2s 2 e 2(T 1s 2 + s + s ] T1 ] T1 2ebn(T2s 2ebn(T1s ] T2 + s + s ] T1 36