FIXED INCOME MICHAEL MONOYIOS

Transcription

1 FIXED INCOME MICHAEL MONOYIOS Absrac. The course examines ineres rae or fixed income markes and producs. These markes are much larger, in erms of raded volume and value, han equiy markes. We firs inroduce zero-coupon bonds ZCBs, he fundamenal fixed income asses, hen ouline ideas of numéraire invariance. We describe some common ineres rae derivaives swaps, and ineres rae opions: caps and floors, and swapions. We describe differen approaches o modelling ineres rae markes, beginning wih shor-rae models, hen forward rae models, characerised by he Heah-Jarrow-Moron HJM framework, and conclude wih LIBOR marke models. The mahemaical ools needed will be Iô calculus, he Girsanov Theorem, and he financial ools will be noarbirage valuaion and hedging, augmened by change-of-numéraire argumens. Conens Useful books 2 1. Bonds and ineres raes Zero-coupon bonds Ineres raes LIBOR raes and ZCB prices Forward raes, spo raes and shor raes 5 2. Valuaion principles and numéraire invariance Numéraire pairs Change of numéraire Forward prices and he T -forward measure Ineres rae derivaives Coupon-bearing bonds Floaing coupon bond floaing rae noe Ineres rae swaps Ineres rae opions Modelling he erm srucure Srucural relaionships Shor rae modelling The erm srucure PDE Classical shor rae models Affine shor rae models Calibraion of shor rae models o ZCB prices Two-facor shor rae models Forward rae models Forward rae dynamics under physical measure HJM drif condiion under P HJM drif condiion under an EMM Q Relaion o affine yield models 31 Dae: March 1, 217. Based parly on noes wrien by Ben Hambly. 1

2 2 MICHAEL MONOYIOS 1 T ime Figure 1. Zero-coupon bond 7. Marke models Black formula compaibiliy LIBOR marke model 33 References 38 Useful books Books you migh find useful o consul include: Björk [1] Chapers 22 28; Shreve [6] Chapers 9 1; Hun and Kennedy [4] good on swap and LIBOR marke models; Filipović [2] good, advanced ex; Musiela and Rukowski [5] Par II, co-wrien by one of he creaors of he LIBOR marke model; Vecer [7] good source of maerial on he subjec of change of numéraire. 1. Bonds and ineres raes A bond is a conrac which makes fixed paymens coupons a regular inervals, o is holder. An agen who issues a bond is borrowing money: he agen receives he bond price a he issue dae, and makes coupon paymens over ime o pay ineres. Typically, he final paymen includes a repaymen of he bond s face value or principal amoun. Governmens and corporaions borrow money by issuing bonds. In he Unied Saes, governmen bonds are known as Treasury bills. In he Unied Kingdom, governmen bonds are known as gils Zero-coupon bonds. A zero-coupon bond ZCB someimes called a discoun bond makes jus one paymen, a is mauriy dae, T. We shall normalise he final paymen o be 1, and will ofen refer o a ZCB of mauriy T as a T -bond. The paymen in a ZCB is shown schemaically in Figure 1. Definiion 1.1 T -bond. A ZCB of mauriy T, or T -bond, is a conrac which pays 1 uni of currency say, dollars o is holder a ime T. Denoe he price of he T -bond a T by P, T. We can hink of a coupon-paying bond as a sequence of ZCBs. Definiion 1.2 Zero-coupon rae. The zero-coupon rae or zero-coupon yield, or simply he zero rae, R, T, seen a ime T for he ime inerval [, T ], is defined by P, T = exp R, T T, T. The zero rae is also called he spo rae a ime for he inerval [, T ]. Assumpion 1.3 Sanding assumpions. We shall assume here exiss a liquid, fricionless marke for T -bonds of all mauriies T, wih he erminal condiion 1.1 P T, T = 1, T >.

3 FIXED INCOME 3 We shall furher assume ha here is no defaul risk in he marke a reasonable assumpion for some governmen bonds, and ha for fixed [, T ], P, T is differeniable wih respec o he mauriy T. Noe he differences compared o modelling equiy markes. Here we are assuming here is an infinie number of asses a coninuum of bonds, for each mauriy. This is of course an idealisaion of realiy, in which here will only be a finie se of bonds corresponding o a finie se of mauriy daes. Our goal is o build models which are arbirage-free and respec he erminal condiion 1.1, and hen o value ineres rae derivaives such as caps and swapions given a model for ineres raes or he underlying bonds which is consisen wih observed bond prices. For fixed, he map T R, T, or equivalenly, he family of zero raes R, T T T, where T is usually 1-15 years, is called he zero-coupon yield curve or jus he zero curve, or he zero-coupon erm srucure a ime. This gives he curren ime ineres raes for all mauriies, and flucuaes randomly. The zero curve is defined wih reference o bonds which are assumed o be defaulfree, such as governmen bonds. If we were discussing bonds issued by a company wih a posiive probabiliy of defaul, hen he yield would be higher as he company has o offer a beer ineres rae o invesors han a governmen, which is assumed o have no chance of defauling on any paymen. Also for fixed, he map T P, T is called he bond erm srucure. For fixed T, on he oher hand, he process P, T T is a sochasic process for he T -bond price Ineres raes. An ineres rae is a parameer ha measures he ime value of money in some way. An example is he zero rae of Definiion 1.2. An annually compounded simple ineres rae r quoed per annum, for insance 5% per annum is defined such ha if we borrow $1 oday ime zero we mus repay $1+r in a year s ime. Suppose we borrow $1 oday ime zero over T years. Define a pariion T = {,..., n } of [, T ] according o = < 1 < n = T, and wrie k := k k 1, for k = 1,..., n. Suppose a sochasic ineres rae r k operaes over he inerval k. A ime T, ha is, afer n compounding periods, we mus repay an amoun $X n, given by n X n = 1 + r k k. k=1 The ineres rae r k, for k {1,..., n}, is a simple ineres rae ha has a compounding frequency m k = 1/ k For example, if k = 1/2 years ha is, 6 monhs hen m k = 2. So we can also wrie n X n = 1 + r k. m k k=1 A coninuously compounded ineres rae process r = r T can be hough of as aking he limi n, or equivalenly, leing he mesh T := max k {,...,n} of he pariion approach zero: T. We hen have X n XT, given by n XT = lim 1 + r k k = exp r d. n k=1

4 4 MICHAEL MONOYIOS We elevae his o a definiion. Definiion 1.4 Bank accoun process. A riskless bank accoun or money-marke accoun, or savings accoun process S, represening he value of 1 uni of money invesed a ime zero wih a coninuously compounded ineres rae or shor rae process r = r, saisfies 1.2 ds = rs d, S = 1, so ha S = exp rs ds, LIBOR raes and ZCB prices. A zero-coupon ZCB value P, T may be associaed wih a simple ineres paymen, for he period [, T ], wih ineres rae L, T saisfying P, T = 1 + L, T T, T. Definiion 1.5 Spo LIBOR rae. The spo LIBOR rae seen a ime < T for he inerval [, T ] is L, T defined by 1.3. The erm spo in finance parlour means oday in broad erms, so he spo price of an asse means oday s price. Noe ha: L, T is se a ime ha is, i is F-measurable bu he paymen associaed wih i is made a ime T. The value a ime T of he LIBOR paymen L, T T made a ime T is ] E [exp Q rs ds L, T T F = P, T L, T T = 1 P, T, where Q P is an equivalen maringale measure EMM and F [,T ] is he underlying filraion associaed wih some probabiliy space Ω, F, P, wih P he physical measure. LIBOR London Inerbank Offered Rae is an indusry sandard which hisorically ses he raes a which banks lend money o each oher. The ime inervals for which LIBOR raes are quoed are ypically 1, 3, 6, 12 monhs Forward LIBOR raes. The assumed exisence of a ZCB marke implies he exisence of given ineres rae reurns over fuure inervals whose end poins are he mauriies of ZCBs, as we now show. Consider wo ZCBs wih mauriies S < T an S-bond and a T -bond, in our abbreviaed erminology, and le S < T. We can use hese bonds o guaranee a fuure rae of reurn over he inerval [S, T ], as follows. Consider he following sraegy: a ime sell he S-bond for price P, S and use he proceeds o purchase P, S/P, T T -bonds. Thus, his sraegy requires no iniial cash invesmen. The fuure cashflows are 1 a ime S and P, S/P, T a ime T ha is, we have borrowed $1 a ime S and we repay $P, S/P, T a T. Hence, a ime we have locked in a simple ineres rae L; S, T over [S, T ], given by L; S, T T S = P, S P, T, S < T. Definiion 1.6 Forward LIBOR rae. The simple ineres rae L f ; S, T L; S, T defined by 1.4 is called he forward LIBOR rae.

5 FIXED INCOME 5 Observe ha he spo LIBOR rae in Definiion 1.5 is he forward LIBOR rae when = S. In oher words, we have L, T L;, T, T. By analogy wih 1.3, he relaion beween spo LIBOR and he spo price of a ZCB, we define he forward ZCB wih price a ime, P f ; S, T P ; S, T, saisfying 1.5 P ; S, T = L; S, T T, S < T. Definiion 1.7 Forward ZCB price. The quaniy P ; S, T defined by 1.5 is called he forward ZCB price a ime for he inerval [S, T ]. From 1.4 and 1.5 we immediaely obain 1.6 P ; S, T = P, T P, S, S < T. For = S, we of course obain he sensible resul ha he forward ZCB price becomes he spo ZCB price: P ;, T = P, T = P, T, T. P, 1.4. Forward raes, spo raes and shor raes. Definiion 1.8 Coninuously compounded forward rae. Le S < T. The coninuously compounded forward rae f; S, T a ime for he inerval [S, T ] is defined by P ; S, T = exp f; S, T T S, S < T, where P ; S, T is he forward bond price a ime for he inerval [S, T ]. Of course, his is a naural generalisaion of he relaionship beween a ZCB price and is yield, P, T = exp yt = exp R, T T. Given 1.6, we have 1.7 exp f; S, T T S = P, T P, S, S < T, and hence 1.8 log P, T log P, S f; S, T =, T S S < T. The forward rae f; S, T seen a ime for he inerval [S, T ] is hus he rae of ineres, fixed a S, payable for an invesmen saring a S and ending a T S. Definiion 1.9 Coninuously compounded spo rae. Le T. The coninuously compounded spo rae R, T a ime for he inerval [, T ] is defined by seing = S in he coninuously compounded forward rae f; S, T From 1.8 we obain or equivalenly R, T := f;, T, T. R, T = log P, T, T, T 1.9 P, T = exp R, T T, T. In oher words, he coninuously compounded spo rae is jus he yield of he ZCB, or he effecive ineres rae implied by he ZCB price, as i should be.

6 6 MICHAEL MONOYIOS R, T R, S ime S T f; S, T Figure 2. Relaion beween zero and forward raes Relaion beween forward raes zero raes. From 1.8 and 1.9 we can relae he forward rae f; S, T seen a ime for he inerval [S, T ], and he zero raes R, S and R, T, o give 1.1 f; S, T T S = R, T T R, SS, S T. This relaionship makes perfec sense: if oday is ime, hen invesing over [, S] a oday s zero rae R, S, followed by an invesmen a he forward rae over he inerval [S, T ], should be equivalen o invesing a he rae R, T over he inerval [, T ]. This is illusraed in Figure 2. Thus, we have expr, SS expf; S, T T S = expr, T T, S T, which is 1.1. Definiion 1.1 Insananeous forward rae. The insananeous forward rae f, T seen a ime T for an invesmen a T is defined by 1.11 f, T := lim S T f; S, T = log P, T, T, where f; S, T is he coninuously compounded forward rae a for he inerval [S, T ], wih S T, and he second equaliy in 1.11 follows from 1.8. Noe ha f, T is F-measurable. For fixed, he map T f, T is called he forward rae erm srucure or forward yield curve. Definiion 1.11 The shor rae. The insananeous shor rae r a ime is defined by r := f, := lim T f, T,. The shor rae of ineres r = r is precisely he process used in defining he bank accoun process S in Definiion Relaion beween forward raes and ZCB prices. If we know f, T for all values of T, we can recover P, T for all values of T by he formula f, u du = log P, T log P, = log P, T, T. Therefore, 1.12 P, T = exp f, u du, T. Thus, here is a one-o-one correspondence beween forward raes and bond prices. From one, we can always deermine he oher. Therefore, a leas in heory, i should no maer which of hese we choose o model.

7 FIXED INCOME 7 2. Valuaion principles and numéraire invariance Here we briefly review he sandard heory for valuing coningen claims in a complee marke, and inroduce he concep of a numéraire. The general se-up is a complee, arbirage-free, coninuous semi-maringale marke, wih raded asse price processes S, S 1,..., S d on a sochasic basis Ω, F, F := F, P, where F is he augmened filraion generaed by a d-dimensional Brownian moion W. We someimes wrie S S 1,..., S d for he vecor of asses excluding S. Remark 2.1 Riskless asse. The zeroh asse S is ypically associaed wih a nonnegaive ineres rae process r = r, as in Definiion 1.4, and one can keep his example in mind in wha follows, hough his is no in fac necessary, and as will be seen he subsequen argumens in Secions 2.1 and 2.2 hold when S is any arbirary raded asse wih sricly posiive price process. In he case when S is indeed he process for he value of a bank accoun, recall ha one usually exhibis he exisence of an equivalen maringale measure EMM Q P by showing ha he discouned asse price vecor S/S is a local Q-maringale and when he marke is complee, Q is unique. As a direc consequence, he wealh process X of any self-financing rading sraegy involving S, S is also a local Q-maringale. Any coningen claim wih FT -measurable payoff H can be replicaed by a wealh process X and, under suiable condiions, he discouned wealh process is a Q-maringale. Given ha X replicaes H, his leads o he classical formula for he value process V of he claim: [ ] H 2.1 V = S E Q S T F, T. Example 2.2 Valuaion formula for a T -bond. If we ake H 1 in 2.1 we obain an alernaive valuaion formula for a T -bond: [ ] P, T = S E Q 1 T ] S T F = E [exp Q rs ds F, T. Example 2.3 Classical Black-Scholes marke. The classical example of he above ideas is he sandard Black-Scholes BS marke. In his case he ineres rae r is consan, so he bank accoun process is S = e r for, and ZCB prices are given by P, T = e rt. The valuaion formula 2.1 hus reduces o he familiar form 2.2 V = e rt E Q [H F] = P, T E Q [H F], T. Remark 2.4 Sochasic ineres raes. If he ineres rae is now aken o be some nonnegaive adaped o he Brownian filraion sochasic process r = r, hen he bank accoun process is given by 1.2 and he valuaion formula 2.1 becomes ] V = E [exp Q rs ds H F, T. If H and r were independen, his formula would simplify according o ] V = E [exp Q rs ds F E Q [H F] = P, T E Q [H F], T, where P, T = E Q [e rs ds F] is he price of he ZCB of mauriy T. This recovers a formula of he same srucure as in 2.2. Bu i is no in general o be expeced ha H and r are independen. Neverheless, we shall see ha we can recover a formula similar o 2.2 using change of numéraire ideas, which we now describe.

8 8 MICHAEL MONOYIOS 2.1. Numéraire pairs. The classical valuaion argumens above used he idea of discouning, ha is, expressing he prices of raded asses in unis of he bank accoun process or, o use he erminology relevan o us, using he bank accoun process S as a numéraire, a sricly posiive raded asse price process which, if we measure all asse prices in unis of he numéraire, we obain local maringales under some measure equivalen o he physical measure P. In fac, using S as a numéraire asse is an arbirary choice. Raher han referring only o Q as he risk-neural measure, i is more accurae o say ha S, Q is a numéraire pair. If we use anoher asse as numéraire, we obain anoher measure which urns all asse prices, when denominaed in unis of he numéraire, ino local maringales, as we shall see. Definiion 2.5 Numéraire and numéraire pair. A numéraire N is a raded asse his includes a porfolio buil from raded asses wih sricly posiive price process, such ha if all asse prices are denominaed in erms of he numéraire, hey are local maringales under some measure Q N P. The pair N, Q N is called a numéraire pair. In paricular, if X is he wealh process of a self-financing porfolio, hen he process X/N is a local Q N -maringale. Assume only for he remainder of his sub-secion ha S is a sricly posiive price process, normalised for simpliciy so ha S = 1. A rading sraegy is a vecor process H = H 1,..., H d, wih each componen an adaped process saisfying H is d[s i ]s < almos surely for all and for each i = 1,..., d, wih H i he process for he number of shares of asse i. Once we choose he vecor H, he wealh placed in he vecor asse S is deermined, and he remaining wealh is placed in asse S. The porfolio wealh process X associaed wih H evolves according o X HS 2.3 dx = H ds + ds, S where HS := d i=1 H is i is he inner produc of vecors H, S. The following proposiion shows ha when we use S as a numéraire, hen he associaed discouned wealh process is a local maringale under Q, where S, Q is a numéraire pair. Proposiion 2.6. Suppose S, Q is a numéraire pair. Le X be he wealh process as in 2.3. Then, defining discouned quaniies X := X/S, S := S/S, he evoluion of X is given by 2.4 d X = H d S = In paricular, X is a local Q-maringale. d H i d S i, i=1 X = X. Proof. Using he Iô formula for coninuous semi-maringales we have d = 1 S S 2 ds + 1 S 3 d[s ]. Using he Iô produc rule, d X X = d S Using 2.3 and 2.5 his convers o exercise! d X = H S 1 = X d + 1 ] [X, S S dx + d 1S. ds S S ds + S S 2 d[s ] 1 S d[s, S ].

9 One also has exercise d S S = d S = 1 S so ha we obain 2.4. FIXED INCOME 9 1 = S d + 1 ] [S, S S ds + d 1S ds S S ds + S S 2 d[s ] 1 S d[s, S ] Theorem 2.7. Suppose ha S has he maringale represenaion propery. Tha is, for any FT -measurable random variable Y, here is a d-dimensional inegrand ψ such ha 2.6 Y = E Q [Y ] + ψ d S. Then he unique no-arbirage value process of a claim wih FT -measurable payoff H is given by [ ] H 2.7 V = S E Q S T F, T, Proof. This is jus he usual replicaion argumen, given in probabilisic erms, as follows. Take Y = H/S T in 2.6, and inerpre his equaion as defining a rading sraegy which replicaes he payoff H. The no-arbirage value of he claim a ime zero is hen he iniial wealh of he replicaion sraegy. This gives [ ] H V = E Q. S T This implies ha he process V/S is a Q-maringale, and his in urn yields he valuaion formula Change of numéraire. Theorem 2.8 Change of Numéraire. Suppose ha N, Q N is a numéraire pair. Le U be a second numéraire. Then U, Q U is a numéraire pair, where he measure Q U Q N P on FT is defined by 2.8 dq U dq N = U/U F N/N, T. Proof. Since N, Q N is a numéraire pair, we have [ ] U E QN = U N N, T. Thus, he process Z, defined by Z := U/U N/N, T, is a Q N -maringale wih iniial value 1, so we can define a measure Q U Q N on FT via he prescripion dq U dq N = Z, T, F which we recognise as 2.8.,

10 1 MICHAEL MONOYIOS Now, a process Y is a local Q U -maringale if and only if ZY is a local Q N -maringale. Take Y = X/U, for an arbirary porfolio wealh process X. Then ZY = N X U N, T, which is indeed a Q N -maringale. Hence, X/U is a Q U -maringale, and herefore U, Q U is a numéraire pair. Example 2.9 Valuaion formula wih general numéraire change. Begin wih a numéraire pair N, Q N, so ha X/N is a local Q N -maringale. Suppose ha X replicaes an FT -measurable claim H. Then he valuaion formula for he claim is V = 1 N EQN [ ] H NT F, [, T ]. Then of course, using he change of numéraire heorem, Theorem 2.8, we have Z := dqu dq N = U/U F N/N, T, for some oher numéraire pair U, Q U. The Bayes rule hen gives [ ] H E QU UT F = 1 [ ] [ ] H Z EQN ZT H UT F = UE QN NT F so ha we obain he anicipaed valuaion formula when using U as numéraire: [ ] H V = UE QU UT F, [, T ]. = V U, Example 2.1 Derivaive valuaion formula wih T -bond price as numéraire. Recall he valuaion formula 2.7 for a derivaive when using S, Q is a numéraire pair, repeaed below: [ ] H 2.9 V = S E Q S T F, T. Now change numéraire according o Theorem 2.8. Selec as numéraire a ZCB of mauriy T a T -bond, wih price process P, T, for [, T ], such ha P, Q P is a numéraire pair for some measure Q P. Wrie Q T Q P for he measure corresponding o using he T -bond as numéraire. We call he measure Q T he T -forward measure his will play a role in laer secions on valuing cerain ineres rae conracs. Le Z denoe he densiy process of Q T wih respec o Q, given by Z = dqt dq = F By he Bayes rule, we have [ ] E QT ZT [H F] = E Q Z H F P, T /P, T S /S and his recovers a formula of he form 2.2: = = S [ H P, T EQ P, T /P, T, T. S ] S T F 2.1 V = P, T E QT [H F], T, = V P, T, T, bu wih measure Q T corresponding o he numéraire pair P, T, Q T. Observe ha his is consisen wih 2.9, because P T, T = 1, so ha H/P T, T = H inside

11 FIXED INCOME 11 he condiional expecaion. Observe also ha if he ineres rae is deerminisic, hen Z = 1, so ha Q T = Q and we recover he formula Forward prices and he T -forward measure. We recall ha a forward conrac on an asse S, sruck a K, wih mauriy T, has payoff ST K a T. Given he numéraire pair S, Q, wih S as usual he bank accoun process, he value of he forward conrac a T is V FWD = S E Q [ ST K S T ] F = S KP, T, T, on using he fac ha S/S is a Q-maringale. The forward price F, T of he asse a ime for delivery a T is he value of K which gives V FWD =, so ha F, T = S P, T, T. By definiion of he T -forward measure, asse prices discouned by he T -bond price are Q T -maringales, so we ge he key fac abou he T -forward measure: Proposiion The T -forward price F, T [,T ] is a maringale under he T - forward measure Q T The Black formula. In he case ha F, T [,T ] is lognormal under he T - forward measure Q T, so has dynamics df, T = σ, T F, T dw T, for some deerminisic funcion σ, hen wriing E T for expecaion under Q T he price process of a call on S is given by 2.11 where 1 d := Σ T V = P, T E T [ST K + F] [ log = P, T E T [F T, T K + F] [ = P, T F, T Nd KNd Σ ] T, F, T K + 1 ] 2 Σ2 T, Σ 2 T = σ 2 u, T du. The formula 2.11 is called Black s formula for a call opion. We shall see laer ha i is used in he conex of valuing cerain ineres rae derivaives, where a suiable choice of numéraire renders he underlying sochasic facor ino a lognormal process. 3. Ineres rae derivaives 3.1. Coupon-bearing bonds. Zero-coupon bonds are usually relaively shor erm insrumens, and for mauriies longer han abou wo years one would observe bond prices of coupon-bearing bonds, which make regular and usually fixed paymens, called coupons, a fixed imes. Definiion 3.1 Fixed coupon-bearing bond. Le T < T 1 < < T n = T. A fixed coupon-paying bond makes coupon paymens c k n k=1 a imes T 1,..., T n, as well as a final paymen of he principal amoun, or noional amoun, N, a he mauriy ime T n =: T, as shown in Figure 3. Denoe he price of he bond a ime by P c, T.

12 12 MICHAEL MONOYIOS c 1 c n + N ime T T 1 T n = T Figure 3. Coupon-bearing bond The bond price a ime is given by n P c, T = exp R, T k T k c k + N exp R, T T = k=1 n P, T k c k + NP, T, T < T 1 <... T n = T. k=1 We observe ha he bond price is a funcion of a number of zero raes wih differen mauriies. Typically, coupons may be quoed in erms of a rae usually he same for each coupon, so ha c k = r k T k T k 1 N for k = 1,..., n, where r k is an ineres rae in pracice quoed wih he same compounding frequency as he paymen frequency for he period [T k 1, T k ], and we have 3.1 n P c, T = N P, T k r k T k T k 1 + P, T, T < < T n = T. k=1 Ofen we have T k T k 1 = δ, a fixed consan, and r k = r, fixed, in which case 3.1 simplifies o n P c, T = N rδ P, T k + P, T, T < < T n = T. k=1 The yield or yield o mauriy of he bond, y y, T, is he single ineres rae ha would give he same bond price as above. Definiion 3.2 Coninuously compounded yield. The coninuously compounded yield o mauriy, y y, T, of he bond wih coupon paymen daes T k n k=1, coupons c k n k=1, noional N, mauriy T n = T, and price P c, T a ime, is defined implicily by n P c, T = c k exp yt k + N exp yt =: By;, T By. k=1 Naurally, his implies ha he coupon-bearing yield y is some form of weighed average of he zero raes R, T k, k = 1,..., n. Observe ha raising ineres raes lowers bond prices, and vice versa. In pracice, zero raes are no direcly observed, bu bond prices and oher ineres rae derivaive prices are, and consrucing he yield curve from observaions of ineres rae derivaive prices is an imporan pracical maer for financial insiuions. For a ZCB, he coninuously compounded yield, y y, T, saisfies P, T = exp yt =: By;, T By, so ha he coninuously compounded yield is jus zero rae corresponding o he bond price: log P, T y y, T = = R, T. T

13 FIXED INCOME 13 Definiion 3.3 Duraion. The duraion of he bond in Definiion 3.2 is Dy;, T Dy given by 1 B Dy := P c, T y y [ n ] 1 = P c T k c k exp yt k + NT exp yt., T k=1 The duraion measures sensiiviy of he bond price o changes in ineres raes ha is, o shifs in he yield curve, since we have For a ZCB we have B y y = P c, T Dy. Dy;, T = T. Definiion 3.4 Convexiy. The convexiy of he bond in Definiion 3.2 is Cy;, T Cy defined by 1 2 B Cy := P c, T y 2 y. For a ZCB we have Cy;, T = T Floaing coupon bond floaing rae noe. Definiion 3.5 Floaing rae noe FRN. A floaing rae noe or LIBOR bond, or floaing rae bond, or floaing coupon bond is a coupon-bearing bond wih floaing coupons usually se a a rae equal o he prevailing LIBOR rae. For simpliciy, normalise he noional amoun o N = 1. Le T < T 1 < < T n = T. The FRN coupon paymens are 1 c k = LT k 1, T k T k T k 1 = P T k 1, T k 1, k = 1,..., n. The ime- value of he coupon paymen made a ime T k is hus P, T k c k = P, T k P T k 1, T k P, T k. To value he FRN we shall need he following resul. Lemma 3.6 Presen value PV of a LIBOR paymen. Le S < T. The value a ime of he LIBOR paymen LS, T T S made a ime T is V LIBOR = P, S P, T, S. Proof. We have 1 LS, T T S = P S, T 1. The ime- value of he fixed paymen 1 is P, T. To value he paymen 1/P S, T, consider he following ZCB sraegy. A ime, purchase one S-bond for P, S, which pays 1 a ime S. Use hese proceeds o purchase 1/P S, T T -bonds a ime S, and hese pay 1/P S, T a ime T. Thus, he value a S of he ime-t paymen of 1/P S, T, is P, S.

14 14 MICHAEL MONOYIOS R Bank C/Pary LIBOR Figure 4. Ineres rae swap Corollary 3.7 Value of FRN. A ime of issue and in fac a all coupon daes, a floaing rae noe is always a par, ha is, is value is equal o he noional amoun. Proof. The FRN value a T is, on using Lemma 3.6, n V FRN, T = [P, T k 1 P, T k ] + P, T = P, T, k=1 so on he issue dae, when = T, we have V FRN T, T = P T, T = Ineres rae swaps. An ineres rae swap is illusraed in Figure 4. Two counerparies swap floaing ineres paymens such as LIBOR for fixed paymens a some rae R, usually on he same noional amoun, usually in he same currency, for a se ime 2 1 years, ypically. Such a swap can be used o aler one s ineres rae exposure from fixed o floaing and vice versa. Le oday be ime, wih T T 1 < T 2 < T n = T, wih T k n k=1 denoing he paymen daes in a swap. Wrie δ k := T k T k 1, for k = 1,..., n, o denoe he ime beween paymens he day-coun fracion. Fix he noional amoun o be N = 1. The floaing paymen a T k is δ k LT k 1, T k. The fixed paymen a T k is δ k R. The ne cashflow a T k, o he pary receiving he fixed paymens, is δ k R LT k 1, T k. Noe in passing ha we can ficiiously adjoin o he swap equal and opposie paymens of N = 1 a he mauriy dae. Thus, he swap can be considered as an exchange of a fixed rae bond for a floaing rae bond. The floaing side is equivalen o a floaing rae noe wihou a final repaymen of principal, wih value oday ime given by n [P, T k 1 P, T k ] = P, T P, T. k=1 The fixed side paymens have presen value a of n Rδ k P, T k. Hence, he swap value a ime o he pary receiving fixed is n 3.2 V swap = R δ k P, T k P, T P, T. k=1 k=1 I < T, his is a forward swap iniiaed a. We shall someimes refer o his as a T T T forward swap, where he firs ime inerval [, T ] denoes he ime unil he firs LIBOR rese dae when he swap begins and he second ime inerval [T, T ] denoes he erm of he swap, he ime up o mauriy from he firs rese dae. Typically, he swap a spo swap, is iniiaed a he firs LIBOR rese dae, T.

15 FIXED INCOME 15 Definiion 3.8 Forward swap rae. The forward swap rae R ; T, T a he iniiaion ime of he swap is he value of he fixed rae R such ha he swap has value zero a he iniiaion ime. The swap rae is hus given by 3.3 R ; T, T = P, T P, T n k=1 δ kp, T k, T < T n = T. The numeraor in 3.3 is he PV of he floaing paymens, while he denominaor is he PV of he fixed paymens, and is someimes called he presen value of a basis poin PVBP, or he accrual facor, denoed by n 3.4 A; T, T := δ k P, T k, T < T 1 < T n = T. k=1 This is he value a T of uni paymens received a T 1,..., T n =: T. If he iniiaion ime is = T, coinciding wih he firs LIBOR rese dae, hen he spo swap rae R, T := R ;, T has he represenaion 3.5 R, T = 1 P, T n k=1 δ kp, T k, T < T n = T. A laer imes his swap does no generally have value zero because he same fixed-side rae is mainained hroughou. For fixed, he map T R, T is called he swap yield curve or he swap curve, or he swap rae erm srucure Ineres rae opions. The sandard ineres-rae opions are caps, floors and swapions Caps. An ineres rae cap allows he holder o pu an upper bound on a floaing usually, LIBOR ineres paymen. I is a series of opions, called caples, each of which can be hough of a a call on he spo LIBOR rae, or as an opion on a forward rae agreemen FRA, or as a pu opion on a ZCB. Le T < T 1 < < T n = T, wih T k n k=1 denoing paymen coupon daes on some loan. Le δ k := T k T k 1, as before. For breviy, wrie L k := LT k 1, T k for he spo LIBOR rae se a ime T k 1 for paymen a T k. Le R be a fixed consan, called he cap rae. A each coupon dae T k, k = 1,..., n, he cap pays a cash amoun δ k L k R +. Remark 3.9 Ineres rae floor. A floor is defined analogously o a cap, bu pays δ k R L k + on each coupon dae. In view of 1.3 we have δ k L k R = P T k 1, T k 1 + δ kr. The value of his paymen a ime T k 1 is P T k 1, T k P T k 1, T k 1 + δ kr = 1 E k P T k 1, T k +, E k where E k := 1/1 + δ k R. Thus, each caple can be viewed as 1/E k pu opions of mauriy T k 1 on a T k -bond. Wih his viewpoin, he caple value a ime T k 1 would be 3.7 V caple = 1 k 1 ] E [exp Q rs ds E k P T k 1, T k + F. E k

16 16 MICHAEL MONOYIOS Reurning o he view of each caple as a call on he appropriae LIBOR rae, he value of he cap a ime T is n [ ] V cap 1 = NE QN NT k δ kl k R + F, k=1 for some numéraire pair N, Q N. I urns ou ha he appropriae choice of numéraire for he caple which pays ou a T k is he corresponding T k -bond. This is a consequence of he following resul. Lemma 3.1 Forward LIBOR rae maringale propery. Le S < T. The forward LIBOR rae L = L; S, T [,S] is a Q T -maringale. Proof. By definiion, L; S, T = 1 T S P, S P, T 1, S. By he definiion of Q T, P, S/P, T [,S] is a Q T -maringale, and his proves he lemma. Because of Lemma 3.1, he value of each caple can be obained via a varian of Black s formula 2.11, if we assume ha he forward LIBOR rae is lognormal see Problem Shee Swapions. A swapion is an opion o ener ino a swap agreemen, a a predeermined fixed-side rae R, saring a a ime T in he fuure. I is a payers swapion if he holder will ener he swap paying he fixed side, and a receivers swapion oherwise. Recall, a T T T forward swap, iniiaed a ime oday begins a ime T he firs LIBOR rese dae and has paymen daes T 1,..., T n, so has erm T T, wih T := T n. From 3.2, he value a ime of he swap o he pary paying fixed is n V swap = P, T P, T R δ k P, T k, δ k := T k T k 1, where R is he fixed rae on he swap. We shall firs show how a swapion may be viewed as a bond opion. A T T T payer s swapion wih swapion srike R is a conrac which, a he exercise dae T, gives he holder he righ o ener a payer s swap wih paymen daes T k n k=1, a fixed swap rae R. The swapion value a T is hus [ + n V swapion T = max[v swap T, ] = 1 R δ k P T, T k + P T, T ]. The payer s swapion is hus a pu opion on a coupon bond wih coupon rae R and mauriy T wih srike 1 and mauriy T. Wih his viewpoin, he swapion value a T can be represened as ] + V swapion = E [exp Q rs ds 1 P c T, T F, k=1 k=1

17 where P c T, T = R FIXED INCOME 17 n δ k P T, T k + P T, T. k=1 The second way o view a swapion is as an opion on he forward swap rae. Using 3.3 and 3.4 he value a ime T of a swap wih fixed rae R, o he pary paying fixed, is V swap = A; T, T R ; T, T R, where A is he accrual facor and R is he forward swap rae. The swapion value a T is hen V swapion T = AT ; T, T R T ; T, T R +. Thus, he payer swapion is a call of srike R on he forward swap rae. Wrie A A; T, T and R R ; T, T, so he swapion payoff is V swapion T = AT R T R +. Now change numéraire, choosing numéraire pair A, Q A, wih he annuiy measure Q A defined in he usual way via he change of numéraire heorem: dq A dq = 1 A F A S, T. Wih his change of numéraire, we ge he swapion value a T as We now need: V swapion = AE QA [ R T R + F ], T. Lemma The forward swap rae R [,T ] is a Q A -maringale. Proof. Recall ha R ; T, T = P, T P, T. A; T, T The numeraor is a porfolio of raded asses, so mus be a Q A maringale when denominaed in unis of A. This resul allows us o develop a Black swapion formula. If R is lognormal under Q A, hen he payer swapion value a T has represenaion [ V swapion = A R Nd RNd Σ ] T, where d = [ 1 Σ log T and where, under Q A, R R + 1 ] 2 Σ2 T, Σ 2 T = σ 2 u; T, T du, dr = σ; T, T R dw A, for a Q A -BM W A, wih σ ; T, T posiive.

18 18 MICHAEL MONOYIOS 4. Modelling he erm srucure The goal is o build models o describe he erm srucure of ineres raes and ZCB prices. We face choices beween he order below reflecs he hisorical developmens: modelling he shor rae r and using his model o hen value ZCBs; modelling he ZCB dynamics; modelling forward rae dynamics and hen obaining ZCB prices from he relaion Since he shor rae, ZCBs and forward raes are all iner-conneced, here will be relaionships beween he parameers used in he above approaches, as we now describe Srucural relaionships. The following relaionships will hold irrespecive of which measure we are working under, bu for concreeness le us suppose we are using he physical measure P. Suppose ha he shor rae, ZCB prices and forward raes saisfy he SDEs dr = a d + b dw, dp, T = m, T P, T d + P, T v, T dw, df, T = α, T d + σ, T dw. The BM W can be vecor-valued, in which case b, v, T, σ, T are row vecors, and all parameers are adaped processes. The parameer funcions m, T, v, T, α, T, σ, T are assumed o be differeniable wih respec o he mauriy parameer T, and all processes are regular enough o allow differeniaion under he inegral sign and reversal of he order of inegraion in, say, double inegrals. Theorem 4.1 Srucural relaionships. 1 If ZCB prices saisfy 4.2 hen forward raes saisfy 4.3, wih α, σ given by α, T = v m, T v, T, T, v σ, T =, T. 2 If forward raes saisfy 4.3 hen he shor rae saisfies 4.1, wih a, b given by a = f, + α,, b = σ,. 3 If forward raes saisfy 4.3 hen ZCB prices saisfy dp, T 4.4 r P, T = + β, T + 12 γ, T 2 d + γ, T dw, wih β, T := α, u du, γ, T := σ, u du. Proof. 1 Problem Shee 3. 2 Se T = in he inegral form of 4.3, and use r = f,, o give 4.5 r = f, + αs, ds + σs, dw s.

19 Now wrie recall f, = r Then 4.5 becomes r = r + + f, = r + FIXED INCOME 19 αs, = αs, s + σs, = σs, s + f, u du + σs, s dw s + s f, u du, s s α s, u du, σ s, u du. αs, s ds + σ s, u du dw s. s α s, u du ds Reverse he order of inegraion in he double inegrals, o ge u u r = r + f, u + αs, u ds + σs, u dw s + αs, s ds + + σs, s dw s f = r + s, s + αs, s ds + which esablishes he second claim. 3 We have log P, T = = f, u du f, u + αs, u ds + σs, s dw s, du σs, u dw s du, [, T ]. Spli he inegrals over [, T ] ino inegrals over [, T ] minus inegrals over [, ]. Thus, we wrie: as well as f, u du = αs, u ds du = f, u du = = s f, u du log P, T αs, u ds du αs, u du ds αs, u du ds u αs, u ds du αs, u ds du αs, u ds du, f, u du, where we have performed one reversal of he order of inegraion, and used he fac ha αs, u is non-zero only if s u. Similarly, we have 4.9 σs, u dw s du = s σs, u du dw s u σs, u dw s du.

20 2 MICHAEL MONOYIOS Using in 4.6, we ge log P, T = log P, T + f, u + Hence, = log P, T + s αs, u du ds P, T = P, T exp X, X := u s αs, u ds + rs + βs, T ds + u σs, u du dw s rs + βs, T ds + σs, u dw s du γs, T dw s. γs, T dw s, and he Iô formula hen gives he required dynamics of he ZCB prices. 5. Shor rae modelling Assume ha under he physical measure P, he shor rae follows 5.1 dr = µ, r d + σ, r dw, wih W a one-dimensional BM. If his is he only exogenously given process, hen he only raded asse whose P-dynamics are uniquely deermined will be he bank accoun process S, following ds = rs d, S = 1. In his framework, ZCBs are reaed as derivaives based on he shor rae. One canno hedge a ZCB using jus he bank accoun, so in his sense he marke is incomplee, unil we inroduce a ZCB say, a T -bond, for some T > as a raded insrumen, which complees he marke. Then, using a T -bond plus cash, we can hedge an S-bond, where S < T, as we show below, and his will also show ha ZCBs of differen mauriies have o saisfy a cerain consisency condiion, in order o preclude arbirage The erm srucure PDE. Assume P, T = F T, r, T, for some smooh funcion F T, x saisfying F T T, x = 1. By he Iô formula, he P-dynamics of he T -bond are given by [ ] 5.2 dp, T = P, T m T, r d + σ T, r dw, where m T, x := σ T, x := 1 F T F T + µf x T + 1, x 2 σ2 F xx T, x, σf x T F T, x. For an S-bond, wih S < T, we have similar dynamics, for funcions m S, σ S given in erms of a funcion F S, x saisfying F S S, x = 1. Now rade he T -bond plus cash so as o replicae he S-bond. The wealh process X of he hedging porfolio follows dx = H dp, T + rx HP, T d, where H is he process describing he number of T -bonds in he porfolio. The replicaion condiion X = P, S, for all [, S], requires dx = dp, S.

21 Equaing he Brownian erms gives FIXED INCOME 21 H = σs, r P, S σ T, r P, T, S. Then, using his and equaing he finie variaion erms gives 5.5 m T, r r σ T, r = ms, r r σ S, S., r Thus, all he ZCBs, regardless of mauriy, mus have he same marke price of risk MPR. The LHS of 5.5 does no depend on S, while he RHS does no depend on T, so he equaion can only hold if boh sides do no depend on any of S or T. We have herefore proven: Theorem 5.1 Common marke price of risk MPR. In an arbirage-free ZCB marke driven by a one facor shor rae process of he form 5.1, here exiss a process λ, r, called he marke price of risk MPR of he ZCBs, such ha an equaion of he form 5.6 m T, r r σ T, r = λ, r, T, holds for every mauriy T >, wih he same universal MPR, and wih m T, σ T given by , and where he T -bond price P, T = F T, r follows 5.2. Using he form of m T, σ T given in we hen obain: Corollary 5.2 Term srucure PDE. In an arbirage-free ZCB marke driven by a one facor shor rae process of he form 5.1, he T -bond price is given by P, T = F T, r, where F T, x saisfies he erm srucure PDE 5.7 F T, x + µ σλ, xf T x, x σ2, xf T xx, x xf T, x =, wih boundary condiion F T T, x = 1. By a Feynman-Kac soluion of 5.7, we ge he anicipaed risk-neural valuaion formula for ZCBs. Tha is, we have ] P, T = F T, r = E [exp Q ru du F, T, where, under Q P on FT, he shor rae dynamics are 5.8 dr = µ, r σ, rλ, r d + σ, r dw Q, wih W Q a Q-BM. The nex observaion is is ha he measure Q is of course an EMM, and ha S, Q is a numéraire pair, as we now demonsrae. Observe firs ha, comparing he P-dynamics 5.1 of r wih is Q-dynamics 5.8, and invoking he Girsanov Theorem, he relaionship beween W and W Q is given by W Q = W + λs, rs ds, T, so ha he P-dynamics 5.2 of he T -bond ransform o Q-dynamics given by dp, T = rp, T d + σ T, rp, T dw Q, on using 5.6. Thus P, T /S is indeed a local Q-maringale.

22 22 MICHAEL MONOYIOS Observe ha he MPR process λ, is no deermined by he P-dynamics of he shor rae: observing he laer hus gives no informaion on he MPR. Insead, he MPR can only be inferred by backing i ou from observed bond prices in some way, by seeking o make he parameers in he Q-dynamics of he shor rae consisen wih observed bond prices. We shall no be delving ino such calibraion issues here. Remark 5.3 Valuing general claims. The same hedging argumens used above can be used o value a general claim wih payoff ΨrT, and he value of he claim a T is V, r, where V, x will solve he erm srucure PDE 5.7 bu wih erminal condiion V T, x = Ψx see Problem Shee 3. Example 5.4 Call opion on a ZCB. In he shor rae modelling se-up of his secion, a call opion of srike K and mauriy T on a T -bond, where T < T, has payoff P T, T K + = F T T, rt K + =: ΨrT, and so has value a T given by V, r = E [exp Q ] ru du ΨrT F, T, and V, x would solve he erm srucure PDE. In cerain models his compuaion, or he equivalen compuaion of he above expecaion, can be dome explicily, usually via a change of numéraire echnique, which changes he valuaion measure o he T -forward measure Q T. An example in he Vasicek model is given in Problem Shee 3. Example 5.5 Valuaion of caple as a bond opion. Consider a caple wih cap rae R, such ha he caple pays he cash amoun δ k L k R + a ime T k, where δ k := T k T k 1, and L k := LT k 1, T k is he spo LIBOR rae se a ime T k 1 for paymen a T k. Recall from 3.6 and 3.7 ha he caple can be viewed as 1/E k pu opions of mauriy T k 1 on a T k -bond, where E k := 1/1 + δ k R. Wih his viewpoin, he caple value a ime T k 1 is given by 3.7, repeaed below: V caple = 1 k 1 ] E [exp Q rs ds E k P T k 1, T k + F. E k In he framework of his secion, we hus have V caple = V, r, where he funcion V, x saisfies V + µ σλv x σ2 V xx xv =, V T k 1, x = 1 E k E k F T k T k 1, x +, wih he funcion F Tk T k 1, x solving he appropriae analogue of he erm srucure equaion 5.7. Equivalenly, as a purely probabilisiic represenaion, we have V, r = 1 E k E Q [ for [, T k 1 ]. e k 1 rs ds E k E Q [ e k T k 1 rs ds ] + k 1] FT F, 5.2. Classical shor rae models. The salien observaion made by he discussion of shor rae models up o now is ha he only relevan dynamics for he valuaion of derivaives are hose under he EMM Q. In he classical models whose SDEs are wrien down below, hese are Q-dynamics.

23 Vasicek model: FIXED INCOME 23 dr = ab r d + σ dw Q, where a >, b and σ > are consans and W Q is a Q-BM. Cox-Ingersoll-Ross CIR model: dr = ab r d + σ r dw Q, where a >, b and σ > are consans and W Q is a Q-BM. Compared o he Vasicek model, he CIR model has he propery ha he ineres rae remains posiive bu i is a less racable model. Dohan model: dr = ar d + σr dw Q, where a and σ > are consans and W Q is a Q-BM. Again, a racable model, bu some of is properies are no realisic. The models above have consan parameers. To fi he model o observed bond prices, i is someimes advanageous o make some parameers ime-dependen, as in he models below. Black-Derman-Toy BDT model: dr = ar d + σr dw Q, where a, σ are deerminisic funcions. Ho-Lee model: dr = a d + σ dw Q, where a is a deerminisic funcion. Hull-Whie exended Vasicek model: dr = a br d + σ dw Q, where a, b, σ are deerminisic funcions. Hull-Whie exended CIR model: dr = a br d + σ r dw Q where a, b, σ are deerminisic funcions. In he Ho-Lee model, he shor rae under he EMM Q follows dr = a d + σ dw Q, where a is a deerminisic funcion, σ > is consan, and W Q is a Q-BM Affine shor rae models. Suppose a shor rae model has dynamics under an ELMM Q given by dr = a, r d + σ, r dw Q, for some Q-BM W Q and deerminisic funcions a,, σ,. The T -bond price a ime is given by ] P, T = E [exp Q ru du F =: F T, r, [, T ], where he funcion F T, solves he erm srucure PDE 5.9 F T, x + a, xf T x, x σ2, xf T xx, x xf T, x =, wih boundary condiion F T T, x = 1.

24 24 MICHAEL MONOYIOS Definiion 5.6 Affine model. An affine model is one in which he funcion in 5.9 akes he form F T, x = exp A, T B, T x, T, for deerminisic funcions A,, B,. The erm srucure PDE 5.9 hen implies ha he A,, B, mus saisfy he equaion 5.1 A, T + B, T x a, xb, T σ2, xb 2, T x =, T, wih erminal condiions AT, T = BT, T =. If we now assume, moreover, ha a, and σ 2, are also affine, ha is, a, x = αx + β, σ 2, x = γx + δ, for deerminisic funcions α, β, γ, δ, hen using his in 5.1 gives ha A,, B, mus saisfy he sysem of equaions 5.11 A, T + βb, T 1 2 δb2, T =, B, T + αb, T 1 2 γb2, T + 1 =, AT, T = BT, T =. Equaion 5.12 is a Riccai equaion for B, which, in some models, can be solved explicily. The resul can hen be used in 5.11 o find A,. Example 5.7 Hull-Whie exended Vasicek model. Here he shor rae SDE under Q is dr = ab r d + σ dw Q, where a, b, σ are deerminisic funcions. The sysem of equaions hen reads as wih soluion A, T + abb, T 1 2 σ2 B 2, T =, A, T = B, T = B, T ab, T + 1 =, AT, T = BT, T =, Bu, T aubu 1 2 σ2 ubu, T du, exp u as ds du, T. A special case occurs when a = a >, b = b R, σ = σ > are consans, and we obain he classical Vasicek model, for which A, T = b σ2 2a 2 T B, T + σ2 4a B2, T, B, T = 1 1 e at, T. a as found via independen mehods in Problem Shee 3.

25 FIXED INCOME Calibraion of shor rae models o ZCB prices. Suppose we have some model for he shor rae process r = r under an EMM Q, wrien as dr = a, r; Φ d + σ, r; Φ dw Q, where Φ represens some vecor of model parameers which migh be consan or imedependen. If we know Φ we obain ZCB prices in he form P, T ; Φ = F T, r; Φ, [, T ], where F T, x; Φ solves he erm srucure PDE. We now have ZCB prices as funcions of he model parameers. The calibraion problem is o find a parameer vecor Φ which is consisen wih observed ZCB prices a, say, some iniial ime. So, suppose we observe marke ZCB prices a ime zero, given as {P, T, T > }. We look for a parameer vecor Φ such ha 5.14 P, T ; Φ = F T, r; Φ = P, T, T >. Wih his iniial calibaraion achieved, we wrie he Q-dynamics for r as dr = a, r; Φ d + σ, r; Φ dw Q, and we can use his model o value derivaives including ZCBs a imes. To achieve easy calibraion one needs ideally closed-form soluions for ZCB price funcions hence he populariy of affine models. Observe ha he calibraion equaions 5.14 consiue an infinie dimensional sysem of equaions if we have P, T for all mauriies T >, or a he very leas a highdimensional sysem if we have marke ZCB prices for a large number of mauriies. Bu we only have a finie parameer vecor if all he parameers are consans, so ha in general i is no possible o perfecly mach he observed iniial ZCB erm srucure wih a consan parameer model. This has spawned aemps o effecively make Φ infiniedimensional, by making one or more parameers ime-dependen as in, for insance, he Hull-Whie models. This hen allows a perfec fi o he iniial erm srucure, as we now show in he Ho-Lee model and an example wih a simplified version of he exended Vasicek model appears in Problem Shee 4. Example 5.8 Ho-Lee model calibraion. The SDE for r under Q is dr = a d + σ dw Q, σ >, where a is a deerminisic funcion. Once again, posulae ha ZCB prices have he affine form 5.15 P, T = exp A, T B, T r, T, for deerminisic funcions A,, B,, which mus saisfy he sysem of equaions , and in his case we obain These have soluion A, T = A, T + ab, T 1 2 σ2 B 2, T =, B, T + 1 =, AT, T = BT, T =. aut u du 1 6 σ2 T 3, B, T = T, T, so ha he ZCB prices are now given in closed form by 5.15.

26 26 MICHAEL MONOYIOS The forward raes are hen given by f, T = log P, T = au du 1 2 σ2 T 2 + r, T. Thus, given an iniial observed marke forward rae curve f, T T, we have f, T = au du 1 2 σ2 T 2 + r, T >, implying f, T = at σ2 T, T >. Hence, he funcion a consisen wih he iniial erm srucure wih σ given, is given by 5.16 a = f, + σ2, Two-facor shor rae models. The nex level of complexiy in modelling he shor rae is o have wo sochasic facors driving he process. Example 5.9 Two-facor Vasicek model. We have wo facors X, Y such ha, under an EMM Q, dx = ax d + dw Q 1, dy = bx + cy d + dw Q 2, a c, r = α + βx + γy,, for consan parameers a >, b, c >, α, β, γ, and independen Q-BMs W Q 1, W Q 2. I can be shown, hough we do no do his here, ha his model can be wrien as one wih a wo-dimensional facor process ξ1 ξ =, following dξ = K Lξ d + Σ db Q, wih consan marices K, L, Σ and a ow-dimensional Q-BM B Q of he form K1 L11 L K =, L = 12 Σ1 B Q, Σ =, B L 21 L 22 Σ Q 1 =, 2 K 2 for Σ 1 >, Σ 1 > and B Q 1, BQ 2 having consan correlaion. The shor rae is hen given as r = ɛ + ɛ 1 ξ 1 + ɛ 2 ξ 2,, for consans ɛ, ɛ 1, ɛ 2. The ZCB prices are hen given as P, T = F T, X, Y, [, T ], such ha he Q-drif of P, is r. Using his and he Iô formula, we find ha F T,, mus saisf he wo-dimensional erm srucure PDE F T, x, y axf x T, x, y bx + cyf y T, x, y + 1 F xx T, x, y + F yy T, x, y α + βx + γyf T, x, y =, 2 wih F T T, x, y = 1. Noe also ha his erm srucure PDE could be derived from a hedging argumen, as we did in he one-dimensional case, bu now one would have o use ξ 2 B Q 2

27 FIXED INCOME 27 wo ZCBs, say a T -bond and a T -bond T > T o hedge an S-bond, wih S < T his is a good exercise. One can ge a closed form for F T,, in his example by assuming an affine form and his is done in Problem Shee 4. Finally, noe ha he model can be made o fi an iniial observed marke erm srucure by making α ime-dependen. 6. Forward rae models Shor rae models are racable bu hey have some drawbacks: hey are somewha unrealisic in he sense ha shor raes and long raes are more highly correlaed han in realiy; one requires a complicaed shor rae model in order o obain a realisic forward rae volailiy srucure; inversion of he yield curve ha is, calibraion o a iniially observed erm srucure ges increasingly difficul as he shor rae model becomes more complex. These issues moivaed he idea of direcly modelling he enire forward rae curve an infinie-dimensional objec. I is hen sraighforward o mach he iniial erm srucure, as his is simply he iniial condiion for he infinie-dimensional sae variable Forward rae dynamics under physical measure. One begins wih posulaed dynamics for f, T T < under he physical measure P. Assume ha f, T T T where T can be very large or even unbounded is known a ime zero from marke observaions of he yield curve. Tha is, we suppose ha 6.1 f, T = f, T, T, where f, T T is he observed forward rae curve a ime zero. Assume he P-dynamics are given by 6.2 df, T = α, T d + σ, T dw, T <,, wih iniial condiion as in 6.1. The BM W is in general muli-dimensional, say n-dimensional, wih n N, and α,, σ, are adaped processes, wih σ, a row vecor. For each fixed T, 6.2 is a separae SDE, so in principle we have an infinie number of such SDEs, one for each f, T, T. Moreover, each forward rae model implies a shor rae model via he srucural relaionship in par 2 of Theorem 4.1, so every shor rae model can be re-cas in forward rae erms, and hen he usual valuaion formula ] V = E [exp Q ru du H F, T, for a claim wih FT -measurable payoff H holds. Moreover, once we have specified α,, σ, and f, T T, so have he forward rae curve, we obain all he ZCB prices via P, T = exp f, u du, T <. As we have an infinie number of ZCBs one for each T and a finie-dimensional BM, we expec ha some sor of consisency condiion involving α, and σ, mus hold in order o preclude arbirage. This is he so-called Heah-Jarrow-Moron HJM drif condiion.