Modeling the Yield Curve Dynamics

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1 FIXED-INCOME SECURITIES Chape 2 Modeling he Yield Cuve Dynamics

2 Ouline Moivaion Inees Rae Tees Single-Faco Coninuous-Time Models Muli-Faco Coninuous-Time Models Abiage Models

3 Moivaion Why do we Cae? Picing (and hedging) of fixed-income secuiies ha pay cashflows which ae known wih ceainy a he iniial dae (e.g., plain vanilla bonds Boils down o a compuaion of he sum of hese cash-flows, discouned a a suiable ae Challenge fo he bond pofolio manage is heefoe limied o being able o have access o a obus mehodology fo exacing implied zeo-coupon pices fom make pices (see Chape 4) Picing and hedging fixed-income secuiies ha pay unceain cash-flows (e.g., opions on bonds) I equies no only he knowledge on discoun facos a he pesen dae, bu also some kind of undesanding of how hese discoun facos (i.e., he em sucue of pue discoun aes) ae going o evolve ove ime In paicula, one needs o accoun fo poenial coelaions beween he discoun faco and he pomised payoff, volailiy of he payoff, ec. Some dynamic model of he em sucue of inees aes is heefoe needed

ng) of fixed-income secuiies ha pay cashflows which ae known wih ceainy a he iniial dae (e.g., plain vanilla bonds Boils down o a compuaion of he sum of hese cash-flows, discouned a a suiable ae

4 Inees Rae Tee Binomial Model Geneal binomial model Given cuen level of sho-em ae, nex-peiod sho ae, can ake on only wo possible values: an uppe value u and a lowe value l, wih equal pobabiliy 0.5 In peiod 2, he sho-em inees ae can ake on fou possible values: uu, ul, lu, ll Moe geneally, in peiod n, he sho-em inees ae can ake on 2 n values > vey ime-consuming and compuaionally inefficien Recombining ees Means ha an upwad-downwad sequence leads o he same esul as a downwad-upwad sequence Fo example, ul lu Only (n) diffeen values a peiod n

5 In peiod 2, he sho-em inees ae can ake on fou possible values: uu, ul, lu, ll Moe geneally, in peiod n, he sho-em inees ae can ake on

5 Inees Rae Tee Binomial Model Peiod 0 Peiod Peiod 2 Peiod 3 Peiod 4 uuuu uuu uu uuul u uul ul uull l ull ll ulll lll llll Recombining ee

6 Inees Rae Tee Analyical Fomulaion We may wie down he binomial pocess as Δ σε whee ε ae independen vaiables aking on values (,-) wih poba (/2,/2) Poblem is aes can ake on negaive values wih posiive pobabiliy Fix ha poblem by woking wih logs Δ ln ln wih pobabiliy (/2,/2) exp ln ( σε ) σε d u exp exp ( σ ) ( ) σ

aes can ake on negaive values wih posiive pobabiliy Fix ha poblem by woking

7 Inees Rae Tee Analyical Fomulaion Moe geneal models (could be wien on log aes) Δ Δ (, Δ, ) ( ) σ, Δ μ, ε Specific case Δ Δ μδ σ Δε Focus and sae- and ime-independen models Squae-oo of ime law is consisen wih he absence of seial coelaion (independen incemens feaue in he andom walk) Coninuous-ime limi (Meon (973)) d d μ d σdw

ime-independen models Squae-oo of ime law is consisen wih he absence of seial

8 Inees Rae Tee Calibaion Calibaion of he model is pefomed so as o make model consisen wih he cuen em sucue We have a dae 0 ln Δ ln Δ 0 u 0 ln ln 0 μδ σε Δ ln l 2σ Δ o exp 2 u l ( σ Δ ) We ake as given an esimae fo σ, he cuen pa yield cuve y, and we ieaively find he values u, l, uu, ul, lu, ll, ec., consisen wih he inpu daa

l 2σ Δ o exp 2 u l ( σ Δ ) We ake as given an esimae fo σ, he cuen pa yield cuve

9 Inees Rae Tee Calibaion Con Conside a 2 peiod ee wih Δ fo simpliciy The pice one yea fom now of he 2-yea pa Teasuy bond can ake wo values: a value P u associaed wih u, and a value P l, associaed wih l l d u u y P y P 00 and Then, aking expecaions a ime 0, we find an equaion ha can be solved fo u and l ( ) 00 2 exp y y y y y y l l σ

value P l, associaed wih l l d u u y P y P 00 and 00 2 2 Then, aking expecaions a ime 0,

10 00 Inees Rae Tee Calibaion Time fo an Example! Conside cuen Teasuy bond pa yield cuve: y 4%, y % We wan o calibae a binomial inees ae ee, assuming a volailiy of % fo he one-yea inees ae We have l exp 4% (.02 ) l 4% 4.3 l u 4.57 % 4.66 %

30% We wan o calibae a binomial inees ae ee, assuming a volailiy

11 Single-Faco Coninuous-Time Models Geneal Fomulaion Geneal expession fo a single-faco coninuousime model (, ) ( ) d σ dw d μ, The em W denoes a Bownian moion, which a pocess wih independen nomally disibued incemens dw epesens he insananeous change. I is sochasic (unceain) I behaves as a nomal disibuion wih zeo mean and vaiance d I can be hough of as dw ε d

wih independen nomally disibued incemens dw epesens he insananeous change.

12 Single-Faco Coninuous-Time Models Popula Models All popula fall ino he following class d Lising of popula models [ ] [ ] α μ μ2 d σ σ 2 dw Model Ω Ω 2 α α 2 ϑ Bennan-Schwaz (980) Φ Φ Φ Cox-Ingesoll-Ross (985) Φ Φ Φ 0.5 Dohan (978) Φ Meon (973) Φ Φ Peason-Sun (994) Φ Φ Φ Φ 0.5 Vasicek (977) Φ Φ Φ

Ω Ω 2 α α 2 ϑ Bennan-Schwaz (980) Φ Φ Φ Cox-Ingesoll-Ross (985) Φ Φ Φ 0.

13 Single-Faco Coninuous-Time Models Wha is a good Model? A good model is a model ha is consisen wih ealiy Sylized facs abou he dynamics of he ems sucue Fac : (nominal) inees aes ae posiive Fac 2: inees aes ae mean-eveing Fac 3: inees aes wih diffeen mauiies ae impefecly coelaed Fac 4: he volailiy of inees aes evolves (andomly) in ime A good model should also be Tacable Pasimonious

Fac : (nominal) inees aes ae posiive Fac 2: inees aes ae mean-eveing Fac 3: inees aes wih

14 Single-Faco Coninuous-Time Models Empiical Facs, 2 and //90 29/6/90 30//90 30/4/9 30/9/9 28/2/92 3/7/92 3/2/92 3/5/93 29/0/93 3/3/94 3/8/94 3//95 30/6/95 30//95 30/4/96 30/9/96 28/2/97 3/7/97 3/2/97 29/5/98 30/0/98 3/3/99 3/8/99 DOW JONES Index Value Fed Fund Rae (in %)

3/2/92 3/5/93 29/0/93 3/3/94 3/8/94 3//95 30/6/95 30//95 30/4/96 30/9/96 28/2/97

15 Single-Faco Coninuous-Time Models Empiical Fac 3 M 3M 6M Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 0Y M 3M M Y Y Y Y Y Y Y Y Y Y Daily changes in Fench swap makes in 998

892 0.953 0.99 7Y 0.228 0.82 0.35 0.66 0.792 0.859 0.924 0.969 0.99 8Y 0.24 0.99 0.35 0.64 0.745 0.826 0.892 0.936 0.968 0.992 9Y 0.238 0.98 0.339 0.

16 Single-Faco Coninuous-Time Models Vasicek Model Vasicek (977) model d a ( b ) d σdw This pocess exhibi a mean-eveing feaue The paamee b may be egaded as he equilibium level of he sho-em inees ae, aound which i sochasically evolves When falls fa below (above) is long-em value b, he expeced insananeous vaiaion of is posiive (negaive) In his case, he sho-em ae will end o move up (down) I will move owads is long-em value quickly when i is fa fom i and when he paamee a (speed of eun o he long-em mean value) is high On he ohe hand, i is no consisen wih facs, 3 and 4

expeced insananeous vaiaion of is posiive (negaive) In his case, he sho-em ae will end o move up (down) I will move owads is long-em value

17 Single-Faco Coninuous-Time Models Cox-Ingesoll-Ross Model CIR (985) model d a ( b ) d σ dw This pocess exhibi a mean-eveing feaue I also pevens inees aes o become negaive I exhibis a sochasic volailiy componen On he ohe hand Inees ae and volailiy isks ae pefecly coelaed Inees aes of diffeen mauiies ae pefecly coelaed (only one souce of andomness)

exhibis a sochasic volailiy componen On he ohe hand Inees ae and volailiy isks ae

18 Muli-Faco Coninuous-Time Models Some Popula Models Fong and Vasicek (99) model Fong and Vasicek (99) ake he sho ae and is volailiy as wo sae vaiables Vaiance of he sho-ae changes is a key elemen in he picing of fixed-income secuiies, in paicula inees aes deivaives Longsaff and Schwaz (992) model Longsaff and Schwaz (992) use he same wo sae vaiables, bu wih a diffeen specificaion Allows hem o ge closed-fom soluion fo he pice of a discoun bond and a call opion on a discoun bond Chen (996) and Balduzzi e al. (996) models Chen (996) and Balduzzi e al. (996) sugges he use of a heefaco model by adding he sho-em aveage of he sho ae These hee sae vaiables can be assimilaed o he hee facos which can be empiically obained hough a PCA of he em sucue dynamics

specificaion Allows hem o ge closed-fom soluion fo he pice of a discoun bond and a call opion on a discoun bond Chen (996) and Balduzzi e al. (996) models Chen (996) and Balduzzi e al.

19 Abiage Models Calibaion of Coninuous-Time Faco Models In pacice, single o muli-faco models ae calibaed in such a way ha he models' paamees ae obained as soluions o a minimizaion pogam of he diffeence (i.e., he squaed spead) beween make pices of efeence bonds and heoeical values geneaed fom he model This is he analogue of he calibaion of an inees ae ee Then he model is used o pice inees aes deivaives (see Chapes 4 and 5) The diffeence beween he deived yield cuve and he obseved cuve, even hough minimized, can no be eniely eliminaed

s ae obained as soluions o a minimizaion pogam of he diffeence (i.e., he squaed spead) beween make pices of efeence bonds and

20 Abiage Models Some Popula Models These models ae buil o be consisen wih cuenly obseved bond pices, which makes hem popula among paciiones Fis example is Ho and Lee (986) Discee-ime binomial seing Discoun bond pices ae diven by a single souce of unceainy Heah, Jaow and Moon (990ab,992) have genealized his appoach by allowing discoun bonds pices o be diven by a muli-dimensional unceainy in a coninuous-ime famewok Makovian vesions of he HJM model ae ofen used Tanslae ino ecombining ees in discee-ime Can be implemened wihou oo much numeical complexiy

(990ab,992) have genealized his appoach by allowing discoun bonds pices o be diven by a muli-dimensional unceainy in a coninuous-ime

Volatility Structures of Forward Rates and the Dynamics of the Term Structure* Peter Ritchken and L. Sanakarasubramanian

Volatility Structures of Forward Rates and the Dynamics of the Term Structure* Peter Ritchken and L. Sanakarasubramanian Volailiy Sucues o Fowad Raes and he Dynamics o he em Sucue* ypesee: RH 1s poo: 1 Oc nd poo: 3d poo: Pee Richken and L. Sanakaasubamanian Case Wesen Reseve Univesiy; Bea Seans & Co Fo geneal volailiy sucues