Multifactor Term Structure Models

Transcription

1 1 Multfactor Term Structure Models A. Lmtatons of One-Factor Models 1. Returns on bonds of all maturtes are perfectly correlated. 2. Term structure (and prces of every other dervatves) are unquely determned by the level of the spot rate. 3. Most Common Term Structure Movements - PCA (Prncpal Components Analyss) One convenent way to analyze the term structure s to use prncpal components analyss or factor analyss. Let y(t,τ) represent a zero coupon bond wth tme to maturty τ. Assume that we are gven the yelds of Treasures zero coupon bonds wth maturtes 3 and 6 months and 1,2,3,4,5,7,10 and 30 years. That s, we have 10 dfferent maturtes.

2 2 y(t,τ 1 )= yeld of the 3-month zero coupon bond at tme t y(t,τ 2 )= yeld of the 6-month zero coupon bond at tme t y(t,τ 10 )= yeld of the 10-year zero coupon bond at tme t Assume that we descrbe the changes on these yelds on the followng way: y(t, τ ) = v 1( τ ) a 1(t) + v 2( τ ) a 2(t) v 10 ( τ ) a 10(t) where a j (t) are 10 ndependent random varables and a j(t) s the change on a j at tme t. a j (t) s called a prncpal component or factor. v j( τ ) s the loadng of the yeld of the bond wth tme to maturty τ on the j th factor. The equaton above mples that f the prncpal component a j (t) ncreases by 1bp s then a j(t) = 1/10000 then the yeld y(t, τ ) changes by bass ponts. v 1( τ )

3 3 The equaton above also mples that the varance covarance matrx of the changes n yelds s gven by: cov( y(t, τ )) = v Λ v ' where v s a 10x10 matrx wth the -th row gven by v 1( τ ), v 2( τ )...v 10 ( τ ) and Λ (the matrx of engenvalues of cov( y(t, τ )) ) s a 10x10 matrx wth the dagonal elements equal to the varances of the factors. The mportance of the factor s measured by the relatve varance of the factor. The varances of the factors have the property that they add up to the total varance of the data (that s, the sum of the varance of the observatons on the three-month rate, sx-month rate and so on ). Therefore the mportance of the factor s measured by: Varance of the factor Total Varance of the Data Ltterman and Schenkman, Journal of Fxed Income, 1991, dd the exercse above. They found the matrx v (factor loadngs) and the matrx Λ for the US rates.

4 4 The frst factor s a roughly parallel shft (83.1% of varaton explaned). Ths motvates the name of the frst factor as the level factor. The second factor s a twst (10% of varaton explaned). The second factor s called the slope of the curve. The thrd factor s a bowng (2.8% of varaton explaned). The thrd factor s usually called the curvature factor.

5 5 B. Multfactor CIR Model 1. Each factor has the CIR dynamcs a. rate s, =1,...,K and the spot K r F. df F dt F dz 1 b. Then zero-coupon bond prces are gven by: F1 b1 t,t FK bk t,t B F,,F,t,T a t,t a t,t e 1 K 1 K c. The yelds are stll affne functons of the factors: 1 %! 1 K 1 K T! % t K K! $ " $ 1 T! t 1 T! t # " y F,,F,t,T Ln B F,,F, t,t A Ln a t,t b t,t F BF d. The estmaton of the model s usually made as n Chen and Scott, "Multfactor CIR Models of the Term Structure", Journal of Fxed Income:

6 6 Assume that the prces of K zero-coupon bonds are observed. We can then perform an nverson of the prcng equatons to obtan a tme seres of the latent state varables, whch s used to estmate the model parameters by maxmum lkelhood. ( ) y F,,F,t,T = A + BF + ε 1 K t t The lkelhood of the observed rates can be computed from the lkelhood of the factors by change n varables. e. Duffe and Sngleton (1997) estmate a 2-factor CIR model usng Lbor and Swap rates data n a way smlar to the one explaned above.

7 7 C. General Affne Models a. Duffe and Kan, Mathematcal Fnance 1996, generalze the CIR and Vascek models to a class of models called affne. b. The zero-coupon bond prces n affne models are gven by: F1 b1 t,t FK bk t,t B F,,F,t,T a t,t a t,t e 1 K 1 K that s, the yelds of zero-coupon bonds s gven by affne functons of the factors F. c. To defne an arbtrage-free affne termstructure model, they prove that t s necessary to: 1. r K 0 δ 1 F = = δ + 2. The drft of the state varables under the rsk neutral measure Q must be an affne functon of the state varables. 3. The dffuson of the state varables must be gven by Σ St where Σ s a matrx of constants and St s a dagonal matrx of affne functons of the state varables F.

8 8 d. Da and Sngleton, JF 2002 emprcally analyze affne models. The analyss of affne models s complcated by the fact that t s not possble to buld a K factor affne model that nests all the other K factor models. So they classfy affne models n terms of the number of factors that affect the volatlty of yelds. And they defne the notaton A m (n) where n s the number of factors n the model and m s the number of factors that affect the volatlty of yelds. For nstance, a 3-factor CIR model s wthn the famly of A 3 (3) models and a 3-factor Vascek model s wthn the famly of A 0 (3) models. Wthn each sub-famly of affne models, they gve condtons to fnd the maxmal model, that s the model n whch all the models of the A m (n) class are nested.

9 9 D. Quadratc Term Structure Models 1. Model the prcng kernel drectly a. Recall that for a zero-coupon bond: T { t u } B t,t E * exp r du r r b. Defne the prcng kernel to be M(.) such that: M T T exp{ r t udu} M t c. Model the dynamcs of M(.) drectly: Assume that there K factors drvng M(.) t And that M(.) has the form: ( ) { } M t exp quadratc(f ) Where F s follow Ornsten-Uhlenbeck processes: df F dt dz

10 10 2. Ths model can be solved explctly for bond and other dervatve prces a. All we need s the expected value of M(T) gven the values of the factors at t. Ths can be done because: Ornsten-Uhlenbeck mples Gaussan dstrbutons There are formulas for the expected value of the exponental of a Gaussan random varable, or ts square. b. Reference: Constantndes, G.M., "A Theory of the Nomnal Term Structure of Interest Rates," Revew of Fnancal Studes, 1992, Ths model can also be put n state-space form to nfer the factors from the observed yelds. Reference: Zheng, C., Unversty of Chcago PhD Dssertaton, 1994 and Ahn et. Al., RFS 2002.

11 11 E. Unobservable Factors: Use Kalman Flter 1. The yelds on bonds wth a varety of maturtes are observable, but we may want to avod prespecfyng the factors. 2. Put the model n state-space form: a. We can flter the values of the factors at each pont n tme based on the observed yelds: Y A BF F F t t t t t 1 t Yt s a vector of yelds at t of bonds of dfferent maturtes Ft s the vector contanng the values of the K factors at tme t A, B, Γ and Ρ are matrces of constant parameters. The parameters n B are called the factor loadngs. η t and ε t are random dsturbances

12 12 b. The Kalman flter produces estmates of the parameters as well as estmates of F t at every date. c. Reference: For the general state-space approach: Harvey, A.C., "Tme-Seres Models," MIT Press, 1993, Chapter 4.