The term structure of interest rates under regime shifts and jumps

Economics Letters 93 (2006) 215 – 221 www.elsevier.com/locate/econbase

The term structure of interest rates under regime shifts and jumps Shu Wu a,*, Yong Zeng b,1 a

Department of Economics, The University of Kansas, Summerfield Hall 213, Lawrence, KS 66045, United States b Department of Mathematics and Statistics, University of Missouri at Kansas City, United States Received 19 October 2005; received in revised form 27 April 2006; accepted 5 May 2006 Available online 12 October 2006

Abstract This paper develops a tractable dynamic term structure models under jump-diffusion and regime shifts with time varying transition probabilities. The model allows for regime-dependent jumps while both jump risk and regime-switching risk are priced. Closed form solution for the term structure is obtained for an affine-type model under log-linear approximations. D 2006 Elsevier B.V. All rights reserved. Keywords: Term structure; Regime switching; Jump diffusion; Marked point process JEL classification: G12; E43

1. Introduction The Fed conducts monetary policy by targeting the short-term interest rate. Many dynamic models of the term structure of interest rates have included a Poisson jump component to reflect the impact of the policy actions.2 A notable feature of the monetary policy behavior is that those discrete changes in the interest rate target of the same direction are very persistent. For example, the Fed decreased the interest rate target 12 times consecutively between January 2001 and November 2002, and since June 2003 there

* Corresponding author. Tel.: +1 785 864 2868. E-mail addresses: [email protected] (S. Wu)8 [email protected] (Y. Zeng). 1 Tel.: +1 816 235 5850. 2

Some recent studies include Ahn and Thompson (1988), Das (2002), Piazzesi (2005) among others.

0165-1765/$ - see front matter D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2006.05.006

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S. Wu, Y. Zeng / Economics Letters 93 (2006) 215–221

have been 11 interest rate hikes by the Fed without a single decrease. Presumably such shifts in the overall monetary policy stance (from accommodative to tightening or vice versa) have more important effects on interest rates than a single interest rate change does. In this paper we propose a regime-dependent jump-diffusion model of the term structure of interest rates to capture the effects of not only discrete jumps in the interest rate target, but also shifts in the policy regime. The paper is based on a strand of recent studies on term structure models with regime shifts, including Landen (2000), Bansal and Zhou (2002), Dai et al. (2003) and Wu and Zeng (2005) among others. The main contribution of the present paper is that it proposes a simple framework to model both discrete jumps and regime shifts in interest rates.

2. The model 2.1. State variables We assume that the first L state variables, X t , are described by the following equation dXt ¼ HðXt ; St Þdt þ RðXt ; St ÞdWt þ J ðXt ; St ÞdNt

ð1Þ

where S t is another state variable following a K-regime continuous-time Markov chain to be specified below, X t and H(X t,S t) are both L  1 vectors; R(X t,S t) is a L  L matrix, W t is a L  1 vector of independent standard Brownian motions; N t is a L  1 vector of independent Poisson processes with L  1 time-varying and regime-dependent intensity d J (X t,S t); J(X t,S t) is a L  L matrix of regimedependent random jump size with a conditional density g( J|X t,S t). Given {X t,S t}, we assume that J(X t, S t) is serially independent and is also independent of W t and N t . To get a convenient representation of S t , we define the mark space E as E ¼ fði; jÞ : iaf1; N ; K g; jaf1; 2; N ; K g; i p jg with r-algebra E ¼ 2E . Let z = (i, j) be a generic point in E and A a subset of E. A marked point process, l(t, A), counts the cumulative number of regime shifts that belong to A during (0,t]. l(t,d ) can be uniquely characterized by its stochastic intensity kernel,3 which is assumed to be cl ðdt; dzÞ ¼ hð z; Xt ÞIfSt ¼ igez ðdzÞdt;

ð2Þ

where h(z, X t) is the regime-shift (from regime i to j) intensity at z = (i, j), I{S t = i} is an indicator function, and e z (A) is the Dirac measure (on a subset A of E) at point z (defined by e z (A) = 1 if z a A and 0, otherwise). Heuristically, for z = (i, j), c l (dt, dz) is the conditional probability of a shift from regime i to regime j during [t,t + dt) given X t and S t = i. The compensator of l(t, A) is then given by Z t Z cl ðt; AÞ ¼ hðz; Xs ÞIfSs ¼ igez ðdzÞds: ð3Þ 0

A

This simply implies that l(t, A)  c l (t, A) is a martingale. 3

See Last and Brandt (1995) for detailed discussion of marked point process, stochastic intensity kernel and related results.

S. Wu, Y. Zeng / Economics Letters 93 (2006) 215–221

Using the above notations, the evolution of the regime S t can be conveniently represented as Z fð zÞlðdt; dzÞ dSt ¼

217

ð4Þ

E

with its compensator given by Z cS ðt Þdt ¼ fðzÞcl ðdt; dzÞ

ð5Þ

E

where f(z) = f((i, j)) = j  i. Without loss of generality, we assume that the instantaneous short-term interest r t is a linear function of the state variables X t given S t rt ¼ w0 ðSt Þ þ w1 ðSt ÞVXt

ð6Þ

where w 0(S t) is a scalar and w 1(S t1) is a L  1 vector. We further specify the pricing kernel M t as4   dMt ¼  rt dt  kVD;t dWt  kVJ ;t dNt  dJ ;t dt  Mt

Z

  kS ð z; Xt Þ lðdt; dzÞ  cl ðdt; dzÞ

ð7Þ

E

where k D,t u k D (X t, S t) is a L  1 vector of market prices of diffusion risk; k J,t u k J (X t, S t) is a L  1 vector of market prices of jump risk conditioning on X t and S t, and k S (z, X t) is the market price of regime-switching (from regime i to regime j) risk given X t. Wu and Zeng (2005) provided a general equilibrium interpretation of the regime-switching risk. 2.2. The term structure of interest rates The specifications above complete the model for the term structure of interest rates, which can be solved by a change of probability measure. Specifically, for fixed T N 0, we define the equivalent martingale measure Q by the Radon–Nikodym derivative dQ ¼ nT =n0 dP where for t a [0,T]  Rt   Rt  Rt Rt  kVD;u dW ðuÞ 12 kVD;u kD;u du kVJ ;u dJ ;u duþ logð1kJ ;u ÞVdNt 0 0 nt ¼ e 0  e 0  Rt R  Rt R kS ðz;Xu Þcl ðdu;dzÞþ logð1kS ðz;Xu ÞÞlðdu;dzÞ E E 0 0  e

4

ð8Þ

Absence of arbitrage is sufficient for the existence of the pricing kernel under certain technical conditions, as pointed out by Harrison and Kreps (1979).

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In the absence of arbitrage, the price at time t of a default-free pure discount bond that matures at T, P(t, T), can be obtained as, Pðt; T Þ ¼

EtQ





e

RT t

ru du

 ð9Þ

with the boundary condition P(t, T) = 1. Moreover, under probability measure Q, ˜ ðXt ; St Þdt þ RðXt ; St ÞdW˜ t þ J ðXt ; St ÞdN˜ t dXt ¼ H dSt ¼

Z

fð zÞl˜ ðdt; dzÞ

ð10Þ ð11Þ

E

˜ t is a L  1 standard Brownian motion where H˜ (X t, S t) = H(X t, S t)  R(X t, S t)k D (X t, S t), W ˜  under Q; N t is a L 1 vector of Poisson processes with intensity d˜J (X t, S t) whose elements are given by d˜i,J (X t, S t) = [1  k i,J (X t,S t)]d i,J (X t, S t) for i = 1,. . ., L;5 l˜(t, A) is the marked point process ˜ (X t) = {h˜ (z, X t)} = {h(z, X t)(1  k S (z, X t))}. The compensator of l˜ (t,A) with intensity matrix H under Q becomes c˜ l ðdt; dzÞ ¼ ð1  kS ðz; Xt ÞÞcl ðdt; dzÞ ¼ h˜ ðz; Xt ÞIfSt ¼ igez ðdzÞdt; Without loss of generality, let P(t, T) = f(t, X t , S t , T). The following proposition gives the partial differential equation determining the bond price. Proposition 1. The price of the default-free pure discount bond f(t,X,S,T) defined in Eq. (9) satisfies the following partial differential equation  2  Z Bf Bf ˜ 1 Bf P H þ tr DS f h˜ ðzÞIfSt ¼ igez ðdzÞ þ d˜ VJ DX f ¼ rf þ RRV þ Bt BX V 2 BX BX V E

ð12Þ

with the boundary condition f(T, X, S, T) = 1. Where D S f = f(t, X t , S t + f(z), T)  f(t  , X t , S t , T), and 0 1 f ðt  ; Xt þ J1 ; St Þ  f ðt  ; Xt ; St Þ A DX f ¼ @ v f ðt  ; Xt þ JL ; St Þ  f ðt  ; Xt ; St Þ R P and DX f ¼ EJQ ð DX f Þ ¼ DX f g ð J jXt ; Xt ÞdJ , i.e. the mean of D X f conditioning on X t and S t under Q.

5

Note that if the market price of jump risk k J,t also depends on the jump size, the conditional density ¯ of the jump size J i (X t,S t) under Q 1k ðX ;S ;J Þ (conditioning on X t and S t ) is given by g˜ ðJi jXt ; St Þ ¼ 1kJ ðXt ;St Þi g ðJi jXt ; St Þ, where EJ is the conditional mean of the jump

size given X and S.

J

t

t

S. Wu, Y. Zeng / Economics Letters 93 (2006) 215–221

219

3. An affine model with regime-dependent jumps In general Eq. (12) doesn’t admit a closed form solution for bond prices. In this section we provide an example of the term structure model that can be solved in analytical form using log-linear approximation. The example falls into the class of affine term structure models. Duffie and Kan (1996) and Dai and Singleton (2000), among others, have detailed discussions of affine term structure models under diffusions. Duffie (2002) introduces a broader class of semi-affine models that have more flexible specifications for the market prices of risk. Duffie et al. (2000) deals with general asset pricing under affine jump-diffusions. Dai et al. (2003), Bansal and Zhou (2002) and Landen (2000) also use affine structure for their regime-switching models. Following this literature, we make the following assumptions: (1) H(X t, S t) = H 0(S t) + H 1(S t)X t where H 0(S t) is a L  1 vector and H 1(S t) is a L  L matrix; (2) p R(X t, S t) is a L  L diagonal matrix with the ith diagonal element given by ½RðXt ; St Þii ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r0;i ðSt Þ þ rV1;i ðSt ÞXt for i = 1,. . ., L; (3) dR J (X t, S t) = d J (S t), which is a L  1 vector; (4) e JVB(Tt,St)g( J|X t, S t)dJ = G(T  t, S t) for some L  1 vector B(T  t, S t); (5) h(z, X t) = e h0(z) + hV1(z)X t; (6) k D (X t, S t) = R(X t, S t)k D (S t) for some L  1 vector k D (S t); (7) k J (X t, S t) = k J (S t), which is a L  1 vector and k J (S t) b 1 for all S t; (8) k S (z, X t) = 1  e k0,S(z)+kV1,S (z)Xt, where k 1,S (z) is L  1 vector. Notice that, among the above assumptions, Eqs. (3) and (4) imply that both the intensity and magnitude of the discreet jump dN t are regime-dependent. This specification can capture the feature of the interest rate movements that jumps on the same direction are persistent. The model can be easily generalized so that the jump intensity also depends on state variable X t. Assumption (5) allows the transition probability of regime shifts to be time-varying. The last three assumptions deal with the market prices of risk. We assume in Eq. (6) that the market price of the diffusion risk is proportional to the diffusion term of the state variable X t as in the conventional affine models. A natural extension would be to use the semi-affine specification of Duffie (2002). We assume in Eqs. (7) and (8) that the market price of the jump risk is a regime-dependent constant and the market price of regime switching risk is an affine function under log-linear approximation. Under these parameterizations, the state variable X t and the Markov chain S t will preserve the affine structure under the risk-neutral measure Q. In particular, let   R0 ðSt Þ ¼ r0;1 ðSt Þ; N ; r0;L ðSt Þ V   R1 ðSt Þ ¼ r1;1 ðSt Þ; N ; r1;L ðSt Þ and 0 KD ðSt Þ ¼ @

kD;1 ðSt Þ O kD;L ðSt Þ

1

0

A;

KJ ðSt Þ ¼ @

1

kJ ;1 ðSt Þ

A

O kJ ;L ðSt Þ

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The drift term H˜ , the regime switching intensity h˜ and the jump intensity d˜J in Eq. (12) of Proposition 1 are given by ˜ 1 ðS ÞX ˜ ð X ; SÞ ¼ H ˜ 0 ðS Þ þ H H ¼ ½H0 ðS Þ  KD ðS ÞR0 ðS Þ þ ½H1 ðS Þ  KD ðS ÞRV1 ðS ÞX

ð13Þ

˜ ˜ h˜ ð z; X Þ ¼ eh 0 ðzÞþh V1 ðzÞX

¼ e½h0 ðzÞþk0;S ðzÞþ½h1 ðzÞþk1;S ðzÞVX

ð14Þ

and d˜ J ðS Þ ¼ ðI  KJ ðS ÞÞdJ ðS Þ

ð15Þ

Using a log-linear approximation similar to that in Bansal and Zhou (2002), we can solve for the term structure of interest rates as follows. Proposition 2. Under the assumption (1)–(8) and that r t = w 0 (S t ) + w 1 (S t )VX t , the price at time t of a default-free pure discount bond with maturity s is given by P(t,s) = e A(s,St)+B(s,St)VXt and the s-period tÞ  Bðs;Sst ÞVXt , where A(s,S) and B(s,S) are determined by the interest rate is given by Rðt; sÞ ¼  Aðs;S s following differential equations 

BBðs; S Þ ˜ 1 ðS ÞVBðs; S Þ þ 1 R1 ðS ÞB2 ðs; S Þ þH Bs 2 Z  D A   ˜ þ e s Ds B þ h˜ 1 ðzÞ  h˜ 1 ð zÞ eh 0 ðzÞ 1ðS ¼ iÞez ðdzÞ ¼ w1 ðS Þ

ð16Þ

E

and h i BAðs; S Þ 1 J VBðs;S Þ ˜ ˜  1 þ H 0 ðS ÞVBðs; S Þ þ Bðs; S ÞVR0 ðS ÞBðs; S Þ þ d J ðS ÞVEJ e Bs 2 Z  DA  ˜ þ e s  1 eh 0 ðzÞ 1ðS ¼ iÞez ð dzÞ ¼ w0 ðS Þ

ð17Þ

E

with boundary conditions A(0,S) = 0 and B(0,S) = 0, where D s A = A(s,S + f(z))  A(s,S), D s B = B(s,S + f(z))  B(s,S), and B 2 (s,S) = (B 12 (s,S),. . ., B L2 (s, S))V. Note that Proposition 2 nests many of the existing regime-switching models of the term structure of interest rates. For example, neither Landen (2000), Bansal and Zhou (2002) nor Dai et al. (2003) include a jump component, therefore E J [e JVB(s,s)  1] = 0. Both Landen (2000) and Bansal and Zhou (2002) assume constant transition probabilities for the Markov process, hence h 1(z) = h˜ 1(z) = 0. Dai et al. (2003) allows for time-varying transition probabilities under the physical measure. But in order to obtain exact analytical solution, they impose that the transition probabilities is constant under the risk-neutral measure, h˜ 1(z) = 0, and assume Gaussian distribution for state variable X t , therefore Ds B = 0.6 The latter specification is also shared by Landen (2000). Moreover, while Landen (2000) is silent on the market 6

Under these assumptions, Eqs. (16) and (17) yield the exact solution for the term structure of interest rates.

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price of risk, Bansal and Zhou (2002) assumes that the risk of regime shifts is not priced, i.e. h˜ 0(z) = h 0(z). The model proposed in the present paper relaxes those restrictions in a tractable way, and it can be estimated by simulation-based method such as Efficient Method of Moments.

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