Linear-price term structure models

Journal of Empirical Finance 24 (2013) 24–41

Contents lists available at ScienceDirect

Journal of Empirical Finance journal homepage: www.elsevier.com/locate/jempfin

Linear-price term structure models C. Gourieroux a,b, A. Monfort a,c,⁎ a b c

CREST, France University of Toronto, Canada University of Maastricht, The Netherlands

a r t i c l e

i n f o

Article history: Received 5 January 2012 Received in revised form 11 June 2013 Accepted 31 July 2013 Available online 9 August 2013 JEL classification: C58 G12

a b s t r a c t We characterize the term structure models in which the zero-coupon prices are linear functions of underlying factors. These models are called Linear-price Term Structure Models (LTSM). We provide two types of LTSM where the observable factors predict regimes which are not observed by the investor. These hidden regimes are represented by a Markov chain, which features either an exogenous, or an endogenous dynamics. We illustrate the possible term structure patterns, their evolutions, in particular their ability to stay close to a zero lower bound. © 2013 Elsevier B.V. All rights reserved.

Keywords: Linear term structure model Hidden Markov chain Finite dimensional dependence Binding floor

1. Introduction The dynamic analysis of the term structure of interest rates is generally based on Affine (yield) Term Structure Models (ATSM), in which the interest rates at different time-to-maturity are affine functions of a finite (or infinite) set of factors. The ATSM includes a number of well-known term structure models such as the Vasicek and Cox, Ingersoll, Ross single factor models, the Langetieg, Quadratic and Wishart multifactor models [see e.g. Dai and Singleton (2000), Ahn et al. (2002), Gourieroux (2006), Gourieroux et al. (2010)]. The ATSM have shown their flexibility to provide various term structure patterns or to capture the effects of default risk [see e.g. Duffie and Singleton (1999), Gourieroux et al. (2006), Monfort and Pegoraro (2007)]. However, there exist alternative flexible specifications. As noted in Siegel (2010), it is a “relativity recent idea to represent bond prices, instead of yields, using a linear factor model”. Thus we get a Linear-Price Term Structure Model (LTSM)1. This new class of term structure models has both interesting technical and practical features. For instance, the no arbitrage restrictions are easy to derive, and the LTSM may be more appropriate than the ATSM to account for regimes hidden to the investors, or for representing special features of interest rates, such as the short rates close to zero recently observed [see Ichine and Ueno (2007), Kim and Singleton (2010)]. Moreover we consider two important classes of LTSM: the Hidden Markov Term Structure Models (HMTSM) and the Finite Dependence Term Structure Models (FDDTSM), which provide quasi explicit formulas for bond prices and European derivatives and which are also easily tractable from an econometric point of view. ⁎ Corresponding author at: Crest 15 boulevard Gabriel Péri, 92240 Malakoff, France. Tel.: +33 141177728. E-mail address: [email protected] (A. Monfort). 1 We have followed the standard terminology of (Linear) Affine Term Structure Model, when the yields are (linear) affine functions of factors. However, the terminology Linear Term Structure Model might appear in the literature for Linear Affine Term Structure Model, when the word Affine is unfortunately omitted. Such an omission appears e.g. in Cochrane (2001), in the title of Section 19.5, which has to be read as “Three Linear Affine Term Structure Models”, as immediately seen from the content. 0927-5398/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jempfin.2013.07.004

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

25

The paper is organized as follows. We first introduce in Section 2, term structure models in which the prices of zero coupon bonds are linear combinations of a finite number of stochastic dynamic factors. We discuss the restrictions on the term structure pattern implied by no arbitrage. Then, we describe two types of LTSM. The Hidden Markov Term Structure Model (HMTSM) is introduced in Section 3. The class is constructed by introducing an exogenous Markov chain which is not observed by the investors, and observed variables, whose evolution depends on the latent exogenous Markov chain. We show that this construction leads to LTSM in bond prices, and we explain how to derive the associated factors. These factors have a complicated nonlinear dynamic with a long memory. We explain how the pricing formula could be extended to any type of interest rate derivatives, leading also to a derivative price, which is linear in the same observable factors. In Section 4 another LTSM is introduced and discussed. This model is based on the notion of Markov process with finite dimensional dependence. We derive the expression of the observable factors. These models are called the Finite Dimensional Dependence Term Structure Models (FDDTSM). In some special cases, these models can be interpreted in terms of hidden Markov chain, where the unobservable Markov chain is now endogenous. An illustration of the two types of term structure models is provided in Section 5. We compare the ability of both models to capture the stylized facts observed when there is a zero binding floor for the short-term rate. We show that the FDDTSM is more flexible for this purpose. Section 6 concludes. The proofs are gathered in. 2. Linear-Price Term Structure Models Let us now focus on Linear-Price Term Structure Models (LTSM) defined by: Bðt; HÞ ¼ aðH Þ′ F t ; ∀H; 1 ≤H ≤H;

ð2:1Þ

where B(t, H) is the price at time t of a zero-coupon bond with residual maturity H and Ft is a set of K linearly independent stochastic factors. One of these factors can possibly be constant in time, and in this case the linear combination includes an intercept. 2.1. The term structure pattern The factor coefficients a(H) cannot be chosen independently, but may be subject to no arbitrage restrictions. We assume below: Assumption A.1. The information available to the investors at date t includes the current and lagged values of the factors (and also due to Eq. (2.1) the current and lagged values of the prices of the zero-coupon bonds). Assumption A.2. The markets of zero-coupon bonds are liquid, that is, any quantity of a zero-coupon bond with maturity H can be traded at date t at the unitary price B(t, H), for any time-to-maturity H; 1 ≤H ≤H. Assumption A.3. There is no short-sell restriction. Under Assumptions A.2–A.3, the investor can trade any coupon bond, with possibly negative coupon for some time-to-maturity. Moreover, under no arbitrage opportunity, the price of such a coupon bond is equal to the combination of the prices of the zero-coupon bonds with the coupons as coefficients. Then, under Eq. (2.1), the prices of the coupon bonds form a vector space, which is included in the space generated by the components of Ft. Let us now explicit the restrictions on coefficients a(H) due to the absence of arbitrage opportunity. We have the following result: Proposition 1. In a Linear-Price Term Structure Model (2.1) and under Assumptions A1–A3, the absence of arbitrage opportunity for a self-financed portfolio of zero-coupon bonds implies that there exists a matrix C such that: aðH Þ′ ¼ að1Þ′C

H−1

:

ð2:2Þ

Proof. See Appendix 1. This type of restriction does not depend on the maximal time-to-maturity H easily tradable on the market. Under conditions (2.2), the KH factor coefficients depend at most on K + K2 underlying parameters, which are the elements of a(1) and matrix C, respectively. Some of these coefficients, namely a(1), are characterizing the short-term rate, whereas matrix C characterizes the pattern of the term structure of zero-coupon prices. 2.2. The multiplicity of factor representations i) The expression of the discount function, that is the sequence of zero-coupon prices with respect to time-to-maturity, implied by Proposition 1 can be written in an equivalent way since the factors are defined up to a one-to-one linear transform. More precisely we can write: Bðt; H Þ ¼ að1Þ′M

−1


 −1 H−1 MCM M Ft ;

26

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

for any invertible matrix M. Let us denote by Δ the triangular Jordan form2 of matrix C (with possibly complex eigenvalues) and M the matrix of the change of basis to get this form, we have: 

H−1

Bðt; HÞ ¼ a ð1Þ′Δ



Ft ;

ð2:3Þ

with F∗t = MFt, a∗(1) = (M′)−1a(1). Thus, the discount function is necessarily a linear combination of exponential functions of H, possibly complex, and multiplied by polynomials. ii) Other factor representations can also be considered. By the Cayley–Hamilton theorem, the power matrices ΔH,H ≥ 0, are linear combinations of the K first power matrices ΔH, 0 ≤ H ≤ K − 1. We deduce the following Proposition: Proposition 2. i) The minimal number of linearly independent stochastic factors is equal to: h i
   K−1 ′  a ð1Þ : r ¼ Rk a ð1Þ; Δ′a ð1Þ; …; Δ ii) Factor Ft in Eq. (2.1) can be chosen as a linear combination of r zero-coupon prices: Bðt; H k Þ; k ¼ 1; …; r; with 1 ≤Hk ≤H; ∀k ¼ 1; …; r: Proof. See Appendix 2.   Thus, the dimension of the space generated by the price of the coupon bonds, Bðt; HÞ; H ¼ 1; …; H that is the spanning dimension of the bond market, is equal to r. Then, the initial set of observable factors Ft can always be replaced by a set of mimicking factors interpretable as prices of r coupon bonds, constructed from the zero-coupon bonds with the K first time-to-maturity with possibly negative coupons at some time-to-maturity (see Assumption A.3). 2.3. The positivity condition Pattern (2.3) is obtained by applying the no-arbitrage restriction to self-financed portfolios including a finite number of zero-coupon bonds. This condition is necessary, but not sufficient for no arbitrage. Other no-arbitrage necessary conditions on the pattern of the discount function are deduced from the positivity of prices. Without loss of generality, let us consider the case where a∗(1) = e = (1,1, …,1)′, i.e. where the price of the short-term zero-coupon bond is B(t,1) = e′F∗t . The positivity conditions are: H−1

Bðt; HÞ ¼ e′Δ

F t ≥0; ∀H; 1≤H ≤H; ∀t:

Let us discuss these conditions in the 1- and 2-factor cases. i) Single factor model The positivity condition becomes: δH−1 F t ≥0; ∀H; 1 ≤H ≤H; ∀t. Thus the positivity condition is equivalent to: δ N 0, Ft ≥ 0, ∀ t. ii) 2-factor model The constraints ensuring the positivity of the discount function in the 2-factor model are summarized in Table 1 below and depend on the eigenvalues δ1, δ2 of matrix Δ, which have necessarily to be real (see Appendix 3 for the proof). The no-arbitrage restrictions imply that the support of the canonical factors is included in an intersection of half-spaces, possibly depending on parameters in the last case. 2.4. Comparison with the literature Term structure models in which the zero-coupon prices are linear functions of underlying factors have already been introduced in the literature in some special cases, even if they are less frequently considered for derivative pricing than affine term structure models (ATSM) of interest rates. For instance, they implicitly appear in the smoothing method by exponential splines largely used by practitioners [Vasicek and Fong (1982), Shea (1985)]. This approach consists of fitting daily a cubic exponential-polynomial on the observed zero-coupon prices:
 Bðt; H Þ ¼ ak;0t þ ak;1t expð−α kt HÞ þ ak;2t exp −2α k;t H
 þ ak;3t exp −3α k;t H ; if Hk b H bH kþ1 ; 2

The Δ matrix has the eigenvalues of C as diagonal elements, 0 or 1 on the diagonal just above the main diagonal, and zero elements anywhere else.

ð2:4Þ

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

27

Table 1 Positivity restrictions in the 2-factor model. Eigenvalues

Discount function

Restrictions

Double Different δ, − δ Different 0 b δ2 b δ1 Different − δ1 b δ2 b 0 b δ1

δH − 1(F1,t + HF2,t) δH − 1[F1,t + −1H − 1F2,t] −1 −1 δH F1,t + δH F2,t 1 2 −1 −1 δH F1,t + δH F2,t 1 2

δ N 0, F2,t ≥ 0, F1,t + F2,t ≥ 0 δ N 0, F1,t ≥ 0, F1,t − F2,t ≥ 0 δ1 N δ2 N 0, F1,t ≥ 0, F1,t + F2,t ≥ 0 δ1 N0 N δ2 N −δ1 ; F 1;t ≥0; F 1;t þ F 2;t N 0; δ1 F 1;t þ δ2 F 2;t N 0:

on each interval between knots, while ensuring the continuity of the exponential spline and of its first and second-order derivatives at the knots. We have explicitly indexed all parameters of the exponential spline by time in order to highlight the standard day per day analysis. Such an ad-hoc smoothing formula is in general not arbitrage free.3 Indeed, let us consider the special case in which αkt = αk is time independent. Then model (2.4) is linear with respect to dynamic factors ak,0,t, ak,1,t, ak,2,t, ak,3t, k varying. However, from Proposition 1, this exponential spline model could only be arbitrage free if there is a single interval in the spline function, that is, if: Bðt; HÞ ¼ aot þ a1t expð−αH Þ þ a2t expð−2αH Þ þ a3t expð−3αH Þ:

ð2:5Þ

This corresponds to a diagonal Δ matrix with the successive powers of exp (−α) ∈ (0, 1) on the diagonal: 0

1 0 B 0 expð−α Þ Δ¼B @0 0 0 0

1 0 0 C 0 0 C: A expð−2α Þ 0 0 expð−3α Þ

The importance of linear factor models written on asset and derivative prices has been already highlighted in a series of papers by Gabaix (2009), Carr et al. (2009), Siegel (2010), Gourieroux and Monfort (2011)]. Such linear (price) factor models are for instance encountered in macrofinance, because they are easy to use and to implement. We will see in the next section, that this linear relationship between prices implies in general a complicated (in particular non-affine) risk-neutral dynamics of the factors. As already mentioned, pattern (2.3) of the discount function and the constraints of Table 1 are necessary, but not sufficient for no arbitrage. Thus, it is important to exhibit risk-neutral factor dynamics able to provide a large set of linear term structures of zero-coupon bonds. Examples of LTSM compatible with Assumptions A.1–A.3 can be derived by looking for risk-neutral distributions of the factors, implying linear expressions (2.1) of the prices of the zero-coupon bonds. Indeed, the existence of a risk-neutral distribution is related to Assumptions A1, A3 and A2: Assumption A2. The markets of derivatives written on (Ft) are liquid. Since these derivative markets include the coupon bond markets, Assumption A2' implies Assumption A2. As seen from the discussion on positivity restrictions, the observable factors have to satisfy inequality restrictions, which will imply a nonlinear factor dynamics. 3. The Hidden Markov Term Structure Model The Hidden Markov Term Structure Model (HMTSM) provides such a set of LTSM. 3.1. The model There are two types of variables yt and Yt, of dimension 1 and L, respectively, observable by the investor. Moreover, there is a latent vector Zt of size K, whose components Zk,t are indicator functions of state k at date t. We assume that the underlying process (Zt) admits an exogenous risk-neutral dynamics and is not by the investor.
observable  Therefore the information of the investor at date t is yt ; Y t , where yt ¼ ðyt ; yt−1 ; …; y1 Þ; Y t ¼ ðY′t ; Y′t −1 ; …; Y′1 Þ′ and all the payoffs at date t are functions of this information. Assumption A.4. The unobservable process (Zt) is exogenous under the risk-neutral distribution Q, that is, h i h i Q Z k;t ¼ 1jZ j;t−1 ¼ 1; Z t−2 ; yt−1 ; Y t−1 ¼ Q Z k;t ¼ 1jZ j;t−1 ¼ 1 :

3 Similarly, the ad-hoc Nelson–Siegel and Svensson formulas [Nelson and Siegel (1987), Svensson (1995)], close to ATSM models, are not dynamically coherent [Filipovic (1999a,b)].

28

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

Therefore, the risk-neutral dynamic of (Zt) is characterized by the transition matrix Π, with elements: h i πk; j ¼ Q Z k;t ¼ 1jZ j;t−1 ¼ 1 ; ∀k; j ¼ 1; …; K:

ð3:1Þ

Assumption A.5. The risk-neutral conditional distribution of yt + 1 given Z tþ1 ; yt ; Y tþ1 depends on Zt + 1 only. The conditional Laplace transform of yt + 1 when Zt + 1 is in state k is denoted by: i Qh   E exp −uytþ1 Z k;tþ1 ¼ 1 ≡ expf−ck ðuÞg; k ¼ 1; …; K:

ð3:2Þ

Assumption A.6. The risk-neutral conditional distribution of Yt + 1, given Z tþ1 ; ytþ1 ; Y t depends on Z tþ1 ; ytþ1 ; Y t only.
 Assumption A.7. yt is the short-term rate between t and t + 1, observed at t, and the investors' information set is yt ; Y t only. Thus the assumptions above correspond to the causal scheme below: Table 1: Causal scheme for the HMTSM →

Zt ↓

→ ðyt ; Y t Þ

→



→

Z tþ1 ↓

ytþ1 ; Y tþ1



→ →

In this scheme the Zt (resp. Yt) are unobservable (resp. observable) by the investors. The price of the zero-coupon bond of residual maturity H is given by: o Qn    Bðt; H Þ ¼ E exp − yt þ … þ ytþH−1 yt ; Y t i Qh    ¼ expð−yt Þ E exp − ytþ1 þ … þ ytþH−1 yt ; Y t ;

ð3:3Þ

By applying the iterated expectation theorem and using Assumptions A.4 and A.7, we get: i Qh    E exp − ytþ1 þ … þ ytþH−1 yt ; Y t  i Q Qh     ¼ E E exp − ytþ1 þ … þ ytþH−1 Z tþH−1 ; yt ; Y t yt ; Y t  Q Q    

Q

 ¼ E E exp −ytþ1 Z tþ1 … E exp −ytþH−1 Z tþH−1 yt ; Y t : Let us denote: Q

E½expð−yt ÞjZ t  ¼ expð−c′Z t Þ ¼ ½Expð−cÞ′Z t ;

ð3:4Þ

where c[resp. exp(− c)] is the vector with components ck(1), k = 1, …, K. [resp. exp(− ck(1))]. The prices of the zero-coupon bonds become:  h i Q Qh    Bðt; H Þ ¼ expð−yt Þ E E exp −c′ Z tþ1 þ … þ Z tþH−1 Z t ; yt ; Y t yt ; Y t  o Q Q     ¼ expð−yt Þ E E exp −c′ Z tþ1 þ … þ Z tþH−1 jZ t yt ; Y t ; The expression of the conditional Laplace transform of the cumulated hidden Markov chain is derived in Proposition a.1 of Appendix 4. We deduce the following Proposition: Proposition 3. The prices of the zero-coupon bonds are given by: Bðt; HÞ ¼ e′½Π ðcÞ

H−1 b

Z t expð−yt Þ; H ≥1;

 Q
 b ¼ E Z y ; Y : with Π ðcÞ ¼ diag½ expð−cÞΠ; and Z t t t t

We get a linear term structure of zero-coupon prices of the form (2.2): Bðt; HÞ ¼ aðH Þ′ F t ;

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

29

bt expð−y Þ, or equivalently a log-affine term structure of the interest rates: with F t ¼ Z t r ðt; H Þ ¼ −

1 1 logBðt; H Þ ¼ − log½aðH Þ′F t ; H H

with positive factors. bt : The components of vector Z   i h i Qh   E Z k;t yt ; Y t ¼ Q Z k;t ¼ 1yt ; Y t ; are the filtered probabilities of the hidden regimes, valued in [0,1] and summing up to 1. In particular the associated linear combination (2.1) includes an intercept. h i bt expð−y Þ At this stage, it is interesting to note that the observable process (yt, Yt) and the observable factor process ð F t Þ ¼ Z t are not Markov. Indeed the model has been constructed by integrating out the unobservable (Zt) history. This implies in general a longer memory for the observable process, and also for the observable factor process, whose components are functions not only on the current values (yt, Yt), but also of all past values of this process. To summarize, the prices would be linear functions of the latent process (Zt), if it were observable, but they are nonlinear functions of the observable processes (yt, Yt) and of their lagged values. The nonlinear dynamic feature is a consequence of this lack of observability. 3.2. Spanning properties In continuous time an affine term structure model of interest rates implies an affine risk-neutral factor dynamic under weak rank conditions [see e.g. Duffie and Kan (1996, p.386), or Duffie (2001 p. 159)]. Can we expect a similar result for the linear term structure model of zero-coupon prices? In other words, can any linear factor model (2.1) be interpreted as a Hidden Markov Term Structure Model? The answer is yes in the 2-factor case. Proposition 4. In the 2-factor case, any LTSM is a Hidden Markov Term Structure Model. Proof. See Appendix 5. This shows the importance of this particular class, even if we can expect other examples, when the number of observable factors is larger (see Section 4). 3.3. Pricing interest rate derivatives The specification of the entire risk-neutral distribution allows to easily compute not only the prices of the zero-coupon bonds, but also the prices of complicated derivatives written on interest rates under the stronger set of Assumptions A.1-A.2'-A.3. Let us focus on European derivatives written on the short-term interest rate. We can consider the derivatives with exponential payoff (possibly complex), since the price of a derivative with another payoff can be deduced by an appropriate Fourier transform [see e.g. Duffie et al. (2000)].   The price at date t of the derivative paying exp −uytþH−1 at date t + H− 1 is: o Qn    pðt; H; uÞ ¼ E exp − yt þ … þ ytþH−1 þ uytþH−1 yt ; Y t  Q Q   

Q

 ¼ expð−yt Þ E E exp −ytþ1 Z tþ1 … E exp −ð1 þ uÞytþH−1 Z tþH−1 yt ; Y t g  o Q Q    ¼ expð−yt Þ E E exp −c′Z tþ1 …−c′Z tþH−2 −cð1 þ uÞ′Z tþH−1 jZ t yt ; Y t ; by following the same approach as in Section 3.1 and by noting cðuÞ ¼ ½c1 ðuÞ; …; cK ðuÞ: Applying the result in Appendix 4 iii), we deduce the proposition below: Proposition 5. The price of the European call with exponential payoff in interest rate is: H−1 b

pðt; H; uÞ ¼ ½Expð−cð1 þ uÞ þ cÞ′½Π ðcÞ

Z t expð−yt Þ; H ≥2:

We deduce the pricing formula for any European derivative written on the short-term interest rate. Let us assume that the   payoff g ytþH−1 admits the Fourier representation:     g ytþH−1 ¼ ∫ exp −uytþH−1 ωðuÞdu;

30

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

where ω is a given measure. Then the derivative price is: pðt; H; gÞ ¼ ∫ pðt; H; uÞωðuÞdu
′ H−1 b ¼ ∫ Exp½−cð1 þ uÞ þ cωðuÞdu ½Π ðcÞ Z t expð−yt Þ; H ≥2:

We deduce the following Corollary. Corollary 6. For the Hidden Markov Term Structure Model, any term structure of derivative prices is a linear function of the same underlying factors as the discount function. Thus, we have just to adjust appropriately the factor coefficients according to the form of the derivative payoff. The space generated by the derivative prices coincides with the one generated by the prices of the coupon bonds if the dimension of the latter is equal to K. 3.4. Updating of observable factors The HMTSM has been constructed by specifying the risk-neutral transition of an extended set of variables (Zt, yt, Yt), including bt expð−y Þ is deduced after components unobservable by the investor. We have seen that the set of observable
factors  Ft ¼ Z t applying the no-arbitrage restrictions based on the available investor's information yt ; Y t only.
  bt ¼ EQ Z t y ; Y t . As noted before, the factors are directly linked to the risk-neutral expectations of the regime indicators Z t

These functions of observable variables are easily updated by applying Kitagawa's algorithm [Kitagawa (1987) and Appendix 6]. Proposition 7. We have:

b Z k;tþ1 ¼

   
 b f y π′k Z t k tþ1 gk Y tþ1 ytþ1 ; Y t K h X

 i  
 b f y ; π′ j Z t j tþ1 g j Y tþ1 ytþ1 ; Y t

j¼1


    where π′k denotes the kth row of transition matrix Π; f k ytþ1 [resp. gk Y tþ1 ytþ1 ; Y t ] is the conditional distribution of yt+1 given Zk,t+1 = 1 [resp. of Yt+1 given Yt and Zk,t+1 = 1]. For the Kitagawa's updating formula to be valid, the conditional risk-neutral densities fk (y) (resp. gk) have to be defined with respect to a common dominating measure, for different values of k. If the variables y,Y are continuous variables, we get the standard probability density functions for continuous variables. If variable y is continuous in some regimes and for instance equal to zero in other regimes, the dominating measure is the sum of the Lebesgue measure and of the unit mass at zero. In this case the density fk (y) has to be interpreted as follows: for a regime k where y ¼ 0 : f k ð0Þ ¼ 1; f k ðyÞ ¼ 0; for y ≠ 0,for the other regimes: fk(0) = 0, and fk(y), y ≠ 0, is the standard pdf of the conditional continuous distribution. bt expð−y Þ is a complicated nonlinear function of the basic factors yt,Yt and of all their lagged values, and the Thus, F t ¼ Z t risk-neutral dynamics of Ft (and of zero-coupon prices B(t,H), which are linear functions of Ft) is complicated too. 4. The Finite Dimensional Dependence Term Structure Model (FDDTSM) Let us now introduce another dynamic model leading to a LTSM. In this model the observable factors are Markov processes with an appropriate nonlinear dynamics. We consider the pricing formulas by means of the historical dynamics and a stochastic discount factor. Finally, we derive and discuss the associated risk-neutral dynamics. 4.1. Markov processes with finite dimensional dependence These processes have been analyzed in Gourieroux and Jasiak (2000). More precisely, they consider (multidimensional) Markov processes (Yt), say, such that the space generated by the predictors E½gðY t ÞjY t−1  ¼ Et−1 ½gðY t Þ; g varying, has a finite dimension K. They show that this condition is satisfied, if and only if the transition of the process can be decomposed as: K X ak ðY t Þbk ðY t−1 Þ lðY t jY t−1 Þ ¼ l0 ðY t Þ k¼1

¼ l0 ðY t Þa′ðY t ÞbðY t−1 Þ; where l0 is a probability density function and b1(Yt − 1),…,bK(Yt − 1) is a basis of the predictor space.

ð4:1Þ

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

31

Special FDD Markov dynamics can be interpreted in term of smooth transitions between hidden regimes. Let us introduce the following conditions: FDD model with smooth transitions: i) ak(Y) ≥ 0, E0(ak) = 1, k = 1, …, K, K

ii) bk ðY Þ≥0; k ¼ 1; …; K; ∑ bk ðY Þ ¼ 1; k¼1

where E0(ak) = E0[ak(Y)] = ∫ ak(Y)l0(Y)dY denotes the expectation with respect to the baseline pdf l0. The first condition provides an interpretation of ak (Y) as a probability density function with respect to the baseline probability measure l0 (Y) dY. The second set of conditions means that, for any given value Yt − 1, the set [b1(Yt − 1), …, bK(Yt − 1)] defines a probability distribution on the state space {1,…,K}. In other words, the model features hidden regimes. At date t − 1, we first draw the regime Z∗t , say, among {1,…,K}, conditional to the observed Yt − 1; once the regime is selected equal to k, the future process value Yt is drawn according to the distribution with pdf l0 (yt) ak (yt). As in Section 3, we get a dynamic model with hidden regime. However, these dynamics differ significantly. In the hidden Markov dynamics the process of regime indicators is exogenous Markov, and process (yt, Yt) is not Markov. In the FDD process with smooth transition, both processes (Yt) and (Z∗t ) are Markov with respect to their own information sets, but (Z∗t ) is endogenous. The causal scheme becomes: Table 2: Causal scheme for the FDDTSM 

→Y t →Z tþ1 →Y tþ1 …

4.2. The term structure Let us now consider a FDD Markov process (Yt) with historical transitions satisfying decomposition (4.1). We assume that the values of Yt are observable for the investor, and we denote by m (Yt + 1) the stochastic discount factor for period (t, t + 1). Proposition 8. Under Assumptions A.1–A.3 and no arbitrage opportunity, the FDD term structure is such that: H−1

Bðt; H Þ ¼ E0 ðma′Þ½E0 ðmba′Þ bðY t Þ; H ≥1; where : E0 ðma′Þ ¼ ∫ mðY Þa′ðY Þl0 ðY ÞdY; E0 ðmba′Þ ¼ ∫ mðY ÞbðY Þa′ðY Þl0 ðY ÞdY:

Proof. See Appendix 7. We get a LTSM (see Proposition 1) with: að1Þ ¼ E0 ðmaÞ; C ¼ E0 ðmba′Þ; F t ¼ bðY t Þ:

The observable factors Ft are functions of the current value of Y only. These factors are Markov with respect to their own information. 4.3. Derivative prices and risk-neutral dynamics Closed form expressions are also easily derived for the prices of European derivatives written on Y. Let us consider a European derivative paying gðY tþH Þ at date t + H. Proposition 9. Under Assumptions A.1–A.3, and no arbitrage opportunity, the price at date t of the European derivative paying gðY tþH Þ is: H−1

pðt; H; gÞ ¼ E0 ðmga′Þ½E0 ðmba′Þ

bðY t Þ; H ≥1:

Proof. See Appendix 7. Therefore, the baseline factors b1(Yt),…,bK(Yt) are spanning the space of all European derivative prices written on observable process Y. This space obviously contains the space spanned by the prices of the coupon bonds, the dimension of which is (r ≤ K) defined in Proposition 2. If moreover r = K, the two spaces are identical.

32

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

We deduce the risk-neutral dynamics of process (Yt). The risk-neutral transition is given by: 

l ðY t jY t−1 Þ ¼ mðY t ÞlðY t jY t−1 Þ=Bðt; 1Þ ¼ l0 ðY t Þ½mðY t Þa′ðY t Þ

bðY t−1 Þ : E0 ½mðY t ÞaðY t Þ′ bðY t−1 Þ

ð4:2Þ

We get a risk-neutral FDD dynamics, 





l ðY t jY t−1 Þ ¼ l0 ðY t Þa ðY t Þ′b ðY t−1 Þ; say;  with ak ðY t Þ ¼ ½ak ðY t ÞmðY t Þ=E0 ½ak ðY t ÞmðY t Þ and :  bk ðY t−1 Þ ¼ fbk ðY t−1 ÞE0 ½ak ðY t ÞmðY t Þg=fE0 ½mðY t ÞaðY t Þ′bðY t−1 Þg: In the special case of a historical FDD model with smoothed transitions, it is easily checked that functions a∗k, b∗k, k = 1, …, K satisfy also the conditions for smoothed transitions. Therefore b∗k(yt − 1) provides the risk-neutral conditional probability of the hidden regime k, and a∗k(Yt − 1) the associated risk-neutral conditional density in this regime. 4.4. Direct risk-neutral modeling Depending on the problem, it is easier to work with the historical distribution and the stochastic discount factor, or to work directly with the risk-neutral distribution.
 Let us assume that we want to particularize the short-term interest rate yt among the e ′t . In the approach by stochastic discount factor developed in Sections 4.2–4.3, functions structural observable factors Y t ¼ yt ; Y m, a, b have to satisfy the pricing relationship: E0 ðma′ÞbðY t Þ ¼ expð−yt Þ; which imposes strong ex-ante restrictions on the form of these functions. Thus, when the short-term rate is introduced among the factors, a direct risk-neutral modeling may be preferable. Specifying a risk-neutral FDD dynamics l∗(Yt|Yt − 1) = l0(Yt)a∗(Yt)b∗(Yt − 1) we get: Q  

Bðt; H Þ ¼ Et exp −yt …−ytþH−1 Q  

¼ expð−yt Þ Et exp −ytþ1 …−ytþH−1 ;

and iH−2

h   ′  b ðY t Þ Bðt; H Þ ¼ expð−yt ÞE0 expð−yÞa ðY Þ E0 expð−yÞb ðY Þa ðY Þ h iH−1  ′  ¼ e′E0 expð−yÞb ðY Þa ðY Þ expð−yt Þb ðY t Þ; h i ′  ′ since: E0 expð−yÞa ðY Þ ¼ e′E0 expð−yÞb ðY Þa ðY Þ :

ð4:3Þ

Therefore, we get a LTSM with: h i  ′ C ¼ E0 expð−yÞb ðY Þa ðY Þ ; 

F t ¼ expð−yt Þb ðY t Þ:

5. An illustration Let us now illustrate the term structure patterns and their risk-neutral dynamic properties in the two types of LTSM, discussed in Sections 3 and 4. 5.1. A binding floor for the short-term interest rate It is useful to get term structure models able to represent short-term interest rates staying close to their zero bound during some period, a stylized fact recently observed [see Ichine and Ueno (2007), Kim and Singleton (2010)], or with a binding floor regularly reached. Since the support of the historical and risk-neutral distributions is the same under no arbitrage, this feature can be incorporated in the risk-neutral framework. A first solution for a zero binding floor has been proposed in Black (1995). The idea is to write the short-term interest rate as rt = max[0, s(yt)], where s(Yt) is a shadow rate process. However, even when the risk-neutral dynamics of the shadow rate is affine, say, the risk-neutral dynamics of the observable rate rt is complicated, and requires numerical methods to derive the term structure pattern [see the discussions in Gorovoi and Linetsky (2004), Kim and Singleton (2010), Section 3.3].

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

33

We will see below that the LTSM are flexible enough to represent such dynamics of the short-term interest rate. 5.2. The FDDTSM model Let us first consider a FDD model, in which the observable variable Yt is bidimensional and where the first component yt of
′ e t is the short-term interest rate. We assume that the dimension of the predictor space is K = 3. The conditional Y t ¼ yt ; Y risk-neutral distribution of (Yt) corresponds to a FDD model with smooth transitions between underlying regimes. The conditional risk-neutral probabilities of the regimes are given by:


  e bk yt−1 ; Y t−1 ¼

e −y yt−1 þ Y t−1 k d

φ 3 X l¼1

φ

!

e −y yt−1 þ Y l t−1 d

! ; k ¼ 1; 2; 3;

ð5:1Þ

where φ is the pdf of the standard normal distribution, y∗k, k = 1, 2, 3, are given values and d is a bandwidth. e t are conditionally independent, with the following distributions: For a given underlying regime k, yt and Y i) For yt: the point mass at zero, if k = 1, and the gamma distribution γ(m2k /σ2k , mk/σ2k ), with mean mk and variance σ2k , if k = 2, or 3. e t : the gamma distribution γ(m22/σ22, m2/σ22). ii) For Y By construction, the short-term interest rate will reach the zero binding value with a positive probability. Moreover, in the period when the short term is staying at zero, the term structure of interest rates will still vary in time due to the second et . observable process Y

The price B(t,H) of a zero-coupon bond is given by formula (4.3). The row vector E0 expð−yÞa∗′ ðY Þ has an explicit form. Its first

νk αk component is equal to 1, whereas its kth component is given by ; for k = 2,3, with αk = mk/σ2k , νk = m2k /σ2k (this αk þ 1 expression is deduced from the explicit formula of the Laplace transform of a gamma distribution).

∗ The matrix E0 expð−yÞb ðY Þa∗′ ðY Þ has no explicit form, but is easily approximated by simulation. More precisely, its first
 S ∗ e s are drawn independently in γ(ν2,d2) and S denotes the number of e s , where the Y column can be approximated by: 1S ∑ b 0; Y s¼1

replications. The other columns for k = 2,3 are approximated by: (  ) S  s  
s s  f ys ; ν k ; α k 1X e ; exp −y b y ; Y S s¼1 f ðys ; ν2 ; α 2 Þ

ð5:2Þ

s

e and ys are drawn independently in the same4 gamma distribution γ(ν2,α2), and f(y;νk,αk) denotes the pdf of the gamma where Y distribution γ(νk,αk), k = 2, 3. The numerical values of the risk-neutral parameters are set to: 





y1 ¼ :03; y2 ¼ :05; y3 ¼ :07; d ¼ :005; m2 ¼ :03; σ 2 ¼ :01; m3 ¼ :04; σ 3 ¼ :02: Assuming, for the sake of simplicity, that the historical and risk neutral dynamics are identical, we then simulate paths of
′ e t and for the interest rates rðt; hÞ ¼ − 1 logBðt; hÞ for h = 5, 10, 20, 100, the initial values length T = 50 for the process Y t ¼ y ; Y t

h

e 1 ¼ :001. The yield trajectories are given in Fig. 1. being y1 ¼ Y The short rate yt is equal to zero in periods 27 to 29 and 36 to 42. Within these periods, the rest of the yield curve is highly variable; in particular within the first period, we observe a change of the slope sign. Fig. 2 gives the term structure at different dates; the term structure can be increasing, decreasing, increasing and then decreasing, or conversely. Fig. 3 gives the values of the r (t, H) as functions of the observed short-term rate yt. For large values of the short rate, these functions are increasing with a slope decreasing with H, but this smoothness is no longer true for smaller values of yt, where sudden changes of monotonicity can be observed. These illustrations show that, even in model with two observable processes and a small dimension K = 3, corresponding to the number of observable factors in formula (2.1), we get very flexible results, including short rates reaching the zero bound and associated volatile long rates [see Kim and Singleton (2010) for a discussion of this point]. 4

An importance function has been introduced in Eq. (5.2) to compensate the drawings in the same distribution γ(ν2,α2).

34

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

Yields

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Trajectories of yields H= 1 (solid lins),5 (daches), 10 (dots),20(dots and dashes), 100 (solid lines)

0

5

10

15

20

25

30

35

40

45

50

Time Fig. 1. Trajectories of yields.

During the simulation steps, we get as a by-product the simulated trajectory of the underlying regimes over the period (see Fig. 4). We see from the causal scheme in Table 2 that the underlying (Zt ) process is a Markov chain with respect to its own ∗ information. The transition matrix of this chain is equal to E0 b ðY Þa∗′ ðY Þ [see Gourieroux and Monfort (2011)], which is easily approximated by simulation. In our case, the transition matrix is: 0

:84 Π ¼ @ :15 :01

1 :07 :08 :46 :30 A: :47 :62

5.3. The HMTSM Let us now consider a similar exercise for the HMTSM. To facilitate the comparison with the FDDTSM studied in Section 5.2, we consider a model, where: i) The exogenous Markov chain is unobservable to the investor, with the same transition matrix Π as in 5.2 and the simulated trajectory given in Fig. 4. ii) The observable process Yt = yt is one-dimensional, and the conditional distributions of yt given the underlying regime k are the same as in 5.2. This model has the same number K = 3 of observable factors.

0.00

Yields

0.01 0.02 0.03 0.04

0.05 0.06 0.07

Term Structures at Various Dates

0

10

20

30

40

50

60

70

Moturity Fig. 2. Term structures at various dates.

80

90

100

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

35

Yields

0.005

0.015

0.025

0.035

0.045

Yields as Functions of the Short Rate H= 5 (dashes),10 (dots),20(dots and dashes), 100 (solid lines)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Ordered Observed Short Rates Fig. 3. Yields functions of the short term.

We display in Fig. 5 the yield trajectories. The yields stay the same for any date corresponding to the regime of zero short-term rate. This is a consequence of the fact that: h i Q Z 1;t ¼ 1jyt ¼ 0; yt−1 ¼ 1 which is easily shown either directly or from Kitagawa's formula in Proposition 7. Therefore, by Proposition 3, we get for any date with yt = 0: 0 1 1 H−1 @ A 0 ; Bðt; HÞ ¼ e′½Π ðcÞ 0 independent of date t. Thus the term structure stays the same within a spell of zero short rate, even if we introduce additional observable processes factors Yt. 6. Conclusions We have introduced term structure models in which the zero-coupon prices are linear functions of a set of underlying observable factors. These models are interesting for their spanning properties, since the prices of any derivatives written on the interest rates are also linear combinations of the same factors.

2 1 0

State

3

4

State of the chain

0

5

10

15

20

25

30

Time Fig. 4. States of the chain.

35

40

45

50

36

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

Yields

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Trajectories of yields H= 1 (solid lins),5 (daches), 10 (dots),20(dots and dashes), 100 (solid lines)

0

5

10

15

20

25

30

35

40

45

50

Time Fig. 5. Yield trajectories.

Some special term structures with this linearity property are especially appealing, since they get interpretations in terms of hidden regimes. We provide two hidden regime term structure models, which differ by the causal chain between the observable factors and the hidden regime. The model with FDDTSM seems much more flexible than the HMTSM to capture the observed term structures in regimes of binding floor. Appendix 1. Proof of Proposition 1 Let us assume B(t,H) = a(H)′Ft and consider at date t a portfolio with allocation α1,…,αH in the H first zero-coupon bonds. The allocation can be of any sign under Assumption A.3. The value at t of this portfolio is: P t ðα Þ ¼

H X

α h Bðt; hÞ;

h¼1

whereas its value at the next date is:

P tþ1 ðα Þ ¼

H X

α h Bðt þ 1; h−1Þ ½with Bðt þ 1; 0Þ ¼ 1:

h¼1

This portfolio is risk-free if and only if, H X

α h aðh−1Þ ¼ 0;

ða:1Þ

h¼2

and then its future value is Pt + 1(α) = α1. When it is risk-free, we get by no-arbitrage the condition: P tþ1 ðα ÞBðt; 1Þ ¼ P t ðα Þ H X ⇔ α h Bðt; hÞ ¼ 0 h¼2

ða:2Þ

H X ⇔ α h aðhÞ ¼ 0: h¼2

Thus, any allocation satisfying Eq. (a.1) has also to satisfy Eq. (a.2). By Farkas's Lemma, [see e.g. Rockafellar (1979), p200], there exists a matrix C of Lagrange multipliers such that: 2 4

3 2 3 að1Þ′ að2Þ′ 5 4 5 ⋮  ¼  ⋮  C; a H−1 ′ a H ′

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

37

or equivalently such that: ∀h : aðhÞ′ ¼ aðh−1Þ′C: By recursive substitution, we deduce that: aðhÞ′ ¼ að1Þ′C

h−1

; ∀h≤H

which is Proposition 1. Appendix 2. Proof of Proposition 2 The vector space spanned by the prices of zero-coupon bonds (equal to the vector space spanned by the price of the coupon bonds) is equal to the vector space spanned by the variables a∗ ð1Þ0 ΔH−1 F ∗t ; H ≤H, or equivalently, by Cayley–Hamilton theorem spanned by the variables a∗(1)'F∗t , …, a∗(1)'ΔK − 1F∗t . Since the factors Ft, and then the factors F∗t are assumed linearly independent, the dimension of the space is equal to the rank of ∗ [a (1), Δ′a∗(1), …, (ΔK − 1)'a∗(1)]. Part ii) of Proposition 2 follows from the definition of the rank. Appendix 3. The positivity conditions Different cases can be distinguished according to the type of eigenvalues and their multiplicity order. • If the eigenvalues are complex conjugates, we have: h i H−1 Bðt; HÞ ¼ δ F 1;t cosðωH Þ þ F 2;t sinðωHÞ ; say; where δ N 0: Due to the periodic function of H, which will take positive and negative values, this situation is not compatible with positive prices of zero-coupon bonds. • If both eigenvalues are equal to δ, we have either B(t,H) = δH − 1Ft (if the matrix can be diagonalized), which is a single factor model, or B(t,H) = δH − 1(F1,t + HF2,t). The condition δH − 1(F1,t + HF2,t) ≥ 0, ∀t, ∀H ≥ 1, is equivalent to: δN0; F 2;t ≥0; F 1;t þ F 2;t ≥0; ∀t:

ða:3Þ

• If the eigenvalues are δ and − δ, with δ N 0, we get: H−1

Bðt; HÞ ¼ δ

n

H−1

F 1;t þ ð−1Þ

o F 2;t :

The positivity condition becomes: δN0; F 1;t þ F 2;t ≥0; F 1;t − F 2;t ≥ 0; ∀t:

ða:4Þ

• Finally, let us consider distinct eigenvalues δ1,δ2 such that |δ1| N |δ2|. The discount function is: H−1

Bðt; HÞ ¼ δ1

H−1

F 1;t þ δ2

F 2;t :

By considering its behavior for large H, we get: δ1 N 0, F1,t ≥ 0, ∀t, since the model is asymptotically equivalent to a 1-factor model. Then, H−1

Bðt; HÞ≥0⇔1 þ ðδ2 =δ1 Þ




 F 2;t = F 1;t ≥ 0; ∀t; H ≥1:

If δ2 N 0, this is equivalent to: F2,t/F1,t ≥ − (δ1/δ2)H − 1, ∀H ≥ 1, or to: δ1 Nδ2 N0; F 1;t þ F 2t ≥0; F 1;t ≥0: If δ2 N 0, we get: ð−1Þ

H−1




H−1  δ F 2;t =F 1;t ≥− 1 ∀t; H ≥1: jδ2 j

ða:5Þ

38

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

The positivity condition becomes: δ1 N 0N δ2 N−δ1 ; F 1;t N 0; F 1;t þ F 2;t N 0; δ1 F 1;t þ δ2 F 2;t N0:

ða:6Þ

Appendix 4. Prediction formulas for a Markov chain i) The conditional Laplace transform: We have, for any vector c:  

Et exp −c′Z tþ1

¼ Et ðExpð−cÞÞ′Z tþ1 0 ¼ ½Expð−cÞ ΠZ t : ii) The conditional Laplace transform of the cumulated chain The times spent in each state between t + 1 and t + H are given by the components of the vector Zt + 1 + … + Zt + H. The joint conditional Laplace transform of these random times is:      ψt;H ðcÞ ¼ E exp −c′ Z tþ1 þ … þ Z tþH Z t : We have the following result: Proposition a.1. We have: H

ψt;H ðcÞ ¼ e′½Π ðcÞ Z t ; H ≥1; where Π ðcÞ ¼ diag ½Expð−cÞΠ; and e′ ¼ ð1; …; 1Þ: Proof. From i), the formula is valid for H = 1. Let us now check the recursion formula. We get:     Et exp  −c′  Z tþ1 þ … þ Z tþH  

 ¼ Et nexp −c′Z tþ1 Etþ1 exp −c′ Zotþ2 þ … þ Z tþH   H−1 ¼ Et exp −c′Z tþ1 e′Π ðcÞ Z tþ1 n

 o 0 H−1 Z tþ1 ; where o ¼ Et ½Expð−cÞ o e′Π ðcÞ denotes the componentwise product. We deduce that: n
o 0

H−1 ΠZ t Expð−cÞ o e′Π ðcÞ

  H−1 0 o½Expð−cÞ ΠZ t ¼ e′Π ðcÞ

ψt;H ðcÞ ¼

H−1

¼ e′Π ðcÞ Π ðcÞZ t H ¼ e′Π ðcÞ Z t ;

which proves the result. QED iii) An extension Let us now consider the following transform:   

  ψt;H ðc; dÞ ¼ E exp −c′ Z tþ1 þ … þ Z tþH −d′Z tþH Z t : A proof similar to the proof of Proposition A.1 leads to the following result: Corollary a.2. 0

H

ψt;H ðc; dÞ ¼ ½Expð−dÞ ½Π ðcÞ Z t ; H ≥1:

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

39

Appendix 5. Proof of Proposition 4 Looking at Table 1, four different cases have to be reached, and are essentially characterized by means of the eigenvalues of matrix Π(c) and the possibility to diagonalize it. In the 2 × 2 case, this matrix can be written as:

Π ðcÞ ¼

γ1 0

0 γ2



p11 1−p11


 1−p22 ; where γ j ¼ exp −c j ∈ð0; 1Þ; j ¼ 1; 2 p22

 1−p22 p11 can be diagonalized with 1−p11 p22 eigenvalues δ1 = γ, δ2 = γ(p11 + p22 − 1). The three last cases appearing in Table 1 are easily reached: the second case is reached with p11 = p22 = 0, the third case is reached when p11 + p22 − 1 N 0, and the fourth case, when p11 + p22 − 1 b 0, as soon as the modulus of the eigenvalues are smaller than 1. ii) The remaining case corresponds to a double eigenvalue for Π(c) and a Π(c) matrix, which cannot be diagonalized. Let us consider another choice of γ1, γ2, that is γ1 = 1, γ2 = γ, say. Matrix Π(c) becomes: i) Let us first consider the case in which γ1 = γ2 = γ, say. The matrix Π ðcÞ ¼ γ

Π ðcÞ ¼

 p11 1−p22 : γð1−p11 Þ γp22

The determinantal equation is: det ½Π ðcÞ−δId ¼ ðp11 −δÞðγp22 −δÞ−γ ð1−p11 Þð1−p22 Þ 2 ¼ δ −δðp11 þ γp22 Þ−γð1−p11 −p22 Þ ¼ 0:

This equation has a double root, if and only if, 2

Δ ¼ ðp11 þ γp22 Þ þ 4γ ð1−p11 −p22 Þ ¼ 0; 22 : and then this double root is equal to the eigenvalue δ∗, say. We get: δ∗ ¼ p11 þγp 2 ∗ Then we can express the condition Δ = 0 in terms of δ . We get:

2

p22 δ



þ 2δ ð1−p11 −p22 Þ−p11 ð1−p11 −p22 Þ ¼ 0:

This equation has real roots if and only if: 

2

Δ ¼ ð1−p11 −p22 Þ þ p11 p22 ð1−p11 −p22 Þ ¼ ð1−p11 −p22 Þð1−p11 Þð1−p22 Þ≥0; which is equivalent to the condition: p11 + p22 ≤ 1. Since the product of the real roots has the same negative sign as − p11p22(1 − p11 − p22), there exists a real root: 

δ ðp11 ; p22 Þ ¼

−ð1−p11 −p22 Þ þ p22

pffiffiffiffiffiffi Δ

≥0:

By making p11, p22 vary such that p11 + p22 ≤ 1, we cover the first case of Table 1 with the positive double eigenvalue δ∗(p11,p22). Appendix 6. Kitagawa's algorithm bt of the regime indicators. More precisely, let us This algorithm provides a recursive formula to compute the predictions Z
 consider the joint density function of Zt + 1, Zt, yt + 1, Yt + 1 conditional on yt ; Y t , denoted by l ztþ1 ; zt ; ytþ1 ; Y tþ1 jyt ; Y t . This density is with respect to the counting measure for the discrete part corresponding to zt + 1, zt, and to a given dominating measure for yt + 1, Yt + 1. This dominating measure can be the Lebesgue, or the counting measure, or the sum of the Lebesgue and point mass

40

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41

at zero for yt + 1, Yt + 1, depending on whether these variables are continuous, discrete, or continuous with mass at zero, respectively. From Bayes formula, and Assumptions A4–A6, we have:
 l ztþ1 ; zt ; ytþ1 ; Y tþ1 jyt ; Y t 
   
¼ l ztþ1 jzt l ytþ1 jztþ1 l Y tþ1 jztþ1 ; yt ; Y t l zt jyt ; Y t : We deduce:
 l ztþ1 jytþ1 ; Y tþ1 ¼

h i    

Σzt l ztþ1 jzt l zt jyt ; Y t l ytþ1 jztþ1 l Y tþ1 jztþ1 ; yt ; Y t h  i : 

Σztþ1 Σzt l ztþ1 jzt l zt jyt ; Y t l ytþ1 jztþ1 l Y tþ1 jztþ1 ; yt ; Y t

The relation above is equivalent to:   
b f y π′k Z t k tþ1 g k Y tþ1 jyt ; Y t b ; Z k;tþ1 ¼ K h i X  
b f y π′ Z g Y jy ; Y j

t

j

tþ1

j

tþ1

t

t

j¼1

by using the notations of the text for the different conditional distributions. Appendix 7. Pricing formulas (Propositions 8 and 9) Let us assume: p(t,H;g) = A(H,m,g)'b(Yt). • For H = 1, we have:    

pðt; 1; g Þ ¼ Et m Y tþ1 g Y tþ1 ¼ E0 ðmga′ÞbðY t Þ; where E0(mga) = ∫ m(Y)g(Y)a(Y)l0(Y)dY. Therefore A(1,m,g) = E0(mga′). • Let us now derive the recursion with respect to time-to-maturity H. We get:           pðt; H; g Þ ¼ Et m Y tþ1 Etþ1 m Y tþ2 …m Y tþH g Y tþH   

0  ¼ Et m Y tþ1 AðH−1; m; g Þ b Y tþ1   

0  ¼ AðH−1; m; g Þ Et b Y tþ1 m Y tþ1 0 ¼ AðH−1; m; gÞ E0 ðmba′ÞbðY t Þ; where E0 ðmba′Þ ≡ ¼ ∫mðY ÞbðY Þa′ðY Þl0 ðY ÞdY: Thus, the recursive formula is: 0

0

AðH; m; g Þ ¼ AðH−1; m; g Þ E0 ðmba′Þ; H ≥2: • We deduce that: 0

AðH; m; g Þ ¼ E0 ðmga′Þ½E0 ðmba′Þ

H−1

:

References Ahn, D., Dittmar, R., Gallant, A., 2002. Quadratic term structure models: theory and evidence. Rev. Financ. Stud. 15, 243–288. Black, F., 1995. Interest rates as options. J. Finance 50, 1371–1376. Carr, P., Gabaix, X., Wu, L., 2009. “Linearity Generating Processes, Unspanned Volatility and Interest Rate Option Pricing ”. New York University D.P. Cochrane, J., 2001. Asset Pricing. Princeton University Press. Dai, Q., Singleton, K., 2000. Specification analysis of affine term structure models. J. Finance 55, 1943–1978. Duffie, D., 2001. Dynamic Asset Pricing Models. Princeton University Press. Duffie, D., Kan, R., 1996. A yield factor model of interest rates. Math. Finance 6, 379–406. Duffie, D., Singleton, K., 1999. Modelling term structure of defaultable bonds. Rev. Financ. Stud. 12, 687–720. Duffie, D., Pan, J., Singleton, K., 2000. Transform analysis and asset pricing for affine jump diffusions. Econometrica 68, 1343–1376. Filipovic, D., 1999a. A general characterization of one-factor affine term structure models. Finance Stochast. 5, 389–412. Filipovic, D., 1999b. A note on the Nelson–Siegel Family. Math. Finance 9, 349–359. Gabaix, X., 2009. Linearity Generating Processes: A Modelling Tool Yielding Closed Forms for Asset Prices. New York University DP.

C. Gourieroux, A. Monfort / Journal of Empirical Finance 24 (2013) 24–41 Gorovoi, V., Linetsky, V., 2004. Black's model of interest rates as options, eigenfunction expansions and Japanese interest rates. Math. Finance 14, 49–78. Gourieroux, C., 2006. Wishart processes for stochastic risk. Econ. Rev. 25, 1–41 (special issue on Stochastic Volatility). Gourieroux, C., Jasiak, J., 2000. State space models with finite dimensional dependence. J. Time Ser. Anal. 22, 665–678. Gourieroux, C., Monfort, A., 2011. Pricing with Finite Dimensional Dependence. CREST-DP. Gourieroux, C., Monfort, A., Polimenis, V., 2006. Affine models for credit risk analysis. J. Financ. Econ. 4, 494–530. Gourieroux, C., Monfort, A., Sufana, R., 2010. International money and stock market contingent claims. J. Int. Money Finance 29, 1727–1751. Ichine, H., Ueno, Y., 2007. Equilibrium Yield Curve and the Yield Curve in a Low Interest Rate Environment. Bank of Japan DP. Kim, D., Singleton, K., 2010. Term Structure Models and the Zero Bound: An Empirical Investigation of Japanese Yields. Stanford Business School DP. Kitagawa, G., 1987. Non Gaussian state space modelling of nonstationary times series. J. Am. Stat. Assoc. 82, 1032–1041. Monfort, A., Pegoraro, F., 2007. Switching VARMA term structure models. J. Financ. Econ. 5, 103–151. Nelson, C., Siegel, A., 1987. Parsimonious modeling of yield curves. J. Bus. 60, 473–489. Rockafellar, R., 1979. Convex Analysis. Princeton University Press. Shea, G., 1985. Interest rate term structure with exponential splines: a note. J. Finance 40, 319–325. Siegel, A., 2010. Arbitrage Free Linear Price Function Models for the Term Structure of Interest Rates. DP Univ. of Washington. Svensson, L., 1995. Estimating forward interest rates with the extended Nelson and Siegel model. Q. Rev. 3, 13–26 (Sveriges Riskbank). Vasicek, O., Fong, G., 1982. Term structure modelling using exponential splines. J. Finance 37, 339–348.

41