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This paper analyzes a simple discrete-time affine multifactor model of the term structure of interest rates in which the pricing factors that follow a Gaussian first-order vector autoregression are observable and there are no possibilities for risk-free arbitrage. We present the theoretical results for the compatible risk-neutral dynamics of observable factors in a maximally flexible way consistent with no-arbitrage under the assumption that the factor loadings of some yields are specified exogenously.

This paper analyzes a simple discrete-time affine multifactor model of the term structure of interest rates in which the factors of the model that follow a Gaussian first-order vector autoregression are observable and there are no possibilities for risk-free arbitrage. Rather than defining latent states indirectly through normalization on parameters governing the dynamics of latent states, a number of recent literatures have instead prescribed observable risk factors. For instance, [

In the history of dynamic term structure model, the pricing factors are treated as shocks of various kinds that are not necessarily designed to be observable. Recent studies show that modeling the factors as observable has enormous computational advantages in the parameter estimation (“calibration”) process (see, for example, [

In this paper, although one of our goals is to classify a family of models that is convenient for empirical work, we are not directly concerned with estimation issues. We refer readers to the empirical studies for such issues, such as [

Contrary to traditional affine latent factor approach in which some dynamics for factors are assumed firstly and then establishing the factor loadings of yields through arbitrage-free restriction, we take the loadings on specific yields as given firstly and then parametrize the

The remainder of this paper is structured as follows. In Section 2 we discuss the general Gaussian affine term structure models (GDTSM) and some assumptions imposed on yields and factors. In Section 3 we discuss the compatible dynamics of observable factors. In Section 4, we give an example. In Section 5, we conclude.

More precisely, for our purposes a useful starting point is the work by [

where

Here, we specify the Jordan form with each eigenvalue associated with a single Jordan block. Thus, when the eigenvalues

and where the blocks are in order of the eigenvalues.

Under Equations (1)-(3), the price of an n-year zero-coupon bond is given by

where

subject to the initial conditions

where

In this setting, the factors are latent variables and the “loadings” (

In other settings, the factor loadings may be specified a priori, for example, the dynamic Nelson-Siegel model

(DNS) was first proposed by [

responding Arbitrage-free model (AFNS) in which the absence of arbitrage and a priori specification of DNS loading actually restrict the coefficient of the process for the factors under

tors are the principal component of zero-coupon bond yields and the factor loadings for the corresponding bond yield are assigned a priori. They show in continuous-time frame that if the factors follow a mean-reverting dynamics, then the pre-specified factor loadings imposes some unexpected constraints on the reversion-speed ma- trix.

In the economy, we observe numerous zero-coupon bond yields with different maturities. We will take out from these yields a set of N yields,

where A an ^{th} element is ^{th} row is

In this paper, we derive a discrete-time arbitrage-free Gaussian affine term structure model under the following assumptions:

Assumption 1: There are N observable factors F_{t} in real economy which can linearly span the latent factors X_{t}.

Assumption 2: The factors F_{t} loadings for our key maturity yields y_{t} are known as a priori. The assumption 1 imply that we can express y_{t} as the affine combinations of F_{t}.

Assumption 2 means that u and U are given exogenously.

In the following, we define the process of

where _{t}. Because F_{t} is observable,

And the risk-free rate can be expressed as the affine functions of the vector of observable factors F_{t}. Without loss of generality we can define

where

In the following, we will take u and U as given exogenously and find compatible dynamics of observable factors. We first derive restrictions on parameters of the process of observable factors F_{t} under risk-neutral measure

Theorem 1. Given key maturities

where

where

with

Proof:

Combining Equations (8) and (7), we have,

Substituting Equation (13) into Equation (1), we have,

We will call Equation (14) the model consistency condition.

From Equations (4) and (6), we observe that

According to [

defined as

Applying this formula to Equation (15), we have

where,

Defining

where

Let us define a block diagonal matrix

where

Now let

where

That is

Substituting Equations (17), (18) and (19) into Equation (16), we have

From Equation (14), we have

In deriving Equation (21), we use

We can check the relation

n

Thus by construction we have shown that with U given,

Since we assume the factors F_{t} are observable and the covariance matrix _{t} under the _{t} to calculate

Theorem 2. Given the choice of key maturities

where

Proof:

From Equations (14), (20) and (21), we have

In addition,

where

which is the first column of matrix G, and then we have

From Equations (5) and (6), we have

where

Then,

Calculating the summation in the above equation, we get

We define

where

n

Theorem 3. Vector

where G is defined by Equation (12) and

Proof:

Substituting Equation (13) into Equation (3), we have,

From Equation (20), we have

For

As before, let

Since

Substituting Equation (28) into Equation (27), we get

n

Theorem 4. Let parameters of the model,

where G and

Proof:

From Equations (26) and (28), we have

Substituting Equation (24) into Equation (30), we get

n

Consider a 3-dimensional observable affine factor model, and let the eigenvalues of the coefficient matrix

where

From Equations (15) and (15*), we have the close form for

A number of previous researchers have discussed the affine term structure with the pricing factors being observable. A distinctive feature of the models with observable factors is its computational advantage over that with latent factors. However, these researches just focus on the computational convenience but do not study such model in depth.

In this paper, our results show that if we treat the pricing factors observable and thus the factors loadings of some key maturity yields are given a priori, the no-arbitrage condition will impose strict restrictions on the risk-neutral dynamics of the observable factors and on the parameters of risk-free rate equation.

We discuss how to impose some important constraints on the

This work is supported by Research Innovation Foundation of Shanghai University of Finance and Economics under Grant No. CXJJ-2013-321. And I am grateful for helpful comments from the anonymous referee and especially to Professor Hong Li for his support and encouragement. All errors are my own.