Affine Term Structure Models

#termstructure #economics #finance #atsm

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.

Main References

Kim, Seung Hyun. (2024). "Asset Pricing and Term Structure Models". WORK IN PROGRESS.

Preliminary Concepts

Assumptions and Models

Refer Fundamentals of Bond Fundamentals of Bond for the basic terminologies.

Assumption (assumptions for ATSM).

Throughout the ATSM (affine term structure models), we assume the following statements unless states otherwise.

there exists a sequence of dimensional latent factors that determines the bond prices and yields at time .
there exists an SDF process SDF process with where .
we assume the SDF process is the empirical SDF empirical SDF defined as where is the risk-free one period ahead rate at , is the dimensional market prices of risk, and is an dimensional random vector that follows standard normal distribution under the physical measure.
by No-Arbitrage Condition > ^33b05b No-Arbitrage Condition > Theorem 16 (Girsanov's theorem), is the dimensional random vector that follows standard normal distribution under the risk-neutral measure.
unless stated otherwise, all expectations at time are conditional on the information up to time , .

Definition (composition of ATSM).

Under ^2fff50 Assumption 1 (assumptions for ATSM), the term structure models are comprised of the following four components:

Short rate dynamics: how the short rate is determined as a function of , written as under the assumption of some function .
-dynamics (risk-neutral dynamics): the dynamics of the factors under the risk-neutral measure, written as where is the conditional mean of under and is the conditional variance.
dynamics (physical dynamics): the dynamics of the factors under the physical measure, written as where is the conditional mean of under and is the conditional variance. note that by No-Arbitrage Condition > ^33b05b No-Arbitrage Condition > Theorem 16 (Girsanov's theorem), and differ only by location.
Market prices of risk: how the market prices of risk is determined as a function of , written as under the assumption of some function .

In practice, it is sufficient to specify the short-run dynamics and only the two of the last three components. This is possible since and we can obtain the dynamics from the dynamics and the market price of risk.
Also we have Therefore, if we specify3 the risk-neutral and physical dynamics, than we automatically obtain the market price of risk.
On the other hand, if we specify the market price of risk and the risk-neutral dynamics, then we obtain the physical dynamics by

In most cases, we choose to specify the risk-neutral and physical dynamics. However, in some case, we can choose to specify the market price of risk and the physical dynamics, if we do not want to involve the risk-neutral measure.

Definition of ATSM

Definition (affine term structure models).

The Affine term structure models (ATSM) are term structure models in which the short rate , risk-neutral condition mean , and the conditional variance are affine functions of the factors .
Following Dai and Singleton (2000), these are typically given as

Note that from the above specification, yields are also given as affine functions of the factors.

Definition (admissible ATSM).

If, from the conditional variance matrix,

each are non negative
is non-singular matrix

then such ATSM model is called admissible.

Definition (Gaussian ATSM).

If is time invariant in ^145b34 Definition 3 (affine term structure models), then the model is called a Gaussian affine term structure model (GASTM).

Note that ^24c2e6 Definition 5 (Gaussian ATSM) is a special case of ATSM when and for any . Thus GASTM is admissible if and only if is non-singular. Usually, we take as the Cholesky factor of the conditional variance, assuming it to be positive definite.

Solution for General Model

The goal of solving ^145b34 Definition 3 (affine term structure models) is to recover bond price , bond yields , and various forward rates and risk premia related to zero-coupon bonds, as functions of the underlying factors .

First, we obtain the bond prices in ATSM, which only requires the risk-neutral measure. Afterwords, we incorporate the physical measure to solve for bond risk premia.

Solve for Bond Prices

Assume the ATSM are given as

Definition 3 (affine term structure models).

and assume that the model is admissible.

Remark that by Duffie and Kan (1996), must be given as an exponential-affine function of the factors , which is where and are functions of the time to maturity .

Proposition (Ricatti equation for ATSM).

In ^24c2e6 Definition 5 (Gaussian ATSM), the solution of bond price is given as an exponential-affine function as where the intercepts and coefficient terms are given as which are called Ricatti equations. By letting the initial conditions and , we can recursively solve for bond prices.
Furthermore, the yields on a zero-coupon bond with -periods is given as which yields are affine functions of the factors.

Proof.Note that since , we have Also, remark that an asset's price is equal to its expected discount payoff, and the payoff of a zero-coupon's payoff is simply its price at . Thus the no-arbitrage equation becomes where the last equality holds by No-Arbitrage Condition > ^51420f No-Arbitrage Condition > Theorem 8 (first fundamental theorem of asset pricing).

Then, we have where the fifth equality holds by the short-rate dynamics and the risk-neutral dynamics.

Remark that the formula for the MGF of normal distribution gives us since follows standard normal distribution under .

Thus, we have by taking logs on the both sides, Now, from the conditional variance, we have where denotes the th element of the dimensional row vector of .

Therefore, which leads to the desired results of and completes the proof. □

Corollary (Ricatti equations for GASTM).

In ^24c2e6 Definition 5 (Gaussian ATSM), we have the closed-form Ricatti equations which are found by Hamilton and Wu (2012).

Proof.Remark that in ^24c2e6 Definition 5 (Gaussian ATSM), we have and for all , and for all . Thus from the previous proof of ^beb7fd Proposition 6 (Ricatti equation for ATSM), we have then we have the Ricatti equations of Using the initial conditions of and , we have and which completes the proof. □

Solve for Bond Risk Premia

Similar to the Solve for Bond Prices, assume the ATSM are given as

Definition 3 (affine term structure models).

Additionally, remark that from Assumption 1 (assumptions for ATSM), we have

One-Period Ahead Risk Premium

Proposition (risk premium of ATSM).

In ^145b34 Definition 3 (affine term structure models), the one-period ahead risk premium is defined as where refers the adjusted risk premium ignoring the Jensen's inequality term.

Proof.From the definition of one-period ahead expected excess rate one-period ahead expected excess rate, and log expression of the rate of return log expression of the rate of return, we have Note that from ^beb7fd Proposition 6 (Ricatti equation for ATSM), we have and Thus we have Note that the first term is referred to as Jensen's inequality term, as it originates from the concavity of the exponential function. By ignoring this term, the adjusted one-period ahead risk premium is Furthermore, as since the conditional variance of under is by ^2fff50 Assumption 1 (assumptions for ATSM). Then, we have which leads to the desired conclusion of where represents the beta term beta term, and is the market price of risk market price of risk. □

Term Premium

Proposition (term premium in ATSM).

In ^145b34 Definition 3 (affine term structure models), the term premium can be expressed as where represents the risk premium unadjusted for the Jensen's inequality term.

Proof.Note that from Fundamentals of Bond > ^7bdabf Fundamentals of Bond > Definition 3 (term premium, risk premium, and forward premium), the term premium for a maturity bond at time is defined as Note that sincewe have and the term premium is Also, note that from the definition of one-period ahead excess return one-period ahead excess return, and log expression of the rate of return log expression of the rate of return, the one-period ahead excess return for the period bond is thus the above term premium can be re-written as a telescoping sum where the third equality holds by the Statistical Proof > ^98e7c2 Statistical Proof > Theorem 18 (Law of iterated expectation). Additionally, the term represents the risk premium unadjusted for the Jensen's inequality term, derived as which completes the proof. □

Note that since , the term premium can also be interpreted as

Forward Risk Premium

Proposition (forward risk premium in ATSM).

In ^145b34 Definition 3 (affine term structure models), the forward risk premium can be expressed as where represents the risk premium unadjusted for the Jensen's inequality term.

Proof.Note that from Fundamentals of Bond > ^7bdabf Fundamentals of Bond > Definition 3 (term premium, risk premium, and forward premium), the forward risk premium is defined as where represents the $h-$period ahead forward rate period ahead forward rate.

Also, from forward rate in terms of returns forward rate in terms of returns, we have and similar to the previous proof previous proof, the forward rate can be re-expressed as a telescoping sum: which completes the proof. □

Equivalence of Expectation Hypotheses

From the previous proofs, we have expressed in terms of risk premium risk premium. This implies the Fundamentals of Bond > ^4f1b61 Fundamentals of Bond > Proposition 2 (expectation hypothesis) are all equivalent.

Proposition (equivalence of expectation hypothesis).

The three forms of assumptions given from Fundamentals of Bond > ^4f1b61 Fundamentals of Bond > Proposition 2 (expectation hypothesis), are all equivalent under ^145b34 Definition 3 (affine term structure models). This means, if the one of the three are assumed, then the other two are implied by the one.

Proof.Here, we will denote the each form of the expectation hypothesis as follows: () Suppose RP EH, i.e. . Then, we have and Thus TP EH and FRP EH hold.

() Suppose TP EH, i.e. . Then by letting , we have If , then implying .

Now consider the case when . Since we have . Thus by induction, RP EH holds. Then by the preceding result, RP EH implies FRP EH.

() Suppose FRP EH, i.e. . By letting , we have since . As we have RP EH, by the preceding result, TP EH also holds. □