Specifying Market Price of Risk

#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.

Completely Affine Models

Definition (Dai and Singleton's model).

The Dai and Singleton's model (completely affine model) is a generalized CIR model (Cox, Ingersoll, and Ross, 1985), which is an admissible ATSM ATSM that belongs to class, which indicates that there are factors whose volatility depends on factors: where is an dimensional nonrandom vector. Note that equals to the volatility of the factors in the sense that the rank of the matrix equals to .

Proposition (Dai and Singleton model is completely affine).

We say ^bbff37 Definition 1 (Dai and Singleton's model) is completely affine since the physical factor also follows a VAR process as the risk-neutral factor does. This means the physical factor dynamics can be written as

Proof.Note that from Affine Term Structure Models > ^2fff50 Affine Term Structure Models > Assumption 1 (assumptions for ATSM), we have Thus the physical factor dynamics are also affine in physical measure: thus the factor also follows a VAR process in physical measure. □

This implies that the physical factor dynamics follow a VAR process under both the risk-neutral measure and physical measure, which considerably simplifies the estimation process.

Remark (identification restriction by Dai and Singleton).

the elements of are all non-negative.
the first factors are non-negative.
we have where are non-negative, and is the th standard basis of , so that .
the last elements of are equal to .
conformably partitioned as where the elements of are non-negative if . note that is a lower triangular matrix if .
the matrix governing the cross correlations of the factors is given as

Advantages and Disadvantages

Under ^bbff37 Definition 1 (Dai and Singleton's model), the market price of risk has the following advantages and disadvantages:

Advantage: affine physical factor dynamics
Advantage: continuity at as the risk goes to as the variance of the factors goes to . this satisfies one of No-Arbitrage Condition > ^c75a75 No-Arbitrage Condition > Assumption 3 (assumptions for existence of strictly positive SDF).
Disadvantage: compensation depends only on factor variance
since is a function only of the factor variances, the second moments of the factors should contain all the necessary information on the risk, which is an unrealistic assumption. i.e. the bond excess returns () can be kept low if and only if the factor variances are small, this make it difficult to replicate stylized facts of the yield curve for the completely affine model.
Disadvantage: each price of risk has a fixed sign
since the sign of the market price of risk entirely depends on the sign of . thus the sign of the risk tends to remain same, which makes it difficult to the stylized fact that bond excess returns often change signs.

Essentially Affine Models

Definition (essentially affine models).

Duffe (2002) proposes the following essentially affine model of , where the specification of the short rate and the risk-neutral dynamics are identical, and only the market prices of risk differs: where is an dimensional nonrandom vector, is an nonrandom matrix and is an diagonal random matrix such that

Note that in ^283a64 Definition 4 (essentially affine models), the market prices of risk also depends on the factors independent of the conditional variances , meaning that the volatility of can be maintained at a high enough level while still keeping its level low.

In addition, the essentially affine model keeps the advantages of the completely affine models.

physical factor dynamics are also affine

Proposition (essentially affine model is essentially affine).

In ^283a64 Definition 4 (essentially affine models), the physical factor dynamics are also affine, meaning that

Proof.similar to the proof of ^5d0ff5 Proposition 2 (Dai and Singleton model is completely affine), we use the identity Note that thus we have where collects the block matrix in the position of . □

the volatility of goes to as the volatility of the factors goes to .

Extended Affine Models

In contrast to ^bbff37 Definition 1 (Dai and Singleton's model) and ^283a64 Definition 4 (essentially affine models), the extended affine model introduced by Cheridito, Filipovic, and Kimmel (2007) specifies physical and risk-neutral factor dynamics first, and then derives the market price of risk.

Definition (extended affine models).

The extended affine model is defined by:
where and are all non-negative, and the last elements of each are equal to .

Here, the market prices of risk are given as

Note that if , then the extended affine model is identical to ^283a64 Definition 4 (essentially affine models), since does not depend on the time and thus is affine function of the factors under the both models. As the case of corresponds to Affine Term Structure Models > ^24c2e6 Affine Term Structure Models > Definition 5 (Gaussian ATSM), the specification of the model using either market price of risk or physical dynamics leads to the same conclusion.