Affine term structure model explained
An affine term structure model is a financial model that relates zero-coupon bond prices (i.e. the discount curve) to a spot rate model. It is particularly useful for deriving the yield curve – the process of determining spot rate model inputs from observable bond market data. The affine class of term structure models implies the convenient form that log bond prices are linear functions of the spot rate[1] (and potentially additional state variables).
Background
Start with a stochastic short rate model
with dynamics:
dr(t)=\mu(t,r(t))dt+\sigma(t,r(t))dW(t)
and a risk-free zero-coupon bond maturing at time
with price
at time
. The price of a zero-coupon bond is given by:
where
, with
being is the bond's maturity. The expectation is taken with respect to the
risk-neutral probability measure
. If the bond's price has the form:
where
and
are deterministic functions, then the short rate model is said to have an
affine term structure. The yield of a bond with maturity
, denoted by
, is given by:
Feynman-Kac formula
For the moment, we have not yet figured out how to explicitly compute the bond's price; however, the bond price's definition implies a link to the Feynman-Kac formula, which suggests that the bond's price may be explicitly modeled by a partial differential equation. Assuming that the bond price is a function of
latent factors leads to the PDE:
where
is the
covariance matrix of the latent factors where the latent factors are driven by an Ito
stochastic differential equation in the risk-neutral measure:
Assume a solution for the bond price of the form:
The derivatives of the bond price with respect to maturity and each latent factor are:
With these derivatives, the PDE may be reduced to a series of ordinary differential equations:
To compute a closed-form solution requires additional specifications.
Existence
Using Ito's formula we can determine the constraints on
and
which will result in an affine term structure. Assuming the bond has an affine term structure and
satisfies the term structure equation, we get:
At(t,T)-(1+B
| |
| t(t,T))r-\mu(t,r)B(t,T)+ | 1 | 2 |
|
\sigma2(t,r)B2(t,T)=0
The boundary value
implies
\begin{align}
A(T,T)&=0\\
B(T,T)&=0
\end{align}
Next, assume that
and
are affine in
:
\begin{align}
\mu(t,r)&=\alpha(t)r+\beta(t)\\
\sigma(t,r)&=\sqrt{\gamma(t)r+\delta(t)}
\end{align}
The differential equation then becomes
| 2(t,T)-\left[1+B |
\delta(t)B | |
| t(t,T)+\alpha(t)B(t,T)- | 1 | 2 |
|
\gamma(t)B2(t,T)\right]r=0
Because this formula must hold for all
,
,
, the coefficient of
must equal zero.
1+B | |
| t(t,T)+\alpha(t)B(t,T)- | 1 | 2 |
|
\gamma(t)B2(t,T)=0
Then the other term must vanish as well.
Then, assuming
and
are affine in
, the model has an affine term structure where
and
satisfy the system of equations:
\begin{align}
1+B | |
| t(t,T)+\alpha(t)B(t,T)- | 1 | 2 |
|
| 2(t,T)&=0\\
B(T,T)&=0\\
A |
\gamma(t)B | |
| |
\delta(t)B2(t,T)&=0\\
A(T,T)&=0
\end{align}
Models with ATS
Vasicek
has an affine term structure where
\begin{align}
p(t,T)&=eA(t,T)-B(t,T)r(t)\\
B(t,T)&=
\left(1-e-a(T-t)\right)\\
A(t,T)&=
| (B(t,T)-T+t)(ab- | 1 | \sigma2) | 2 |
| - |
a2 |
\end{align}
Arbitrage-Free Nelson-Siegel
One approach to affine term structure modeling is to enforce an arbitrage-free condition on the proposed model. In a series of papers,[2] [3] [4] a proposed dynamic yield curve model was developed using an arbitrage-free version of the famous Nelson-Siegel model,[5] which the authors label AFNS. To derive the AFNS model, the authors make several assumptions:
- There are three latent factors corresponding to the level, slope, and curvature of the yield curve
- The latent factors evolve according to multivariate Ornstein-Uhlenbeck processes. The particular specifications differ based on the measure being used:
dx=KP(\theta-x)dt+\SigmadWP
(Real-world measure
)
(Risk-neutral measure
)
- The volatility matrix
is diagonal
- The short rate is a function of the level and slope (
)
From the assumed model of the zero-coupon bond price:The yield at maturity
is given by:
And based on the listed assumptions, the set of ODEs that must be solved for a closed-form solution is given by:
where
\rho=\begin{pmatrix}1&1&0\end{pmatrix}T
and
is a diagonal matrix with entries
. Matching coefficients, we have the set of equations:
To find a tractable solution, the authors propose that
take the form:
Solving the set of coupled ODEs for the vector
, and letting
l{B}(\tau)=-{1\over{\tau}}B(\tau)
, we find that:
Then
reproduces the standard Nelson-Siegel yield curve model. The solution for the yield adjustment factor
l{A}(\tau)=-{1\over{\tau}}A(\tau)
is more complicated, found in Appendix B of the 2007 paper, but is necessary to enforce the arbitrage-free condition.
Average expected short rate
One quantity of interest that may be derived from the AFNS model is the average expected short rate (AESR), which is defined as:where
is the
conditional expectation of the short rate and
is the term premium associated with a bond of maturity
. To find the AESR, recall that the dynamics of the latent factors under the real-world measure
are:
The general solution of the multivariate Ornstein-Uhlenbeck process is:
Note that
is the
matrix exponential. From this solution, it is possible to explicitly compute the conditional expectation of the factors at time
as:
Noting that
, the general solution for the AESR may be found analytically:
References
- Duffie. Darrell. Kan. Rui. 1996. A Yield-Factor Model of Interest Rates. Mathematical Finance. en. 6. 4. 379–406. 10.1111/j.1467-9965.1996.tb00123.x. 1467-9965.
- Christensen. Jens H. E.. Diebold. Francis X.. Rudebusch. Glenn D.. 2011-09-01. The affine arbitrage-free class of Nelson–Siegel term structure models. Journal of Econometrics. Annals Issue on Forecasting. en. 164. 1. 4–20. 10.1016/j.jeconom.2011.02.011. 0304-4076.
- Christensen. Jens H. E.. Rudebusch. Glenn D.. 2012-11-01. The Response of Interest Rates to US and UK Quantitative Easing. The Economic Journal. en. 122. 564. F385–F414. 10.1111/j.1468-0297.2012.02554.x. 153927550 . 0013-0133.
- Christensen. Jens H. E.. Krogstrup. Signe. 2019-01-01. Transmission of Quantitative Easing: The Role of Central Bank Reserves. The Economic Journal. en. 129. 617. 249–272. 10.1111/ecoj.12600. 167553886 . 0013-0133.
- Nelson. Charles R.. Siegel. Andrew F.. 1987. Parsimonious Modeling of Yield Curves. The Journal of Business. 60. 4. 473–489. 10.1086/296409. 2352957. 0021-9398.
Further reading
- Book: Bjork, Tomas . Arbitrage Theory in Continuous Time, third edition. 2009 . . 978-0-19-957474-2.