Affine term structure model

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 inverting the yield curve – the process of determining spot rate model inputs from observable bond market data.

Background

Start with a stochastic short rate model $r(t)$ 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 $T$ with price $p(t,T)$ at time $t$ . If

p(t,T)=F^T(t,r(t))

and $F$ has the form

F^T(t,r)=e^{A(t,T)-B(t,T)r}

where $A$ and $B$ are deterministic functions, then the short rate model is said to have an affine term structure.

Existence

Using Ito's formula we can determine the constraints on $\mu$ and $\sigma$ which will result in an affine term structure. Assuming the bond has an affine term structure and $F$ satisfies the term structure equation, we get

A_t(t,T)-(1+B_t(t,T))r-\mu(t,r)B(t,T)+\frac{1}{2}\sigma^2(t,r)B^2(t,T)=0

The boundary value

F^T(T,r)=1

implies

\begin{align} A(T,T)&=0\\ B(T,T)&=0 \end{align}

Next, assume that $\mu$ and $\sigma^2$ are affine in $r$ :

\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

A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)-\left[1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)\right]r=0

Because this formula must hold for all $r$ , $t$ , $T$ , the coefficient of $r$ must equal zero.

1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)=0

Then the other term must vanish as well.

A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)=0

Then, assuming $\mu$ and $\sigma^2$ are affine in $r$ , the model has an affine term structure where $A$ and $B$ satisfy the system of equations:

\begin{align} 1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)&=0\\ B(T,T)&=0\\ A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)&=0\\ A(T,T)&=0 \end{align}

Models with ATS

Vasicek

The Vasicek model $dr=(b-ar)\,dt+\sigma \,dW$ has an affine term structure where

\begin{align} p(t,T)&=e^{A(t,T)-B(t,T)r(T)}\\ B(t,T)&=\frac{1}{a}\left(1-e^{-a(T-t)}\right)\\ A(t,T)&=\frac{(B(t,T)-T+t)(ab-\frac{1}{2}\sigma^2)}{a^2}-\frac{\sigma^2B^2(t,T)}{4a} \end{align}

References

Bjork, Tomas (2009). Arbitrage Theory in Continuous Time, third edition. New York, NY: Oxford University Press. ISBN 978-0-19-957474-2.