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 ${\displaystyle r(t)}$  with dynamics

${\displaystyle dr(t)=\mu (t,r(t))\,dt+\sigma (t,r(t))\,dW(t)}$ 

and a risk-free zero-coupon bond maturing at time ${\displaystyle T}$  with price ${\displaystyle p(t,T)}$  at time ${\displaystyle t}$ . If

${\displaystyle p(t,T)=F^{T}(t,r(t))}$ 

and ${\displaystyle F}$  has the form

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

where ${\displaystyle A}$  and ${\displaystyle B}$  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 ${\displaystyle \mu }$  and ${\displaystyle \sigma }$  which will result in an affine term structure. Assuming the bond has an affine term structure and ${\displaystyle F}$  satisfies the term structure equation, we get

${\displaystyle A_{t}(t,T)-(1+B_{t}(t,T))r-\mu (t,r)B(t,T)+{\frac {1}{2}}\sigma ^{2}(t,r)B^{2}(t,T)=0}$ 

The boundary value

${\displaystyle F^{T}(T,r)=1}$ 

implies

${\displaystyle {\begin{aligned}A(T,T)&=0\\B(T,T)&=0\end{aligned}}}$ 

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

${\displaystyle {\begin{aligned}\mu (t,r)&=\alpha (t)r+\beta (t)\\\sigma (t,r)&={\sqrt {\gamma (t)r+\delta (t)}}\end{aligned}}}$ 

The differential equation then becomes

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

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

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

Then the other term must vanish as well.

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

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

${\displaystyle {\begin{aligned}1+B_{t}(t,T)+\alpha (t)B(t,T)-{\frac {1}{2}}\gamma (t)B^{2}(t,T)&=0\\B(T,T)&=0\\A_{t}(t,T)-\beta (t)B(t,T)+{\frac {1}{2}}\delta (t)B^{2}(t,T)&=0\\A(T,T)&=0\end{aligned}}}$ 

Models with ATS

Vasicek

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

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

References

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