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 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 (and potentially additional state variables).

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

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

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

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

The differential equation then becomes

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

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:

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

Models with ATS

Vasicek

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

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

References

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