Bahcall–Wolf cusp

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

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

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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