# 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 ${\displaystyle r(t)}$ 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}}}
{\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

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