Template:Expert-subject
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
with dynamics
![{\displaystyle dr(t)=\mu (t,r(t))\,dt+\sigma (t,r(t))\,dW(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5abc96eb3f53af21d0d36fc8b50904da808fc3eb)
and a risk-free zero-coupon bond maturing at time
with price
at time
. If
![{\displaystyle p(t,T)=F^{T}(t,r(t))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b02d499edb6cbf47674ad7ad255598e06ad59fe5)
and
has the form
![{\displaystyle F^{T}(t,r)=e^{A(t,T)-B(t,T)r}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12d563df9cebc328f16589a04d7b7cbe021c35b)
where
and
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
and
which will result in an affine term structure. Assuming the bond has an affine term structure and
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](https://wikimedia.org/api/rest_v1/media/math/render/svg/47400552fa98a5c0c5f3a283865b3b41427dd6df)
The boundary value
![{\displaystyle F^{T}(T,r)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34cd2fcd4607d368ab099efb7150e9bd17df2143)
implies
![{\displaystyle {\begin{aligned}A(T,T)&=0\\B(T,T)&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37437ad5d385a412f0f60140816229cc6bdd9913)
Next, assume that
and
are affine in
:
![{\displaystyle {\begin{aligned}\mu (t,r)&=\alpha (t)r+\beta (t)\\\sigma (t,r)&={\sqrt {\gamma (t)r+\delta (t)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30b78fca5d8d0ed29755a3b57fdf77fdb6cdc93c)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a473073de2594439fc0529c6b8cf3013b9a13f2a)
Because this formula must hold for all
,
,
, the coefficient of
must equal zero.
![1+B_{t}(t,T)+\alpha (t)B(t,T)-{\frac {1}{2}}\gamma (t)B^{2}(t,T)=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86a16a161b7002b0a13378eb1f69025a147b890)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c810ba1ad81213abf08ed8ae85a3eda5ddff4d9)
Then, assuming
and
are affine in
, the model has an affine term structure where
and
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01529d7379e4ef0ad94307b648641e0c9eb41efb)
Models with ATS
Vasicek
The Vasicek model
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80f54418f69f161361506335b37ace408bc9e6be)
References
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