Bahcall–Wolf cusp

2013-11-13T08:30:12Z

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{{expert-subject|date=December 2012|reason=Confirmation, details on the 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 <math>r(t)</math> with dynamics

: <math>
dr(t)=\mu(t,r(t)) \, dt + \sigma(t,r(t)) \, dW(t)
</math>

and a risk-free zero-coupon bond maturing at time <math>T</math> with price <math>p(t,T)</math> at time <math>t</math>. If

: <math>p(t,T)=F^T(t,r(t))</math>

and <math>F</math> has the form

: <math>F^T(t,r)=e^{A(t,T)-B(t,T)r}</math>

where <math>A</math> and <math>B</math> 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 <math>\mu</math> and <math>\sigma</math> which will result in an affine term structure. Assuming the bond has an affine term structure and <math>F</math> satisfies the [[term structure equation]], we get

: <math>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</math>

The boundary value

: <math>F^T(T,r)=1</math>

implies

: <math>
\begin{align}
A(T,T)&=0\\
B(T,T)&=0
\end{align}
</math>

Next, assume that <math>\mu</math> and <math>\sigma^2</math> are affine in <math>r</math>:

: <math>
\begin{align}
\mu(t,r)&=\alpha(t)r+\beta(t)\\
\sigma(t,r)&=\sqrt{\gamma(t)r+\delta(t)}
\end{align}
</math>

The differential equation then becomes

: <math>
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
</math>

Because this formula must hold for all <math>r</math>, <math>t</math>, <math>T</math>, the coefficient of <math>r</math> must equal zero.

: <math>
1+B_t(t,T)+\alpha(t)B(t,T)-\frac{1}{2}\gamma(t)B^2(t,T)=0
</math>

Then the other term must vanish as well.

: <math>
A_t(t,T)-\beta(t)B(t,T)+\frac{1}{2}\delta(t)B^2(t,T)=0
</math>

Then, assuming <math>\mu</math> and <math>\sigma^2</math> are affine in <math>r</math>, the model has an affine term structure where <math>A</math> and <math>B</math> satisfy the system of equations:

: <math>\begin{align}
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{align}</math>

== Models with ATS ==

=== Vasicek ===

The [[Vasicek model]] <math>dr=(b-ar)\,dt+\sigma \,dW</math> has an affine term structure where

: <math>
\begin{align}
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^2B^2(t,T)}{4a}
\end{align}
</math>

== References ==

*{{cite book | author=Bjork, Tomas | title=Arbitrage Theory in Continuous Time, third edition| year=2009 | publisher = New York, NY: [[Oxford University Press]] | isbn = 978-0-19-957474-2}}

[[Category:Finance theories]]
[[Category:Interest rates]]
[[Category:Mathematical finance]]
[[Category:Fixed income analysis]]
[[Category:Stochastic processes]]
[[Category:Short-rate models]]

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Bahcall–Wolf cusp