Affine term structure model: Difference between revisions

Revision as of 02:54, 21 December 2012

Revision as of 20:10, 22 December 2012

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{{Multiple issues|{{expert|date=December 2012|reason=Confirmation, details on the Affine Term Structure Model.}}}}

An '''affine term structure model''' is a specific type of financial model whichthat 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 ==

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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'''.

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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\\

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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)\\

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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>

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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>

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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\\

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=== 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)}\\