Affine term structure model

An affine term structure model is a specific type of financial model which 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 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.