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 deriving the yield curve – the process of determining spot rate model inputs from observable bond market data. The affine class of term structure models implies the convenient form that log bond prices are linear functions of the spot rate (and potentially additional state variables). 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 deriving the yield curve – the process of determining spot rate model inputs from observable bond market data. The affine class of term structure models implies the convenient form that log bond prices are linear functions of the spot rate (and potentially additional state variables). Start with a stochastic short rate model r ( t ) {displaystyle r(t)} with dynamics and a risk-free zero-coupon bond maturing at time T {displaystyle T} with price p ( t , T ) {displaystyle p(t,T)} at time t {displaystyle t} . If and F {displaystyle F} has the form where A {displaystyle A} and B {displaystyle B} are deterministic functions, then the short rate model is said to have an affine term structure. Using Ito's formula we can determine the constraints on μ {displaystyle mu } and σ {displaystyle sigma } which will result in an affine term structure. Assuming the bond has an affine term structure and F {displaystyle F} satisfies the term structure equation, we get