Bahcall–Wolf cusp

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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 r(t) with dynamics

dr(t)=μ(t,r(t))dt+σ(t,r(t))dW(t)

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

p(t,T)=FT(t,r(t))

and F has the form

FT(t,r)=eA(t,T)B(t,T)r

where A and B 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 μ and σ which will result in an affine term structure. Assuming the bond has an affine term structure and F satisfies the term structure equation, we get

At(t,T)(1+Bt(t,T))rμ(t,r)B(t,T)+12σ2(t,r)B2(t,T)=0

The boundary value

FT(T,r)=1

implies

A(T,T)=0B(T,T)=0

Next, assume that μ and σ2 are affine in r:

μ(t,r)=α(t)r+β(t)σ(t,r)=γ(t)r+δ(t)

The differential equation then becomes

At(t,T)β(t)B(t,T)+12δ(t)B2(t,T)[1+Bt(t,T)+α(t)B(t,T)12γ(t)B2(t,T)]r=0

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

1+Bt(t,T)+α(t)B(t,T)12γ(t)B2(t,T)=0

Then the other term must vanish as well.

At(t,T)β(t)B(t,T)+12δ(t)B2(t,T)=0

Then, assuming μ and σ2 are affine in r, the model has an affine term structure where A and B satisfy the system of equations:

1+Bt(t,T)+α(t)B(t,T)12γ(t)B2(t,T)=0B(T,T)=0At(t,T)β(t)B(t,T)+12δ(t)B2(t,T)=0A(T,T)=0

Models with ATS

Vasicek

The Vasicek model dr=(bar)dt+σdW has an affine term structure where

p(t,T)=eA(t,T)B(t,T)r(T)B(t,T)=1a(1ea(Tt))A(t,T)=(B(t,T)T+t)(ab12σ2)a2σ2B2(t,T)4a

References

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