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# Vasicek model

In the Vasicek model for the short rate, r(t) satisfies the stochastic differential equation

dr(t)=κ(θ-r(t))dt+σdW(t),

r(0)=r_{0},

which has the solution

r(t)=θ-e^{-κt}(θ-r_{0})+_{0}∫^{t}e^{-κ(t-s)}σdW(s).

The solution is also called an Ornstein-Uhlenbeck process. The Vasicek model is in the class of affine term structure models, which gives that the value of a ZCB is

p(t,T)=e^{A(t,T)-r(t)B(t,T)},
where A and B are given as

B(t,T)=(1-exp(-κ(T-t)))/κ,

A(t,T)=(B(t,T)-(T-t))(θ-1/2(σ/κ)^{2})-(σB(t,T))^{2}/(4κ).

dr(t)=κ(θ-r(t))dt+σdW(t),

r(0)=r

which has the solution

r(t)=θ-e

The solution is also called an Ornstein-Uhlenbeck process. The Vasicek model is in the class of affine term structure models, which gives that the value of a ZCB is

p(t,T)=e

B(t,T)=(1-exp(-κ(T-t)))/κ,

A(t,T)=(B(t,T)-(T-t))(θ-1/2(σ/κ)

Questions: Magnus Wiktorsson

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