## Glossary

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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)=r0,
which has the solution
r(t)=θ-e-κt(θ-r0)+0te-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)=eA(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).