# Finding B(t) in the Vasicek model relating to the bond equation, more specifcally from the initial condition

In the Vasicek model for derving bond prices, we have the ODE $$\frac{dB}{dt}=\gamma B-1$$ which gives rise to the general solution $$B(t)=C_1 e^{\gamma t}+C_2$$My problem is that we have the "initial" condition $$B(T)=0$$, but apparently this one initial condition is sufficient for us to arrive at $$B(t)=\frac1 \gamma(1-e^{-\gamma (T-t)})$$and I cannot see how this one condition allows us to realise both unknown constants.

• Please check the equations. How the 1st equation follow from Vasicek model? Are you sure the 2nd equation satisfies the 1st? The bond price at maturity is face value, thus check the initial condition. Apr 20 '19 at 16:45

## 1 Answer

B(T,T)=0 implies the constant is zero. Here are the steps. Notice I have suppressed dependence of B on t and T in the beginning for clarity, and also ignore the dt in the 4th step: