Applying the Terminal Condition in the Vasicek Bond Equation

Summary

The document asks how the Vasicek bond-pricing function can be determined from the condition that its value is zero at maturity. It gives a first-order differential equation and its general solution, which appears to contain two constants, then asks how the maturity condition leads to the familiar expression for the time-dependent coefficient. The key point is that the equation is solved with maturity as a fixed parameter: the boundary condition is applied at the terminal time, and the coefficient is expressed in terms of the time remaining until that date.

The supplied answer is only a brief pointer to another explanation and says that the terminal condition fixes a constant; it does not show the derivation in the document itself. As presented, the material is an incomplete question rather than a full worked solution. It provides no numerical example, model calibration, or discussion of assumptions, so readers seeking the complete algebra or broader bond-pricing context will need additional material.

Key ideas

• The Vasicek bond coefficient is specified by a differential equation with a terminal boundary condition.
• The time to maturity is fixed while solving for the coefficient as a function of the current time.
• The document does not include the detailed algebra needed to reconcile the general solution with the stated formula.

Tags

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


# 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.

## Answer by Magic is in the chain (score 0, accepted)

https://quant.stackexchange.com/a/45220

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:

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.