# Deriving solution to bond pricing equation

Consider the Vasicek model for the spot model:

$$ππ = (πΌ β πΎπ)ππ‘ + βπ½ππ$$

Suppose $$πΎ = 0.1, πΎ = 0.1$$, and the volatility of the process is 0.02. The spot rate is 10%.

Assume the form of solution to the BPE (Bond Pricing Equation) is $$Z(t, T; r) = e^{A(t, T) - rB(t,T)}$$ and derive equations for $$A$$ and $$B$$.
Solve the equation and obtain the form of $$Z(t, T; r)$$, hence price the zero coupon bond.

How do I go about this problem? Do I need to substitute Z into the BPE and get the partial derivatives from there?

• Yes. They are giving you a hint (or Ansatz) for the BPE solution, so you calculate its derivatives $\frac{\partial Z}{\partial t},\frac{\partial Z}{\partial r},\frac{\partial Z}{\partial r^2}$ and plug the Z and its 3 derivatives into the BPE to check that it works and find out what A(t,T) and B(t,T) actually are. Commented May 6 at 11:36
• That makes sense. Do you know of any resources that would help me navigate this problem? I did find this paper, but most of it went over my head.
– user72282
Commented May 6 at 11:53
• The only skill required is taking derivatives and manipulating the resulting expressions carefully. Only tools needed are plenty of blank paper and a pencil with eraser. Commented May 6 at 12:00
• Thanks. I'll give it a try.
– user72282
Commented May 6 at 12:03

It is very easy to do it by direct calculation; it is a bit easier for the dynamics

$$dr = a(b-r)dt + \sigma dW_t$$

1. Integrate it using the integrating multiple $$e^{at}$$ and moving $$-are^{at}dt$$ to the left side
2. Compute $$\int_0^T {r dt}$$, for which you will need to change order of integration, to end up with $$\int_0^T{ ... dW_t}$$. By Ito isometry, this will be a normal variable with known mean and variance.
3. Take expectation of the exponential of that normal variable

Everyone has to do this calculation once :) Good luck!

• Thanks for the clarification. Helps out a lot!
– user72282
Commented May 6 at 12:19