In this section, we will take a look at pricing a callable bond. We assume that the bond to be priced is a zero-coupon paying bond with an embedded European call option. The price of a callable bond can be thought of as:
The value of a zero-coupon bond with a par value of 1 at time 
and prevailing interest rate 
is defined as:
Since the interest rate is always changing, we will rewrite the zero-coupon bond as:
Now, the interest rate is a stochastic process that accounts for the price of the bond from time t to T, where T is the time to maturity of the zero-coupon bond.
To model the interest rate we can use one of the short rate models as discussed in this chapter as a stochastic process. For this purpose, we will use the Vasicek model to model the short rate process.
The expectation of a log-normally distributed variable 
is given by:
Taking moments of the log-normally distributed variable X:
