Pricing a callable bond option

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 follows:

Price of callable bond = price of bond with no option − price of call option

Pricing a zero-coupon bond by the Vasicek model

The value of a zero-coupon bond with a par value of 1 at time t and a prevailing interest rate, r, is defined as follows:

Since the interest rate, r, is always changing, we rewrite the zero-coupon bond as follows:

Now, the interest rate, r, is a stochastic process that accounts for the price of the bond from time t to T, where T is the...