Callable bond pricing embedded call option

Consider the following example. We assume to have two hypothetical bonds, a treasury bond and a callable bond. Both bonds have the same maturity of 5 years and pay semiannual coupons, respectively, of 2.4% and 5.5%. We perform a valuation in which we assume a credit spread of 300 basis points and an OAS spread of 400 basis points above the yield curve. Table 11.1 illustrates the prices of a treasury bond, conventional bond and callable bond. In particular, considering only the credit spread we find the price of a conventional bond or option-free bond. Its price is 106.81. To pricing a callable bond, we add the OAS spread over the risk-free yield curve. The price of this last bond is 99.02. We can now see that the OAS spread underlines the embedded call option of the callable bond. It is equal to 106.81-99.02, or 7.79. In Section 11.2.3, we will explain the pricing of a callable bond with the OAS methodology adopting a binomial tree. 

In this section, we illustrate the pricing of bonds with embedded options. The price of a callable bond is essentially formed by an option-free bond and an embedded option. In fact, it is given by the difference between the value of an option-free bond and a call option as follows ... 

As noted, a bond may contain an embedded option which permits the issuer to call or retire all or part of the issue before the maturity date. The bondholder, in effect, is the writer of the call option. From the bondholder s perspective, there are three disadvantages of the embedded call option. First, relative to bond that is option-free, the call option introduces uncertainty into the cash flow pattern. Second, since the issuer is more likely to call the bond when interest rates have fallen, if the bond is called, then the bondholder must reinvest the proceeds received at the lower interest rates. Third, a callable bond s upside potential is reduced because the bond price will not rise above the price at which the issuer can call the bond. Collectively, these three disadvantages are referred to as call risk. MBS and ABS that are securitized by loans where the borrower has the option to prepay are exposed to similar risks. This is called prepayment risk, which is discussed in Chapter 11. 

The difference between the price of the option-free bond and the callable bond at any time is the price of the embedded call option. The behavior of the option element depends on the terms of the callable issue. 

The value of a callable bond, and therefore of a call option, depends on the interest rate path. Thus, a callable bond has a lower price than the one of a conventional bond due to the embedded option. If the value of a call option increases, the value of a callable bond decreases and vice versa. This happens when interest rates are lower than the ones at issue. 

The pricing of the conventional bond is the same than the one exposed for callable bonds in Figure 11.9. Therefore, the option-free bond is always equal to 106.13. The main difference consists in the estimation of the embedded option (put option rather than call option) and pricing of the putable bond. Thus, we illustrate these two steps ... 

To calculate the modified duration of a bond with an embedded option, the bondholder must assume a fixed maturity date based on the bond s current price. When it is unclear what redemption date to use, modified duration may be calculated to both the first call date and the final maturity date. This is an unsatisfactory compromise, however, since neither date, and so neither measure, may be appropriate. The problem is more acute for bonds that are continuously callable or putable from the first call or put date until maturity. 

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