Bonds with embedded options callable bond pricing

The option-adjusted spread (OAS) is the most important measure of risk for bonds with embedded options. It is the average spread required over the yield curve in order to take into account the embedded option element. This is, therefore, the difference between the yield of a bond with embedded option and a government benchmark bond. The spread incorporates the future views of interest rates and it can be determined with an iterative procedure in which the market price obtained by the pricing model is equal to expected cash flow payments (coupons and principal). Also a Monte Carlo simulation may be implemented in order to generate an interest rate path. Note that the option-adjusted spread is influenced by the parameters implemented into the valuation model as the yield curve, but above all by the volatility level assumed. This is referred to volatility dependent. The higher the volatility, the lower the option-adjusted spread for a callable bond and the higher for a putable bond. 

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 ... 

Effective duration recognizes that yield changes may effect the future cash flow of a bond and so its price. For bonds with embedded options the difference between traditional duration and effective duration can be significant. The effective duration of a callable bond, for example, is sometimes half its traditional duration. As noted in chapter l4, for mortgage-backed securities, the difference is sometimes greater still. 

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. 

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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