Callable bond pricing putable bonds

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. 

To explain the pricing methodology, we suppose a putable bond with the same characteristics of the callable bond. The putable bond can be given back to the issuer with the following put schedule shown in Table 11.4. 

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 ... 

Determine the Value of an Embedded Put Option Conversely to a callable bond, the embedded option of a putable bond is a put option. Therefore, the value is estimated as the maximum between 0 and the difference between the strike price and bond s price. The strike price is defined according to the put schedule, while the bond s price is the value of the option-free bond at each node as shown in Figure 11.9. The value at maturity of a putable option if exercised is given by Formula (11.10) ... 

Determine the Value of a Putable Bond As exposed in Formula (11.4), the value of a putable bond is the sum of an option-free bond and an embedded put optimi. Therefore, conversely to a callable bond, the embedded option increases the value of the bond. When the option is deeply in the money, the bond matches the values defined in the put schedule. When the option has no value, option free and putable bonds have the same price. The value of our hypothetical putable bond is 106.13 + 0.33 or 106.45. This is illustrated in Figure 11.14. 

It is now possible to complete the price tree for the callable bond. Remember that the option in the case of a callable bond is held by the issuer. Its value, given by the tree in figure 11.11, must therefore be subtracted from the conventional bond price, given by the tree in figure 11.10, to obtain the callable bond value. For instance, the current price of the callable bond is 105.875 — 0.76, or 105.115. FIGURE 11.12 shows the tree that results from this process. A tree constructed in this way, which is programmable into a spreadsheet or as a front-end application, can be used to price either a callable or a putable bond. 

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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