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Introduction to Pricing Approach

A callable bond is defined as a debt security that gives the issuer the right but not the obligation to repay the face value of the security at a given price at specified dates prior to maturity. In many cases the call privilege can only be exercised on coupon anniversary dates, a 'lock-out' period may apply to the first few years after issue that prevents a call, and the redemption price may decline on the call dates closer to bond maturity. With these characteristics, the embedded bond option is seen to be Bermudan style.

The callable bond holder has sold a call option to the bond issuer that gives the issuer protection against decreases in market interest rates. For example, assume that an issuer sells 10-year bonds with a 10% annual coupon, but with the right to call the issue at par on the coupon payment date in year five, and on every subsequent coupon date prior to maturity. If interest rates have declined to, say, 6% on the fist callable date, then the value of the bond will exceed the par value and the issuer's call option will be well in-the-money. That means that the issuer can effectively retire the expensive debt costing 10% and re-issue bonds at the prevailing rate of 6%.

The holder of a callable bond is effectively long an equivalent conventional bullet bond and short a call option on the same bond. That is, the value of a callable bond can be decomposed into 2 parts:

 

Equation Template

where,

the time 0 price of a callable bond that matures at time

the time 0 price of an equivalent vanilla bond that matures at time

the time 0 value of a call option written on the vanilla bond

Puttable bonds are similar to callable bonds in many ways, except they give the bond holder the right but not the obligation to 'put' (sell) the bond back to the issuer at a given price at specified dates prior to maturity. For these bonds, the embedded option is held by the investor and this means that the bond value will exceed that for an otherwise equivalent bond with no put provision. Like callable bonds, the market value of a puttable bond can be decomposed into two parts:

 

Equation Template

where,

 

the time 0 price of a puttable bond that matures at time

the time 0 price of an equivalent vanilla bond that matures at time

the time 0 price of a put option written on the vanilla bond

On the face of it, the simplest way to value callable bonds would be to deal with each of the components in th above equations separately. The value of the vanilla bond is straight forward and can be derived from the appropriate implementation of the general ISMA formula, while the put or call option value can be independently calculated from one of the supported term structure models.

However, an equivalent, internally consistent approach would be to directly value the callable or puttable bond within the selected term structure model. Recall that all of the models are constructed to fit the existing term structure of zero-coupon rates by definition, and if the model is carefully constructed it should therefore be possible to recover the value of the bond directly from the interest rate tree. The impact of the rights and obligations that flow from the embedded option can also be directly applied at the appropriate nodes in the tree and the value of the option thereby automatically reflected in the bond value that is derived from the term structure model.

Looking at the problem in another way, the constructed interest rate tree for all of the various models can be used to value both a bond without the embedded options (the vanilla equivalent bond) and a bond that includes the embedded option (the callable or puttable bonds themselves). The value of the embedded option can then be simply determined as the difference between the derived vanilla bond value and callable / puttable bond value. That is:

 

Equation Template

Equation Template

This deconstruction of the callable and puttable bond values also clearly demonstrates the indirect valuation approach that can be used to value the call and put options themselves.

 

See Also

Term Structure Models

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