Valuing Bonds With Embedded Options, Part 1
What you'll learn
- LOS 25.cDescribe how the arbitrage free framework can be used to value a bond with embedded options.
- LOS 25.fCalculate the value of a callable or putable bond from an interest rate tree.
The basic process for valuing a callable (or putable) bond is similar to the process for valuing a straight bond. However, instead of using spot rates, one period forward rates are used in a binomial tree framework. This change in methodology computes the value of the bond at different points in time; these checks are necessary to determine whether the embedded option is in the money (and exercised).
When valuing a callable bond, the value at any node where the bond is callable must be either the price at which the issuer will call the bond (i.e., the call price) or the computed value if the bond is not called, whichever is lower. This is known as the call rule. Similarly, for a putable bond, the value used at any node corresponding to a put date must be either the price at which the investor will put the bond (i.e., the put price) or the computed value if the bond is not put, whichever is higher.
This is known as the put rule.
Consider a two year, 7% annual pay, $100 par bond callable in one year at $100. Also consider a two year, 7% annual pay, $100 par bond putable in one year at $100.
The interest rate tree at 15% assumed volatility is as follows:
Value the embedded call and put options.
Solution
Value of the straight (option free) bond:
Consider the value of the bond at the upper node for Period 1, :
Similarly, the value of the bond at the lower node for Period 1, , is:
Now calculate , the current value of the bond at Node 0.
The completed binomial tree is shown as follows:
Valuing a Two Year, 7.0% Coupon, Option Free Bond
Value of the callable bond:
The call rule (call the bond if the price exceeds $100) is reflected in the boxes in the completed binomial tree, where the second line of the boxes at the one year node is the lower of the call price or the computed value. For example, the value of the bond in one year at the lower node is $101.594. However, in this case, the bond will be called, and the investor will only receive $100.
Therefore, for valuation purposes, the value of the bond in one year at this node is $100.
The calculation for the current value of the bond at Node 0 (today), assuming the simplified call rules of this example, is:
The completed binomial tree is shown as follows:
Valuing a Two Year, 7.0% Coupon, Callable Bond, Callable in One Year at 100
Value of the putable bond:
Similarly, for a putable bond, the put rule is to put the bond if the value falls below $100. The put option would therefore be exercised at the upper node in Year 1 and hence the $99.830 computed value is replaced by the exercise price of $100.
Value of the embedded options: