## The Binomial Interest Rate Tree

An issuer’s bonds can be valued with a binomial interest rate tree. In order to do this, the analyst will need to:

- Calculate the spot rate curve for the borrower based on that company’s most recently issued debt.
- Use the spot rate curve to calculate forward rates for the issuer.
- Applying forward rates when discounting cash flows is arbitrage free valuation.

- Develop a higher and lower interest rate scenario around each projected forward rate on the tree, based on an estimation of interest rate volatility.
- If necessary, adjust the tree so it properly values the currently issued bonds.
- If the tree leads to a bond value that is above the market price, then interest rates will need to be increased; if the tree leads to a bond value that is below the market price, then interest rates will need to be lowered.

- For bonds with embedded options, cash flows will need to be forecasted along all of the interest rate paths; these cash flows will need to be discounted to the present value to create a value for the bond.

## Valuing an Option Embedded Bonds

### Valuing a Callable Bond from a Binomial Interest Rate Tree

- When valuing a callable bond with a binomial interest rate tree, the analyst must assume that the bond will be called back by the issuer when the strike price is exceeded.
- At any nodes where the calculated bond price exceeds the call price, the calculated price is replaced with the call price.

### Valuing a Putable Bond from an Binomial Interest Rate Tree

- When valuing a putable bond with a binomial interest rate tree, the analyst must assume that the bond will be put back to the issuer when the price falls below the put floor.
- At any nodes where the calculated bond price is below the put price, the calculated price is replaced with the put price.

## The Value of an Embedded Option

**Price**

_{callable bond}= Price_{ncb}– Call_{option value}

The price of a callable bond is lower than that of a comparable bond that does not contain an embedded option.

**Price**

_{putable bond}= Price_{npb}+ Put_{option value }

The price of a putable bond is higher than that of a comparable bond that does not contain an embedded option.

- Put options increase the cost of the bond to the investor as they serve as an “insurance policy” that the value of the bond cannot drop below the floor established by the put.
- Call options reduce the cost of the bond to the investor as the call serves creates a ceiling for the borrower. By embedding a call option in its bond issuance, the borrower is preserving the ability to refinance its debt on more favorable terms, should interest rates drop significantly.
- The Effect of Interest Rate Volatility on the Value of Embedded Options
- As interest rate volatility increases, so do the value of embedded options.
- For example, in the case of a putable bond, the value of the embedded put goes up when interest rate volatility increases. This increases the purchase price of the putable bond to the investor.

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