Valuation of Risky Callable and Putable Bonds

Bonds have risk. At least, a lot of them do. If you're working with something that isn't default-free, you'll need to account for this. There are a couple ways to do it, and if you have a spot curve specifically for the issuer, great. Just add a uniform change to each rate, which is called the ___Z_-spread__, and you'll have your lower price to account for risk.

It's like this. $$\displaystyle V_0 = \frac{5}{1 + 0.0230 + 0.005} + \frac{5}{(1 + 0.0251 + 0.005)^2} + \frac{105}{(1 + 0.0262 + 0.005)^3} $$ What is the _Z_-spread here?

No. Remember that the _Z_-spread is a uniform value added to the spot rates. This describes the spot rates but not the _Z_-spread.

You're right! The uniform value added to each discount rate here is the value of 0.005, which is 0.5%, or 50 basis points. That's the _Z_-spread.

No. The _Z_-spread is larger than that.

Most of the time you won't have this information, and you'll need to use market spot curves. That's fine. Then instead of adding a uniform rate to the issuer's spot rates, you'll need to model risk with an interest rate tree based on the market rates, and then just add the uniform values to each rate there. That's called an __option-adjusted spread (OAS)__. How might you relate this to the _Z_-spread?

No. This doesn't have to be the case. But the OAS would equal the _Z_-spread with zero volatility.

Exactly!

Remember that a volatility input of 0% just turns the interest rate tree into a single interest rate branch, which represents forward rates. These can be used to value a bond as well, so the OAS would be the _Z_-spread in this case.

As you add a small _Z_-spread to a tree, all forward rates go up. This means that all discounted values come down. Once they reach the market price of the bond, you have your OAS for that bond. How would this affect the number of call adjustments you would need to make on a callable bond, as you're performing backward induction with your new OAS-adjusted tree?

You got it. Call adjustments are made when prices come in above the call price. Adding an OAS to the interest rate tree raises all forward rates, lowering all prices. This will likely lead to fewer call adjustments.

No. The number of call adjustments couldn't increase from the inclusion of an OAS. Consider what this does for rates and node values in the tree.

No. Actually, changes will be quite possible. The OAS changes all rates, which then affects the valuation at each node.

Also, it might help to remember that "volatility is volatility." When you have a bond, the risk of the bond has to be modeled somewhere to allow the price to fall to market value. So if you model an interest rate standard deviation of 10% in the interest rate tree, then a certain OAS will be needed to find the market price. If 15% is modeled in the tree, then there's more volatility in the tree itself. What do you expect will happen to the OAS needed?

Absolutely.

No. It will be smaller, actually.

More volatility modeled in the tree means a smaller adjustment will be needed for the OAS. Higher interest rate volatility relates to a lower OAS, and this is the logic behind that relationship.

To summarize: [[summary]]

Five basis points

50 basis points

From 2.30% to 2.62%

The OAS must be higher than the _Z_-spread

The OAS would equal the _Z_-spread with zero volatility

No change possible

Possibly more adjustments

Possibly fewer adjustments

It will be bigger

It will be smaller

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