Valuation of Callable and Putable Bonds with Interest Rate Volatility
For a bond with spot rates (and therefore forward rates) that are assumed fixed, valuation of a callable or putable bond involves discounting the final cash flows back to the present, and then adjusting any price that suggests exercise to the exercise price, and recalculating.
But when there's interest rate volatility, that needs to be incorporated into the calculation. If the standard deviation of interest rate values rises from 0% to 15%, how might you use this in your valuation estimate?
No.
That's essentially an assumption of zero volatility, when single rates are used for each period. But an interest rate tree will work.
Yes!
Using average forward rates reflects an assumption of zero volatility, when single rates are used for each period. But an interest rate tree will work.
Here's an example: A three-year, Bermudan-style callable bond with 6% coupons. The spot rates for today through Year 2 are 4.0%, 4.6%, and 5.2%. This, along with a 15% volatility measure, translates into the following binomial (and lognormal) tree:
Again, these values are derived implicitly through calibration to the spot rate curve. There's no way to explicitly calculate them from the rates given.
Through backward induction, you can first calculate the option-free values at each node, working back to today's value, and adjusting where needed for the embedded option. Start with the three Year 2 values. What's the largest value you find?
No. Note that you shouldn't include the Year 2 coupon here.
That's right.
Just discount the ending cash flows of 106 at each of the three Year 2 forward rates, and you get
$$\displaystyle \frac{106}{1 + 0.084936} = 97.7017$$
$$\displaystyle \frac{106}{1 + 0.062922} = 99.7251$$
$$\displaystyle \frac{106}{1 + 0.046614} = 101.2790 $$.
Not quite.
That's one of the three, but it's the smallest value.
If the bond is callable with an exercise price of 100, what adjustments would you make at this point?
That's not right. Some adjusting will be required, since not all of the prices are on the same side of 100.
Not exactly.
That would be the case for a putable bond with an exercise price of 100.
That's right!
Over 100, and the bond will be called. Adjust that large price down to 100, and continue the backward induction. In Year 1, the two average discounted prices (with coupons) are
$$\displaystyle 0.5 \times [\frac{97.7017 + 6}{1 + 0.059847} + \frac{99.7251 + 6}{1 + 0.059847}] = 98.8004 $$
and
$$\displaystyle 0.5 \times [\frac{99.7251 + 6}{1 + 0.044336} + \frac{100 + 6}{1 + 0.044336}] = 101.3683 $$.
After making one more adjustment down to 100 for the call in Year 1, the final current value is calculated as
$$\displaystyle V_0 = 0.5 \times [\frac{98.8004 + 6}{1 + 0.040000} + \frac{100 + 6}{1 + 0.040000}] = 101.3464 $$.
Suppose you went through these same numbers again, assuming that the bond was putable instead of callable. The prices would adjust in the other direction toward 100 to account for the put. What effect would that have on the final current price of the bond?
Absolutely.
No, it would be higher.
Nothing new here; a putable bond will always have a higher value than a callable bond with the same features. Higher prices at later nodes will discount to a higher present value.
To illustrate, consider the same Year 2 values, this time adjusting two of them up to 100 for the Year 1 calculations.
$$\displaystyle 0.5 \times [\frac{100 + 6}{1 + 0.059847} + \frac{100 + 6}{1 + 0.059847}] = 100.0144 $$
$$\displaystyle 0.5 \times [\frac{100 + 6}{1 + 0.044336} + \frac{101.2790 + 6}{1 + 0.044336}] = 102.1123 $$
What does this result tell you about the bond potentially being put?
Probably not.
This is a possibility, but consider what an interest rate tree is really showing you.
Actually, that's not right.
Notice that both of the Year 1 values are in excess of 100. The bond will not be put in Year 1.
Exactly!
Nothing is certain. Even assuming that the modeling is accurate, the prices in Year 1 are both over 100, so it will survive that year. Adjustments were made in the Year 2 calculations for two of the nodes, but not the third. If interest rates follow a central or high path, then the bond will be put. But if it follows the "LL" path down to 4.6614%, then prices will remain sufficiently high, and the bond won't be put. In expectation, however, the bond is more likely to be put in Year 2.
And just to complete the example, today's price of the putable bond would be modeled as
$$\displaystyle V_0 = 0.5 \times [\frac{100.0144 + 6}{1 + 0.040000} + \frac{102.1123 + 6}{1 + 0.040000}] = 102.9455 $$.
A higher price for this greater investor value. Valid.
To sum up:
[[summary]]
Use an interest rate tree
Use average forward rates
97.70
101.28
107.28
None
Adjust two up to 100
Adjust one down to 100
It would be lower
It would be higher
It may not be put
It will be put in Year 1
It will be put in Year 2
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