Binominal option pricing - When to discount to coupon and when not?

CFA Level II
#1

I understand the logic behind the binominal option pricing model, but there seems to me to be slight differences in the calculation. Some add the coupon without discounting, and in other cases we add the coupon but also discount it. I provide two examples from the book below:

  1. On page 288-289 example 3 we price a three-year, annual-pay bond with a coupon rate of 5%. At time year 2 the value at the various nodes is 0.5 × [(105/1.08 + 105/1.08)] + 5 = 102.2222. –> The coupon is added but not discounted.

  2. Then on page 330 exhibit 12 there is a Three-Year 4.25% Annual Coupon Bond Callable at Par One Year and Two Years from Now at 10% Interest Rate Volatility. So this bond has call option. In year one the value is 0,5×(99.738+ 4.25 /1.031681)+(100+ 4.25 )/1.031681) = 100.922. –> The coupon is added and discounted.

The difference is that with a normal bond we seem to add a coupon each year (the coupon that is paid), but this coupon is not discounted. With the embedded option we add the coupon, but we also discount it. Any reason for this difference?

#2

All bonds are treated the same way.

Suppose that you have a 4-year bond. Using a binomial interest rate tree you will discount the principal plus the final coupon from year 4 to year 3 using the year 3 forward rates in the tree; that gives you the present value (as of year 3) of the future cash flows.

You then discount that present value plus the coupon (payable at year 3) from year 3 to year 2 using the year 2 forward rates in the tree; that gives you the present value (as of year 2) of the future cash flows.

You then discount that present value plus the coupon (payable at year 2) from year 2 to year 2 using the year 1 forward rates in the tree; that gives you the present value (as of year 1) of the future cash flows.

Finally, you discount that present value plus the coupon (payable at year 1) from year 1 to today using today’s forward rate in the tree; that gives you the value of the bond today.

This is true for _ any _ bond. Fixed-rate, floating-rate, callable, putable, prepayable, whatever. _ Any bond _. Period.

If the bond has embedded options, you may change the present value of the future cash flows at any given node (if the option were exercised), but the methodology doesn’t change.

I wrote a series of articles on binomial interest rate trees that may be of some help, starting here: http://www.financialexamhelp123.com/binomial-trees-for-fixed-income/

Full disclosure: as of 4/25 I’ve installed the subscription software on my website, so there’s a charge for viewing the articles.

#3

Actually, what i observed is that, when you are dealing with a straight bond, you add the coupon to the discounted cash flow to arrive at the value at the current node.

However, for a bond with option, you ignore the coupon in computing the value at each node, because if you add the coupon, it will undervalue the put option and overvalue the call option.

SO just follow those rules, and you will be fine.

#4

Yes: the coupon is paid at each node irrespective of whether the option is exercised or not. So the decision to exercise is based on the present value of the future cash flows (as I computed above), but the value at the node is the value of the bond (possibly adjusted because of an option) plus the value of the coupon.

#5

I agree with everything you write, but if you look at the two examples I provided there is a difference in the calculation. Why is that so? Anyway, it seems to me you will get the same value in year 0 regardless of what approach you use.

#6

It the SAME METHODOLOGY!!!

In your example: 105 is the next nodes CFs. You discount them back and then add the coupon.

In the embedded option example: 99.738+4.25 if the value of the next nodes CF. So is 100+4.25. The 100 just comes from the fact that in the next node the option was assumed to be exercised. You discount these back to the current node and then take the average.

With bonds without embedded options there is no need to adjust for put/call rules - you just discount the next nodes CF’s take the average then add the coupon.

For bonds with embedded options you MUST ADJUST for put/call rules thus the PV of the next nodes CF is adjusted for a call rule Max(Call Price, PV of Next Node’s CF) or a put rule Max(Put Price, PV of Next Node’s CF). Then take the average and add the coupon.

We have gotten really bogged down on this topic. Follow the above steps and you will never be lost. I swear.

“This is true for _ any _ bond. Fixed-rate, floating-rate, callable, putable, prepayable, whatever. _ Any bond _. Period.” - BINGO! ANY BOND!!! ANY BOND!!!

#7

LOL! I have started this query 2-3 months ago… cool Maybe the guy who wrote this part of curriculum should ask himself is he able to explain what he meant to wider auditory. We are not all skilled mathematicians as well as we are all not accountants.

#8

Hi there, could you explain how if you add the coupon it will undervalue the put option and overvalue the call option?

Thanks,

#9

First: are you really a Level I candidate? (That’s what it says under your username.)

Second, I don’t understand your question. Can you give an example of what you’re thinking?

#10

Thanks Magician. I’m L2 candidate.

My question is twofold.

1) Why does the curriculum use two different formula to price the bond at each node in the binomial tree? This is the same question I have as the OP but this thread still doesn’t quite clear this up for me.

One formula discounts both the bond value and the coupon in the next node to arrive at the prior node, BUT does not add the value of the coupon in the current node to get the bond price. This is the formula that’s used in pricing bonds with embedded options.

The other formula discounts both the bond value and the coupon in the next node to arrive at the prior node, AND also add the value of the coupon in the current node to get the bond price. This formula is used in pricing option-free bonds.

The text does not explicitly state when to use which formula. So my question is which formula should be applied to which scenario?

2) From the third comment in this thread (copied below) I seem to be getting an answer that would help me clear this confusion but I’m not 100% sure. How does adding the coupon from the current node overvalue the call option and undervalue the put option?

"Actually, what i observed is that, when you are dealing with a straight bond, you add the coupon to the discounted cash flow to arrive at the value at the current node.

However, for a bond with option, you ignore the coupon in computing the value at each node, because if you add the coupon, it will undervalue the put option and overvalue the call option.

SO just follow those rules, and you will be fine."

Is the reason for not adding the coupon in the current node to the bond value for a bond with embedded option because adding the coupon will inflate the bond’s price and, therefore, make the call option more attractive (from the issuer perspective) thereby overvaluing the call option? The opposite goes for a bond with embedded put.