# Callable bonds not path dependent?

1. I understand that bonds with options (say call options) do not experience refinancing burnout like MBSs, which means they are not as path dependent. But if a bond is called in say year 3, then it’s no longer outstanding which does effect future cash flows…so doesn’t any future cash flow thus depend on prior rates (and whether or not the bond was called) thereby making the value of a callable bond path dependent?

In this case, how is backward induction meaningful. That is, how can you go backwards from say year 5 if the bond is/would’ve been called in year 2? Any susequent cash flows wouldn’t exist bc the bond has already been called, no?

1. And is it that because binomial models cannot incorporate refinancing rates, this is why we use monte carlo models to value MBSs and other related securities.

2. Also, the values of MBSs (and related CMOs trenches, etc) are affected by prepayments bc it effects the amount of cash flows, the timing of cash flows as well as the interest rates used to value (discount) them. Is this correct?

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\$100 \$7

\$99.391 \$99.391 \$7 7.6559%

\$101.735 \$100 4.8755% \$7

\$100.966 \$100 \$7 5.9755%

\$100 \$7

1. If you look at the above tree. The Price at T1 has been calculated at U node through UU and UL. Whereas the price at L node at T1 has been calculated through LU and LL. The price if the bond has not been called is \$100.966. Since the bond is callable and it is quite sure that the call option buyer (issuer) will call the bond at \$100 at T1 that’s why the effective price of \$100 has been taken to calculate the price of bond at T0.

[(\$100 + \$99.391)/2 + \$7] / (1+4.8755%) = \$101.735

This implies that if the bond is called at T1 then the third node (T2) becomes irrelevant and backward induction is used at T1 to calculate the price of bond at T0 as the effective price is the call price (if any) that’s why it is used.

1. In MBS we use monte carlo simulations to have scenario analysis I think where we can maneuver multiple variables to judge the prepayment rates and infact discount rates etc.

2. I am not sure about the 3rd statement but I think for an investor investing in a tranch would consider all these variables. Judging the quality of servicer and/or the insurance of collateral (CDS) may also be part of valuing MBS.

Are you saying that if you have an interest rate tree with say 5 periods (years) and the bond is callable after the second year, and assuming it should be called at that time, that you would start the backward induction from the 3rd period and disregard the fourth and fifth periods ?? Or do you always start at the end and work backwards (in this case at node 5) ?

I guess im just confused how later periods are considered / should be factored in if the bond is called in a prior period. Just seems like backward induction, by working backwards from future years, ignores the actual progression of time. For example, say you have a bond that can be called at anytime after the second year and it’s called at year 3. If you have a 5 period model, and the 4th period also suggests that you should call the bond, then it seems that the bond would’ve already been called before you even get to the 4th year. But this is not the case since you start from the back…in effect you are ignoring how time actually unfolds, no? Seems like you should first figure the value of the bond at year 3 instead of working back from year 5?

if it is called at any period - you have the a different figure applying there… you would not take the calculated figure from the 2 future nodes there.

since bond is callable at par - when it is called the 100 would apply there… not the 101.xxx or whatever you calculated there.

They are not path dependent because the future path taken by the interest rates NO LONGER APPLIES now, unlike how it does if it were a non-callable bond.

Here the 100 (or whatever the bond is called at) always applies.

If the option is exercised in period two, how could it also be exercised in period 4 if it’s already been called? By going backwards, it seems like it ignores events that might preceed it. Doesn’t this mean the model value will reflect the exercise of multiple call options – perhaps that’s the point ?

in the binomial model with the tree - you start at the last node work your way backwards to arrive at the price. you then decide at the point of the time of the call option - whether the price calculated by the binomial tree will apply or would the call price apply.

you are evaluating at point in time. for that the future length of life of the bond does hold relevance. (And using the prices you are evaluating at which point, e.g. after year 2 does the bond become callable).

Remember - when a company issues a callable bond - it is doing so in the anticipation of a fall in rates, and so that they do not want the fall in rates to hurt them - because they will owe their bond holders a much higher price. That does not happen in a vacuum when the company originally prices the bond for issuance. The structure remains the same. What the current price at time 0 would be - is in anticipation of the bond getting called e.g. in period 2. If the bond were not called - its price at time 0 would be different (I forget if it would be higher or lower). With the call being exercised at some point - what happens is that the earlier path of the rates get snapped. That is what the statement means. (at least to my understanding).

The bond is stated as being callable ANY TIME AFTER YEAR xxx…

so on your binomial model - once it has been called e.g. in year 4 - that price would be used to trace backwards thro’ the tree, and the call price would still apply. because on some other tree path - the bond may not have as yet been callable. What the price at a node is depends on the price on the next node and the interest rate on the current node.

Thanks cpk123, one more thing -

In the CFAI book it gives the example (pg 278) that the market price of a 4-year 6.5% callable bond is \$102.218 yet the model at 10% volatility gives a value of \$102.899. My question: if the model is arbitrage free, how could the model give you anything other than the mkt price of \$102.218 ?? So how will the model help if it’s merely giving you the price the mkt is already telling you it will be?

Oh, and in your second last response, by “earlier path gets snapped” are you referring to the first values calculated (that is, the values at the end of the model)? “Earlier” is confusing bc backward induction starts at the end.

I’ve stared at this too long today. Thanks for your help!

earlier is backwards… you look at the bond from maturity backwards. And since you backward induction model works from back to front - an earlier calculated value - is what I meant to say.

as regards your first statement - model is using an estimated volatility. Volatility in the market vs. what is in the model may not be identical.

so in that example, we’re assuming that the correct volatility is 10%, and at that rate the value is \$102.899. So are we (the model used) saying that we think the correct volatility is 10% and if so, that the market has mispriced the bond, and there is an arbitrage opportunity??

Guess just a little confused bc i thought that we adjust the volatility until the model = market price, so clearly 10% would not be correct, right?

Or is the process of adjusting the volatility, when we’re basically calibrating the model against a treasury to make sure it lines up with the treasury mkt price. And then using those interest rates (based on the same model) to value other bonds such as an MBS ?

the volatility goes a long way in determining the upper and lower rates at any point. (This is not discussed in the curriculum. There used to be a snippet in Schweser - talking something about e^something related to volatility.

our model’s assumed volatility - is 10% - based on which our up and down rates are estimated. so our entire model in the book (which you are looking at) is an estimation. What the actual was - is not known. But given the difference - it is not way off.

Arbitrage opportunity - you are going way beyond - do not think that is ever the intent. The process here is to make you aware of the process used, that is all I would think.

(and your statement of adjusting volatility to get model = market price – not sure where you are getting it from. since the volatility - we know it exists, but how would you calculate it?)

I guess i was thinking that the model value should equal the market price. So in the example i gave, which is from the book, the model does not equal the market price. On page 271, it discusses constructing a binomial interest rate tree and the underlying principle is that the value from the model should equal the market price. Does this mean that the model is incorrect bc the market price does not equal our model value?

I guess im just not understanding why the model value would not equal the market price, in which case what good is the model if its merely confirming the the price that you already know (the market price) ?

SHOULD is the key word there.

model is doing a prediction.

does every prediction work the way it should. you hope it would. but there are many other factors… not all of which every model accounts for. That is where subjective things comes in.

In any case - I think we are digressing with regards to this…

Sorry, i don’t mean to digress. I guess i am just now wondering what the purpose of the binomial model is if it should simply give us the value which is equal to the market price. Do we need the model to tell us / confirm a price that we already know (ie the market price) ? So what are we getting out of it ?

doesn’t someone do some analysis, some due diligence before hand?

how does a sack of rice cost x\$ e.g.? Same principle applies here.

Someone arrives at a price. If it sells well, someone else goes tries to derive a model to understand what factors drive the price to be what it is… a market equilibrium price. The model price might work in some cases, might not in some … then someone goes builds a better model, and this process continues.

If you look at the tree which I have made there is a 50% probability that the interest rates will rise or fall. Based on those the price is calculated. Once the bond is on a price where it could be called the call price is used and the calculated is ignored. If ypu look at the tree at T1 and Lower node L The price is calculated using LU and LL but the price at T0 is calculated using the call price of L node and the calculated price of U node. Backward induction is used to calculate the price of a bond at T0 with a tentative interest rate tree which we think will be there.