I’m really confused about the OAS. In L1 it seemed to make perfect sense, now I just don’t get it.

What I don’t get is that there seems to be conflicting definitions about what OAS actually is.

Definition 1: You have a callable bond with say a Z-spread of 40bps over a risk free curve benchmark. My understanding of their explanation is that the OAS represents the Z-spread of that same bond if the call option wasn’t there. So the purpose of the OAS is to normalize a set of bonds you might be looking at that might have different call strike prices.

Definition 2: This is where I get confused. In the chapter on valuing bonds with options, the OAS becomes a spread that you need to add to your interest rate tree in order to get from your theoretical valuation of the callable bond to the price that matches the market price of the bond. So if your model tells you the callable bond is worth $101 and the market is selling at $102, the OAS is the spread you need to add to all your rates so that your calculated price matches the market.

Can somebody explain to me how these two definition of OAS are the same? I get both of them on their own but I can’t undertand how they’re somehow the same thing.

okay. I will attempt to provide my take on it. I just read this a few days ago, and was confused as well.

In general, the OAS is a spread measure that can be used to account for the difference between the theoretical value and the actual price of a bond (to ensure arbitrage free condition).

As per definition one, the OAS is the spread over some benchmark with the option removed. Assuming a bond with a call option, OAS=Z-spread-cost of call. And Z-spread is the spread over the entire spot rate curve that makes the theoretical value of the bond (PV of the bond`s CF) equal to its market price.

In definition two, when you are valuing a bond with a call option using an interest rate tree, you are adjusting for the call option at each node. However, this may not give you an arbitrage free value. So the OAS is the spread that you add to your rates to make your theoretical value equal to your market price.

That`s my understanding of it, but I might be completely off… would appreciate addtional clarification.

This is in line with your Def 1: OAS = spread - spread due to optionallity

This is given: Bond yield = Benchmark yield + spread

–> Bond yield = Benchmark yield +OAS +spread due to optionallity

So for your Def 2:

You observe a price of 102 this is equal to the Bond yield. For this to hold true, your model must then from your price of 101 give: benchmark yield (which should be observable) + spread due to optionallity. The OAS part would then be the yield that you add on top of that to change the value from 101 to 102.

I haven’t read this material yet so I don’t know if I missed something important. You talk about a valuation tree, I don’t know what is meant by that (yet).

The reason that you get confused is actually pretty simple: nobody . . . nobody . . . ever explains the connection of OAS and a binomial interest rate tree correctly, and nobody . . . nobody . . . ever draws the z-spread vs. OAS curves correctly. It turns out that it’s pretty simple.

You’ve been told that OAS is the spread for the bond with the option removed (makes sense: option-adjusted spread: the spread’s been adjusted for the option), but when you calculate it using a binomial interest rate tree you’re told to add the spread to each node, then figure out whether the option would be exercised at each node, then discount and sum and fiddle with the spread until you get the market price and voilà! you have your OAS.

And you’re saying to yourself: how can that be a spread with the effect of the option removed when I explicitly used the option in the tree, deciding when to exercise it and when not to?

The answer lies at the left side of the equation: the market price itself includes the effect of the option. By explicitly including the effect of the option on the right side of the equation (the value you get from the tree), you have neutralized - eliminated - the effect of the option.

As for drawing the curves, the problem there is that the Z-spread is calculated by adding a constant spread to the (zero-volatility) Treasury spot curve, assuming that the option is never exercised, and fiddling with the spread until the value comes out to the market price, whereas, as we’ve seen, the OAS is calculated by adding a constant spread to each node in a (nonzero-volatility) binomial interest rate tree. The problem with the drawing is that they draw the OAS curve as if the OAS were added to the (zero-volatility) Treasury spot curve; it isn’t. (If you tried adding the OAS to the Treasury spot rates and discounting the cash flows using those rates, you wouldn’t get the market price of the bond.)

I think I’ve reached the lightbulb moment and it’s clicked in now.

To recap, here is my understanding of what’s been said and what I’ve found digging a bit deaper.

It all has to done with the building of the IR tree. You build the tree using your benchmark on the run issues and your assumed volatility. Now, using that tree as your reference you value the callable bond you see in the market. Since you are valuing it with your benchmark rates though the price you calculate using your benchmark tree does not equal the price you see in the market. The reason for this is because the bond trades at a spread relative to your benchmark, this is the OAS spread. So all you need to do now is shift all the rates in your benchmark tree until the theoretical (i.e. using the IR tree calculation) matches what you see in the market. That bump you’ve put into on top of you benchmark rates is the OAS spread.

Thanks everyone!

Now on to figuring out the next piece of the CFA L2 thousand-piece puzzle.