OAS and Z spread

Hi all,

I’m having trouble grasping the z spread and OAS spreads clearly. It seems that a z-spread > OAS means that the issuer has a call option. My confusion is that wouldn’t the lower OAS mean that the bond is discounted at a lower rate, thus a higher present value than without the call option. Why would somebody be willing to pay more when there is a call option?

Thanks in advance and let me know if my explanation needs any clarification.

You’re mixing up the spreads: the Z-spread applies to the value of the bond with the option included; the OAS applies to the value of the bond with the option removed. Thus, a higher Z-spread than OAS would say that the price of a callable bond is lower than the price of an identical, noncallable bond.

OAS, as you know, abbreviates _ O ption- A _djusted _ S _pread. The “option-adjusted” part of that phrase means “adjusted for the value of the option”; i.e., with the value of the option removed: it’s the spread that applies to a bond without the option in question.

I just wanna say this question is one of the most confusing in curriculum at all.

Is there any mnemonics to remember all this?

If you want merely a mnemonic, perhaps you could use something like this:

  • Callable bond: _ lower _ price than a noncallable, _ lower _ OAS than Z-spread
  • Putable bond: _ higher _ price than a nonputable, _ higher _ OAS than Z-spread

However, I, for one, would prefer understanding why this is true, rather than simply memorizing it. Furthermore, the understanding isn’t all that difficult.

See my post above for why it’s true.

Price Suggestion to Mnemonic

Callable lower compared to noncallable CLeaN

Putable lower compared to nonputable PLaiN

Suggestion to Mnemonic

Callable lower OAS than Z-spread CLOsed

Putable lower OAS than Z-spread PLOsed

Magician, it makes sense.

Putable bond is more expensive because it gives the bond holder the right to sell, thus an advantage to the bond holder, thus a higher OAS than Z Spread. If we remove this option, the bond holder does not have this benefit, thus the price of the bond drops and yields go up (higher OAS). If we include this option, the spread is lower (price is higher) because the option is an advantage to the bond holder.

Good to hear.


You’re a saviour. Couldnt have understood this better. The mnemonics is the icing. Thank you.

You’re too kind.

It’s my pleasure.

To help me engrain this: magician, if most investors are using arbitrage-free valuations, have you experienced that most OAS spreads, all else fixed, are roughly equal? Or, do the different methods of modeling short-term rates cause an OAS spread dispersion?

I spent the bulk of my time at PIMCO developing prepayment models for mortgage-backed securities. The objective was to determine for ourselves the correct OAS, rather than rely on OAS estimates from outside sources (e.g., Bloomberg). Ultimately, our numbers sometimes came very close to those from outside sources, and sometimes differed substantially (especially for CMO tranches).

Appreciate you sharing–really cool to hear.

Great explanation. I was refreshing my memory and I think many people are tripped up about this (judging by the number of posts on this topic) because Schweser’s explanation makes it sound as though the Z spread does not capture the option cost with this sentence, “While the Z spread is based on the on-the-spot rate curve instead of a single YTM benchmark, it still does not consider embedded option features.

Then when it talks about OAS next, we might think that the OAS is the one that captures embedded option features and, therefore, the OAS must be higher than the Z spread in order to make the callable bond trade at a lower value.

Yeah I can see how that’s confusing. The missing part (I think) is that the market price already factors in an option cost. That is, the market will price a callable lower than an option free bond. From there, Z spreads are added when rates follow the forwards, which will never consider the option being exercised (realistically, clients would never buy callables structured in such a way that the bond is expected to be called when rates follow the forwards). OAS is calculated using Monte Carlo simulations, by taking lots and lots of potential rates paths and adding the spread just as in the Z spread case. However, with random rates paths you would inevitably come across many paths where the bond gets called. That is why the CFA explanation is true and makes sense, but you have to dig in a little bit.

By the way, the above intuition isn’t entirely correct for MBS (because even along the forwards there’s a certain expectation for prepayments), but the general intuition is similar and really the curriculum doesn’t treat MBS deeply enough to care about this nuance.

Thanks so much for this. I’m on the brink of understanding it. I don’t know why but every time I revisit for the next level it scrambles my head again!

Writing this to try to get it straight in my head conceptually. Please do correct/improve:

What the OAS is doing is essentially trying to strip OUT the impact of the option (volatility) to get a read on the underlying yield of the bond on an apples-to-apples basis vs. similar bonds without embedded options. What it is NOT doing is adding IN the ‘extra’ value that the option represents - this is where my confusion has always been, with the (errant) logic being: “The put option value will increase the bond price. Therefore the yield will be lower. Therefore the OAS must be lower than the Z-spread”.

Wrong - here’s why:

The current market price of bonds includes two components:

  1. The value corresponding to the underlying bond assuming zero volatility (“spread” here represents compensation for credit risk premium, term premium, etc.)
  2. The value corresponding to the embedded option on the bond, which varies with volatility (using a put option for example: higher volatility → higher value → put options have value to the investor → higher premium added to the price of the bond (i.e. investor has to pay more for it))

The Z-spread is just a constant number that’s being applied to the periodic cash flows to ‘solve’ for the market price. It is ‘blind’ to the difference between 1 and 2 above. That’s fine if volatility has no impact (i.e. nonputable bond, #2 above = 0) but not when volality is in the mix.

Taking a putable bond:

  1. The value of the option is positive to the investor (since they can put the bond if rates go up high enough)
  2. The market price for the putable bond is therefore HIGHER than the nonputable equivalent
  3. The Z-spread (i.e. the backsolved x-factor) is therefore LOWER than the nonputable equivalent, since a lower discount rate is needed to equate the future cash flows the (higher) present value
  4. So if you are simply comparing the putable bond vs. a nonputable bond on the basis of Z-spread alone, you may end up making the wrong choice from a non-volatility risk/return perspective.

But really for the apples-to-apples comparison, you need to strip out the value of option and compare only the yields on the underlying bonds themselves.

So for the putable bond above:

  1. The OAS essentially REMOVES the present value of the option from the market price (adjusting it downwards, because the option value is positive) - note, this is the fundamental reason why my logic in the first para is wrong
  2. This in turn increases the required spread to equate the future cash flows back to the lower market price*
  3. This higher spread IS the OAS i.e. I am adjusting the spread the remove the impact of the option so I can compare like-for-like on an option-free basis

In this circumstance, you might find the putable version actually offers a higher underlying yield for the non-volatility risk exposures you are taking on, and therefore choose that instead.

Hope this makes sense?

*Added afterwards: in this instance, am I right in thinking that the future cash flows are also theoretically adjusted downwards to remove the put feature, but given the PV of the put option will be priced at greater than zero by the market, the implied discount rate (OAS) has to go up on a net basis?

Dear Fireside:

It all looks good except for the “added afterwards” bit.

For a putable bond, the future cash flows are adjusted upward, not downward. The bondholder isn’t going to exercise the put to make themselves worse off (lower cash flows); they’ll exercise the put to make themselves better off (higher cash flows). The way that the OAS removes the value of the option is by having the higher (than for the straight bond) average cash flows on the right side of the equation, and the higher price (than for the straight bond) on the left side of the equation; if you have modeled the option correctly, those two exactly cancel each other, and you’re left with a spread that corresponds to the underlying option-free bond.

Thank you for your response - that is very clear. I think I worded my post script a little lazily. What I meant by adjusting the cash flows ‘downwards’ to remove the put is the following:

If removing the PV of the put option from the left hand side of the equation, then the equivalent for the right hand side of the equation (the cash flows) would be to remove the incremental benefit of the put option in those cash flows. For example, if the exercise price of the put option 100 and the price drops to 99, the incremental value of the put option is +1 in that cash flow period. Removing that from the right hand side replicates the cash flows of the option-free bond.

Unsurprisingly, your way of explaining it is clearer i.e. the higher cash flows on the right hand side are the required adjustment to offset the impact of the present value of the option on the left hand side.


You seem to have grasped this. Good job: it’s not easy.