How Z-Spreads and Option-Adjusted Spreads Differ

# What is the difference between Option Adjusted Spread (OAS) and Z-spread?


# What is the difference between Option Adjusted Spread (OAS) and Z-spread?












I am preparing for the CFA level 2 exam, I got confused by the concept Z-spread and OAS.

When a call option is added to a bond, since it is not favorable to the bond buyer, they would require more spread (which is the OAS) for this instrument in order to get more discount on the bond price.

So to me, Z spread should be less than the OAS.

But this is not what has been discussed in the book. Can someone help with this?

## Answer by SRKX (score 34)

https://quant.stackexchange.com/a/3321

To sum up what is discussed in the CFA curriculum, it discusses 3 types of spreads. They are used to compare a risky bond to a Treasury bond (assumed to be risk-free).

Nominal spread

Simply computes the difference between the YTM of the risk-free bond and the YTM of the risky bond.

The major problem of this measure is that it doesn't take into account the shape of the spot yield curve.

Zero-volatility Spread or Z-Spread

This spread is actually a single value that needs to be added to every spot yield of the curve in order to make the present value of the risky bond equal to the present value of the risk-free bond.

That is, if the present value of the risky bond is $v_b$, then the Z-spread $z$ is the value such that:

$$v_b = \sum_{i=1}^N \frac{C}{(1+R_f(0,i)+z)^i}+\frac{FV}{(1+R_f(0,N)+z)^N}$$

where $R_f(0,t)$ is the spot rate for maturity $t$ for a risk-free bond (not annualized).

This still doesn't take into account any embedded option to the risky bond.

Option-Adjusted Spread (OAS)

This last spread is used to measure the impact of the optionality in the bond. It is defined as follows:

$$\text{OAS}=z-o$$

where $o$ is the the price of the embedded option.

For callable bonds, the option benefits the issuer (it allows him to buy back the bonds if rates go down, i.e. bond prices go up), and $o>0$ hence $\text{OAS}<z$.

For putable bonds, the option benefits the bond owner (it allows him to sell back the bonds if rates go up, i.e. bond prices go down), and $o<0$ hence $\text{OAS}>z$.

It there is no embedded option, then $o=0$, $\text{OAS}=z$.

I believe spreads are to be understood as being used to measure the risk inherent to the core bond, and not to the option that are embedded to it. Hence, if the bond was callable, you required more yield for the bond, but the "core" bond only required a spread equal to the OAS and not really the spread computed by the Z-Spread approach.

## Answer by Matt L (score 5)

https://quant.stackexchange.com/a/9626

Z-spread is the all-in spread, meaning spread from the risk profile AND from the call risk. The OAS factors out (subtracts) the additional spread associated with the embedded option, so the OAS will be lower. It's more useful than Z-spread because it allows for apples-to-apples comparisons between bullet maturities & callables. Remember, Z-spread is the spread that, when be added to the discount rate, will equate cash flows to price so this will naturally include spread from all sources, whereas OAS will not include the spread specifically from the option cost.

## Answer by Miles (score 4)

https://quant.stackexchange.com/a/8077

The Z Spread is the constant spread above the treasury curve that compensates the bond holder for credit, liquidity and option risk.

The Option Adjusted spread is simply the Z- Spread excluding the premium to compensate for the option risk.

Thus, the OAS is the spread above the treasury curve that compensates for credit and liquidity risk only.

## Answer by David Nehme (score 3)

https://quant.stackexchange.com/a/3320

The z-spread and OAS both are measures of the difference in price between an ABS and a zero-risk bond. The OAS and z-spread are not spreads that a bond with and without options should require, they are two ways of looking at the same bond.

The CFA material states that the z-spread is equal to the OAS when

- There is no prepayment option

- There is a prepayment option, but the borrowers tend not to exercise that option.

The z-spread essentially ignores the option, so a larger spread is needed to explain the price of a bond if there is an option that has a good chance of being exercised.

## Answer by Roberto (score 3)

https://quant.stackexchange.com/a/9988

Sorry for interefering in the discussion as a guest, but I'm studying this topic at the moment for FRM Part II exam, in the context of MBS valuation through Monte Carlo simulation, and at the beginning I had exactly the same doubt. This is the explanation I gave myself and I don't know if it can help:

When you use Z-spread you are not simulating cash flows taking into account the option (in my case, mortgage prepayment by mortgagor), so the resulting sum of the "scheduled" cash flows is higher and, in order to reconcile it with the market price of the security you need, from a purely mathematical point of view, a smaller discount factor (for each time bucket, more precisely), i.e. a higher spread to put in the equation. When, instead, you calculate cash flows using Monte Carlo simulation and take into account interest rate volatility (by means of some assumptions in the model), this means you are considering teh option possibility and this automatically reflects in a smaller value of the scheduled cash flows. This in turn implies that you need a higher discount factor, i.e. a smaller spread, to reconcile this value with the market value of the security.

So I think you should interpret the spread not as what an investor demands, but as what an investor actually earns, over the treasury (or LIBOR, depends on the model) curve by holding the security to maturity. The OAS is smaller because you earn less than holding a security without the embedded option, and this is because the cash flows are actually less.

So the conclusion is that z-spread has to take into account the option risk (so it has to be higher) because this is not taken into account in the structure of the cash flows. Instead, this is not necessary with MT simulation for the reason explained above, i.e. optionality is accounted for in the structure of cash flows and not in the OAS, so in this sense it is an "option adjusted" spread, and I think it more realistically reflects, to an investor, the real value of a security especially if compared with securities without optionality. In this sense we can say, quoting Miles answer above, that OAS "compensates for credit and liquidity risk only", and for this reason this is the spread that should be used if one wants to compare the MBS with, for example, a security (like a traditional bond) that embeds itself only credit and liquidity risk. The presence of the "compensation" for option risk in the z-spread, instead, distorts this commparison and is neither a realistic representation of what the MBS really earns.

Sorry if I've been too verbose, but I hope this clarifies your doubt.

## Answer by Tom (score 3)

https://quant.stackexchange.com/a/11502

Lots of answers, but I'll throw in something very simple and practical.

Bonds are priced based the spread investors are willing to pay above treasuries. When they determine the price on a bond with optionality, aka the spread, this price is reflected in the Z Spread. The Z Spread is the complete spread required by investors to compensate for all of the risk, including the embedded optionality, of the bond. OAS simply implies, what would the spread on this bond be if there wasn't any optionality as priced in by the market.

## Answer by 291890964 (score 1)

https://quant.stackexchange.com/a/36990

eh, just bumped into this. the answer is quite simple - z-spread is based on a curve that is not dynamic. there are different ways to set up your prepayment model, but generally you'd use forward rates. so, for each future period, your prepayment model would use this static curve & a current coupon (index) model to calculate your refinancing incentive and come up with CPR. Once this is done, you have cash flows for each period until your product matures. if a price is already given, what the computer/software will do is calculate the required spread to be added to each point in the static discount spot curve so that the resulting price equals your given price (input).

OAS is a bit more complicated, but in essence you have many (250, for instance) paths, instead of one. so you have 250-sets of cash flows that need to be discounted. depending on how your term structure model is set up, this might affect your valuation (or risk) dramatically. generally, simulating the curve gives you increased prepayments (there is a long answer for this), which drives your cash flows shorter. now these shorter cash flows need to produce the same price, which is given as an input. how do you do that? the simulation discounts less, i.e., adds smaller spread (OAS vs static) to come up with the same final price.

cheers.

d.

## Answer by Victor (score 0)

https://quant.stackexchange.com/a/8038

The question remains unanswered...

What you basically state by OAS = z - o is that OAS is less than the z-spread for callable bonds. Now if you compute the PV of the bond by discounting it by respective yields (Treasury yield + spread) you will find that the the bond discounted using the z-spread will be cheaper than the one discounted by OAS. It does not make sense because as a rational investor I demand a larger discount for the call option that I provide the issuer with.

## Answer by Learn4Ever (score 0)

https://quant.stackexchange.com/a/8098

Very insightful discussion. Let me try if I can address this issue. What victor has challenged makes perfect sense, but I think we when calculate z-spread, we just ignore the option. So when we actually calculate call embedded bonds, investors are looking for larger discount to compensate, so the spread is larger which reflect on the market price. And this price is what we use to calculate z-spread. So you will see larger z-spread version put embedded one with smaller z-spread. And Z = OSA + Option. OSA is a ideal spread which is measuring the spread between the fair value and market value. Like if you don't have the call option, the bond price will be higher and the discount rate will be lower, and the spread is smaller. The difference is the price(spread) of option. For call is positive and for put is negative. I hope this can clear some misconceptions.

## Answer by user6571 (score 0)

https://quant.stackexchange.com/a/9521

I think you are just inverting the stuff. (Like I did) You consider Z-spread what, in fact, is OAS.

OAS: discount to add to treasury excluding the option price. (Z-spread of an equivalent bond with no option)

Z-spread: discount to add to treasury for the option bond (thus accounting for the option).

It is misleading because "Option Adjusted" calls for a spread adjusted to reflect the value of the option in the Bond and we instinctively go for it as the actual spread to use for the bond having the option. And, off course, it should reflect a higher discount for a call option, reducing the price of the issue.

They should call it: "Option Excluded Spead" or something like this...

Take care,

## Answer by preparingcfa2 (score 0)

https://quant.stackexchange.com/a/17991

Lots of answers, but not totally clear... Let me try to explain the I understood... Take 3 bonds -> 1) treasury bond 2) risky bond 3) risky bond (same as above) with embedded call option

For bond 2, Z-spread = 2% (200 basis points) For bond 3, Z-spread = 3% (which includes the optionality) -> if you adjust the optionality(that is with out option) or OAS, it will be 2% -> which is same as above.

So Z-spread for bond 3 = Z-spread for bond 2(=OAS) + option cost Not sure whether I confused more or make it little clearer.

## Answer by Parth Thakkar (score 0)

https://quant.stackexchange.com/a/33247

For an Option Free bond

Option Adjusted Spread = Z spread

For a bond with embedded Call options

Option Adjusted Spread + Option Spread = Z spread ( For Callable Bonds) ... Why Option Spread is Positive ? Ans - This is because Investors are to be compensated for additional risk of having callable bonds in their portfolio.

For Bonds with Embedded Put Options

Option Adjusted Spread - Option Spread = Z spread Why option spread is Negative ? - This is because the issuer is to be compensated for the risk of issuing bonds with put option.

So in exam

Z spread > OAS - The bond has an embedded Call option Z spread < OAS - The bond has an embedded Put option Z Spread = OAS - The bond has no embedded options, it is a straight bond

## Answer by Yatin Bisht (score 0)

https://quant.stackexchange.com/a/68765

Well many answer here says that Z spread includes option risk. Z spread is zero volatility spread, so at zero volatility option cost is zero that means z spread cannot accommodate a option cost. When valuing a option free risky bond the spread we add to risk-free bond is Z spread. But the same cannot be done when value a risky option embedded bond. First we will have to arrive a riskfree option embedded bond price, which will require volatility assumption and then adjust the value by adding a spread to accommodate riskyness of the bond this spread will be OAS (option adjusted spread)

So after all this if one has to arrive a relation between OAS and Z spread OAS = Z spread +/- option cost depending upon type of option OAS = Z spread - call option OAS = Z spread + put option

For the question asked why OAS will be lower OAS is option adjusted spread so what is the spread if already adjusted for option cost obviously the spread will be lower than the Z spread as the spread is not changing and much of the spread will be eaten up option cost.

Also if your argument is since call option is added to the bond and you should be compensated for that then you are right just you are asking the question at wrong place. OAS will not give u answer whether you are getting compensated correctly it will help u to compare with another option embedded bond and then if both are identical in risk and the former is lower you can safely say the former is overvalued and may be not compensating me properly for option cost

## Answer by Hritabrata Das (score 0)

https://quant.stackexchange.com/a/85753

Think of OAS as a metric used to find numerical differences between real market price vs. modelled price. By modelled price, I mean you can , for example , use a one factor model like 6 month yield for 2 years already assumed , from there you use risk-neutral probability to price all steps in backwardation.

Now , you observe the market and notice that market price of the bond is different. What do you do ? Well, using the same cash flows in each node , you argue , even after adjusting for payoffs and drifts of interest rates (p's), there might be some additional return that's hidden as a whole. That's your OAS.

For Z spread, the concept is relatively simple. You do that assuming it was not adjusted for option payoffs, meaning your Z spread > OAS.

## Answer by user7915 (score -1)

https://quant.stackexchange.com/a/11083

As previously stated, the OAS is really an Option Excluded Spread.

The Z-Spread is the spread that includes option risk and is therefore higher. Higher spread means higher discount rate which means lower Price.

The optionality of a callable bond, which benefits the issuer, results in a lower price and higher yield for the investor. This makes perfect sense to me.