Z Spread vs Option Adjusted Spread

Question (Kaplan book): An investor purchases a bond that is putable at the option of the holder. The option has value. He has calculated the Z-spread as 223 basis points. The options adjusted spread will be:

Answer:: greater than 223 basis points

I dont understand if OAS = z Spread - Option Cost., and the option has value to the investor, why would it be greater (instead of smaller).

If anyone can further expand on this, and the below LOS in particular, id appreciate it. I gues I just don’t understand the concept behind it.

For callable bonds, zspread > OAS and option cost > 0

Forr putable bonds, z spread < OAS and option cost < 0

As you quoted

For putable bonds, zspread < OAS and option cost <0

option cost < 0 means that option cost is negative which implies that

OAS = Zspread - (-option cost)

OAS = Zspread + option cost

So OAS > ZSpread

To explain this. Option has a premium named as option cost. For a callable bond the bond holder (investor) is the option writer and he has received premium which is equal to the option cost. The borrower or bond issuer in this case is the buyer of the option and he has paid the option premium (cost) by issuing the bond at a price less than the price of an option free bond. For this particular reason

Price of Callable bond = Price of Option free bond - Option Cost

Conversly for a putable bond the price of the bond is higher than an identical option free bond because the option writer in this regard is the Bond Issuer and the buyer of the bond is the Bond Holder. The Buyer purcahses the bond and also the option which gives him the right of returning the bond at the price higher than the market price.

Price of a Putable bond = Price of Option free bond + Option Cost

Apologies for the wordy response, wanted to make the explanation as explicit as possible:

A call option on a bond is an option for the issuer, written by the bondholder. It gives the issuer the option to buy redeem the bonds prior to maturity. The bondholder receives payment for giving this option to the issuer, therefore Option Cost > 0.

A Put Option is an option for the bondholder, written by the issuer. It gives the bondholder the option to demand early repayment of principle at the exercise date(s). The bondholder makes a payment for having this option from the issuer, therefore Option Cost < 0.

Note these “payments” are not separate, rather they are priced into the coupon.

So these Option Cost values are both from the perspective of the bondholder and from that perspective make complete sense.

Now for this statement:

OAS = Z-Spread - Option Cost


Option Cost = Z-Spread - OAS

Remember the Z-Spread alone includes the effect of embedded options, so think what effect you’d expect embedded options to have on yields.

Callable Bond - Increase yields (compared to identical option-free bond), to compensate bondholder for the fact issuer can decide to redeem bonds prior to maturity.

Puttable Bond - Decreased yields, to “penalise” bondholder for the fact he/she can demand repayment of principle prior to maturity.

Now think about what the OAS is doing, it’s removing the effect of the embedded option

On the callable bonds OAS < Z- Spread:

The Call Option increases the yield (+ve), so when we remove the effect of this option to get the OAS, it will be less than the Z-Spread (which included that effect).

On the puttable bonds OAS > Z- spread:

The Put Option decreases the yield (-ve), so when we remove the effection of this option to get the OAS, it will be greater than the Z-Spread (which included that effect).


I think this is more of a CFA Level 2 question as one must know binomial rate trees and backward induction to understand OAS. I just finished reading the chapter on the option bonds and I too was confused by the fact that OAS for putable bonds are always greater than the Z-spread! Luckily, some number game-play revealed the reason. I saw this post after figuring out the reason and though previous answers do the job too, I thought of adding my two cents. Here we go:

Z-spread encompasses the effect of liquidity, credit & options. It is the +ve calibration needed in the rates of the binomial rate tree to adjust the bond-value down to the market value. If we use the options-adjusted binomial tree then the bond value is already adjusted for the option and only additional adjustment required is for the liquidity and credit. That adjustment is called OAS.

We can put this relation in form of various equations:

a) (Z-spread - option-spread = OAS)

b) (Z-spread - OAS = option-spread)

c) (Z-spread = OAS + option-spread)

Let’s take the case of a bond with call option. Say the Z-spread is 21 basis points. It is composed of 5 basis points for liquidity, 6 for the credit and 10 for the call options. By the above equation :

Z-spread (21) = OAS (11) + option-spread (10)

If we use an option adjusted binomial rate tree then the option effect will be stripped out and we will need to raise the rates in the binomial tree further only by (5+6 = 11) basis points to push down the calculated bond value to the market value.

In a bond with a put option, the Z-spread will be 1 basis point (5+6-10=1) as the option component is negative. Why? A put option increases the value of a bond. So instead of raising the rates in the rate tree one would lower them to approach the market price. Liquidity and credit difference move the Z-spread to positive direction while the put option move the Z-spread in negative direction. By the above equation :

Z-spread (1) = OAS (11) + option-spread (-10)

You can see that by definition, Z-spread for a putable bond is OAS “less” option-spread. So Z-spread will “always” be smaller than the OAS.

The easiest way to think of this, in my opinion, is to realize that the option cost is the additional yield that the issuer has to pay for the option.

  • For a callable bond, the issuer buys the call option, so he has to pay for it with a higher yield; the option cost is positive.
  • For a putable bond, the issuer sells the put option, so he is paid for it with a lower yield: the option cost is negative.

Because OAS removes the value of the embedded call/put option,

Z-spread − option cost = OAS

Thus, the OAS for a callable bond is lower than the Z-spread – the option cost is positive – and the OAS for a putable bond is higher than the Z-spread – the option cost is negative.

For whatever it’s worth, I just wrote an article on OAS that may be of some help here: http://financialexamhelp123.com/option-adjusted-spread-oas/

I don’t go into Z-spread vs. OAS in this article, but I may add that.

1 Like