Z Spread vs Option Adjusted Spread

[Znieh15]

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

 

[mohammad.belaal]

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

 

[Kman2001]

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
or
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).