Z Spread vs Option Adjusted Spread 223, understand, particular, adjusted

cyber21

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

whew1110

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