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