Z spread and Option Adjusted Spread

We have 3 Bonds:

Bond A Bond B Bond C

Z-spread 130 150 135

OAS 140 125 135

Why Bond A is more likely to have put option embedded than Bond B?

Is Z-spread = OAS - Call Option, and Z-spread = OAS + Put Option.

The higher the OAS, the lower the price of Bond, isn’t it?

OAS spread = Z-spread - option cost

In the case of put options, the option cost is “negative” in the sense that it is the investor’s option and thus increase the value of the bond. Using the numbers from above, 140 = 130 - (-10), with -10 being the cost of the put option.

In the case of call options, the option cost is positive because it is the issuer’s option and thus reduces the value of the bond. Using the numbers from above, 125 = 150 - 25, with 25 being the cost of the call option.

But shouldn’t High OAS imply low bond price, and since call option’s adjusted spread is lower, how come its price is higher than putable bond’s?

Higher coupon rate, perhaps?