I don't see the option either but it also depends on what apps you have installed. For example .. my options are Bluetooth, GMail, EMail & Google Drive. I tried installing a 3rd party SMS app but had no luck.. it would not show up as an option.
These are the instructions/features for contact sharing:
You can share a contact with someone in two ways: by sending them the contact in vCard format, or by using Android Beam
To share a contact in vCard format:
1. Open People.
2. Touch the contact to view its details.
3. Touch Menu > Share contact.
4. Touch a method for sharing the vCard: Gmail, Bluetooth, and so on. The list depends on the applications you have installed.
Use the application you picked to finish sharing the .vcf file.
To share a contact using Android Beam, first make sure both devices are unlocked, support Near Field Communication (NFC), and have both NFC and Android Beam turned on. Then follow these steps:
1. Open People.
2. Touch the contact to view its details.
3. Move the back of your phone toward the back of the other device.
When the devices connect, you hear a sound, the image on your screen reduces in size, and you see the message Touch to beam.
4. Touch your screen anywhere.
Your friend's device displays the transferred contact information