As you all know, PS4 runs Orbis OS, that is highly based on FreeBSD (which is a opensource operating system) so let's start!
> Exploit Details
Console: PlayStation 4
Type: Privilege Escalation/Buffer Overflow (allows to run assigned code)
Created: 25 November 2013
Published: 2 December 2013
Jailbreak Package (exploit):
You need the DAY ONE Update to jailbreak the PS4
1. Create a folder on your USB storage device. This is where you'll put the exploit.
2. Create a "SANDERPS4" folder. Inside that folder, create another folder named "EXP."
3. Extract the PSORBISEXP.PUP file from the package, and save it in the EXP folder.
4. Make sure your PlayStation 4 is turned off.
5. Connect the USB storage device to your PlayStation 4, and press the power button for at least 7 seconds. The PlayStation 4 will start in Safe Mode.
6. Select "Update System Software."
7. Follow the on-screen instructions to install the jailbreak.
8. If your PlayStation 4 doesn't recognize the jailbreaking file, make sure that the folder and file names are correct.
You now have the ability to run unassigned/assigned code and pirated games on your PS4, but do NOT go online, if you do go online, your console will be immediately banned.