How to define a "global variable" from an "ansible-playbook extra variable" in Ansible?

I'm using Ansible 2.1.1.0. In Ansible, is there a way to define a variable in group_vars/all file with a dynamic variable set as an extra var in ansible-playbook command?

I have my play.sh file as follows.

ansible-playbook site.yml -i hosts -e "home_dir=$1"

Then I want to use it as a global variable in group_vars/all file as follows.

my_dir: {{ home_dir }}/sub_dir/my_dir

I know the syntax above is wrong for the all file, and this can be done by using the variable directly in the task yaml file. Is there any way I can use a dynamic 'ansible-playbook extra variable' for defining a 'global variable'?