how to update and display an option without reloading the page

Hi mate this is a really good tutorial for using Ajax in WordPress. Everything you’ll need for what you want is basically covered. https://premium.wpmudev.org/blog/using-ajax-with-wordpress/

Essentially Ajax is just a custom jquery function in your JavaScript file that listens for an action on the page. It fires to the server to run the php function you specify, and then returns data from the server.