add function to saving change on Options Pages

To execute a task after a option has been updated, you can use updated_option action hook:

add_action( 'updated_option', 'updated_option_callback', 10, 3 );
function updated_option_callback( $option, $old_value, $value ) {
     //Do something
}

Also, you can use update_option_{option-name} specifically for each option. For example:

add_action( 'update_option_new_option_name', 'update_new_option_name_callback', 10, 2 );
function update_new_option_name_callback( $old_value, $value ) {
     //Do something
}