In the past where I've converted options it's been directly with...

In the past where I've converted options it's been directly with the company.
Contact the company secretary for their standard forms to make the conversion and BPay the funds directly. Takes a few days end to end.

Call options are something different again; they are not issued by the company but rather option writers who choose to offer them.
ASX options are offered only by the companies in question.Calls and Puts on the US exchanges can be written by anyone and it's in lots of 100 shares with a premium paid for buying them + the strike price (exercise) to be paid at or before expiry (European vs American options).If you close the option prior to expiry you're buying or selling it to someone else.

Hong Kong SE has similar but they call them warrants, and they're issued by companies (subscription warrants) or financial institutions (derivative warrant).