I recently asked for (and was awarded) a raise. I have weekly 1-on-1's with my boss (CEO) - they usually go for about an hour & we simply talk about everything happening in the business/team.
At the end, I simply said 'Something I want to bring up is that I've looked at the market and I don't feel I am being paid adequately for the value I'm creating'. I'd be lying if I said it wasn't an awkward conversation. I didn't mention any number at all & was actually met with a slightly negative response, but a week later at the next 1-on-1, it was brought up again & I was awarded a 10% cash raise (I negotiated some ISO's too).
In your scenario, I wouldn't tell them that you interviewed at a competitor either. You can simply tell them that you've looked at other roles & determined market rate as being higher than what you're currently paid. Only use another offer as leverage if you are prepared to leave & take that offer.
Modesty aside, I'll also admit that I'm a high performer in our team & so I already knew that I had leverage. Something I've learned though is that unless you bring it up, it's never going to happen - there is no-one looking out for you unfortunately, so take the initiative & help yourself where you can - it equates to hundreds of thousands of dollars over your career.
I think the dynamic at your company (where you're a direct report to the CEO) is a totally different animal than your standard corporate dev job. When you're working at a place that employs 1000+ people, these types of conversations are much less likely to go in the direction of the employee.
That's true - we're a small (15 person) team & I lead Design & Product so the CEO is my "direct report".
However, just because you have a lower likelihood of success, it doesn't mean you shouldn't ask. If you receive a 'No', it will probably speed up the inevitable decision of whether you actually want to stay with this company (knowing that they aren't going to incentivise your performance).
I would also keep in mind that you should only be asking if you either feel you're being paid lower than market or that you've taken on more responsibility so now your wage doesn't adequately match your role. Just asking periodically without a reason isn't going to work for anyone.
At the end, I simply said 'Something I want to bring up is that I've looked at the market and I don't feel I am being paid adequately for the value I'm creating'. I'd be lying if I said it wasn't an awkward conversation. I didn't mention any number at all & was actually met with a slightly negative response, but a week later at the next 1-on-1, it was brought up again & I was awarded a 10% cash raise (I negotiated some ISO's too).
In your scenario, I wouldn't tell them that you interviewed at a competitor either. You can simply tell them that you've looked at other roles & determined market rate as being higher than what you're currently paid. Only use another offer as leverage if you are prepared to leave & take that offer.
Modesty aside, I'll also admit that I'm a high performer in our team & so I already knew that I had leverage. Something I've learned though is that unless you bring it up, it's never going to happen - there is no-one looking out for you unfortunately, so take the initiative & help yourself where you can - it equates to hundreds of thousands of dollars over your career.