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I agree with your security team. The "security question" is not secure on its own: cities can be found over ip, names can be googled and nicknames of pets can be guessed.
However, you could combine both methods: first send the mail, and when the user clicks the link, ask the question. This way you would add at least a bit more security to the procedere. Also think about a lock (e.g. 12 hours) after three bad attempts to ensure bruteforcing isn't possible.
I agree with your security team, these kind of questions are too easy to lift from social media, online research or just guesswork. It can be made a lot better then 'what is your dogs name', but its still antiquated.
Depending on your needs, in most situations I recommend an third choice. Required mobile phone number registration and SMS to the mobile phone with a verification code on password change.
Sadly, some users have the same/similar password everywhere. And what if this customer of yours have had an attacker start with taking his email account? Then finding he have an account on your product from reading his emails. Then taking over his account with your company, then .. changing the password since you email the password changing link to him/her ..
Can anyone direct me to a list of sites that allow password resets through security questions only? I would like to check my sites against those and put in unique responses for any sites I care about, as I assume (100%) that my true security question answers are available for any moderate hacker or even just an averagely persistent person.