1. Sleep: Teens love to stay up late, but if you establish regular sleep times and try to help them stick with them, it will help them TREMENDOUSLY to be in a better mood. You can talk to them about it, and tell them that studies show that teens need the same amount of sleep as toddlers do.
2. Exercise: Regular exercise calms those hormones and gets that serotonin level up. My daughter was like two different people. When she exercised she was happy and calm, and when she didn't she was up and down a lot. So we tried to encourage her to enroll in school swim team and track, in order to help her mood swings. Many boys also need physical exercise as an outlet for their raging testosterone at this age, which can trigger aggression or picking on family members if not given an outlet.
3. Time with family: Even though teens think they don't have time for this, they need to be bonding with parents individually and the family collectively to stay stable. This means you will have to establish times for this. We have a regular family night, a sit down family dinner where we all talk together, and I take them on weekly one on one dates, and try to talk to them individually each day, mostly at bedtime. Also family traditions and vacations help the teens stay bonded.
4. Verbal communication is key. When teens have sadness, anger or are experiencing a strong emotion, they need to talk about it, so it doesn't come out in their actions instead. Know what is going on in your teen's life and find ways to talk about it.
5. Relationship with a higher Power: Whatever your religion, it will help your teen to be taught how to pray and read scriptures that will help them in their daily challenges. We do this with our family and have them do it individually before bedtime.
6. Love Languages: Make sure you know your teen's love language and are speaking it to them. (although they need all five love languages regularly).
These six things will go a long way in helping your teen be more stable and less moody during their adolescent years. The emotional behavior will not be gone, but things will be less up and down for you and your teen.