Really tough to give good strategic advice without all the detail, but the first thing I would focus on is doing some deep competitive research. If someone is cutting prices by 45% and the lowest you can go to is 20%, that's a pretty big gap.
Possible scenarios I can think of...
- They are either procuring better than you by finding a supplier that provides for better margins
- They have a better relationship with the manufacturer or possibly are the manufacturer, providing them with better margins.
- They are taking a loss to push you out.
- Their customer acquisition costs are substantially lower which could be due to numerous factors. One example off the top of my head, they have places they are acquiring their customers from that are much lower cost than where yours are coming from (e.g. SEO). Say this comprises 70% of their acquisition activity. The 30% that they are competing with you on (e.g. PPC), they can afford to take a slight loss for the sake of volume because of this channel mix. Their conversion rates, AOVs (incentivizing multiple unit purchases), etc vs. yours could also play a role.
- Retention rates could be factoring in. Maybe they are willing to acquire a loss on acquisition because they know they will make it up over the lifetime value of the customer.
Just some ideas, but I would really try and understand the mechanics of what is going on before you craft a strategic battle plan on how to combat this. If it's simply a matter of procuring more effectively, that's one thing. If it's someone trying to push you out, it's another. And if it's a manufacturer doing it, well that's probably the most challenging of the 3 because you will never be able to compete on price (which is pretty much the single thing that drives most gray market products). Hard to create differentiation in that space I would imagine.
I also concur with bizzy's idea above about positioning a "premium" brand against an "inferior" one. Excellent advice.
Hope some of that is helpful.