Is doing a PhD a good way to develop a deeper understanding of quantitative finance and statistics? I work in business and mainly deal with quant finance and statistics + do a lot of programming to implement ideas from quant finance and statistics. I have a master of science in economics.

My problem is: whenever I am trying to dig deeper (by reading articles from scientific journals) to understand some theoretical foundations of a certain idea from statistics and quant finance (ranging from pricing of exotic options to kriging based methods) to modify and customize this idea for my situation, I discover that my knowledge of math is not enough to penetrate deeper in a subject.

Is it a sign that I need to get a PhD in quant finance/ statistics/ math or it would be an overkill and I better go for a less time consuming (bloody) option (if there is any)?
No, this is not a sign that you need a PhD. A PhD is to prove to other people that you can do novel research to a significant depth, with significant rigour.

What you want is to learn some mathematical techniques. Now, the only way to do that, is to practice using those techniques. You need some exercises that will take you from where you are now, to where you want to be. You need to do those exercises, and find out which ones you got wrong, and why you got them wrong; and then do them correctly.

A PhD isn't an efficient or cost-effective way to do that. But a taught  Masters degree (e.g. in Quant finance) might be. A free online course might be. A study-buddy who understands this stuff might be. A private tutor might be. Finding out which textbooks an appropriate Masters course uses, and working your own way through them, doing the exercises in each chapter, might be. The exact recipe will be whatever works for you. But what you need is to build up your toolkit of maths knowledge and skills, and you can only learn that by using those tools.