When you say "asian", I'm not sure if you mean East Asian or Asian as a whole.
It depends your bloodline and where the previous few generations grew up (geographical adaptation).
For example I'm from Bangladesh (south asia) and my hair is like a slightly thicker, curlier version of Sandara Park's (2NE1, S. Korea) natural hair. Sorry that's the best comparison I could find haha.
Many people in my family have that stereotypical sleek and shiny Asian hair, like my sister and cousin (whom people'd think is actually from South China). However I inherited my hair from both my mom (who has very frizzy, curly hair) and someone, probably my maternal grandmother who is originally Indian, who had the sleek, shiny hair.
I think curly hair is more common in Western Asia than it is in Eastern Asia. But it's not non-existant. I've met a few chinese and koreans with wavy, frizzy or and even curly hair. :)