Does the notion like "correlation between option and its underlying" or "correlation between option and option" exist?If yes,where could I learn about it?

Linear correlation might not be the proper tool due to option non-linearity.

Search for "the greeks", especially "delta".

As already mentioned, the relationship, while obviously positive, is non-linear due to the convexity of option payoffs.Robert Merton wrote about the relationship between option and underlying returns in his characteristically thorough fashion some time ago.It might be these papers:Merton, Robert C., Myron S. Scholes, and Matthew L. Gladstein. "The Returns and Risk of Alternative Put Option Portfolio Investment Strategies." Journal of Business 55 (January 1982): 183-242. Merton, Robert C., Myron S. Scholes, and Matthew L. Gladstein. "The Returns and Risk of Alternative Call Option Portfolio Investment Strategies." Journal of Business 51 (April 1978): 183-242.

as DavidJN wrote the relationship is positive but non-linear, i.e. the 'beta' is convex (because delta of option changes). the term used to describe relationship between option and underlying is option 'elasticity'. collector mentions it in know your weapon part I.