Calls and puts are two different instruments

A call gives its holder the right to acquire the underlying at the strike. A put gives its holder the right to dispose of the underlying at the strike. Same contract architecture, opposite direction of the transaction.

The symmetry is real in the definitions and breaks in the numbers, for a reason worth understanding. A call's value has no ceiling, because the underlying's price has no ceiling. A put's value has a ceiling, because the underlying cannot fall below zero — a $50 strike put can never be worth more than $50 per share. So the two instruments have fundamentally different value distributions even at identical distances from the current price.

There is a second asymmetry that shows up in pricing. Markets do not treat upside and downside moves as equally likely or equally fast. Sharp declines have historically been faster and more correlated than sharp advances, and option prices reflect that: a put a given distance below the price is often priced with higher implied volatility than a call the same distance above. Track 3 covers this as the volatility smile or skew.

One naming point that trips people constantly. There are four positions, not two, because each instrument can be held or written: long call, long put, short call, short put. The words call and put describe the instrument. The words long and short describe which side of it you are on. Being unclear about which of the four is under discussion makes any conversation about options unresolvable.