Please provide steps to justify the below.
1) If the stock prices, strike and other price related parameters are scaled by the same constant, will the derivative price scale accordingly? I would think this is different from using a constant as a numeraire, since a numeriance does not affect the strike price. Please confirm.
2) Would scaling by a constant lead to any issues (or a different price), when pricing vanilla options?
3) Are there any derivatives (exotics) whose price is affected when scaling by a constant? One example, is a variance swap as suggested by user @Mats Lind
Related Question: Using a Constant as a Numeraire
The rest of standard Geometric Brownian Motion and Black Scholes assumptions apply.