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.

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    $\begingroup$ For question 1), it depends on the payoff format. For example, $(aS_T-aK)^+=a(S_T-K)^+$, the price is also changed. On the other hand $(aS_T/(aK)-1)^+=(S_T / K-1)^+$, the price will not change. $\endgroup$ – Gordon Aug 17 '16 at 15:01

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