Let $C(t) = C(t; S,K,T)$ the price at time $t$ of a plain vanilla call option with maturity $T$ and strike $K$ on an underlying $S$; if for $t_1<t_2$ we have $C(t_1) > C(t_2)$, it could not be true that (choose the correct answer):

  1. The value of $S$ has declined ($S_{t_1} > S_{t_2}$);
  2. The volatility of $S$ has significantly increased;
  3. The owner of a short position on this contract is making more money than if the position were naked (no long position on $S$ too);
  4. The contract is now more in-the-money;
  5. If a Value-at-Risk $X$ with confidence interval $\alpha$ were forecasted in the time horizon from $t_1$ to $t_2$ for a long position on this contract, then a risk manager expected that $C(t_2) - C(t_1) > X$ with probability $\alpha$.

My thoughts:

  1. Price decreases, stock decreases: it can be true;

  2. Other things being equal, a significant increase in the volatility would cause a significant increase of the price of the call (going against $C(t_1) > C(t_2)$). However, movements in volatility can always be counterbalanced by movements in the underlying, resulting in a decrease of the price of the stock: it can be true;

  3. if the price of the call decreases and you went short, you are making money: it can be true;
  4. as said before, movements in the stock can always be counterbalanced by movements in volatility: it can be true;
  5. This cannot be true: it is going against the definition of VaR, which states that the loss cannot be greater than a specific value given a confidence interval
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    $\begingroup$ Your fifth point is missing something I think, expects $C(t_2)-C(t_1)$... what exactly? $\endgroup$ Feb 25 '20 at 11:18
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    $\begingroup$ You are right! I have just edited the question, sorry for the inconvenience $\endgroup$ Feb 25 '20 at 11:24
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    $\begingroup$ @PietroScaglione Perhaps you could to share your own ideas, attempts, thoughts about these questions? $\endgroup$
    – Kevin
    Feb 25 '20 at 13:10
  • $\begingroup$ I am not an expert of VaR, but I would have excluded both 1 and 2 as they seem obviously wrong to me, but I'm not able to identify the correct one $\endgroup$ Feb 25 '20 at 13:16

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