# Vanilla option pricing at different points in time

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 $$t_1 we have $$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
• Your fifth point is missing something I think, expects $C(t_2)-C(t_1)$... what exactly? – Daneel Olivaw Feb 25 '20 at 11:18
• You are right! I have just edited the question, sorry for the inconvenience – Pietro Scaglione Feb 25 '20 at 11:24
• @PietroScaglione Perhaps you could to share your own ideas, attempts, thoughts about these questions? – Kevin Feb 25 '20 at 13:10
• 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 – Pietro Scaglione Feb 25 '20 at 13:16