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I was going through a use case where

At time $t_{t}$, the price of a call option is $C1$ and the price of underlying stock is $S1$

At time $t_{t+1}$ day, the price of a call option is $C2$ and the price of underlying stock is $S2$

Now $S2 > S1$ which means that $C2 > C1$ but on the contrary the $C2 < C1$ ,it turned out that the volatility at $S2$ was lower than the volatility at $S1$.

Say if we have observations of price of call options for various different Strike prices like $K1, K2, K3$ etc, fundamentally, what does the volatility of a stock have to do with the change in Strike prices and time for an option to expire (term structure).

The above information is used to plot a vol surface but I am trying hard to understand how would traders use a vol surface at all ? Please explain in simple terms.

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    $\begingroup$ Think of it this way: if the Black Scholes Merton model was perfectly accurate the Volatility Surface would be flat and stay the same from day to day. So the actual Volatility Surface on any day is very informative about how option prices in the real world differ from the BSM theory. Both in terms of shape and how it moves. Very valuable for all kinds of applications... $\endgroup$ – noob2 May 23 at 12:08
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    $\begingroup$ Would you prefer to trade options based on a theoretical article from the Journal of Political Economy 1971 or this plus data about the actual option market? $\endgroup$ – noob2 May 23 at 12:32