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Why call option price increases with higher volatility
I agree that high volatility just means the underlying stock price fluctuates more, and it does not imply if the stock is going up or down. But a high volatility in the price of an underlying also means that there is a higher chance that the underlying price could reach extreme prices (albeit in either direction). However, if you purchased a call option then if the underlying price reached an extremely high value, then you will be richly rewarded. But if the underlying price reached an extremely low value, you won't lose any more than the initial premium that you paid. There is no additional risk on your side, it's capped to the premium that you paid for the call option. It's this asymmetric outcome (Heads - I win, Tails - I don't lose) combined with high volatility that means that call options will increase in value when the underlying price becomes more volatile. If the optionality wasn't there then the price wouldn't be related to the volatility of the underlying. But that would be called a Future or a Forward :-)