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Explain volatility associated to the standard deviation

How is volatility associated to the standard deviation of the underlying’ return?

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The real rate at which the underlying grows on average doesn’t influence the value. Certainly, the volatility, associated to the standard deviation of the underlying’s return, does issue. During practice, it’s generally much, much harder to estimate such average growth than the volatility; therefore we are rather spoiled in derivatives which we only require to estimate the relatively stable parameter, volatility2.

The purpose that this is true is that by hedging an option with the underlying we remove any exposure to the direction of the stock, whether this goes up or down ceases to matter. Through eliminating risk in this way we also remove any dependence upon the value of risk. End result is that we may as well see we are in a world wherein no one values risk at all, and all trade-able assets grow at the risk-free rate upon average.

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