Because options prices can be modeled mathematically with a model such as the Black-Scholes, many of the risks associated with options can also be modeled and understood. This particular feature of options actually makes them arguably less risky than other asset classes, or at least allows the risks associated with options to be understood and evaluated. Individual risks have been assigned Greek letter names, and are sometimes referred to simply as "the Greeks."
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Volatility also increases the price of an option. This is because uncertainty pushes the odds of an outcome higher. If the volatility of the underlying asset increases, larger price swings increase the possibilities of substantial moves both up and down. Greater price swings will increase the chances of an event occurring. Therefore, the greater the volatility, the greater the price of the option. Options trading and volatility are intrinsically linked to each other in this way.
The price at which you agree to buy the underlying security via the option is called the "strike price," and the fee you pay for buying that option contract is called the "premium." When determining the strike price, you are betting that the asset (typically a stock) will go up or down in price. The price you are paying for that bet is the premium, which is a percentage of the value of that asset.