Valuation

The premium for an option contract is ultimately determined by supply and demand, but is influenced by five principal factors:

The price of the underlying security in relation to...

The strike price. Options will be in-the-money when there is a positive intrinsic value; when the strike price is above/below (put/call) the security's current price. They will be at-the-money when the strike price equals the security's current price. They will be out-of-the-money when the strike price is below/above (put/call) the security's current price. Options at-the-money or out-of-the-money have an intrinsic value of zero.

The cumulative cost required to hold a position in the security (including interest + dividends).

The time to expiration. The time value decreases to zero at its expiration date. The option style determines when the buyer may exercise the option. Generally the contract will either be American style - which allows exercise up to the expiration date - or European style - where exercise is only allowed on the expiration date - or Bermudan style - where exercise is allowed on several, specific dates up to the expiration date. European contracts are easier to value. Due to the "American" style option having the advantage of an early exercise day (i.e. at any time on or before the options expiry date), they are always at least as valuable as the "European" style option (only exercisable at the expiration date).

The estimate of the future volatility of the security's price. This is perhaps the least-known input into any pricing model for options, therefore traders often look to the marketplace to see what the implied volatility of an option is -- meaning that given the price of an option and all the other inputs except volatility you can solve for that value.
Pricing models include the binomial options model for American options and the Black-Scholes model for European options. Even though there are pricing models, the value of an option is a personal decision, requiring multiple trade offs and depending on the investment objective. See the Excel model for the metrics of a call option.

Because options are derivatives, they can be combined with different combinations of
other options
risk free T-bills
the underlying security, and
futures contracts on that security

to create a risk neutral portfolio (zero risk, zero cost, zero return). In a liquid market, arbitrageurs ensure that the values of all these assets are 'self-leveling', i.e. they incorporate the same assumptions of risk/reward. In theory traders could buy cheap options and sell expensive options (relative to their theoretical prices), in quantities such that the overall delta is zero, and expect to make a profit. Nevertheless, implementing this in practice may be difficult because of "stale" stock prices, large bid/ask spreads, market closures and other symptoms of stock market illiquidity. If stock market prices do not follow a random walk (due, for example, to insider trading) this delta neutral strategy or other model-based strategies may encounter further difficulties. Even for veteran traders using very sophisticated models, option trading is not an easy game to play.