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In the case of a plain-vanilla (non exotic) option issued at fairmarket value (i.e., strike = stock price. If stock > strike, it’s adiscounted option. If strike > stock, then it’s a premium option),the holder of a call option has the right, but not the obligation, toexercise and receive the asset (stock):

Noticesix factors impact the Black-Scholes-Merton stock option value(technically, five incorporate into the original model and dividendsare treated by adjusting the model). Also,

• European option: can only be exercised at maturity
• American option: can be exercised prior to maturity (but remember: in theory, it is never optimal to exercise a American option on a non-dividend paying stock before maturity. If the stock pays dividends, it might be optimal just before the stock goes ex-dividend).

Atany given point in time, prior to expiration, we can decompose the fairvalue of the option into two pieces: intrinsic value and a time value.Intrinsic value is the immediate value, if the option were exercised: =MAX[0,S-X]. Time value is the value that derives from the potential forfurther upside (or, you may think of it as the value you would get ifyou were to re-price the option, in the moment, as if the strike pricewere reset to the current stock price. You would be holding a "new"fair value option with no intrinsic value and a stub period forremaining life - it is worth something as long as there is liferemaining).

Hereis a classic chart of the factor influences on the Black-Scholes optionvalue. For example, more time increases the value of American calls andputs.

Mostof these are likely intuitive. The relationship between time to expireand option value is given by theta. As mentioned, theta is negative forAmerican options (because the passage of time leads to less valuableoptions) but it’s not always negative for European options. It’s usually negativefor European options, but if you hold an at-the-money European putoption on a non-dividend paying stock, you can imagine circumstancesunder which holding the option saw an increase in option value even asmaturity approaches.
Here is our tip for remembering the risk-free rate and the dividend yield:
Rememberthat put-call parity formula is sort of embedded inside theBlack-Scholes model. Specifically, if you ignore the two cumulativenormal distribution functions, you’ve got the minimum value function.In words, the minimum value (lower bound) of the call option is the stock price minus the discounted exercise price:
c = S - Ke-rt
Ifyou know this formula, you can see that a higher interest rate producesa lower discounted exercise price which increases the call optionvalue.
For the dividend yield, you don’t need to store anythingnew in your memory banks! Just go back to the cost of carry readings.The dividend yield is like a convenience yield: the opposite of afinancing cost and, therefore, it works in the opposite direction ofthe risk-less rate.
Dividend yield = Convenience yield (i.e., benefit of owning the asset)
which works in the opposite direction of:
Risk-free rate = Financing cost