In finance, an option is a contract which conveys to its owner, the holder, the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified date, depending on the style of the option.

## Value¶

In Black–Scholes (a) pricing of options, omitting interest rates and the first derivative, the Black–Scholes equation reduces to $$\Theta = - \Gamma$$, “(infinitesimally) the time value is the convexity”. That is, the value of an option is due to the convexity of the ultimate payout: one has the option to buy an asset or not (in a call; for a put it is an option to sell), and the ultimate payout function (a hockey stick (a) shape) is convex – “optionality” corresponds to convexity in the payout. Thus, if one purchases a call option, the expected value of the option is higher than simply taking the expected future value of the underlying and inputting it into the option payout function: the expected value of a convex function is higher than the function of the expected value (Jensen inequality). The price of the option – the value of the optionality – thus reflects the convexity of the payoff function.

## Styles¶

### American options¶

An American option […] may be exercised at any time before the expiration date.

### European options¶

A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time.