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.

(“Option (Finance)” 2022)



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.

(“Convexity (Finance)” 2022)


American options

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

(“Option Style” 2022)

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.

(“Option Style” 2022)