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.

## Types

## 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

optionto 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 ishigherthan 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.

## Bibliography

*Wikipedia*, July. https://en.wikipedia.org/w/index.php?title=Convexity_(finance)&oldid=1101317587.

*Wikipedia*, September. https://en.wikipedia.org/w/index.php?title=Option_(finance)&oldid=1110213428.

*Wikipedia*, November. https://en.wikipedia.org/w/index.php?title=Option_style&oldid=1119699571#American_and_European_options.