Editing Option style (section)

===Difference in value===
Assuming an arbitrage-free market, a [[partial differential equation]] known as the [[Black-Scholes equation]] can be derived to describe the prices of derivative securities as a function of few parameters. Under simplifying assumptions of the widely adopted [[Black model]], the Black-Scholes equation for European options has a closed-form solution known as the [[Black-Scholes formula]]. In general, no corresponding formula exist for American options, but a choice of methods to approximate the price are available (for example Roll-Geske-Whaley, Barone-Adesi and Whaley, Bjerksund and Stensland, [[binomial options model]] by Cox-Ross-Rubinstein, [[Black's approximation]] and others; there is no consensus on which is preferable).<ref>{{cite web|url=http://www.global-derivatives.com/index.php?id=14&option=com_content&task=view|title=global-derivatives.com|website=www.global-derivatives.com|access-date=12 April 2018}}</ref> Obtaining a general formula for American options without assuming constant [[Stochastic volatility|volatility]] is one of [[List of unsolved problems in finance|finance's unsolved problems]].

An investor holding an American-style option and seeking optimal value will only exercise it before maturity under certain circumstances. Owners who wish to realise the full value of their option will mostly prefer to sell it as late as possible, rather than exercise it immediately, which sacrifices the time value. See [[exercise (options)#Exercise Considerations|early exercise consideration]] for a discussion of when it makes sense to exercise early.

Where an American and a European option are otherwise identical (having the same [[strike price]], etc.), the American option will be worth at least as much as the European (which it entails). If it is worth more, then the difference is a guide to the likelihood of early exercise. In practice, one can calculate the Black–Scholes price of a European option that is equivalent to the American option (except for the exercise dates). The difference between the two prices can then be used to [[calibrate]] the more complex American option model.

To account for the American's higher value there must be some situations in which it is optimal to exercise the American option before the expiration date. This can arise in several ways, such as:

* An [[in the money]] (ITM) [[call option]] on a [[capital stock|stock]] is often exercised just before the stock pays a [[dividend]] that would lower its value by more than the option's remaining time value.
* A [[put option]] will usually be exercised early if the [[underlying]] asset files for bankruptcy.<ref>{{Cite web| title=American versus European Options | url=http://www.bus.lsu.edu/academics/finance/faculty/dchance/Essay16.pdf | archive-url=https://web.archive.org/web/20100721033458/http://www.bus.lsu.edu/academics/finance/faculty/dchance/Essay16.pdf | archive-date=2010-07-21}}</ref>
* A deep ITM [[currency]] option (FX option) where the strike currency has a lower interest rate than the currency to be received will often be exercised early because the time value sacrificed is less valuable than the expected depreciation of the received currency against the strike.
* An American [[bond option]] on the [[dirty price]] of a [[Bond (finance)|bond]] (such as some [[convertible bond]]s) may be exercised immediately if ITM and a [[Coupon (bond)|coupon]] is due.