Exercise Styles: European, American and Bermudan

Front-office staff at financial institutions that trade options, and staff who handle swaptions or bonds with early redemption clauses over the counter, look at the exercise style when they check the terms of a trade and the exercise procedure.

Three styles defined by when you can exercise

The times at which the buyer of an option can use the right are set for each product. An option that can be exercised only on the expiry date is a European option, and one that can be exercised at any time from the start of trading to the last trading day is an American option¹. In between is the Bermudan option, which can be exercised on several dates fixed in advance²³.

Consider a hypothetical example in which an option is started one year before expiry.

The Bermudan style takes its name from Bermuda, which lies between Europe and America. If the set of exercise dates holds only the expiry date, the option is European; if it holds every date, the option is American. A Bermudan option lies between the two⁴.

Figure description

This figure lays out the exercise dates of a Bermudan option started one year before expiry. The start date is not an exercise date. The spacing of the events is not proportional to elapsed time.

The style sets only when the option can be exercised. It is decided separately from other terms, such as whether the option is a call option (call) or a put option (put) and what the strike price is. How calls and puts work is covered in Option Basics: Calls and Puts.

Examples among listed options

On Japanese exchanges, options on stock indexes and on individual stocks are European. On the Osaka Exchange, the options on three indexes (Nikkei 225 options, TOPIX options and JPX-Nikkei Index 400 options) and gold futures options can be exercised only on the day after the last trading day of each contract month⁵. An index option that is exercised is settled by paying or receiving the difference between the strike price and the SQ value (the special quotation, calculated from the opening prices of the component stocks on the day after the last trading day)⁶. Securities options, which are options on individual stocks, are also European, and exercising one leads to buying or selling the underlying shares⁷.

JGB futures options are American. They can be exercised at any time up to the last trading day, and exercise creates a trade in JGB futures⁸. However, according to a primer on JGB futures options published by Japan Exchange Group, they are rarely exercised before expiry, and some practitioners regard them as European options in practice⁹. The idea that exercising before expiry gives up time value is covered later, in the section on exercising early or waiting, with an example of stock options.

In the United States, products on the same exchange can have different styles depending on the underlying asset. Cboe's S&P 500 Index options (SPX) are European and, as a rule, can be exercised only on the expiry date¹⁰. Equity options, on the other hand, are American and can be exercised on any business day up to the expiry date¹¹. Japan's securities options and US equity options both have individual stocks as the underlying asset, yet their exercise styles differ.

OTC options and the Bermudan style

For options traded as OTC derivatives, the two parties can agree on the exercise style between themselves. A swaption is an option that gives the right to enter into an interest rate swap³. The exercise styles for swaptions include European, American and the Bermudan style in between, and investors can choose one to suit their needs².

Early redemption clauses attached to bonds can also be exercisable on several fixed dates. One example is a bond that includes the issuer's right to redeem early on each semiannual coupon date. In this bond, the investor sells the right to redeem early to the issuer and receives an option premium in return¹².

Because its exercise dates are fixed in advance, the right to redeem early on each coupon date is Bermudan by definition³. Naming differs across sources, however. A New York University lecture note calls a bond right that can be exercised on a set range of dates, such as coupon dates, American¹³.

Exercise early or wait

For the same underlying asset, strike price and expiry, an American option is worth at least as much as a European one. The buyer of an American option can wait until expiry and receive the same payoff as with a European option, and also has the choice of exercising early¹⁴¹⁵. A Bermudan option is the same as a European one if its only exercise date is expiry, and it comes closer to an American option as the exercise dates are spaced more finely¹⁵.

Whether exercising before expiry pays off depends on the underlying asset and the type of option. For an American call on a stock that pays no dividends, selling the option in the market brings in more than exercising it before expiry¹⁴. Consider a hypothetical example in which the strike price is 100 yen and the stock price is 120 yen.

In this example, exercising brings in the intrinsic value (what the option is worth if exercised now) of 20 yen, while selling brings in 23 yen. This is because, when interest rates are positive and there are no dividends, a call is priced above its intrinsic value before expiry¹³. The 3-yen difference is time value, and exercising before expiry gives it up. For this reason, there is no reason to exercise a call on a stock without dividends early, and the American and European options have the same value¹⁴.

In some cases, however, exercising early is the better choice. One is a call on an underlying asset that pays dividends or interest. Exercising the call means paying the strike price early and giving up the interest it could have earned, but in exchange the buyer receives the dividends or interest paid before expiry. Early exercise can pay off when those dividends or interest are larger than the interest given up and the price of the underlying asset is well above the strike price (the call is deep in the money)¹³.

The other is a put. What a put can bring in is capped at the strike price, so when the stock price has fallen sharply, there is little left to gain by waiting. For example, with a put at a strike price of 100 yen and a stock price of 5 yen, exercising now brings in 95 yen. Waiting until expiry brings in at most 100 yen, so the most there is to gain is 5 yen. The 95 yen received now earns interest until expiry, so early exercise can be the better choice even on a stock without dividends¹⁴.

Binomial Trees shows how to follow the exercise decision in numbers, using the valuation of an American put as an example.

Exercise procedures and assignment

For options on Japanese exchanges, the buyer does not need to take any special step to exercise at expiry. Option series that have not been sold or bought back by the last trading day are exercised automatically if they are in the money (exercising now would make a profit). The exception is when the buyer abandons the option, deciding not to use the right. If the option is out of the money (exercising now would cause a loss), the position lapses unless the buyer exercises it⁶. The same treatment applies to index options, securities options and JGB futures options⁷⁸.

When an American JGB futures option is exercised before expiry, the seller to be assigned is chosen by lottery. The exchange's rules call the sellers' positions short open interest; these are split into trading units and picked at random⁹. For US equity options, an exercise notice given on a business day results in delivery of the stock on the next business day¹¹.

Notes

1. Japan Exchange Group, "ヨーロピアンタイプオプション（用語集）" (Glossary: European type option, in Japanese). The source uses the terms "ヨーロピアンタイプ" (European type) and "アメリカンタイプ" (American type). ↩

2. Takahiro Hattori, "債券（金利）オプション入門―スワップションについて―" (An introduction to bond (interest rate) options: swaptions, in Japanese), Finance (Ministry of Finance), August 2021, page 51. The source uses the terms "ヨーロピアン・タイプ" (European type), "アメリカン・タイプ" (American type) and "バミューダ・タイプ" (Bermuda type). ↩ ↩²

3. Bank of England, "The Technical Standards (EMIR Reporting and Data Quality and Miscellaneous Amendments) Instrument 2023," Annex A, Table 2, fields 132 and 133 (page 37), and the same fields in Annex B (page 97). ↩ ↩² ↩³ ↩⁴

4. Martin Schweizer, "On Bermudan Options," in K. Sandmann and P. J. Schönbucher (eds.), "Advances in Finance and Stochastics" (Springer, 2002), "0. Introduction." ↩

5. Japan Exchange Group, "よくあるご質問" (Frequently asked questions, in Japanese), futures and options, Q9 "【OSE】権利行使はいつできますか。" ([OSE] When can options be exercised?). ↩

6. Japan Securities Clearing Corporation, "指数オプション取引の決済方法" (How index option trades are settled, in Japanese). ↩ ↩² ↩³

7. Japan Securities Clearing Corporation, "有価証券オプション取引の決済方法" (How securities option trades are settled, in Japanese). ↩ ↩²

8. Japan Securities Clearing Corporation, "国債先物オプション取引の決済方法" (How JGB futures option trades are settled, in Japanese). ↩ ↩² ↩³

9. Takahiro Hattori, "国債先物オプション入門" (An introduction to JGB futures options, Japan Exchange Group, 2022, in Japanese), page 19. ↩ ↩² ↩³

10. Cboe, "S&P 500 Index Options Product Specifications," "Exercise Style." ↩

11. Cboe, "Equity Options Specifications," "Exercise Style" and "Settlement of Option Exercise." ↩ ↩²

12. Center for Advanced Financial Technology, Financial System and Bank Examination Department, Bank of Japan, "仕組商品投資のリスク把握と管理" (Understanding and managing the risks of investing in structured products, in Japanese), the worked example in "２．商品の仕組みとリスク特性" (2. Product structure and risk characteristics), pages 4–7. The source describes the right as "ﾏﾙﾁｺｰﾗﾌﾞﾙ：発行体が利払（半期）毎に期限前解約できる権利" (multi-callable: a right of the issuer to terminate early on each (semiannual) interest payment; our translation) and calls it "期限前解約権" (early termination right). ↩

13. Jennifer Carpenter, "American Options, Callable Bonds, and Swaptions" (lecture notes, New York University Stern School of Business), pages 3, 6 and 7 of the PDF. ↩ ↩² ↩³

14. "American Options" (Lecture 8), lecture notes, Courant Institute of Mathematical Sciences, New York University. For calls, Section 1.1, items 3 and 4 with notes (a) and (b) (pages 1–2); for puts, item 3 of the same section with its note (page 3); and item 1 of the Results in Section 1.3.2 (page 10). ↩ ↩² ↩³ ↩⁴

15. Steven P. Lalley, "Lecture 15: American Options" (lecture notes, University of Chicago), the end of Section 3.1 and Section 3.2 "Bermuda Options" (page 3 of the PDF). ↩ ↩²