Overview of Exotic Options

Exotic options come up when a company chooses an option to manage its currency exposure, or when a financial institution values and hedges an option it has sold to a client. For those who check valuations, they also come up when explaining why a price changes with the model.

How exotic options differ from vanilla options

Among options, the right to buy the underlying asset at the strike price is a call option (call), and the right to sell it is a put option (put). These standard forms are called vanilla options. An exotic option differs from a vanilla option in at least one of the following: how the price moves before expiry, the shape of the payoff, the number of underlying assets it refers to, or how the payoff is converted into the currency of payment.

The Federal Reserve Board's manual for examiners (the February 1998 edition) lists exotic options as instruments traded over the counter (OTC derivatives). OTC options are generally tailored to meet the user's needs¹².

For a vanilla call that is a European option, which can be exercised only at expiry, the payoff is the difference between the price of the underlying asset at expiry and the strike price (0 if the difference is negative). Whatever path the price takes before expiry, the payoff is the same as long as the price at expiry is the same³. The basics of calls and puts are covered in Option Basics: Calls and Puts.

Example: payoffs that change with the path

An option whose payoff depends not only on the price at expiry but also on the prices the underlying passed through before expiry is called a path-dependent option³. One whose payoff is set by the average price over the period is called an Asian option, and one whose payoff is set by the highest or lowest price over the period is called a lookback option¹.

Consider a hypothetical example in which a call with a strike price of 100 and a term of 1 year has the price of the underlying asset observed four times, once every 3 months (the last observation is on the expiry date). Paths A and B both end at 108 at expiry, but they move differently along the way. The Asian option pays the average of the four observations minus the strike price. The lookback option is the type with a fixed strike price, and it pays the highest of the four observations minus the strike price⁴.

The vanilla option pays 8 on either path, because the price at expiry is 108. The Asian option pays 6 on path A and 5 on path B, because the average is 106 on path A and 105 on path B. The lookback option pays 15 on path A, which uses the high of 115 reached along the way, and 8 on path B, whose highest price is the 108 at expiry.

How exotic options are grouped

Exotic options can be grouped by which part of the vanilla option they change.

Asian options are also called average price options¹. The average is usually an arithmetic average, and in practice it is taken over the prices on set dates⁵.

Besides the type with a fixed strike price, there is also a lookback option that, in the case of a call, gives the right to buy at the lowest price over the period⁶.

A barrier option is an option whose right comes into existence or disappears depending on whether the price of the underlying asset touches a set level (the barrier)¹.

A digital option (binary option) pays a fixed amount set in advance if the price at expiry is above the strike price, and nothing if it is below. The amount is the same no matter how far above the strike price it ends up¹.

A basket option is an option on, for example, the sum of the prices of several underlying assets⁷. There are also options on the maximum or minimum of several prices, and options on the difference between two prices⁸.

A quanto option is an option on an underlying asset priced in a foreign currency, whose payoff is converted into the domestic currency at an exchange rate set at the start of the trade⁹¹⁰.

In addition, a Bermudan option, which can be exercised on several dates set in advance, is sometimes counted as an exotic option¹. Exotic products specific to interest rates, foreign exchange, and equities are not included in the types listed here.

Why they are used to fit users' needs

One reason exotic options are used is that they can keep the premium down. A barrier option that can disappear before expiry has a lower premium than a vanilla option with the same strike price and expiry. An Asian option also tends to have a lower premium, because using an average narrows the range of price moves¹. A 2007 Bank of Japan commentary describes a view in the market from a period when exchange rates moved little. The view was that importers and exporters were increasingly buying currency options that disappear when the exchange rate touches a set level, to save on premiums¹¹. When the rate touches that level, however, the option disappears and the firm loses the effect of its hedge. To cover itself after the option disappears, the firm has to arrange its currency position again, for example by buying another option or entering into a forward exchange contract¹².

Matching the payoff to the shape of the user's income or costs is another reason. For a firm that receives sales proceeds or pays for purchases at regular intervals, an Asian option set by the average over the period fits the item being hedged more closely than a vanilla option set by the price at a single point in time⁵. A quanto option lets the buyer receive, in the domestic currency, a payoff linked to the price moves of foreign stocks or commodities without being affected by exchange rate changes⁹. Because the exchange rate is fixed, any gain from a favorable move in the exchange rate is not included in the payoff either.

Investors can also take the selling side of options. Structured products combine loans, deposits, or bonds with swaps or options, and many of them aim for a higher return using the premium the investor receives in return for selling options¹³. In exchange for receiving the premium, the investor takes on a corresponding risk¹¹. Many structured products have very low market liquidity, and the price when selling before maturity can be well below the theoretical price¹⁴. How structured products are put together is covered in What Are Structured Products?.

Why valuation is difficult

Exotic options are hard to value because market prices cannot be used as they are. Many exotic options trade in markets less liquid than those for vanilla options. A quoted price may not be one at which the position can actually be closed out¹. Barrier options also have lower market liquidity and less price transparency than vanilla options¹⁵.

Instead, the price of an exotic option is estimated with a model fitted to the volatility derived from the prices of liquid vanilla options. The price depends on the model used¹⁶. A vanilla option's payoff depends only on the price at expiry, so models that give the same distribution for the price at expiry give the same vanilla option prices. An exotic option's payoff depends on the path, so models with different assumptions about how the price moves before expiry give different prices¹⁷.

The calculation method depends on the type of exotic option. In a 2002 paper from the Institute for Monetary and Economic Studies, practice chooses analytic solutions that can be written as formulas first, then lattices (trees) or finite difference methods, and the Monte Carlo method last. The Monte Carlo method generates many paths for the price of the underlying asset and averages the present values of the payoffs on each path¹⁸.

Lattices and finite difference methods are not usually used for path-dependent options that can be exercised only at expiry. For options on several underlying assets, lattices and finite difference methods can handle three underlying assets at most. For basket options on many underlying assets, the Monte Carlo method is the usual choice¹⁸. On the other hand, some path-dependent types can be solved with formulas, such as barrier and lookback options that watch the underlying price all the time (continuous monitoring)¹⁹.

Relationships between several values also enter the valuation. For a basket option, the paths of the underlying prices are generated using the correlations between the price moves of the stocks (the variance-covariance matrix)⁷. The price of a quanto option changes with the correlation between the exchange rate and the price of the underlying asset²⁰.

Why hedging is difficult

A financial institution that sells exotic options buys and sells options to meet its clients' needs, and then hedges the risk of the resulting position¹¹. The risk of exotic options is usually managed by trading the underlying asset and vanilla options²⁴.

Figure description

The client (a firm or investor) pays the premium to the financial institution acting as the dealer, and the dealer sells the exotic option to the client. To reduce the risk of the option it sold, the dealer buys and sells the underlying asset and vanilla options with counterparties such as other financial institutions. The "Hedges" in the figure are the underlying asset and vanilla options, and the figure shows the case where the dealer buys them from the market and pays for them. When it sells them, the directions are reversed.

There are two ways to build a hedge: dynamic hedging and static hedging. Using barrier options as an example, both have problems. Dynamic hedging recalculates the option's sensitivities whenever the market moves and changes the position in the underlying asset to match. It requires frequent rebalancing, and trading costs become large¹⁵.

Static hedging builds a combination of vanilla options at the start of the trade and does not rebalance it, except to unwind it when the barrier is touched and the right disappears¹⁶. A combination built on the assumption that volatility does not change leaves those changes out, so the hedge error becomes large. A combination that also takes changes in volatility into account requires many illiquid options, which makes it hard to put in place at all¹⁵.

For a knock-out barrier option, which disappears when the barrier is touched, the amount of hedge trading grows as the price of the underlying asset approaches the barrier, and the direction of trading can reverse sharply²⁵.

Notes

1. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual" (February 1998), Section 4330.1 "Options," "Exotic Options" (page 2; page 40 of the PDF). ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹

2. The same manual, Section 4330.1, "Description of Marketplace" (page 3; page 41 of the PDF). ↩

3. Mark Broadie and Jérôme Detemple, "Option Pricing: Valuation Models and Applications" (Management Science 50(9), 2004), Appendix A.2 "Path-Dependent Options" (page 1170). The paper defines options whose payoff depends not only on the price at the time of exercise but also on earlier prices. ↩ ↩²

4. Steven Kou, "Discrete Barrier and Lookback Options" (2008), Section 1 "Introduction" (page 343) and Section 1.1 (page 344). These are the payoff formulas for the Asian and lookback calls. ↩

5. Broadie and Detemple, "Option Pricing: Valuation Models and Applications," Appendix A.2.3 "Asian Options" (page 1171). ↩ ↩²

6. Kou, "Discrete Barrier and Lookback Options," Section 1.1 "Barrier and lookback options" (page 344). ↩ ↩²

7. Tatsuya Ishikawa and Yoshihiko Uchida, "モンテカルロ法によるプライシングとリスク量の算出について" (Pricing and risk measurement with the Monte Carlo method), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (June 2002, in Japanese), Section 5 (1) "バスケット型オプションのプライシング" (Pricing basket options) (page 66). The paper uses a call on the sum of the prices of 32 stocks as an example. ↩ ↩²

8. Broadie and Detemple, "Option Pricing: Valuation Models and Applications," the appendix section "Multiasset Options" (page 1172). ↩

9. Martin Haugh, "Foreign Exchange, ADR's and Quanto-Securities" (Columbia University, IEOR E4707 lecture notes, 2013), Section 4 "Quanto-Securities" (page 8). ↩ ↩²

10. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual," Section 4345.1 "Equity Derivatives," note 1 (page 1; page 57 of the PDF). ↩

11. Bank of Japan Financial Markets Department, "通貨オプション取引の増加と市場取引への影響" (The increase in currency option trading and its effect on market transactions) (Bank of Japan Review, November 2007, in Japanese), page 2, "2007年前半にかけての取引への影響" (Effects on trading through the first half of 2007), and Chart 2. ↩ ↩² ↩³

12. The same paper, pages 3–4. ↩

13. 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) (reference material 2 for the basic seminar, July 2019), page 3, "１．仕組商品とは" (1. What structured products are). ↩

14. The same material, page 30, "（１）購入前の検討（続き）" ((1) Considerations before purchase (continued)). ↩

15. Yuki Shinbara, "確率局所ボラティリティ・モデルのもとでのヘッジ戦略" (Hedging strategies under stochastic local volatility models), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2011, in Japanese), Section 1 "はじめに" (Introduction) (page 126). ↩ ↩² ↩³

16. The same paper, Section 2 (1) ロ "エキゾチック・オプションの時価評価と静的ヘッジ" (Fair valuation and static hedging of exotic options) (pages 131–132). ↩ ↩² ↩³

17. The same paper, Section 2 (1) (pages 133–134) and note 15. ↩

18. Tatsuya Ishikawa and Yoshihiko Uchida, "モンテカルロ法によるプライシングとリスク量の算出について" (in Japanese) (2002), Appendix 3 "派生商品のプライシングを行う際の数値計算手法" (Numerical methods for pricing derivatives) (pages 85–86). The paper gives its examples of using the Monte Carlo method for European-type options. ↩ ↩²

19. Kou, "Discrete Barrier and Lookback Options," Section 1.1 (page 345). The chapter says that many models assume continuous monitoring because it gives analytic solutions, and it cites the literature that gives formulas for continuously monitored lookback and barrier options. ↩

20. Haugh, "Foreign Exchange, ADR's and Quanto-Securities," "Quanto Options" (page 9). The correlation between the exchange rate and the price of the underlying asset enters the dividend yield term of the pricing formula. ↩

21. Broadie and Detemple, "Option Pricing: Valuation Models and Applications," Section 5.1.1 "Path-Dependent Options" (pages 1165–1166). ↩

22. Yuki Shinbara, "確率局所ボラティリティ・モデルのもとでのヘッジ戦略" (in Japanese), Section 2 (1) イ "オプションのデルタ・ヘッジ" (Delta hedging of options) (page 128). ↩

23. The same paper, note 11 (page 132). ↩

24. The same paper, Section 1 "はじめに" (Introduction) (page 127). ↩

25. Bank of Japan Financial Markets Department, "通貨オプション取引の増加と市場取引への影響" (in Japanese), the box "金融機関のノックアウト・オプションに関するデルタ・ヘッジ" (Delta hedging of knock-out options by financial institutions) on page 4. ↩