Barrier Options

Barrier options come up in the work of trading foreign exchange and equity options and of building structured notes. In the seller's hedging work, the handling of the area near the barrier becomes the main issue.

Where the barrier is and what touching it does

The position of the barrier, and whether touching it cancels or creates the right, divide barrier options into four types. A barrier option is an option that sets a price level called the barrier in addition to the strike price. As with standard calls and puts (vanilla options), the payoff at expiry is the difference between the price at expiry and the strike price. Whether that right is in force, however, depends on whether the price of the underlying asset crossed the barrier before expiry¹. Where they sit among other exotic options is covered in Overview of Exotic Options.

The four types combine two distinctions. A barrier above the current price is "up," and one below it is "down." An option whose right disappears when the barrier is touched is a knock-out ("out"), and one whose right comes into existence only when the barrier is touched is a knock-in ("in"). A knock-out that reaches expiry without touching the barrier pays the same as a vanilla option with the same strike price and expiry, and pays nothing if the barrier is touched. A knock-in works the other way round: it becomes a vanilla option only if the barrier is touched¹.

Each of the four types in the table exists as a call option (call) and as a put option (put), so there are eight combinations¹. The examples below take the form of a European option, which can be exercised only at expiry.

Some contracts set a fixed amount to be paid to the buyer when the barrier is touched. This amount is called a rebate. In principle it can be attached to any type, but in practice rebates are attached only to knock-outs, as compensation to the buyer whose right has disappeared¹.

A rebate is normally paid when the barrier is touched, but some contracts defer the payment to expiry². Adding a rebate raises the premium the buyer pays by the value of the rebate³. The value of a rebate paid when the barrier is touched is found as the value of an option that pays a fixed amount when the barrier is touched⁴. How options that pay a fixed amount work is covered in Digital Options.

Figure description

The payments between the buyer and the seller of a knock-out with a rebate. The buyer pays the premium to the seller at the start of the trade. If the barrier is not touched before expiry, the seller pays the buyer the same amount as a vanilla option at expiry. If the barrier is touched and the right disappears, the seller pays the buyer the agreed rebate instead of the payoff at expiry. The rebate is normally paid when the barrier is touched, but some contracts defer it to expiry.

The payoff at expiry depends on the path to expiry

The payoff of a barrier option can differ even when the price at expiry is the same, depending on how the price moved before expiry (the path). An option whose payoff depends on the path in this way is called a path-dependent option⁵.

Consider a hypothetical example in which an up-and-out call and an up-and-in call have the underlying at a current price of 100, a strike price of 100, a barrier of 120, 1 year to expiry, and no rebate. The payoffs at expiry along four paths are as follows.

Paths A and B both end at 110, but B touched 120 along the way, so the up-and-out's right has disappeared and the up-and-in's right has come into existence. On path C, the price at expiry is below the strike price, and either option pays 0. On path D, the up-and-out's right has disappeared, and it pays 0. An up-and-out call pays only when the price at expiry is below 120, so its payoff is always less than 20 (120 − 100) per unit. A knock-out like this one, with the barrier on the side where exercising would make a profit (the in-the-money side), is also called a reverse knock-out⁶.

In-out parity

A knock-in plus a knock-out equals a vanilla option. Suppose you hold both a knock-in and a knock-out that can be exercised only at expiry (European), of the same kind (call or put), with the same strike price, barrier, and expiry. Whether or not the barrier is touched, one of the two remains as a vanilla option. The sum of their two prices therefore equals the price of a vanilla option with the same terms⁷. This relationship is called in-out parity⁸. In the table in the previous section too, the payoffs of the up-and-out and the up-and-in add up to the payoff of the vanilla call on every path. The relationship holds when there is no rebate⁹.

$$\text{Price of the knock-in} + \text{Price of the knock-out} = \text{Price of the vanilla option}$$

Take the example from the previous section (current price 100, strike price 100, 1 year to expiry), add an interest rate of 1% and volatility of 20%, and assume the barrier is checked all the time. The prices are as follows¹⁰.

The up-and-out call is priced at 1.12, less than one-seventh of the vanilla call's 8.43. That is because its right disappears on the paths where the price of the underlying rises and the call becomes more valuable. The down-and-out call is priced at 6.88, close to the vanilla option. Its right disappears only if the price of the underlying falls to 90, on paths where the call was worth little to begin with⁷. In both examples, the knock-out and the knock-in are cheaper than the vanilla option, and the difference is the value of the paths on which the right disappears (or never comes into existence)³.

What barrier options are used for

According to a 2011 paper from the Bank of Japan's Institute for Monetary and Economic Studies, barrier options are traded in the foreign exchange and equity markets, both on their own and embedded in structured notes, and are thought to have the largest trading volume among exotic options¹¹. The Federal Reserve Board's 1998 manual for examiners lists barrier options as one of the main types of exotic OTC derivatives¹².

The paper by Derman and Kani gives reasons for using them such as the following. The buyer can keep the premium down by not paying for paths that the buyer considers unlikely. And by choosing a form whose right disappears when the hedge is no longer needed, the buyer can match the option more closely to the purpose of the hedge³.

Consider a hypothetical example in which Company B bought the underlying asset at 100 and holds it. It has decided to sell the underlying if the price rises to 105, and until then it wants protection against a fall below 95. Instead of a vanilla put with a strike price of 95 and 1 year to expiry, Company B buys an up-and-out put with a strike price of 95 and a barrier of 105, because once it sells the underlying at 105, it no longer needs the put either³. Under the same conditions as the previous section (interest rate 1%, volatility 20%, barrier checked all the time), the premium is 5.11 for the vanilla put and 2.71 for the up-and-out put¹⁰. The profit or loss per unit on the underlying and the put combined varies by path as follows. The premiums are rounded to two decimal places, and interest on the premium is left out.

On the first three paths, the up-and-out put comes out ahead by 2.40, the amount of premium saved. On the third path, the result for the vanilla put leaves out the remaining value of the put. On the other hand, if Company B keeps holding the underlying after the price touches 105 and the price then falls, the put has already disappeared, and the company bears the whole loss from the fall. On the fourth path, the result is −22.71, where a vanilla put would have held it to −10.11.

On the seller's side, one use is to sell barrier options that require a payment only on paths the seller considers unlikely, and to collect the premium³. If such a path does occur, the seller owes the same payment as if it had sold a vanilla option.

Continuous and discrete monitoring

Checking whether the barrier has been touched at all times until expiry is called continuous monitoring, and checking it only at set times (such as each day's closing price) is called discrete monitoring. According to a 1997 paper by Broadie, Glasserman, and Kou, many pricing formulas assume continuous monitoring, while a considerable share of actual contracts specify that the barrier is checked at set times⁵. The same paper also notes that when a contract does not fix the monitoring times, the barrier may be deemed touched by a price in a less liquid market while the main markets are closed¹³.

Under discrete monitoring, a path that crosses the barrier and comes back between monitoring times counts as not having touched it. The barrier is therefore touched less often than under continuous monitoring, so a knock-out is priced higher and a knock-in lower. Even with daily monitoring, the price can differ substantially from the continuous-monitoring price¹³. The Broadie-Glasserman-Kou correction approximates the discrete-monitoring price with the continuous-monitoring formula. It shifts the barrier away from the current price of the underlying, multiplying it by $e^{0.5826\,\sigma\sqrt{\Delta t}}$ for an up barrier and by $e^{-0.5826\,\sigma\sqrt{\Delta t}}$ for a down barrier, and then prices the option⁵. Here $\sigma$ is the volatility and $\Delta t$ is the interval between monitoring dates (in years).

The up-and-out call from the table in the in-out parity section (strike price 100, barrier 120) is priced at 1.12 with continuous monitoring. With 50 monitoring dates before expiry, the correction raises the barrier to 121.99, and the price becomes 1.46. With 200 monitoring dates it is 1.29, so the more monitoring dates there are, the closer the price gets to the continuous-monitoring price¹⁰. In the paper's numerical examples, the accuracy of the corrected approximation declines slowly as the number of monitoring dates decreases¹⁴.

Hedging is hard near the barrier

The seller of a barrier option can offset its risk by trading the underlying asset. How much the value of an option moves for a change in the price of the underlying is called delta, and a hedge that holds the underlying in line with the delta is called a delta hedge¹⁵. Near the barrier, the delta changes sharply, so the hedging trades become large.

With an up-and-out call, a rise in the price of the underlying makes a larger payoff at expiry more likely and, at the same time, makes it more likely that the right disappears³. Under the conditions of the in-out parity section, raising the price of the underlying from 100 to 110 raises the price of the vanilla call from 8.43 to 14.94, but lowers the price of the up-and-out call from 1.12 to 0.74¹⁰. As the price nears the barrier, the delta turns from positive to negative¹⁶. The seller then has to sell the underlying it had bought for the hedge and go on to take a short position¹⁵. When the price of the underlying is just short of the barrier, a small move across the barrier makes the delta jump, so the position cannot be hedged continuously¹⁷.

When the delta changes a lot, the hedge has to be rebalanced often, and trading costs mount¹⁷. In practice, trades cannot be made at arbitrarily short intervals, so hedging errors also grow when rebalancing cannot keep up¹⁸. One approach studied to ease this problem is static hedging, which holds a combination of vanilla options set up at the start of the trade and does not rebalance it afterward¹⁹.

The seller's trading can also affect market prices. A 2007 Bank of Japan commentary explains this with the example of a financial institution that sold an option to sell yen and buy dollars, with a strike price of 115 yen and a knock-out barrier of 125 yen. As the yen weakens, the institution at first buys more dollars, but as the price approaches the barrier, it switches to selling dollars. When the barrier is touched and the option disappears, the institution rushes to close that short position. When many options share the same barrier, this could amplify short-term swings in the exchange rate¹⁵. The shorter the time to expiry, the more the delta changes and the larger the hedging trades become²⁰.

Notes

1. Emanuel Derman and Iraj Kani, "The Ins and Outs of Barrier Options: Part 1" (Derivatives Quarterly, Winter 1996), "BARRIER OPTIONS DEFINED" and Exhibit 1 (page 56). ↩ ↩² ↩³ ↩⁴

2. Emanuel Derman and Iraj Kani, "The Ins and Outs of Barrier Options: Part 2" (Derivatives Quarterly, Spring 1997), page 78. The explanation of rebates paid when the barrier is touched (non-deferred) and rebates deferred to expiry (deferred). ↩ ↩² ↩³

3. Derman and Kani, "The Ins and Outs of Barrier Options: Part 1," "WHY USE BARRIER OPTIONS?" and "SPECIAL FEATURES OF BARRIER OPTIONS" (pages 56–57). ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

4. Derman and Kani, "The Ins and Outs of Barrier Options: Part 2," page 74. It values a USD 1 rebate as the value of a binary up-and-in call or a binary down-and-in put. ↩

5. Mark Broadie, Paul Glasserman, and Steven Kou, "A Continuity Correction for Discrete Barrier Options" (Mathematical Finance, October 1997), the abstract and Section 1.1 (page 325). ↩ ↩² ↩³

6. 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), note 5 (page 6). ↩

7. Derman and Kani, "The Ins and Outs of Barrier Options: Part 1," pages 58–59. ↩ ↩²

8. Jr-Yan Wang (National Taiwan University), lecture notes "Ch 8. Barrier Option," Chapter 8, page 4 ("In-out parity"). The relationship holds for European options when the knock-in and the knock-out have the same barrier. ↩

9. Derman and Kani, "The Ins and Outs of Barrier Options: Part 2," page 80. The explanation of a knock-in and a knock-out with zero rebate. ↩

10. Values computed with the Black-Scholes model and the continuous-monitoring formula, with no dividends, and rounded to two decimal places. The formulas are in the appendix from page 77 of Derman and Kani, Part 2, and discrete-monitoring values shift the barrier. ↩ ↩² ↩³ ↩⁴

11. 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：最尤経路を利用したバリア・オプションの静的ヘッジ" (Hedging strategies under stochastic local volatility models: static hedging of barrier options using the most likely path), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2011), "1. はじめに" (1. Introduction), page 126 (in Japanese). ↩

12. 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). ↩

13. Broadie, Glasserman, and Kou, "A Continuity Correction for Discrete Barrier Options," Section 1.1 (pages 325–326). ↩ ↩² ↩³

14. Broadie, Glasserman, and Kou, "A Continuity Correction for Discrete Barrier Options," Section 2 (page 331). Table 2.2 (number of monitoring points and accuracy of the approximation). ↩

15. 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) (page 4) and the main text (pages 3–4). ↩ ↩² ↩³

16. Derman and Kani, "The Ins and Outs of Barrier Options: Part 1," the explanation of Exhibit 10 (page 63). ↩ ↩²

17. Derman and Kani, "The Ins and Outs of Barrier Options: Part 1," "INTERPRETING THE PLOTS" (page 60). ↩ ↩² ↩³

18. 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：最尤経路を利用したバリア・オプションの静的ヘッジ" (in Japanese), the explanation of Assumption A in Section 3 (1) (page 147). ↩

19. 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：最尤経路を利用したバリア・オプションの静的ヘッジ" (in Japanese), Section 3 (2) "静的ヘッジ" (Static hedging) (pages 148–149). ↩

20. 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), page 5 of the box. ↩