FX Barrier Options

Importers and exporters meet this product when they want to lower the premium on an FX hedge. Financial institutions meet it when they manage the price and the hedge of a barrier trade they have sold.

Double knock-outs

A double knock-out is a barrier option with one barrier above and one below the current rate, and the right disappears if either is touched¹. The four barrier types and in-out parity are covered in Barrier Options. The clauses in the FX trading confirmation also treat the type with two barriers alongside the single-barrier types². Conversely, an option that comes into existence only when one of the two barriers is touched is a double knock-in. For the same strike price, maturity, and pair of barriers, the prices of the knock-in and the knock-out add up to the price of the vanilla option³.

The payoff of a double knock-out is never larger than that of a knock-out with the same terms and only one of the barriers. Each added barrier adds paths on which the right disappears. As with a single barrier, the premium is lower by the amount of the right that can disappear⁴. Under the assumptions of the Black-Scholes model, a knock-out with one barrier has a closed-form solution. The solution for the double knock-out is an infinite series in which most terms quickly approach 0¹.

Consider a hypothetical example in which an importer, Company K, plans to buy USD 1 million in 6 months. Company K buys a dollar call (a yen put) with a strike price of 148 yen, a lower barrier of 140 yen, and an upper barrier of 160 yen. The current spot rate is 1 dollar = 150 yen, and the comparison below leaves out the premium and looks only at the payoff.

The third and fourth rows are paths on which a vanilla call would pay 17 million yen and 7 million yen. After the barrier is touched there is no right, so Company K cannot protect itself against a weaker yen. It buys more cheaply than with a vanilla call at the same strike price, in return for giving up the payoff on these paths.

Knock-out forwards

A knock-out forward is a transaction like an FX forward, which buys or sells a currency at a set rate on a set date, with the added condition that the transaction is cancelled if the exchange rate touches the barrier even once. If the barrier is not touched, the currency is bought or sold at the forward rate. If it is touched, there is no settlement.

The idea of removing the premium payment is the same as in zero-cost options⁵. A specific way of building one from knock-out options is derived here from the definition of zero-cost and the barrier option formulas.

The construction is to buy a knock-out call and sell a knock-out put with the same strike price, barrier, and maturity, and to choose the strike price at which the two premiums are equal. Take Company K buying dollars in 6 months. Under the conditions below, with the barrier at 160 yen above the current rate, the strike price comes to about 143.2 yen⁶.

With a strike price of 143.2 yen, the premium per dollar is 2.29 yen for the knock-out call and 2.30 yen for the knock-out put, which nearly cancel each other⁶. The price at which Company K buys is better than the ordinary forward by about 4.2 yen, or about 4.2 million yen on USD 1 million. This advantage corresponds to the value given up if the barrier is touched. At the same strike price, the vanilla call is 6.52 yen and the vanilla put is 2.33 yen. The part that disappears when the barrier is touched is 4.23 yen for the call and 0.03 yen for the put⁶. The 4.23 yen for the call less the 0.03 yen for the put, about 4.2 yen, is almost the same as the improvement in the strike price.

This advantage holds under these conditions with the barrier at 160 yen, above the current rate. If the barrier is placed at 140 yen, below the current rate, the zero-cost strike price is about 153.74 yen, which is worse than the ordinary forward⁶. Also, the values shown here are theoretical values from a closed-form solution with continuous monitoring, not prices of actual trades.

Figure description

This chart shows, per dollar, the profit or loss at maturity of Company K's agreement to buy dollars at 143.2 yen when the 160 yen barrier is never touched during the period. The profit or loss compared with buying dollars in the spot market is a straight line that rises by the same amount as the rate rises. It is 0 at 143.2 yen, a profit if the yen is weaker, and a loss if the yen is stronger. If the 160 yen level is touched during the period, the transaction is cancelled and this chart does not apply.

For Company K's purchase of USD 1 million, the payments by spot rate at maturity and by whether the barrier was touched are as follows. The ordinary forward is at about 147.4 yen, so Company K pays 147.4 million yen.

On paths where the barrier is not touched, the obligation to buy at 143.2 yen remains even if the rate moves toward a stronger yen, and the loss against buying in the spot market grows as the yen strengthens. On paths where it is touched, Company K may need to hedge again after the transaction is cancelled⁷.

A similar structure that uses a knock-in is the forward extra, which combines buying a vanilla option with selling a knock-in option. If the barrier is touched, the option that was sold comes into existence, and the seller is obliged to buy the currency at the strike price⁸. Whether the option disappears or comes into existence on touching the barrier decides in which situations the right or the obligation remains. The TARF, which ends when the accumulated gain reaches a target, is covered in Target Redemption Forward (TARF).

Reverse knock-outs

A reverse knock-out is, for a call, an up-and-out call with the barrier above the strike price¹. Note 5 of the Bank of Japan article deals with reverse knock-outs, which have the barrier on the side where the right can be exercised (the in-the-money side)⁹. The call bought in the knock-out forward has its barrier at 160 yen against a strike price of 143.2 yen, so it is this type.

With this type, the payoff grows as the rate moves in the favorable direction, and drops to 0 the moment the barrier is touched. In Company K's example, holding a reverse call with a strike price of 148 yen and a barrier of 160 yen pays 11.9 yen per dollar if the rate at maturity is 159.9 yen, but pays 0 once 160 yen has been touched during the period. Without the barrier, the payoff would keep growing as the rate rises further from 159.9 yen.

The premium is not refunded as a principle. Some confirmations provide for the premium to be rebated on a reverse knock-out¹⁰.

How barriers are monitored

In the 2005 explanation by the Foreign Exchange Committee (FXC), the clauses used in FX barrier option confirmations fall into two types. In the continuous type (non-discrete, or "spot market"), no particular rate source is specified, and whether the barrier has been touched is decided from the rate derived from actual trades in the global spot market. The decision covers the period when the spot market is open continuously, which begins at 5 a.m. Monday in Sydney and ends at 5 p.m. Friday in New York¹¹.

In the other type, the discrete type, the confirmation specifies a rate source (the Barrier Event Rate Source), and the decision is made only on days when that rate is published. Examples include the reference rate of the currency for major currencies, and the rate used to settle non-deliverable trades for emerging market currencies. If the rate was due to be published but was not, the market practice is to treat that day as a decision day as well and to look for a rate in a liquid spot market¹². With either type, the decision days can be limited to particular days of the week¹².

One-touch payoffs

A one-touch is an option that pays an agreed amount if the rate touches a set level even once during the period. It has no strike price and its payoff is a fixed amount, so among first-generation FX exotics it is classed separately from barrier options that have a strike price¹³. A double one-touch, which pays if either of two levels is touched, can be derived from buying a call spread and a put spread on the double knock-in¹³.

For FX barriers and digital options (one-touch and no-touch), the clauses of the confirmation are shared, and the rules for deciding whether the barrier was touched (continuous or discrete, the rate source, the direction) are the same². A no-touch is an option that pays an agreed amount if the rate never touches the level during the period. The condition is the same as the condition under which a knock-out barrier option does not disappear.

The probability of not touching the barrier, which corresponds to the probability that a knock-out right survives, equals the price of that no-touch divided by the discount factor¹⁴. This probability is used to weaken the smile adjustment in the valuation.

The smile and barrier prices

The Black-Scholes model assumes that volatility is constant until maturity, but in actual markets it changes. According to the revised 2010 paper by Bossens et al., in the FX market the price from this model is used as a reference quote. It lets the parties to a trade confirm they are talking about the same option¹⁵. For barrier options, the widely used method is to add a smile adjustment to the Black-Scholes price¹⁶. The barrier price changes with the method of adjustment.

One such method is the vanna-volga method. It uses the market prices of a highly liquid combination, the ATM option, the risk reversal, and the butterfly. For the combination that cancels the vega, vanna, and volga of the barrier option, it finds the difference between the market price including the smile and the Black-Scholes price. It adds that difference to the Black-Scholes price of the barrier option¹⁷. Vanna is the change in vega when the spot rate moves, and volga is the change in vega when volatility moves. The risk reversal carries mainly vanna, and the butterfly carries mainly volga¹⁷.

When the barrier is near the current rate, vanna is large and volga is small¹⁸. The adjustment is found by multiplying the differences calculated from the risk reversal and butterfly quotes by the option's own vanna and volga and by a weakening factor, and adding the result. Adding the adjustment unchanged makes it too large, so the market practice is to weaken it using the probability of not touching the barrier or the expected time to first exit from the barrier¹⁹. A simple way of setting the strength of this weakening to fit only the one-touch prices agreed well with the result of fitting it jointly to the other products as well²⁰.

The method of valuation changes barrier prices. In the same paper, in the chart comparing one-touch prices, neither local volatility nor the Heston stochastic volatility model gave a satisfactory result. The paper also made a simple count of which model came closer for each product. With its market data, local volatility came closer for more of the products in USD/JPY, which has a large skew. Stochastic volatility did for more of the products in USD/CHF, which has a small skew²¹. A model fitted to the vanilla option market does not always get barrier prices right, because the vanilla market has no information on how the smile will move from here.

Hedging around a knock-out

In FX, the hedging trades are themselves trades in the spot market. In a 2007 Bank of Japan article, a financial institution that sold a customer a reverse knock-out option to sell yen and buy dollars changes the direction of its spot hedge as the barrier nears. When the barrier is touched and the right disappears, it unwinds that position in a hurry⁷.

The actions of customers after the right has disappeared also affect the market. When customers who have lost the right arrange new hedges (buying options or FX forwards), it again works to amplify market moves⁷. If many customers like Company K, who need to redo their trades after the barrier is touched, are at the same level, they may briefly amplify moves around that level.

The size of the effect depends on the time to maturity. The longer the time to maturity, the smaller the swings in the financial institution's spot hedging trades and the more gradual the customers' cover after the right disappears, so the impact on the market is thought to be smaller²³. The same article says that in the first half of 2007 some options to sell yen and buy dollars, with the barrier in the direction of a weaker yen, had apparently disappeared after the barrier was touched. It also reports market participants saying little effect on spot trading was seen²⁴.

Notes

1. Frédéric Bossens et al., "Vanna-Volga methods applied to FX derivatives: from theory to market practice" (arXiv:0904.1074, version 3, May 2010), Section 2 (page 3). Definitions of the reverse and double knock-outs, and the series solution for the double barrier. ↩ ↩² ↩³

2. Foreign Exchange Committee (FXC), "Practice Notes to the 2005 Barrier Option Supplement" (2005 Annual Report), pages 161 and 165, Section 3.9(e)(f)(k)(q). The double-barrier types and the definitions of the upper and lower barrier. ↩ ↩²

3. Bossens et al., Section 2 (page 3) and Section 5 (page 12). The relation that a knock-out plus a knock-in equals the vanilla, and the case of double barriers. ↩

4. Bank of Japan, "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. The premium is lower in return for accepting that the right can be lost. ↩

5. Financial Futures Association of Japan, "金融先物取引の知識" (Knowledge of financial futures trading, in Japanese), April 2015, page 117, "ｄ．ゼロコスト・オプション" (d. Zero-cost options), and page 88. The definition of a trade with no premium exchanged. ↩

6. Calculated with the FX form of the Black-Scholes model, with volatility of 10%, a yen interest rate of 0.5%, a dollar interest rate of 4%, a maturity of 0.5 years, and the closed-form solution for continuous monitoring (no rebate), rounded to two decimal places. Interest is continuously compounded, and the ordinary forward is 150 × e^((0.005 − 0.04) × 0.5) = about 147.40 yen. ↩ ↩² ↩³ ↩⁴

7. Bank of Japan, "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, in Japanese), pages 3–4 and the box on page 4. The seller's change in hedge direction, and the amplification of moves by customers who hedge again after losing the right. ↩ ↩² ↩³

8. John Mark Caruana et al., "Built-in Premium Effects within Exotic FX Options" (2017), "Constructing a Forward Extra" (page 40) and page 41. The combination of a bought vanilla and a sold knock-in. ↩

9. Bank of Japan, "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 6, note 5. The treatment of reverse knock-outs with the knock-out rate set on the in-the-money side. ↩

10. FXC, "Practice Notes to the 2005 Barrier Option Supplement" (Foreign Exchange Committee, 2005), page 163, Section 3.4(c). The principle that the premium is not refunded, and an example of a confirmation rebating it on a reverse knock-out. ↩

11. FXC, the same explanation, pages 164–165, Section 3.9(c)(ii), 3.9(n)(ii), and 3.9(p). The rate used for the non-discrete type and the hours when the spot market is open continuously (5 a.m. Monday in Sydney to 5 p.m. Friday in New York). ↩

12. FXC, the same explanation, pages 163–165, Section 3.9(c)(i), 3.9(d), and 3.9(n)(i). The rate source and decision days of the discrete type, the treatment of a day when the rate was not published, and an example of limiting the days to a weekday (every Wednesday). ↩ ↩²

13. Bossens et al., Section 2 (page 3). The classification of first-generation FX exotics, the definitions of the one-touch and double one-touch, and the derivation of the double one-touch price from spreads on the double knock-in. ↩ ↩²

14. Bossens et al., Section 5.1 (page 13), equations (37) and (38). The relation between the survival probability and the no-touch price divided by the discount factor. ↩

15. Bossens et al., Section 1 (page 1). The statement that, at the time of the paper, the Black-Scholes theoretical price is used only as a reference quote. ↩

16. Bossens et al., Section 7 (page 23). The description of the vanna-volga method as widely used for pricing FX exotic options ("a popular pricing tool"). ↩

17. Bossens et al., Section 1 (page 2) and Section 4 (pages 8–10). How the vanna-volga adjustment is found, and the definitions of vanna and volga. ↩ ↩²

18. Bossens et al., Section 5 (page 12). The relation of vanna (large) and volga (small) when the barrier nears the current rate. ↩

19. Bossens et al., Section 4.1 (page 10) and Section 5 (pages 12–14). The attenuation factor based on the survival probability or the expected first exit time. ↩

20. Bossens et al., Section 6.3 (pages 22–24). The good agreement ("good agreement") between a setup fitted to the one-touch prices only and fitting to all products. The double knock-outs outside the shaded range in Figure 5. ↩

21. Bossens et al., Section 6.2 (pages 20–21). The model comparison in Figure 4, the closeness by product in Table 2, and the point that the vanilla market has no information on how the smile moves. ↩

22. Bossens et al., Section 5.5 (pages 16–17). For a 1-year USD/CHF one-touch (theoretical price about 4%, barrier 1.55), an error of 0.5% in σBF25 changes the price by 3%. ↩

23. Bank of Japan, "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, in Japanese), pages 4–5. The view that the longer the remaining maturity, the smaller the swings in spot hedging trades and the more gradual the customers' rebuilding of positions. ↩

24. Bank of Japan, the same article, page 4, "2007年前半におけるノックアウト・オプションの影響" (The effect of knock-out options in the first half of 2007), and page 6. Options with the barrier in the direction of a weaker yen had apparently expired, and market participants saw little effect on spot trading. ↩