Chapter 24
American Barrier Options
Standard American barrier options are one of the most frequently traded exotic FX derivative contracts. American barrier options have a vanilla payoff at expiry plus they also have a single American-style barrier. There are two main variations: Regular barrier options have the American barrier positioned out-of-the-money compared to the option payoff and reverse barrier options have the American barrier positioned in-the-money compared to the option payoff. As described in Chapter 20, American barriers are monitored continuously against the spot level in the market.
Regular American Barrier Options
A regular American knock-out (KO) call option structure is shown in Exhibit 24.1. Note that the barrier is positioned out-of-the-money.
Exhibit 24.1 CCY1 call knock-out barrier option structure
Consider a CCY1 call knock-out barrier option with the barrier positioned so far below the strike that there is zero chance of spot hitting the barrier and then ending up back in-the-money at expiry. The pricing and risks of this knock-out option will be identical to the equivalent vanilla option:
- Far barrier knock-out option TV = vanilla priced using ATM volatility
- Far barrier knock-out option vega = vanilla vega
- Far barrier knock-out option TV adjustment = vanilla zeta
As Exhibit 24.2 shows, even with a barrier (1.2000) much closer to the strike (1.3000), as spot goes higher, the downside barrier is less likely to be touched and the knock-out TV converges to the vanilla TV. If spot goes down through the barrier, the barrier option knocks out and becomes worthless.
Exhibit 24.2 Knock-out barrier and vanilla TV profiles
The delta profile of the knock-out option versus the equivalent vanilla option is shown in Exhibit 24.3.
Exhibit 24.3 Knock-out barrier and vanilla delta profiles
As spot goes higher, the knock-out delta converges to the vanilla delta. If spot goes down through the barrier, the knock-out barrier has triggered and there is no delta exposure or indeed any other Greek exposures. Therefore, American barrier options have barrier delta gaps like touch options (see Chapter 23).
Exhibit 24.4 shows how the theoretical value of a knock-out barrier option changes as the barrier level changes. As the barrier level moves closer to spot, the knock-out barrier TV reduces because there is an increased chance of the barrier knocking prior to maturity.
Exhibit 24.4 TV of knock-out barrier option with different barrier levels
| Barrier Level | Stopping Time | Knock-out Barrier TV (CCY1%) |
| 0.5000 | 100.00% | 3.781% |
| 1.0000 | 99.94% | 3.781% |
| 1.1000 | 98.18% | 3.781% |
| 1.2000 | 80.52% | 3.638% |
| 1.2500 | 52.38% | 2.782% |
| 1.2900 | 12.85% | 0.762% |
- Contract details: EUR/USD 1yr 1.3000 EUR call/USD put knock-out option.
- Spot: 1.3000.
Exhibit 24.5 shows the vega profiles for knock-out barrier options with these different barrier levels. As the barrier moves higher, the peak vega also shifts higher and the increasing chance of knocking out causes the premium and the vega exposure to reduce. The closer barrier also causes increased vanna 
exposures at current spot because vega falls sharply into the barrier level.
Exhibit 24.5 Vega profiles of knock-out barrier option with different barrier levels
As mentioned in Chapter 23, when American barriers are close to spot, the option is not expected to live to expiry and therefore vega exposures are to implied volatility at maturities closer than the expiry. Again, it is important to view bucketed Greek exposures when risk managing any option contracts with American barriers.
The volga 
profile of a knock-out barrier option changes dramatically as the barrier moves higher, as shown in Exhibit 24.6.
Exhibit 24.6 Volga profiles of knock-out barrier option with different barrier levels
With barriers far from spot (0.50 and 1.00) the knock-out barrier volga profile looks like the vanilla volga profile, as expected. As the knock-out barrier moves closer to spot, volga gets generally less positive and eventually creates an area of negative volga. In general, regular American knock-out barriers produce negative volga, which leads to more negative TV adjustments. Intuitively, the reason for this is:
- Volatility higher brings the knock-out barrier relatively closer to spot → option is more likely to knock and less likely to pay out.
- Volatility lower reduces the expected option payout.
It is therefore preferable that implied volatility does not move, hence producing a negative volga exposure.
Regular Knock-in Barrier Options
Knock-out barrier options have a vanilla payoff at expiry providing the American barrier has not triggered prior to expiry, whereas knock-in (KI) barrier options have a vanilla payoff at expiry, providing the American barrier has triggered prior to expiry.
Providing all contract details (expiry, strike, barrier, cut, and notional) are the same:
- Knock-out barrier + knock-in barrier = vanilla option
It follows that:
- Knock-out TV + knock-in TV = vanilla TV
- Knock-out barrier TV adjustment + knock-in barrier TV adjustment = vanilla zeta
Exhibit 24.7 shows the vega of a knock-in barrier option with a 1.2000 barrier. Note the vega symmetry around the barrier with peak vega at the barrier level. The vega beyond the knock-in barrier must be equal to the vanilla vega.
Exhibit 24.7 Vega profile of a vanilla option versus a knock-in barrier option with the same strike
Looking at the vega profile, it is clear that the knock-in barrier option will be longer volga around 1.3000 spot since vega increases to the downside. This is shown in Exhibit 24.8. Again, note the symmetry around the barrier level.
Exhibit 24.8 Volga profile of a vanilla option verses a knock-in barrier option with the same strike
Regular Barrier Option Pricing
The main trading risks on knock-out and knock-in options are vega and gamma exposures, which can be hedged with vanilla options. Over time, either the knock-out option triggers or the risk becomes more vanilla as the chance of the barrier triggering decreases. For the knock-in option, the option either triggers and becomes a vanilla option or the risk dies away as the chance of triggering decreases. Regular American barrier options therefore become easier to risk manage over time.
Local volatility pricing models will generate a price above the market price for knock-out options and below the market price for knock-in options due to local volatility undervaluing (short) convexity. A well-calibrated mixed-volatility model will generally give good pricing for knock-out and knock-in options in liquid currency pairs, but as always, the interest rate effect must be additionally quantified on longer-dated trades.
As a quick sense-check for knock-out TV adjustment, it will usually be lower (due to the volga) than vanilla zeta × stopping time (i.e., the cost of the strike on the smile multiplied by the length of time the option will be alive).
In terms of bid–offer spread, knock-outs and knock-ins are generally spread slightly wider than the equivalent vanilla because barrier options have only marginally higher trading risks than the equivalent vanilla options.
Finally, barrier delta gaps must be considered when pricing knock-out and knock-in options. Most often, market participants do not want stop-loss spot orders since they represent a larger P&L risk than take-profit orders. The pricing of regular American barrier options reflects that preference: Knock-out options become relatively more attractive to buy and less attractive to sell, while knock-in options become relatively less attractive to buy and more attractive to sell. This idea is covered in more detail in Chapter 25.
Reverse American Barrier Options
Reverse knock-out (RKO) and reverse knock-in (RKI) barrier options have a vanilla payoff at expiry plus a single American barrier that is positioned in-the-money versus the payoff, as shown in Exhibit 24.9.
Exhibit 24.9 CCY1 put reverse knock-out structure
Reverse barrier options have additional trading risks that arise from the in-the-money barrier. At expiry, a reverse knock-out option goes from being worth the intrinsic value (see Chapter 22) with spot just before the barrier to being worth nothing if spot trades through the barrier level: the same P&L dynamic as a touch option. When the reverse knock-out option is bought, one-touch risk is effectively sold because P&L changes negatively as spot goes through the barrier level.
Exhibit 24.10 shows the TV profiles of a reverse knock-out barrier option versus the equivalent vanilla option.
Exhibit 24.10 Vanilla call and reverse knock-out TV profiles
The unlimited upside from the vanilla option is curtailed by the in-the-money barrier. For this reason, reverse knock-out options are often significantly cheaper than the equivalent vanilla options and they are therefore an effective way to express the view that spot will move in a certain direction but not very far. For institutional clients, short-dated RKOs are an attractive product: They can be bought cheaply, transacted live (i.e., no delta hedge), and then left until expiry to hopefully generate a payoff.
Traders and clients often compare the value of the reverse knock-out option with the equivalent vanilla option to get a measure of how much discount the barrier provides. Plus they compare the value of a reverse knock-out option with the equivalent European knock-out option to ascertain the value of the “American-ness” of the barrier. Another popular analysis compares the maximum payoff from the option (i.e., spot just inside the knock-out barrier) with the cost. This is sometimes called the leverage of the option.
Within a delta hedged options portfolio, reverse knock-out options with large notionals can be challenging to risk manage due to the touch risk at the barrier. As with European barriers (see Chapter 22), the size of the risk at the barrier is the intrinsic value (IV). For example, USD100m of USD/JPY 1mth 95.00 reverse knock-out 98.00 has 100m × (98.00 − 95.00)/98.00 = USD3.06m of touch risk embedded within it.
The trading risk on a long reverse knock-out option can be decomposed into two parts: the long strike and the short one-touch (recall that single barrier touch risk is always traded in the market via a one-touch contract). The vega profile of a reverse knock-out shown in Exhibit 24.11 confirms this.
Exhibit 24.11 Long 1yr 1.3000 strike/1.5000 barrier CCY1 call reverse knock-out vega profile
Specifically, a long reverse knock-out option in notional N with strike K and American knock-out barrier B can be approximately decomposed into these two elements:
- Long vanilla call in notional N with strike K
- Short one-touch in payout IV with barrier B
A quick sense-check for the reverse knock-out TV adjustment can therefore be obtained by calculating:
- The “Strike Zeta × Stopping Time” gives the cost of the strike on the smile, weighted by how long the option is likely to exist for.
- The “Intrinsic Value × One-touch TV Adjustment” gives the TV adjustment on the one-touch risk at the barrier level. This quantity is subtracted because long a reverse knock-out barrier gives short one-touch risk.
Example: EUR/USD 6mth 1.3500 RKO 1.4500 with spot 1.3430, ATM volatility 8.15% and RKO TV 0.73 EUR%. Working to the nearest basis point in CCY1% terms:
- Strike Zeta from EUR/USD 6mth 1.3500 vanilla = +0.02% (a small value since the strike is close to the ATM)
- Stopping Time = 93%
- Strike Zeta × Stopping Time = 0.02%
- Intrinsic Value = (1.45 − 1.35)/1.45 = 6.9%
- One-touch TV Adjustment = –2.85%
- Intrinsic Value × One-touch TV Adjustment = –0.19%
These results suggest that the majority of the smile risk on this reverse knock-out comes from the barrier (-0.19%) rather than the strike (0.02%).
The reverse knock-out TV adjustment approximation of 0.02% − (−0.19%) = +0.21% is close to the mixed volatility model TV adjustment of +0.19%. Exhibit 24.12 shows these options within a pricing tool.
Exhibit 24.12 Reverse knock-out “replication” in pricing tool
In practice, this means that when pricing a reverse knock-out it is important to know the intrinsic value at the barrier and the equivalent TV of a one-touch at the barrier level. Then, knowing whether the one-touch option trades over or under TV gives quick guidance on whether the reverse knock-out will trade over or under TV in the opposite direction.
Recall that one-touch vega moves toward the barrier over time but does not reduce. An RKO vega exposure shows similar behavior in Exhibit 24.13. At the 1mth expiry, the risk has started noticeably splitting into strike risk and one-touch risk at the barrier. This long strike versus short one-touch risk spread creates a vega profile that is quite similar to a risk reversal. Trading a vanilla spread with strikes at the RKO strike and RKO barrier level to the RKO expiry date is often an effective way to hedge reverse knock-out vega.
Exhibit 24.13 Long 1.3000 strike/1.5000 barrier CCY1 call reverse knock-out vega profile over time
Reverse Knock-in Barrier Options
A familiar pricing identity exists for reverse barrier options. Providing all contract details (expiry, strike, barrier, cut, and notional) are the same:
- Reverse knock-out + reverse knock-in = vanilla
A long reverse knock-in has similar trading risk to a long one-touch at the barrier level, providing the strike and barrier are far enough apart. This is shown in the vega profiles in Exhibit 24.14.
Exhibit 24.14 Long reverse knock-in and long one-touch vega profiles
The reverse knock-in and equivalent one-touch vega profiles are very similar in front of the barrier. At the barrier, the reverse knock-in becomes a vanilla so the reverse knock-in and vanilla vega profiles intersect at that point.
Reverse Barrier Option Pricing
From a risk management perspective, the major risks on reverse barrier options are initially vega Greeks, which can be approximately hedged with vanillas. For a reverse knock-out option, the main risk is a strike versus barrier vega spread. The pricing of this risk on the smile will be heavily impacted by the skew in the volatility surface. For a reverse knock-in option, the main vega risk comes from the barrier. Over time, touch risk at the barrier becomes the major risk and this can be hedged like a standard one-touch option.
In freely floating currency pairs, a well-calibrated mixed volatility model will usually give reverse barrier option prices that match the market well, especially since reverse barrier options are generally quite short-dated hence minimal interest rate risk and have low premium hence low vega exposures.
In terms of bid–offer spread, reverse knock-out and reverse knock-in options usually derive the majority of their spread from their touch risk in the same way that European barrier options derive the majority of the spread from their European digital risk. Exhibit 24.15 shows a reverse knock-in and equivalent one-touch option within a pricing tool. In approximate terms:
Exhibit 24.15 Reverse knock-out and equivalent one-touch option within a pricing tool
Reverse barrier bid–offer spread = intrinsic value × equivalent one-touch bid–offer spread.
However, a reverse knock-out generally has less vega than a one-touch (due to the strike versus barrier risk offset), so the reverse knock-out may be quoted relatively slightly tighter.
One-touch bid–offer spreads are monitored and directly observed in the interbank broker market so exotics traders tend to describe reverse barrier bid-offer spreads in terms of the embedded one-touch bid-offer. Saying “I showed an AUD/USD 6mth RKO 1.5% wide on the one-touch” is far more useful than “I showed an AUD/USD 6mth RKO 0.20% wide.”
Finally, traders need to be careful when clients request prices in reverse knock-in options with barriers close to the strike. These options have a virtually identical theoretical value to the vanilla option with no barrier. However, since vanilla = reverse knock-out + reverse knock-in, the equivalent reverse knock-out is very low TV and would often trade higher than a pricing model suggests. Therefore, the reverse knock-in should be priced correspondingly lower.
Double American Barrier Options
Double knock-out (DKO) and double knock-in (DKI) barrier options have a vanilla payoff at expiry plus two American barriers: one barrier positioned in-the-money versus the payoff and one positioned out-of-the-money versus the payoff as shown in Exhibit 24.16.
Exhibit 24.16 CCY1 call double knock-out structure
The primary risk on double American barrier options usually comes from the in-the-money barrier. Therefore, similar pricing and spreading methodologies are used for double knock-out options as reverse knock-out options.
Traders often simplify the product in order to identify the most important risks on a double-barrier option. By removing each barrier in turn, a small TV change implies an unimportant barrier and a large TV change implies an important barrier. Therefore, pricing and risk management can be focused around the most appropriate element of the structure.
Finally, the presence of two barriers can sometimes cause double knock-out options to have trading risks initially similar to a double-no-touch, specifically, larger convexity exposures.
Knock-in/Knock-out Option Replication
Knock-in/knock-out (KIKO) options have a vanilla payoff at expiry plus two American barriers: one barrier positioned in-the-money and one positioned out-of-the-money. One of the barriers is knock-out while the other is knock-in. There are two variations of knock-in/knock-out options:
- Knock-out until expiry
- Knock-out until knock-in
A long knock-out until expiry KIKO option with knock-in barrier B1 (B1 > K) and knock-out barrier B2 (B2 < K) can be replicated with the following standard American barrier options with the same strike, payoff, and notional:
- Long knock-out with barrier B2
- Short double knock-out with barriers B1 and B2
Thinking through the possible scenarios:
- Spot gets to maturity without having touched either KIKO barrier level: The knock-out and double knock-out payoffs within the replication offset and there is no payout.
- Spot touches the KIKO knock-out barrier level B2 first: The knock-out and the double knock-out both expire and there is no payoff at maturity.
- Spot touches the KIKO knock-in barrier level B1 first: The short double knock-out expires and only the long knock-out risk remains; the structure has been “knocked in.” If spot then touches the KIKO knock-out barrier level B2, the knock-out option will expire; otherwise the option has a vanilla payoff at maturity.
Therefore this variation is called “knock-out until expiry.” Exhibit 24.17 shows the legs of the replication. Note that this replication works no matter the payoff.
Exhibit 24.17 Knock-in/knock-out replication in pricing tool
Strike-out Options
Strike-out options are American knock-out barrier options such that strike = barrier. Consider an AUD/USD 1yr 0.8000 AUD call/USD put with American knock-out barrier at 0.8000. Assume initially that AUD and USD interest rates are zero and hence spot = forward as shown in Exhibit 24.18.
Exhibit 24.18 AUD/USD strike-out option structure with 0% AUD rates and 0% USD rates
With spot at 0.9000, the TV of this option is simply the intrinsic, that is, (0.9000 − 0.8000)/0.9000 = 11.11%. There is 100% delta, zero vega, and zero gamma over all spots (above the barrier). This trade is equivalent to a long spot position in the notional amount that is closed out once the barrier level trades. The important thing to note is that there is no optionality coming from the strike due to the barrier. Exhibit 24.19 shows this option in a pricing tool.
Exhibit 24.19 AUD/USD strike-out option with 0% AUD rates and 0% USD rates
AUD (CCY1) rates are now increased to 5% and USD (CCY2) rates are kept at 0%. The forward moves to the left (to 0.8571) and the option value drops to 7.31%. Importantly the option value is not equal to the intrinsic any more: (0.8571 − 0.8000)/0.8571 = 6.66%. The option payoff using this new market data is shown in Exhibit 24.20. As shown in the pricing tool in Exhibit 24.21, the option is now long vega, and it only has 80.52% delta, so there must be another mechanism at work. The vega profile of the strike-out with higher CCY1 interest rates is shown in Exhibit 24.22.
Exhibit 24.20 AUD/USD strike-out option payoff with 5% AUD rates and 0% USD rates
Exhibit 24.21 AUD/USD strike-out option with 5% AUD rates and 0% USD rates
Exhibit 24.22 AUD/USD 0.8000 strike-out vega profile (AUD rates = 5%/USD rates = 0%)
Now, AUD (CCY1) rates are put back to 0% and USD (CCY2) rates are increased to 5%. The forward moves to the right (to 0.9465), the option TV increases to 15.07%, delta is 110%, and vega is now short. The option payoff using this new market data is shown in Exhibit 24.23. The strike-out option is shown priced with this new market data in Exhibit 24.24. The vega profile of the strike-out with higher CCY2 interest rates is shown in Exhibit 24.25.
Exhibit 24.23 AUD/USD strike-out option payoff with 0% AUD rates and 5% USD rates
Exhibit 24.24 AUD/USD strike-out option with 0% AUD rates and 5% USD rates
Exhibit 24.25 AUD/USD 0.8000 strike-out vega (AUD rates = 0%/USD rates = 5%)
Given there is no optionality coming from the strike, where is this vega exposure coming from? The key element within strike-out options is the interest rate carry.
Intuitively, the hedge for this strike-out option is to sell spot now and buy it back if spot trades at or through the barrier. If CCY1 rates are larger than CCY2 rates, holding the short cash position on the hedge will cost money over time. Therefore, ideally spot trades through the barrier level as quickly as possible and hence the hedge can be unwound. This explains why the higher AUD (CCY1) rates case is long vega; higher volatility will increase the probability of knocking out earlier.
Alternatively, if CCY1 rates are lower than CCY2 rates, holding a short cash position on the hedge will earn money over time. It is therefore preferable that spot trades through the barrier level as late as possible, or ideally not at all. This explains why the higher USD (CCY2) rates case is short vega; again, higher volatility will increase the probability of knocking out earlier.
When pricing and hedging strike-out options, it is important that:
- The full interest rate curves are used. The single interest rate to expiry may be drastically different from the short-dated interest rates, which generate the positive or negative return from holding the spot hedge.
- The full ATM curve is used. If the ATM curve is steeply upward or downward sloping and the stopping time is short, this could have a significant impact on pricing and Greeks.
- Spot versus interest rate correlations are taken into account. Changes in the knock probability versus carry earned on hedge will have a significant impact on the TV adjustment. Over time, the vega decays away linearly with time (since interest rate carry is interest rate differential × time) and moves toward the barrier as shown in Exhibit 24.26. Therefore, any vanilla hedges put on to offset the strike-out vega will need to be rebalanced over time.
Exhibit 24.26 AUD/USD 0.8000 strike-out vega over time (AUD rates = 0%/USD rates = 5%)