# Chapter 12 Barrier Options

This chapter has been written using several books, namely: Frans de Weert's book - Exotic Option Trading (2008), Bouzoubaa and Osseiran's book - Exotic Options and Hybrids (2010), Encyclopedia of Quantitative Finance (2010).

You can price and analyze the underlying risks of barrier options using our barrier options pricer. We used it to retrieve most of the graphs from this chapter.

## 12.1 Description

Barrier options are path-dependent options, that is, their payoff is not only a function of stock level relative to option strike but also dependent upon whether or not the stock reaches certain prespecified barrier level at or before maturity.

Barrier options are very popular amongst retail investors as the barrier feature provides the investor with additional protection or leverage.

The two most common kinds of barrier options are *Knock-Out* and *Knock-In* options.

**Knock-Out (KO)** options are options that expire worthless when the underlying's spot crosses the prespecified barrier level.

**Knock-In (KI)** options are options that only come into existence if the prespecified barrier level is crossed by the underlying asset's price.

The barrier observation can be at any time during the option's life (American style) or at maturity only (European style).

Depending upon the barrier level relative to the initial underlying asset level, we can have an 'up' or a 'down' barrier.

Together, we can therefore have four types of barrier options:

- Up-and-In options (UI)
- Up-and-Out options (UO)
- Down-and-In options (DI)
- Down-and-Out options (DO)

## 12.2 Knock-Out Options

Knock-Out (KO) options are options that expire worthless when the underlying's spot crosses the prespecified barrier level.

In the case of KO options, an additional feature called a **rebate** can be added to the contract specifications. The rebate is a coupon paid to the holder of a KO option in case the barrier is breached.

The leverage effect of an KO option can be much more attractive than the leverage of a comparable vanilla option for an investor who believes the spot will not reach the outstrike during the investment period. He gets more profit for bearing the risk of knocking out.

### 12.2.1 Some sensitivities

#### 12.2.1.1 Barrier Level

The closer the barrier level is to the initial spot, the cheaper the KO option would be. This is quite intuitive: the closer the barrier level, the higher the probability of the option expiring worthless.

#### 12.2.1.2 Barrier Observation

For a KO option, the higher the number of barrier observations, the higher the probability of the option knocking out and the cheaper the KO option. A KO option having an annually monitored barrier would be more expensive than a similar KO option having a quarterly monitored barrier.

#### 12.2.1.3 Volatility

A KO option is less sensitive to volatility than a vanilla option carrying the same features. Indeed, a higher volatility increases the probability of expiring ITM but also increases the probability of reaching the barrier and ending with no value. The Vega of a KO option is generally lower than the Vega of a comparable vanilla option.

#### 12.2.1.4 Gap risk

When KOs are defined with the barrier placed in such a way that the option vanishes when it is OTM, we call these **regular** KO options. In these, it is easier for traders to hedge the associated risks.

Otherwise, KO options are classified as **reverse** and they present higher trading difficulty and risks. For example, they are subject to a *discontinuity risk* or a *gap risk*. We will discuss how does a trader handle this gap risk in more details in this chapter.

## 12.3 Knock-In Options

Knock-In (KI) options are options that only come into existence if the prespecified barrier level is crossed by the underlying asset's price.

The leverage effect of a KI option can be much more attractive than the leverage of a comparable vanilla option for an investor who believes the spot will reach the outstrike during the investment period. He gets more profit for bearing the risk of not knocking in.

### 12.3.1 Some sensitivities

#### 12.3.1.1 Barrier Level

The nearer the barrier level to the initial spot, the more expensive the KI option would be. This is quite intuitive: the closer the barrier level, the higher the probability of the option coming into existence.

#### 12.3.1.2 Barrier Observation

For KI options, the higher the number of barrier observations, the higher the probability of the option being activated, the more expensive the KI option. A KI option having a quarterly monitored barrier would be cheaper than a similar KI option having a monthly monitored barrier.

#### 12.3.1.3 Volatility

Unlike a KO option, a KI option is more sensitive to volatility than a vanilla option carrying the same features. Indeed, a higher volatility can benefit the holder of the option because it increases not only the probability of maturing ITM but also the probability of reaching the barrier and being activated. The Vega of a KI option is then higher than the Vega of a comparable vanilla option.

#### 12.3.1.4 Gap risk

When KI options are defined with the barrier placed in such a way that the options are activated when it is OTM, then we call them **regular** KI options since it is easier for traders to hedge the associated risks.

Otherwise, KI options are classified as **reverse** and they present greater trading difficulties and risks. For example, they are subject to a *discontinuity risk* or a *gap risk*. We will discuss how does a trader handle this gap risk in more details in this chapter.

## 12.4 In-Out Parity for barrier options

Being long a KO option and a KI option with the same features is equivalent to owning a comparable vanilla option independently from the behaviour of the spot with respect to the barrier level.

\(\boxed{\text{Knock-In (K,T,B) + Knock-Out (K,T,B) = Vanilla (K,T)}}\)

It is very easy to see that, for any given scenario of the underlying asset path before maturity, the portfolio (KI + KO) will always have the same payoff as the corresponding vanilla option. This relationship holds for both the put and call options in the absence of rebates.

## 12.5 Review of Payoffs

\(\text{UI Call}_T = max[S_T - K, 0] * \mathbb{1}_{\{(max_{t \in [0,T]} S(t)) \; \ge \; B\}}\)

\(\text{UI Put}_T = max[K - S_T, 0] * \mathbb{1}_{\{(max_{t \in [0,T]} S(t)) \; \ge \; B\}}\)

\(\text{UO Call}_T = max[S_T - K, 0] * \mathbb{1}_{\{(max_{t \in [0,T]} S(t)) \; < \; B\}}\)

\(\text{UO Put}_T = max[K - S_T, 0] * \mathbb{1}_{\{(max_{t \in [0,T]} S(t)) \; < \; B\}}\)

\(\text{DI Call}_T = max[S_T - K, 0] * \mathbb{1}_{\{(min_{t \in [0,T]} S(t)) \; \le \; B\}}\)

\(\text{DI Put}_T = max[K - S_T, 0] * \mathbb{1}_{\{(min_{t \in [0,T]} S(t)) \; \le \; B\}}\)

\(\text{DO Call}_T = max[S_T - K, 0] * \mathbb{1}_{\{(min_{t \in [0,T]} S(t)) \; > \; B\}}\)

\(\text{DO Put}_T = max[K - S_T, 0] * \mathbb{1}_{\{(min_{t \in [0,T]} S(t)) \; > \; B\}}\)

## 12.6 Deeper into Risk Analysis

We went into little detail about the risk analysis of barrier options. Let us go deeper into this subject as, from a risk management perspective, barrier options are very interesting because the risks are discontinuous arround the barrier. Therefore the Greeks become less predictable and often change sign around the barrier.

It is essential to understand the risks embedded in barrier options as those risks will have to be taken into account in the price. Again, the price of an option should reflect the cost of hedging it!

We will not go through the 8 types of barrier options but will use the DI put as a leading example to get across all the risks within a barrier option.

The reason we use a single leading example is because the underlying causes for barrier option risk are generic and once the drivers of barrier option risk are understood for a DI put, one will be able to derive the risks for the other types of barrier options.

The reason we choose the DI put is because many structured products use it to obtain enhanced yields or increased participation. The investor accepts the risk from selling the DI puts to generate extra funding that is used in the structure to increase the yield or participation.

### 12.6.1 Leading Example : Down-and-In Put (DIP)

#### 12.6.1.1 Delta Change over the Barrier

Traders on the sell side are usually long the DIP and have to hedge the risks associated with this position accordingly.

The trader taking a long position in the DIP is short the forward and will need to buy delta in the underlying stock at inception and adjust dynamically his delta hedge to remain delta neutral.

Assume the trader buys a 100/80% DIP on Total from an investor. The payoff of this position is represented in the above graphic.Assume the delta at inception is 0.4 so that the trader hedges himself by buying Total shares in a ratio of 0.4 shares per option.

When Total stock price breaches the 80% barrier level, the DIP goes instantaneously from not being an option to being a 20% ITM put. As a result, the value of the DIP increases significantly over the barrier, even if the stock price barely moves down from 80.1% to 79.9% of its initial value. This means that the absolute delta of DIP becomes extremely large when the underlying price gets closer to the barrier. This absolute delta often becomes greater than 1.

Since the absolute delta of a vanilla option can never be greater than 1, the trader will accumulate too many Total shares when the stock approaches the barrier. As soon as the stock price breaches the barrier, the trader would need to sell any excess shares. In practice, it is very likely that he will not be able to sell these excess shares exactly at the barrier level. Indeed, the share price is already going down for the barrier to be breached at the first place, and the fact that the trader needs to sell a potentially large quantity of shares will push the price further down. So the trader will probably sell these excess shares below the barrier level and incurs a loss on this sale.

Clearly, the risks of a barrier option near the barrier level can be difficult to manage. The delta of a barrier option can jump near the barrier causing hedging problems. So near the barrier, the Gamma (= the sensitivity of Delta to a movement in the underlying stock price) can be very large. To make things worst, it actually changes sign around the barrier. A long DIP position goes from being long gamma to short gamma as the share price approaches the barrier.

One method to smooth out the risks to make them manageable is to apply a barrier shift. To avoid the loss of selling the excess shares below the barrier level, the trade will give himself a cushion. To do that, he will price and risk manage a slightly different option where the barrier is shifted downwards in such a way that the trader has enough room to sell the excess shares without incurring a loss. If he believes that he needs a cushion of 2% to sell these shares over the barrier, he will price and risk manage a 100/78% DIP rather than a 100/80% DIP. It makes the option cheaper for the trader.

#### 12.6.1.2 Magnitude of the Barrier Shift

In practice, the size of the barrier shift (2% in the above example) is not a random choice and depends on several factors:

**The size of the transaction**. The larger the size, the more shares will have to be sold over the barrier, therefore the trader is more likely to move the stock price against him. To sum up, the larger the transaction size, the larger the barrier shift.**The difference between strike price and barrier level**. If this difference is large, the DIP goes from not being a put to a put that is far ITM when the underlying pruce breaches the barrier. To sum up, the larger the difference between strike price and barrier level, the larger the barrier shift.**The daily volume of the underlying share**. If the daily traded volume is low, it might be difficult for the trader to sell the excess shares over the barrier and he will likely sell them at a bad price, incurring a higher loss. To sum up, the lower the daily traded volume of the underlying asset, the larger the barrier shift.

These three first factors form a liquidity-based barrier shift and account for the discontinuity in the Delta near the barrier.

**The volatility of the underlying stock**. The more volatile the underlying stock, the large is the risk to the trader of the stock price approaching the barrier level. In other words, the larger the volatility, the larger the barrier shift needed to protect the trader against a larger move. This can be easily seen through our previous example. You are the trader long 100.000 100/80% DIP on Total. Total is currently trading at 81% of its initial level. Suppose you priced this DIP as a 100/78% and you are risk managing it as such. Assume the absolute delta for this 100/78% DIP is currently 2 so that you are long 200.000 of Total stocks. If Total gaps down 10%, the DIP knocks in and its absolute delta is 1 (far ITM put). It means that you need to be long a 100.000 of Total shares. Well, you were long 200.000 of these shares so you need to sell for 100.000 of them over the barrier. As the stock gapped down, you can only sell them at 71% of the initial stock price! Since you managed this DIP as a 100/78% DIP, your loss is slightly reduced but still very painful!

This kind of gap down can bring situations where the trader would actually need to buy stocks when the barrier is breached. Using the DIP as above, assume the stock is trading far away from the barrier, at 87% of its initial level for example. At this point, the absolute delta is not particularly huge and the trader has not accumulate any excess delta. If the stock price goes through the barrier in a gap move down, the absolute delta will increase and the trader will need to buy shares (and will be able to do so at a lower pricer).

**The barrier level**. Since lower stock prices tend to go hand in hand with higher volatilities and higher volatilities result in larger barrier shift, the lower the barrier level, the higher the barrier shift.**The time left to maturity**. The closer to maturity, the larger the absolute delta just before the barrier and therefore the larger the change in delta over the barrier. This translates into a higher risk and a higher barrier shift for shorter maturities.

Note that certain types of barrier options do not require a barrier shift. A long position in a UO put is a good example. The delta hedge of such position is always a long share position so that the trader will need to sell shares over the barrier. The trader does not need to shift the barrier as he can always let a limit order just before the barrier. If the stock gaps up, he will be pretty happy to sell them at a higher price anyway.

#### 12.6.1.3 Types of Barrier Shift

When traders quote barrier options, they can be more or less conservative on their prices depending on the magnitude of the barrier shift but also on the way this barrier shift is applied.

Until now, we have only considered constant barrier shift but the it could also be an increasing function of time. It is quite intuitive if you think about it. In the first days of an option's life, under normal levels of volatility, it is quite unlikely to see the underlying breach the barrier level. If one were to simulate paths and monitor the points in time at which the barrier was breached, it is quite obvious that the KI events occur more frequently down the line. That is the reason why traders often apply a barrier shift that is an increasing function of time.

Assume an investor is willing to sell a 100/70 KI call. 3 IBs are competing for this trade and will all take the same commissions and apply the same pricing parameters (volatility, skew, etcâ€¦). They all want to apply a shift of 2% but they have different ways of shifting the barrier.

Trader 1 is very conservative and applies a **constant barrier shift** of 2%.

Trader 2 is less conservative and decides to apply a **linear barrier shift**. At inception date, there is no barrier shift since there is no expected risk around the barrier. He believes that the maximum shift to be applied would be 2%, which is the shift value at maturity. The shift grows linerarly from zero at inception to 2% at maturity.

Trader number 3 uses a **curvy barrier shift** in time which is computed from evaluating knock-in scenarios. he is the most aggressive in his barrier shift. Therefore, his bid is the highest and he wins the trade in this case.

#### 12.6.1.4 Barrier at maturity only

While closed-form solutions exist for continuously monitored barriers, other types of barrier observation require a Monte Carlo process. This is the case when barrier options are only live at maturity or on specific days. However, barrier options that are only live at maturity can also be priced as a combination of European options.

Let us keep our example of a trader long a 1Y 100/80 DIP on Total where the option can only knock in at maturity. It is obviously worth less than the DIP that can knock in anytime before maturity and can be replicated as follows:

- The trader buys 10 times a 1Y 80% European put
- The trader sells 10 times a 1Y 78% European put
- The trader buys one 1Y 80% European put

The fact that the trader buys a 10x leverage put spread 80/78 is an overhedge and therefore a conservative way to replicate the DIP's payoff at maturity. The 2% wide put spread can be seen as a barrier shift. The tighter the put spread the more the replication converges to the actual price of the DIP with KI at maturity only. The gearing of the put spread can be calculated by dividing the size of the discontinuity (the strike/barrier differential) by the width of the put spread.

This discussion should make you think about what has been said in the chapter about digital options that can be seen as a limit of a call/put spread! This is exactly the same here as the discontinuity around the barrier corresponds somehow to a digital option.

Now that you have seen how you can replicate a barrier option with barrier observation at maturity only, you should easily see why these barrier options are sensitive to skew!

#### 12.6.1.5 Volatility and Skew

We have already seen that a knock-in option is more sensitive to volatility (higher vega) than a vanilla option carrying the same features. Indeed, a higher volatility can benefit the holder of the option because it increases not only the probability of maturing ITM but also the probability of reaching the barrier and being activated.

The trader buying a DIP is therefore long volatility. This long vega position can be hedged, at leat partially, by buying vaniall put options on the same underlying stock with strikes between the barrier and the spot.

Risk wise one can compare a KI barrier to a long option position at that barrier and a KO option to a short option position at the specific barrier. It is clear then that the owner of DIP is long the skew as his position is similar to being long a downside option (option with lower strike). In the presence of skew, the volatility around the barrier is higher than the ATM volatility, which makes the probability of crossing the barrier higher.

From a model point of view, we will need to calibrate a model to the IVs of options on the underlying asset across strikes with specific attention to the downside skew.

If the barrier is monitored continuously, we will need a model that gives a smooth calibration through all ends of the surface between short maturities and up to the option's maturity. It means we will need to calibrate to both skew and term structure. The reason is that a continuously monitored barrier option can be triggered at any time up to maturity, therefore it has vega sensitivity through the different time buckets. So European options with different maturities must be calibrated so that the model shows risk against them. You must understand that the vega sensitivity will change as the underlying moves if the underlying stock gets closer to the barrier level, then the short term vega will increase and the long term vega will decrease.

If the barrier is only monitored at maturity, then getting the skew corresponding to that maturity correct is the primary concern and we would use the exact date-fitting model.

### 12.6.2 Counter-example : Call Up-and-Out (CUO)

Let us highlight some differences in the risk analysis of a Up-and-Out call with respect to a Down-and-In put.

#### 12.6.2.1 Delta Hedge

Assume a trader is short a up-and-out call. The delta at inception actually depends on how far is the barrier relatively to the spot. This clearly depends on the volatility of the underlying asset. For example, for a 1y CUO, 20% can be seen as far from the strike if the volatility level is low and quite close if the volatility is high.

Initially, the trader usually needs to buy shares as a delta hedge. However, when the stock price gets closer to the upper barrier level, there is an inflection point where the trader will need to go short shares to be delta hedged! So delta can actually change sign before the barrier level for an up-and-out call.

Once the call knocks out, the trader will need to buy back these shares as it is no longer a hedge against anything since the option does not exist anymore.

#### 12.6.2.2 Volatility

A long position in an up-and-out call is not necessarily long vega. In fact, more often than not it is short vega, meaning that when volatility goes up, the CUO becomes less valuable. The higher IV results in a higher probability of the call knocking out and a lower chance of CUO having a payout at maturity.

The vega position also depends on how far is the barrier relatively to the spot. For example, if the barrier is very upside, then the probability of knock-out is very low and an increase in IV has a higher impact on the option part than on the fact that it increases the probability of the option knocking out.

To sum up, an up-and-out call will be long vega for very upside barriers and short vega for lower barriers. How far is the barrier is relative and depends on the barrier/strike differential and the level of volatility.

## 12.7 Double Barrier Options

A double barrier option is another variation that has two barriers, typically one up barrier and the other down barrier. There are typically ised within structured products to achieve a specific type of payoff. These options are priced using a Monte Carlo process. Therefore, from a trading perspective, the magnitude and direction of the barrier shift will be of great interest.

The first type of double barrier option features dependency between the trigger of the first and second barrier. In other words, the second barrier can only trigger if the first barrier has been triggered. The Monte Carlo modeling is slighly more complex as there is an additional condition but it is fact more transparent from a risk perspective than if the barriers were completely independent. Assume a trader sells an ATM call with a knock-in barrier at 90% and a knock-out barrier at 120%. Assume the call can only knock out after it has knocked in. Since the 120% barrier is dependent on the 90% barrier, one should first focus on determining the shift on the 90% one and then the shift on the 120%. The direction and magnitude of these barrier shifts have already been discussed by now.

The second type of double barrier option shows complete independence between the two barriers. When a specific barrier is breached, it triggers regardless of whether the other barrier has been triggered. The Monte Carlo modeling is easier but, from a risk perspective, it is actually less transparent. Let us take the same example of an ATM call with a 90% KI barrier and 120% KO barrier to show why it is less transparent. This option is worth less than the previous conditional one as it can already knock before it has even become a call option. This also means that the shift applied on the 120% barrier can be less if the barrier breaches before the 90% barrier has been breached. However, if the 90% has been breached first then you would require the same barrier shift as a regular CUO. As you can see, there is clearly more ambiguity regarding the magnitude of the barrier shift to apply.

You can price and analyze the risks of the second type of double barrier options using our double barrier options pricer.