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- Volatility Derivatives 1
- The world of Structured Products 4
- Library of Structured Products 0
- Table of Contents
- Vanilla Options
- Volatility, Skew and Term Stru
- Option Sensitivies: Greeks
- Option Strategies
- Correlation
- Dispersion Options
- Barrier Options
- Digitals
- Autocallable Structures
- The Cliquet Family

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- Barrier Options

# Barrier Options

__10. Barrier Options__

Barrier options are options that have a payoff contingent on crossing a second strike known as the barrier or trigger.

There are two kinds of barrier option: **knock-out options** and **knock-in options**. KO options are options that expire when the underlying’s spot crosses the specified barrier. KI options are options that only come into existence if the barrier is crossed by the asset’s price. The observation of the barrier can be at any time during the option’s life (**American style**) or at maturity only (**European style**).

**10.1. Barrier Option Payoffs**

**10.1.1. Knock-out Options**

KO options are **path-dependent options** that are terminated if a specified spot’s price reaches a specified trigger level at any time between inception and expiry. In this case, the holder of the option gets zero payout.

Therefore, **the closer the barrier level is to the initial spot, the cheaper the KO option** would be. Moreover, it is interesting to note that 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.

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.

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 the up-and-out call (down-and-out put) can be much more attractive than the leverage of a comparable vanilla call (put) 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.

**10.1.2. Knock-in Options**

KI options are **path-dependent options** that are activated if a specified spot rate reaches a specified trigger level between the option’s inception and expiry. If such a barrier option is activated, the option then becomes essentially European-style and so these options also have lower premiums.

Therefore, **the nearer the barrier level to the initial spot, the more expensive the KI option would be**. Moreover, it is interesting to note that 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.

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.

The leverage effect of the down-and-in put (up-and-in call) can be much more attractive than the leverage of a comparable vanilla put (call) for an investor who believes that the spot will touch the in barrier during the investment period. He gets more profit for bearing the risk of not knocking in.

**10.2. Discussion**

**10.2.1. Parity Relationships**

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.

**Knock-in (K,T,H) + Knock-out (K,T,H) = Vanilla (K,T) **

**10.2.2. Discrete Barriers**

In the case of KO options, the higher the number of barrier observations, the higher the probability of observing the barrier being breached and the option knocking out. A KO option having an annually monitored barrier would be more expensive than a similar KO option having a bi-monthly monitored barrier.

For KI options, this is the opposite. If the number of barrier observations increases, the price of the option is more expensive since the probability of activating it is higher. A KI option having a monthly monitored barrier would be cheaper than a similar KI option having a weekly monitored barrier.

For up-and-out options, a higher barrier decreases the probability of knocking out and thus increases the up-and-out option price. The seller of this option is short the barrier since a lower barrier would decrease the price of the sold asset.

In the case of down-and-in options, a lower barrier level would decrease the probability of knocking in and thus decreases the option price. One buying the down-and-in options would be long the barrier since a higher barrier will increase the value of his holdings.

**10.3. Hedging Down-and-in Puts**

Many structured products use down-and-in puts to obtain enhanced yields or increased participation. The view is non-bearish in the sense that one would not expect the puts to knock in and will just receive the high coupon or participation. These products are good examples of structured products not offering capital protection since the buyer of the structure is in fact selling down-and-in puts at maturity and can lose the money invested. The investor accepts the risk from selling the down-and-in puts to generate extra funding that is used in the structure to increase the yield or participation.

**10.3.1. Monitoring the Barrier**

**Traders on the sell side are usually long the down-and-in put at maturity**, and have to hedge the risks associated with this position accordingly.

If the forward price of the underlying share goes down, then the price of the down-and-in put goes up for two reasons. Firstly, the potential payoff of the put is higher and, secondly, the probability of activating the option increases. The trader taking a long position in the down-and-in put is then short the forward, and will need to buy Delta in the underlying asset on day 1, and adjust dynamically through the life of the trade to remain Delta neutral. The seller will be short interest rates but is long dividends and long borrowing costs.

The risks of a barrier option near the barrier can be difficult to manage. **The Delta of a barrier option can jump near the barrier causing hedging problems**. When near the barrier, a small move in the underlying can have a large impact on the price of a barrier option. **So near the barrier, the Gamma can be very large**; a small move in the underlying will change the value of Delta significantly. **One method to smooth out the risks to make them manageable is to apply a barrier shift.**

If the knock-in barrier is near the initial spot level, this makes the option more expensive because the probability of crossing this barrier is higher. The trader buying the down-and-in put is then long the barrier since a higher barrier level increases the option price. Therefore, the trader will apply a shift to the initial barrier when pricing the option in order to compensate for the associated risk. For instance, if one is about to make a bid on a 100 put/KI 60, he can price this option as if the option was a 100 put/KI 58%; in other words, he can apply a barrier shift equal to 2%, which makes the option cheaper for the trader since it is less risky.

The shift will take into account the size of the digital around the barrier and the maximum volume of underlying share that can be bought or sold during one day. This is referred to as a **liquidity-based barrier shift**, and accounts for the discontinuity in the Delta near the barrier. Depending on the trader’s position with respect to the option, he might need to buy or sell a large amount of underlying stock if the barrier is reached. **Therefore, the barrier risks are higher if the digital size is high or the daily traded volume is low**.

**In the first days of an option’s life, under normal levels of volatility, the underlying is unlikely to breach the barrier level**. If one were to simulate paths and monitor the points in time at which the barrier was breached, it is obvious that the knock-in events occur more frequently down the line. **One can thus apply a barrier shift that is not constant but is in fact an increasing function of time**.

Assume an investor is willing to sell an american 100 call/KI 60. 3 IBs are competing for this trade and will all take the same CC and apply the same pricing parameters (volatility, skew, etc…). **They all want to apply a shift of 2% but they have differents way 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 Shiftmax 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 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.

**10.3.2. Volatility and Down-and-in Puts**

A higher volatility increases not only the probability of knocking in but also the potential payoff of down-and-in puts. The trader buying this option is long volatility. This long Vega position can be hedged, at least partially, by buying vanilla put options on the same underlying asset **with strikes between the barrier and the spot**.

With regards to the skew effect, the volatility around the barrier is higher than the ATM volatility, which makes the probability of crossing the barrier higher. Then **the price of the down-and-in put is higher because of the skew**. Since the trader buying this option wants its price to go up, he is then long skew.

From the model point of view, and in order to capture the skew effects, we will need to calibrate a model to the implied volatilities of options on the underlying, across strikes, with specific attention to the downside skew.

If the barrier is monitored continuously we will need to apply a model that gives a **smooth calibration through all ends of the surface** between short maturities and up to the maturity. This means that one must **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, and therefore has **Vega sensitivity through the different time-buckets**. European options with different maturities must now be calibrated so that the model shows risk against them.

**The Vega sensitivity will change as the underlying moves**: if a barrier event is close to happening, i.e. **the underlying is trading close to the barrier, then the short-term Vega will increase and sensitivity to the long-term volatility 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.

**10.3.3. Dispersion effects on WO Down-and-in Puts**

The KI event can be triggered by any one of the assets. It is the case more often than not, given the short maturity, that the underlying to trigger the barrier is the worst performing at maturity. Nonetheless, if knocked in, the payoff at maturity is that of a put option on the worst performing asset in the basket, irrespective of which element triggered the knock-in. Also note that a worst-of down-and-in put is more expensive than a down-and-in put with the same strike, barrier level and maturity since the potential payoff is much higher. This makes it more effective when used in the context of yield enhancement or for generating income for a higher upside participation, but this obviously involves the investor bearing additional risk.

We must now account for dispersion. The buyer of the option, typically the sell-side desk, is no longer buying volatility on one asset, but on several, and is short the correlations between these underlyings. This is the consequence of the buyer of the worst-of down-and-in put being long dispersion. The individual Vegas can be hedged by turning to the vanilla market and buying OTM puts on the various underlyings, with strikes between the barrier and the spot. This can lead to additional transaction costs.

Depending on the individual volatilities, the Vegas to each of the underlyings can be different. The price of the option is sensitive to all the individual volatilities but will be so much more sensitive to the more volatile underlyings owing to the increased probability that they end up as the worst-of. Therefore, the vanilla puts used in the volatility hedges will be of different notionals.

Being long a worst-of down-and-in put results in a long skew position on the different assets.

The choice of model breaks down to the same case as the single asset down-and-in put. Also, because this is a multi-asset option we will need to do this for each underlying, and run the simulations based on a correlation matrix and taking into account that the seller of this derivative is short correlation. Correlation skew risk may be exhibited by the option, especially if the barrier is far from the spot.

**10.4. Barriers in Structured Products**

**10.4.1. Multi-asset Shark**

The shark note is a product based on a basket of underlying stocks. It is composed of a ZC bond and an up-and-out call option with rebate.

The holder of a shark note has a bullish view on the underlying stocks and believes that the performances of the stocks will not be above a specific level. The up and out call is cheaper than a vanilla call option with same strike and maturity.

The delta of the up-and-out call is positive as the spot is below the KO barrier but becomes negative as the spot approaches the barrier. At inception, the trader is therefore short the forwards but will become long the forward if the spot starts to be close to the outstrike.

The position in volatility is a function of the parameters of the contract. On one hand, a higher volatility raises the payoff of the call, but at the same time it increases the probability of hitting the barrier. Therefore the position in volatility depends on the strike, the barrier and the time to maturity.

As this is a basket option, correlation has its main effect on the volatility. Higher correlation means a higher overall volatility for the basket and the analysis of the position in correlation is the same as the volatility.

The skew position is somewhat awkward. The seller is short the ATM volatility but long the volatility near the barrier. The magnitude of the volatility sensitivity near the barrier is a function of the size of the rebate and the location of the barrier. Thus the overall position in skew is not immediately clear and needs to be checked.

**10.4.2. Single asset Reverse Convertible**

The Reverse Convertible is an extremely popular product. Many variations exist but the main one is the combination of a ZC bond, a DIP and a fixed coupon while the buyer of the reverse convertible is short the DIP. This holds a downside risk as the performance of the stock could potentially be large and negative and wipe out both the coupon and part/all of the capital invested. The investor is willing to accept a downside risk in exchange for an above market coupon. This structure fits an investor who is moderately bullish.

**10.4.3. WO Reverse Convertible**

It is similar to the reverse convertible but here the equity exposure is on the worst of a basket instead of just one underlying.

The worst-of feature makes the DIP more valuable and will allow the investor to receive a higher coupon.

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