# Chapter 14 Certificates

## 14.1 General Description

Certificates are among the most popular participation products. There are characterized by a wide variety of risk-return profiles. This diversity of profiles can be achieved because the investor usually foregoes, with the purchase of a certificate, the dividends paid by the underlying asset during the life of the certificate. For that reason, certificates on shares that pay a high dividend often have a particularly attractive risk/reward profile.

Some advantages about investing in certificates:

- Opportunity to implement sophisticated investment strategies through the purchase of a single product.
- They are listed in regulated markets.
- They are very liquid instruments.
- Favorable taxation.

In this chapter, we will only describe the most well-known variants of certificates.

## 14.2 Tracker Certificates

The Tracker Certificate is a structured product that replicates (tracks) the performance of an underlying asset or often a basket of securities simply, inexpensively and with high liquidity. As such, there is no optionality involved in this product category. It is the reason one this simple product is not directly available in our exotic derivatives prices. However, you can easily price it with the **certificates pricer** by selecting ‘Discount Certificate’ and setting the Cap to 999. You can also use the **Classic option pricer** and price a zero-strike call option.

Excluding the issuer risk, its risk is identical to the underlying asset it tracks.

*Why do they exist then? Why not simply investing in the underlying asset?*

The purpose of trackers is their ability to allow investments otherwise not possible or not economically feasible.

Trackers are generally used to diversify the investor’s risk exposure across a wide range of individual stocks. It provides a simple and cost-effective mean of investing in an entire stock market without having to buy each of the individual shares. With trackers, you can easily get access to exotic markets, for which you would not have access otherwise via your bank/broker.

*What is the difference with a share index then?*

Major share indices are priced in thousand of points. In other words, you need thousand of the denominated currency to buy a share index. That is clearly an obstacle for many investors. A tracker makes it possible to invest in the same index or any another baskets of stocks with smaller amounts of money. Exchange traded funds (ETFs) may also be a good alternative.

Tracker certificates are one of the few structured products that may have no maturity.

### 14.2.1 Payoff

Tracker certificates are constructed using a zero-strike call on the underlying asset. If the certificate tracks an index on which a liquid future is available, the issuer often buys the future instead of a zero-strike call. The issuer may even physically hold the underlying individual shares if the certificate is based on a basket of shares.

The below graph shows the payoff of a 1y tracker certificate on an underlying at a price of 100 with a dividend yield of 3.5% and interest rates hypothetically set to 2%.

The price of this tracker certificate is 96.56. This is simply the price of a zero-strike 1y call option but could also be split into:

- Long 1y Zero-Coupon Bond: 98.02 (assuming 2% interest rate for the sake of the example).
- Long 1y ATM Call : 7.04 (assuming 20% implied volatility and 3.5% of dividends).
- Short 1y ATM Put : 8.50 (assuming 20% implied volatility and 3.5% of dividends).

## 14.3 Discount Certificates

Discount certificates are yield-enhancement products that allow the investors to buy an underlying instrument for less than its current market price (at a discount). As there is no free lunch, the investor is now limited to a predetermined amount (the cap). The lower the cap, the more limited the upside, the greater the discount.

Discount certificates are typically short term products with maturity ranging from one to three years.

At maturity, the investor will incur a loss only if the price of the underlying asset has decreased so far that the discount has been fully eroded. The good thing is that, if you incur a loss with a discount certificate, it will always be less than the amount you would have lost if you had invested directly in the underlying asset.

Where you place the cap really depends on your market view. If you expect a sideways market, you would place the cap closer to the current price of the underlying instrument than if you expect a modestly rising market.

### 14.3.1 Payoff

Discount certificates are constructed as a combination of a long zero-strike call and a short cap-strike call on the underlying asset.

The below graph show the payoff of a 2y discount certificate with a cap at 115 on an underlying at a price of 100 with a dividend yield of 3.5% and interest rates hypothetically set to 2%.

The price of this discount certificate is 88.42. This price can be split into :

- Long 2y 0 Call : 93.24.
- Short 2y 115 Call : 4.81.

but it could also be split into :

- Long 2y Zero-Coupon Bond: 96.08 (assuming 2% IR for the sake of the example).
- PV(guaranteed coupon of 15 in 2y) : 14.41 (assuming 2% IR for the sake of the example).
- Short 2y 115 Put : 22.07 (assuming a 20% IV and a 3.5% dividend yield).

### 14.3.2 Risk Analysis : The Greeks

The delta of the product is obviously positive. This positive delta position also implies that the investor is short dividends on the underlying stock. How positive is the delta depends on the position of the cap. The higher the cap, the closer to a tracker certificate is the product and the closer to 1 is the delta. In the above example, the delta is 63%. It corresponds to the delta of the combination of a long position in a 100/115 call spread (18%) and a short position in an ATM put (45%).

As the investor has more downside than upside, he will be short volatility. As the price of the underlying gets closer to the cap level, the investor becomes more sensitive to volatility. Think about the fact that the investor is short an option striking at the cap level and remember that the vega of an option is higher around its strike. On the contrary, as the price of the underlying asset decreases, the sensitivity to volatility also decreases (in absolute value).

Since the trader selling a discount certificate is buying an (OTM) option, not only he is buying volatility but he is also long gamma and short theta. He will get longer gamma as the underlying price gets closer to the cap level (the option strike) and the time to maturity gets smaller.

### 14.3.3 Deep-discount Certificates

Now that you understand a bit more the payoff of a discount certificate, let me introduce you to the extremely defensive version of it with a cap far below the current level of the underlying asset. This version is called the deep-discount certificates and can be seen as a substitute to a savings account since it provides much security.

Let us take a numerical example to show you the principle of the deep-discount strategy. Imagine you buy a discount certificate at 63.5 with a cap at 65 when the value of the underlying asset is at 100. Taking this into consideration, you will make a profit as long as the underlying does not drop below 63.5 (a loss of 36.5%). In addition, you will make a profit of 1.5 (2.36%) in any scenario in which the underlying does not lose more than 35 (a loss of 35%).

## 14.4 Bonus Certificates

Discount certificates can be a bit frustrating during bull markets since their performances lag those of the market. At the same time, investors keep on willing some safety buffer against price declines. Bonus certificates seem a good solution for the investors not willing to miss on the upside.

Bonus certificates are also typically short term products with maturity ranging from two to four years.

They guarantee a specified bonus level as long as the underlying asset has not breached an established barrier level during the term of the certificate.

In contrast to discount certificates, the payoff is not capped to a fixed amount so that the investor keeps on participating in the price gains above the bonus level.

However, if the safety barrier is breached, then the product changes into a tracker certificate and the right to a guaranteed payout of the bonus level no longer exists. If the underlying asset goes back up afterwards, the investor can still participate 1:1 in the potential gains but the guaranteed payout of the bonus level will not be reinstated after such move!

Hence, bonus certificates tend to perform well in both sideways and rising markets.

### 14.4.1 Payoff

Bonus certificates are constructed as a combination of a long zero-strike call and a long Down-and-Out put with striking at the bonus level on the underlying asset. The zero-strike call provides the participation, and the DOP option generates the barrier and the bonus (determined by the DOP strike). In case of stocks, the dividends are used to buy the DOP option. Therefore, higher implied dividends generate a higher bonus level, a greater contingent protection level or a mix of both.

The below graph show the payoff of a 2y bonus certificate with a bonus level at 112.5 and a barrier level at 75 on an underlying at a price of 100 with a dividend yield of 3.5% and interest rates hypothetically set to 2%.

Note that this payoff is only valid at maturity; the mark-to-market price of the bonus certificate during its lifetime differs strongly from its final payoff. It may be stated that in general, the contingent protection and bonus will only “grip” after most of the product’s maturity has elapsed.

The price of this bonus certificate is 98.94. This price can be split into :

- Long 2y 0 Call : 93.24 (assuming a 3.5% dividend yield).
- Long 2y American DOP with barrier at 75 and strike at 112.5 : 5.70 (assuming a 20% IV and a 3.5% dividend yield).

### 14.4.2 Risk Analysis : The Greeks

The dynamics of the greeks are largely explained by the long DOP position. Since barrier options have already been discussed, you should be able to understand it and explain it by now. As always, the sensitivities depend on the product’s parameters.

Understanding the greeks and their dynamics is probably the most important and the most complicated aspect about derivatives. In this regard, I believe the exotic option pricer is an infinite source of information. The very large majority of interview questions related to equity derivatives asked by any financial institution can be answered using the exotic option pricer. To analyze the greeks of a bonus certificate, use the **certificates pricer** and select ‘Bonus Certificate’ as the type of certificate.

Remember, as soon as you see discontinuities in the payoff, think about the way the trader will smooth it on his side.

## 14.5 Capped Bonus Certificates

This is a particular version of the bonus certificate with limited participation in rising markets up to the cap level. As a bonus certificate, if the barrier has not been breached during the lifetime of the product, investor receives a minimum redemption equal to the bonus level. Should the barrier level be breached during the lifetime of the product, the capped bonus certificate turns into a tracker certificate with a cap.

### 14.5.1 Payoff

Bonus certificates are constructed as a combination of a long zero-strike call, a long Down-and-Out put with striking at the bonus level and a short cap-strike call on the underlying asset.

The below graph show the payoff of a 2y capped bonus certificate with a bonus level at 113, a barrier level at 70 and a cap level at 125 on an underlying at a price of 100 with a dividend yield of 3.5% and interest rates hypothetically set to 2%.

The price of this capped bonus certificate is 98.85. This price can be split into :

- Long 2y 0 Call : 93.24 (assuming a 3.5% dividend yield).
- Long 2y American DOP with barrier at 70 and strike at 113 : 8.24 (assuming a 21% IV and a 3.5% dividend yield).
- Short 2y 125 Call : 2.63 (assuming a 19% IV and a 3.5% dividend yield).

As you can see from the difference in implied volatilities between the down-and-out put and the OTM call, we assumed a bit of a volatility skew.

By limiting his upside to the cap level, the investor benefits from more protection and a slightly better bonus when comparing this capped bonus certificate with the bonus certificate previously discussed.

### 14.5.2 Risk Analysis : The Greeks

Since the product is a variant of the bonus certificate, its sensitivities to the various market parameters are very similar.

Its delta is also positive but smaller than the delta of the bonus certificate since the upside potential is capped. Selling an OTM call implies a negative delta on this position and therefore reduces the overall positive delta of the capped bonus certificate.

Its vega position is also negative and is larger in absolute value than the vega of the bonus certificate. An increase in volatility increases the probability to breach the downside barrier and lose the bonus. Since the upside potential is capped, the investor has more to lose than he has to win when volatility increases.

As for the bonus certificate, skew also has an impact on the pricing as both products include an down-and-out put option. The skew sensitivity is likely to be smaller for the capped bonus certificate as the skew will have a counterbalancing effect on the sold OTM call option (a stronger skew will tend to decrease the price of the long DOP position and decrease the price of the sold OTM call).

## 14.6 Airbag Certificates

Airbag certificates fully participate to the upside price performance of an underlying asset and have a certain amount of downside protection down to which the investor does not incur any losses. In other words, as long as the underlying asset does not lose more than a predefined level (the airbag level), the capital is 100% protected. Below the airbag level, the product starts to decrease in value at a leveraged pace compared to the underlying asset. The leverage is the inverse of the airbag level. There are many similarities with a bonus certificate with the one big difference that the Airbag does not knock-out and stays until maturity no matter what. For this reason, there is no discontinuity in the payoff as in the bonus certificate.

### 14.6.1 Payoff

Airbag certificates are constructed as a combination of a long leveraged zero-strike call, a short leveraged airbag-strike call and a long ATM call. The leverage is the inverse of the airbag level.

Let us take a specific example to make everything clear. The below graph shows the payoff of a 2y airbag certificate with an airbag at 85% of the current price of the underlying asset. This underlying asset is assumed to have a 3.5% dividend yield and interest rates are hypothetically set to 1%. In this example, the leverage will then be of 1/0.85 \(\approx\) 1.1765. It represents the speed at which the product’s payoff will decrease below the airbag level.

As previously highligthed, the protection is bought the expected dividends of the underlying asset. The higher the dividend, the larger the protection.

The reason why you see ‘Put Strike’ in the legend of this graph is because you can also see the left side of the graph as a short position in a leveraged OTM put.

The price of this airbag certificate is 99.27. This price can be split into :

- Long leveraged 2y 0 Call : 93.24 * 1.1765 = 109.694 (assuming a 3.5% dividend yield).
- Short leveraged 2y 85 Call : 16.11 * 1.1765 = 18.953 (assuming a 21% IV, a 3.5% dividend yield and 1% IR).
- Long 2y ATM Call : 8.53 (assuming a 20% IV, a 3.5% dividend yield and 1% IR).

As you can see from the difference in implied volatilities between the OTM and the ATM calls, we assumed a bit of a volatility skew.

### 14.6.2 Risk Analysis : The Greeks

You can analyze the greeks using our **certificates pricer** and selecting ‘Airbag Certificate’ in the certificate type. I do not think there will be much of a surprise for this instrument. In any case, feel free to ask me any questions if you find the dynamics of a product strange for whatever the reason.

## 14.7 Outperformance Certificates

Outperformance certificates enable you to participate disproportionately in price advances in the underlying instrument if it trades higher than a specified threshold value. To that purpose, these certificates are equipped with a strike price and a participation factor. Depending on the product maturity, the participation factor usually lies between 120% and 200%.

While the outperformance certificate does not appear directly in the list of certificate types available in our **certificates pricer**, you can easily price it using the ‘Outperformance bonus certificate’ and setting the downside put barrier and the bonus level at the same level. Note that this pricer will not let you enter a downside put barrier/strike higher than the spot level.

### 14.7.1 Payoff

Outperformance certificates are constructed as a combination of a long zero-strike call and a long (Participation level - 1) ATM Call. The participation level represents the speed at which the product’s payoff will increase above the strike.

The below graph shows the payoff of a 2y outperformance certificate with a 165% participation level on an underlying asset at 100 with a 3.5% dividend yield and with interest rates hypothetically set to 2%.

The price of this outperformance certificate is 99.27. This price can be split into :

- Long 2y 0 Call : 93.24 (assuming a 3.5% dividend yield).
- Long Leveraged 2y ATM Call : 0.65*9,29 = 6.03 (assuming a 20% IV, a 3.5% dividend yield and 2% IR).

### 14.7.2 Risk Analysis : The Greeks

Risk analysis should be very straightforward with a delta larger than 1 (delta-1 + partial investment in an ATM call), a bit of vega, gamma and theta due to the partial investment in an ATM call. If you are not very confortable with this, please refer to **chapter 5** for a fresh reminder.

## 14.8 Outperformance Bonus Certificates

As its name indicates, it combines the strengths of both outperformance and bonus certificates by protecting the investor on the downside by a bonus level while still offering the opportunity to participate disproportionately in upside gain in the underlying asset. Obviously, there is no free lunch and the bonus level, the barrier level and the participation level will not be as attractive in an outperformance bonus certificate as they would be in a bonus certificate and an outperformance certificate respectively.

### 14.8.1 Payoff

Outperformance Bonus certificates are constructed as a combination of a long zero-strike call, a long donw-and-out put striking at the bonus level and a long (Participation level - 1) OTM Call striking at the bonus level.

The below graph shows the payoff of a 2y outperformance bonus certificate with a 112.5 bonus level, a 80 barrier level and a 150% participation level on an underlying asset at 100 with a 3.5% dividend yield and with interest rates hypothetically set to 2%.

The price of this outperformance bonus certificate is 98.87. This price can be split into :

- Long 2y 0 Call : 93.24 (assuming a 3.5% dividend yield).
- Long 2y American DOP with barrier at 80 and strike at 112.5: 2.93 (assuming a 21% IV and a 3.5% dividend yield).
- Long 2y Leveraged OTM call: 0.5*5.4 = 2.7.

### 14.8.2 Risk Analysis : The Greeks

While the delta of the outperformance bonus certificate is generally larger than 1, the other greeks can take quite different values depending on the product’s parameters.

For example, the above example has a slightly negative vega at inception. If you take the same structure with a barrier level at 77, a bonus level a 103 and a participation level of 145%, the vega will now be slightly positive at inception. While this result is not surprising, it is still something you need to remind yourself about as you could quickly be fooled.

## 14.9 Twin-Win Certificates

As the ‘twin-win’ indicates, the investor can actually win it both ways. This product has 100% participation to the upside (not capped) and 100% participation to the downside in absolute terms as long as the barrier was not breached during its lifetime. If the barrier is breached, the twin-win transforms itself in a tracker. In other words, the losses recorded down to a specified barrier level are converted into profits. For that to occur, the barrier level may never be so much as touched during the certificate’s entire term to maturity. If by expiration there has never been a breach of the barrier, you will receive cash payment of the difference between the closing price of the underlying instrument and the level where it stood on the issuance date. In any case, you can rest assured that your twin-win certificate will never perform worse than the underlying instrument.

### 14.9.1 Payoff

Twin-win certificates are constructed as a combination of a long zero-strike call and long two DOP, where strike is set ATM.

The below graph shows the payoff of a 2y twin-win certificate with a 70 barrier level on an underlying asset at 100 with a 3.5% dividend yield and with interest rates hypothetically set to 2%.

The price of this twin-win certificate is 98.83. This price can be split into :

- Long 2y 0 Call : 93.24 (assuming a 3.5% dividend yield).
- Long two 2y American DOP with barrier at 70 and strike at 100: 2*2.795 = 5.59 (assuming a 24% IV and a 3.5% dividend yield).

### 14.9.2 Risk-Analysis : The Greeks

The maturity of the product plays an important role as the twin-win delta usually amounts to a bit less than 1 at inception. Hence, the positive performance that should be reflected in the mark-to-market price of the product in a bearish market only ‘grips’ when around 70% of the time to maturity has expired. For example, a delta of 0.98 means that if our stock price goes from 100 to 99, the twin-win price will go down by approximately 0.98. However, the payoff is larger when the stock price is at 99 rather than 100. Since there is a large time left to maturity, this decrease in price has increased the probability of breaching the downside barrier, explaining the negative effect on the twin-win price.

I advice you to play with the ‘maturity’ and ‘stock price’ parameters of the **Twin-win certificates pricer** to observe the interesting behaviour of the greeks when the product is getting closer to maturity.

For example, the delta of the above twin-win will not be of the same sign if you are at 105 or at 95 close to maturity. You will see that delta and gamma can get quite large in absolute value as you get closer to the barrier near to maturity. Depending on the relative distance of the stock price with respect to the barrier, the vega position can also take different signs.

Well, there are so many things that could be said about the greeks analysis of structured products. I have developped and shared the pricer exactly for this reason. It sums it all with graphical representations that will help you developing your understanding and your intuition behind the dynamics behaviour of the greeks.