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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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- Autocallable Structures

# Autocallable Structures

__12. Autocallable Structures__

Autocallables are very popular in the world of structured products. In this chapter, we look at standard autocallables and several variants on them.

**12.1. Single Asset Autocallables**

**12.1.1. General Features**

*Payoff Description*

Consider an autocallable note based on a single asset S, a structure which pays coupons depending on the underlying’s performance reaching two triggers H and B, and has a payoff defined as follows: at each observation date t_{i} , (i = 1 . . . n) we have:

Since the wrapper is a note, the holder receives back 100% of the notional except that, in this case, the time of payment is not fixed. The notional redemption can be at any observation date, not necessarily at maturity.

**The autocallable structure doesn’t have a fixed maturity.** What we call maturity is in fact the maximum duration this product can stay alive.

H is called the autocall trigger or threshold. The threshold can be fixed during the life of the option or can be variable. In some cases, the threshold can be increasing or decreasing as time goes by.

B is the coupon trigger, also called coupon level.

*Risk Analysis*

In order to price the autocallable digitals effectively, one can compute the undiscounted **conditional probabilities** of receiving the coupons. These values can enable one to quickly check whether the pricing makes sense. After these probabilities are computed, they should be discounted and multiplied by the coupons to be received. This gives us the price of the autocallable digitals.

The first digital option is a classical European digital. The undiscounted probability of striking the second year is conditional on not autocalling at the end of the first year. The undiscounted probability of striking the third year is conditional on not autocalling at the end of the first and second years. **As time goes by, the probabilities of coupons being paid decrease and the value of the last path-dependent digitals can be very small because the conditional probabilities of striking would be low**. In this case, the seller must be careful when offering a very large digital with a low probability.

The risks associated with a single asset autocallable structure are similar to those associated with single asset digitals. **The seller of an autocallable is short the underlying’s forward (short interest rates, long dividends and long borrowing costs) and short the skew. **

**The position in volatility depends on the coupon level and the forward price of the underlying**.

The Vega of digitals is positive if the underlying’s forward price is lower than the trigger; otherwise, Vega is negative. The Vega hedge will consist of a set of European options with strikes matching (as closely as possible) the autocall trigger dates. The **overall volatility sensitivity is split over these Vega buckets**, and each of these sensitivities will change as the market moves. **If an autocall event is about to happen, the short-term Vega will increase and the Vega in the other buckets will decrease**, in line with the higher probability of autocalling. A Vega hedge set at inception will need to be readjusted if the market moves significantly in relation to the autocall triggers.

**12.1.2. Interest Rate / Equity Correlation **

The autocall is an excellent example of a structure where the correlation between equity and IR has an effect on the price.

The autocallable is redeemed at a time in the future that is a function of the path the underlying equity takes. Assume the investor is paying the bank LIBOR in exchange for this equity exposure, then the duration of the swap is dependent on the equity and thus the structure is sensitive to the correlation between interest rates and the equity underlying. In the case where an autocallable is structured to provide equity exposure as part of a note, then the investor is in this case implicitly short the floating leg of an interest rate swap and the same thing holds.

In the case of the autocallable, the pricing of this correlation effect is typically done by taking a small margin. This can be specified by first deciding on the level of correlation and also the length of the trade. To get an idea about this effect we use a hedging argument. Setting aside the equity component that will be Delta hedged using the underlying equity, we look at the interest rate hedge of the seller. The seller of a 2y annual autocallable will go long ZC bonds with respective maturities of 1 and 2 years.

*First case: Assume that the equity/interest rate correlation is positive. *

**If the underlying increases**, then the probability of early redemption at the first autocall date increases, and to adjust the interest rate hedge accordingly the seller will increase the amount of 1-year bonds held and sell some of the 2-year bonds. Because of the positive correlation in this case between the underlying equity and interest rates, we expect that interest rates will also increase on average, and thus the price of the zero coupon bonds will decrease. Since the bond with the longer maturity decreases more in value than the bond with the shorter maturity (using simple bond maths) the seller nets a loss on the rebalancing of this hedge because the seller is buying one bond but selling the one that decreased more in value.

**If the underlying decreases**, then the probability of the structure autocalling early de- creases. In this case the seller must adjust the interest rate Delta hedge by selling some of the 1-year zero coupon bonds held and buying more of the 2-year bonds. On average we expect that interest rates will also decline because of the positive correlation. This implies that the 2-year bond will increase in value more than the 1-year bond, and again the seller thus nets a loss on the rebalancing of this hedge.

*Second case: Assume that the equity/interest rate correlation is negative. *

**If the underlying increases**, the opposite happens. We expect the interest rate in this case to decline, on average, and the same rebalancing as the case of increased possibility of early redemption as above will in this case net the seller a profit.

**If the underlying decreases**, then again the opposite happens: negative correlation means that we expect rates to go up and thus reduce the price of the 2-year bond more than the 1-year bond. The decreased probability of early redemption means the seller will need to buy more of the 2-year bond and sell some of the 1-year bond, thus netting a profit.

The upshot of this analysis is that the autocallable’s price should be higher when a positive correlation is assumed between interest rates and the underlying equity, and lower if the correlation is negative.

The question thus arises as to whether we should employ a model that includes stochastic rates, and thus be able to enter a value for this correlation and include its impact in the price.

Arguments in favour of the use of such models are discussed by Giese (2006) for example, and the impact on pricing is discussed and concluded to be important. However, although using such models allows one to see this impact, they do not give us additional information regarding the hedging of the equity interest rate correlation. Specifically, since the sign of this correlation governs whether there is a cost or a benefit, deciding on which correlation to use and adding a cost accordingly can be done without having to employ a stochastic rates model.

The magnitude of this cost will be a function of the maturity of the structure. If we had assumed a 3y annual autocallable then the same hedging argument holds, only the Delta hedge for IRs will include the 1- and 3-year bonds. The impact of a move in IRs is greater on a 3-year bond than on a 2-year bond and thus the impact of the correlation is greater the longer the maturity of the autocall structure.

The use of stochastic interest rate models and the importance of the correlation between rates and equities becomes more significant when pricing hybrid products. Generally we can say here that since this correlation cannot be hedged in a straightforward manner, and perhaps not hedged at all, the best thing to do is decide on the level for this correlation and add a cost accordingly. To trade this correlation, and thus hedge this correlation risk, one would need a liquid structure involving equity and interest rates, from which one could extract this correlation by hedging away the other parameters.

Note that, from the investor’s point of view, a positive correlation would imply that he is likely to get his above market autocall coupon and his money back in a high interest rate environment. This is a good scenario for the investor.

**12.2. Autocallable Participating Note**

The autocallable participating note (APN) is an interesting structure that offers 100% capital protection and can be used to take advantage of a bull market. Let’s consider a share Alpha that is near an all time high. An investor may consider converting a portion of his Alpha portfolio into an autocallable participating note, thus locking in the current gains yet retaining the ability to profit from continuing appreciation, via an autocallable structure with 250% participation in case the note has never been autocalled.

It’s interesting to note that the APN offers 100% capital protection, multiple lock-in profit levels as well as an uncapped 250% participation in the appreciation of Alpha shares if not autocalled. Compared to holding these shares outright, **the investor loses his dividends, in return for 100% capital protection plus autocall coupons at roughly three times the USD interest rate (assuming USD rates are roughly 3%) and 250% participation in the stock upside if not autocalled. **

**12.3. Autocallable with Down-and-in Puts**

**12.3.1. Adding the Put Feature **

If the investor believes that the underlying index will not be lower than a specific level at maturity, she can **add a put feature to the autocallable structure to increase the potential coupon received**. This means that the capital is no longer protected as the holder is short a put option at maturity T.

The put option can be a vanilla at-the-money European put option whose maturity is the maturity of the autocall. But most of the time, the buyer is short a down-and-in at-the-money put option that can be either European or American style.

**When a trader sells an autocall with a put feature at maturity, he is short the autocallable digitals and long a path-dependent put option**. In order to price this structured product, one should price the autocallable digitals as described above and deduct the price of the path- dependent put option.

To briefly summarize the risk analysis of the DIP we did in chapter 10, **a higher forward price of the underlying decreases the down-and-in put price. Volatility and skew increases its price. **

The seller of an autocall would definitely be short the underlying’s forward price and long the triggers (autocallable trigger, coupon trigger and DI put barrier level). However, his **overall position in volatility and skew are not immediately clear owing to potentially offsetting effects from the two components**.

If the forward price is higher than the coupon level, then volatility decreases the price of the autocallable digitals. A trader selling an autocallable is then usually long volatility with respect to the digitals, and always long volatility with respect to the put.

As for the skew position, **the seller of an autocallable with a put feature has a short skew position with respect to his short position in digitals, but has also a long skew position with respect to his long position in the down-and-in put**.

**12.3.2. Twin-Wins **

It is an autocall structure with a down-and-in put, with the potential of capturing the absolute performance of the underlying at maturity. The name Twin-Wins comes from the fact that this note enables the holder to get a participation in both the upside and the downside movements of the underlying asset.

Twin-Wins is an interesting structure in the case where no early redemption has occurred during the life of the note. Indeed, investors can still capture the absolute performance of the underlying at maturity if no knock-in event occurred during the life of the product.

+ discussion grosse digit, sensi etc..

**12.4. Multi-Asset Autocallables**

**12.4.1. Worst-of Autocallables**

The payoff is exactly the same except that we observed the performance on the worst performing share of the basket.

In terms of risk analysis, it is quite similar except that there is an **additional dispersion dimension**. The **holder of a WO autocall is long correlation as it increases the probability of receiving the coupons and being autocalled and decreases the probability of being into the DIP**.

**12.4.2. Snowball Effect and Worst-of put Feature **

*Payoff Description*

If the product is still alive at year i, this means that the investor didn’t receive the previous periodic coupons. **The snowballing structure enables the investor to receive a coupon equal to the sum of all previous coupons if the trigger is reached**. The investor is then recovering his losses due to the non-received previous coupons.

One can also add a worst-of down-and-in put feature to the autocallable structure. This happens when the investor is willing to increase the potential coupons received and he believes that all the shares composing the underlying basket will perform above a specific level.

Assume that we start with a basket composed of n assets S_{1},S_{2},...,S_{n}, then a worst-of autocallable note based on this basket has the following payoff:

At each observation date t_{i}:

If the structure has not autocalled, the redemption at maturity is determined as follows:

(a) If all the assets composing the underlying basket close at or above the knock-in level, the note is redeemed at 100%;

(b) If one of the underlyings closes below the knock-in level, the note is converted into physical shares at strike.

*Risk Analysis *

In the case of snowballing coupons, the digital size increases with time since the potential coupons are higher. Therefore, one must be careful when offering a very large digital with a very low probability of striking

A worst-of autocallable note with a worst-of down-and-in put is composed of zero coupon bonds, worst-of autocallable digitals and a worst-of down-and-in put. We are already familiar with these structures, and therefore we can easily analyse the risks associated with the full structure.

If one is **short a worst-of autocallable structure with snowball effect and worst-of down-and- in put, he would definitely be short the forward prices, long the triggers, long volatility and short correlation**.

The **position in skew is more complex** to determine since the **skew makes the sold worst-of digitals more expensive** but also makes the **bought worst-of down-and-in put more expensive**. **Even if the short skew position in digitals has usually more effect on the price than the long skew position in the put at maturity, it is not always the case. **

**12.4.3. Outperformance Autocallables**

Let’s consider two assets S_{1 }and S_{2}. An outperformance autocall based on the outperformance of S_{1} vs S_{2} is typically a European style option that pays a coupon C at each observation date if S_{1} − S_{2} outperforms a specific level called a cushion. This outperformance structure has an autocall feature that pays a coupon upon early redemption. This note payoff is as follows:

At each observation date t_{i}:

Outperformance autocallable options are composed of path-dependent outperformance digitals.

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