# Chapter 16 Autocallables

Autocallables are probably the most popular structures in the world of structured products. In this chapter, we will look at standard autocallables and some of their variants.

## 16.1 Description

The standard autocallable is a note that only pays a coupon if the underlying asset (S) is above a certain coupon barrier level (CB) and the note automatically redeems early if it breaches an autocall barrier level (AB), which can be the same or different as the coupon barrier level, at an observation date. The autocallability feature can be whether daily, monthly, yearly, or based on any schedule determined in accordance with the client. The coupon barrier level is always lower or equal to the autocall barrier level, otherwise the product does not make much sense. When the CB = AB, I will call the structure an Autocall and when CB < AB, I will call the structure a Phoenix.

An autocallable is clearly a yield enhancing strategy. The investor receives an above market yield and in turn takes the risk not to receive any coupon if the underlying asset is below the coupon barrier level on the observation date.

In this low-yield environment, it is impossible to offer attractive coupons with such a structure. Therefore, to get an enhanced coupon the investor typically also sells an option embedded in the note (usually a down-and-in put or a leveraged OTM put). We will start with analyzing the most basic autocallable structure and then add the most frequent features observed in the market.

Because of the autocall feature, this structure does not have a fixed maturity. What we call maturity is in fact the maximum duration this product can stay alive.

## 16.2 Payoff

At each observation date $$t_i$$ with i = 1, ..., n, we have :

$$\boxed{\text{Coupon}(t_i) = \text{Notional * C% * } 1_{\left\{ \frac{S_{t_i}}{S_{t_0}} \; \ge \; CB \right\}} * 1_{\left\{ max_{j=1,...,i-1} \frac{S_{t_j}}{S_{t_0}} \; \le \; AB \right\}}}$$

and

$$\boxed{\text{Redemption}(t_i) = \text{Notional * } 1_{\left\{ \frac{S_{t_i}}{S_{t_0}} \; \ge \; AB \right\}} * 1_{\left\{ max_{j=1,...,i-1} \frac{S_{t_j}}{S_{t_0}} \; \le \; AB \right\}}}$$

Once the autocall barrier is breached on one of the observation dates, the note dies and there is no future payment afterwards.

Note that the autocall barrier level can be fixed during the life of the option as well as variable. It is not rare to see it decremental for example. This makes sense as the conditional probability of breaching a fixed autocall barrier is getting lower at each observation date. These conditional probabilities are shown in our autocallables pricer.

As you can see, the above payoff description does not contain any short position in option. This will be added and discussed later on. For now, it basically looks like a stream of conditional coupons (digitals) with an autocall feature.

## 16.3 Risk Analysis

As usual, the price of the product is nothing but the present value of the expected future cashflows. To calculate the expected future cashflows, we start by computing the undiscounted conditional probabilities of receiving the coupons. It will help you to develop a feeling whether the pricing makes sense or not. Once these probabilities are computed, they should be multiplied by the coupons and then discounted. These undiscounted conditional probabilities of receiving the coupons are also made available in our autocallables pricer

The first coupon is a classical European digital. All the next coupons are conditional on not autocalling at any previous observation dates. As time goes by, the probabilities of coupons being paid decrease. The value of the last path-dependent coupons can therefore be very small. As a seller, you should really be careful when offering a very large digital with a low probability.

It is no surprise that the risks associated with this basic autocallable structure is similar to those associated with digitals. That is 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 relative position of the forward price of the underlying with respect to the coupon barrier level. The overall vega is split over the observation dates, we speak about vega buckets. Each of these vega sensitivities will change as the market moves. If an autocall event is about to happen, the ST vega will increase while the other vega buckets will decrease. The vega hedge will therefore be dynamic and readjusted depending on the evolution of the underlying asset.

As the structure has a zero coupon bond embedded in it, it also has an important first order sensitivity to interest rates moves. More precisely, the structure has an overall negative exposition to interest rates.

Whenever the autocallable structure is denominated in a currency that is different from the underlying's currency, there is an exposition to the correlation between the underlying's return and the forex return. We have studied this FX exposure in chapter 10 and have seen that there is an impact on the dynamic delta hedging.

### 16.3.1 Equity / Interest Rate Correlation

This aspect of the pricing is often neglected but the correlation between equity and IR has an effect on the autocallable price. This cannot be seen from our pricer but will often be tackled during interviews.

Since the note is redeemed at an unknown time in the future that is dependent on the performance of the underlying stock, the autocallable structure is sensitive to the correlation between IR and the equity underlying. To get an idea about this effect, we use a hedging argument since the price of a product should reflect the cost of hedging it.

Let us start by assuming that this correlation is positive so that if the underlying increases, then interest rates also tend to increase. Since an increase in the underlying will increase the probability of early redemption after 1y, the autocallable seller will sell some of the 2y bonds to buy more of the 1y bonds. We know that the 2y ZC bond will be more impacted by the IR increase than the 1y ZC bond (as it has a longer duration). All in all, this means that the seller will net a loss on his IR rebalancing since he will be selling the bond that decreased more in value to buy the one that decreased less.

In the same way, if the underlying decreases, then interest rates also tend to decrease. Since a decrease in the underlying will decrease the probability of early redemption after 1y, the autocallable seller will sell some of the 1y bonds to buy more of the 2y bonds. We know that the 2y ZC bond will be more impacted by the IR decrease than the 1y ZC bond (as it has a longer duration). All in all, this means that the seller will net a loss on his IR rebalancing since he will be selling the bond that increased less in value to buy the one that increased more.

Let us now assume that this correlation is negative so that if the underlying increases, then interest rates tend to decrease. Since an increase in the underlying will increase the probability of early redemption after 1y, the autocallable seller will sell some of the 2y bonds to buy more of the 1y bonds. We know that the 2y ZC bond will be more impacted by the IR decrease than the 1y ZC bond (as it has a longer duration). All in all, this means that the seller will net a profit on his IR rebalancing since he will be selling the bond that increased more in value to buy the one that increased less.

In the same way, if the underlying decreases, then interest rates tend to increase. Since a decrease in the underlying will decrease the probability of early redemption after 1y, the autocallable seller will sell some of the 1y bonds to buy more of the 2y bonds. We know that the 2y ZC bond will be more impacted by the IR increase than the 1y ZC bond (as it has a longer duration). All in all, this means that the seller will net a profit on his IR rebalancing since he will be selling the bond that decreased less in value to buy the one that decreased more.

To sum up, the autocallable price should be higher when a positive Equity/IR correlation is assumed and lower if this correlation is assumed to be negative. It makes sense when you think about the investor's position as a positive correlation means that he is likely to get his money back in a high interest rate environment, which is a pretty good scenario for him.

Knowing this, you could employ a model that includes stochastic rates in order to enter a value for this correlation and include its impact in the price. Our pricer does not go that far as using such model does not give us additional information regarding the hedging of this Equity/IR correlation. This correlation cannot be hedged in a straightforward manner. To 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. Most often than not, it cannot be hedged at all. You could therefore decide on which correlation to us and add a cost accordingly without having to employ such a stochastic rate model. The magnitude of this cost will clearly be a positive function of the maturity of the structure. You can understand why by thinking about the above hedging argument. If the maturity of the autocallable would have been 3 years, then the hedging argument would have implied rebalancing 1y and 3y ZC bonds. Since the duration of the 3y ZC bond is longer than the 2y ZC bond, the impact of an IR move would have been more pronounced. So the longer the maturity of the autocallable structure, the greater the impact of the Equity/IR correlation.

## 16.4 Memory Feature

### 16.4.1 Autocall Incremental

In a standard Autocall in which the coupon and autocall barriers are at the same level, the coupon has a memory feature. We say that the coupon is incremental. It means that once the product is redeemed, the investor receives a coupon equal to the sum of all previous coupons.

At each observation date $$t_i$$ with i = 1, ..., n, we have :

$$\boxed{\text{Redemption}(t_i) = \text{Notional * } \text{(1 + i*C%)}_{\left\{ \frac{S_{t_i}}{S_{t_0}} \; \ge \; AB \right\}} * 1_{\left\{ max_{j=1,...,i-1} \frac{S_{t_j}}{S_{t_0}} \; \le \; AB \right\}}}$$

In this payoff, the digital size increases with time and, as a seller, one must be careful when offering a very large digital with a very low probability of striking.

### 16.4.2 Phoenix memory

In a Phoenix, the coupon barrier level is smaller than the autocall barrier level. In such a case, the memory feature is optional. In case of memory feature, we speak about Phoenix memory and the payoff becomes:

#### 16.4.2.1 Coupons

At the first observation date $$t_1$$, we have :

$$\boxed{\text{Coupon}(t_1) = \text{Notional * C% * } 1_{\left\{ \frac{S_{t_i}}{S_{t_0}} \; \ge \; CB \right\}}}$$

At each observation date $$t_i$$ with i = 2, ..., n, we have :

$$\boxed{ \text{Coupon}(t_i) = \left( \text{Notional} * \left[ \text{(i*C%) } - \sum_{i}^{t-1} Coupons \right] \right) * 1_{\left\{ \frac{S_{t_i}}{S_{t_0}} \; \ge \; CB \right\}} * 1_{\left\{ max_{j=1,...,i-1} \frac{S_{t_j}}{S_{t_0}} \; \le \; AB \right\}}}$$

#### 16.4.2.2 Redemption

At each observation date $$t_i$$ with i = 1, ..., n, we have:

$$\boxed{\text{Redemption}(t_i) = \text{Notional * } 1_{\left\{ \frac{S_{t_i}}{S_{t_0}} \; \ge \; AB \right\}} * 1_{\left\{ max_{j=1,...,i-1} \frac{S_{t_j}}{S_{t_0}} \; \le \; AB \right\}}}$$

## 16.5 Put Feature

In this low-yield environment, it is impossible to offer attractive coupons with such a structure. Therefore, to get an enhanced coupon the investor typically also sells a put option embedded in the note: a down-and-in put or a leveraged OTM put whose maturity is the maturity of the autocallable structure. This means that the investor's capital is no longer protected.

To price this structured product, one should you deduct the price of the put option from the previously explained price.

Adding a put feature does not change the seller's position with respect to the forward price of the underlying asset: he is still short the forward. However, his overall position in volatility and skew are not straightforward as there are potentially offsetting effects from the digitals (coupons) and the put.

While a trader selling an autocallable is always long volatility with respect to the put, his vega with respect to the autocallable digitals depends on the relative position of the forward price with respect to the coupon barrier level. Most of the time, it results in a long position in volatility. We have seen that it was relevant to analyze the specific vega sensitivities across time until maturity. In this way, we should rather evaluate the vega for each time period by progressively shifting parts of the volatility term structure. This process makes it possible to clearly conclude on the local exposition to volatility across varying maturities. Moreover, not only Vega exposure can differ across volatility maturities, but it can also be ambiguous for a single volatility maturity. Anyway, despite these local variations across the term structure, the investor is globally selling volatility: the put has the biggest impact on the Vega.

As for the skew position, the seller of an autocallable is short skew with respect to the digitals but long skew with respect to the long position in the downside put. Most of the time, it results in a short skew position.

I invite you to review the risk analysis of the down-and-in put option as all the risks associated with it are present in most of the autocallable structures.

## 16.6 Smoothing the autocall barrier

It is important for you to understand that each barrier (DIP barrier, coupon barrier, autocall barrier) has some sort of digital associated with it. Therefore, everything that has been said in chapter 12 about the importance of smoothing discontinuities applies. There are many discontinuities linked to the autocallable's payoff: around the DIP barrier, around the coupon barrier, around the autocall barrier. These discontinuities have crucial consequences in terms of risk management and hedging as they imply potentially unstable and explosive Greeks.

The autocall barrier at each observation date represents an up-and-out barrier that can be approximated by a call spread with spread between AB-$$\epsilon$$ and AB. The thing is that it is often impossible to determine ex-ante the direction of the discontinuity in the payoff. Because of this, you would not know now in what direction you should shift the autocall barrier.

If continuation value < exit value --> $$epsilon$$ > 0 Fig: 16.1 : Barrier shift when continuation value < exit value

If continuation value > exit value --> $$epsilon$$ < 0 In theory, the exit cashflow received in a knock-out event is known at the barrier's observation date $$t_i$$. At $$t_i$$, the exit value is simply this exit cashflow.

On the contrary, the continuation cashflows received in the absence of a knock-out event depend on the underlyings' trajectory after the barrier's observation date $$t_i$$. At $$t_i$$, the continuation value is the expected value of these continuation cashflows. Evaluate this continuation value implies evaluating the expected value of future cashflows at each observation date $$t_i$$ for each simulated trajectory so that $$|S_{t_i} - \text{AB}| < \epsilon$$. The available methods such as Monte Carlo of Monte Carlo or American Monte Carlo are not much robust.

To apply the optimal smoothing (most conservative price), it is not necessary to know the continuation value. It is enough to know the sign of the discontinuity Exit value - Continuation value.

By definition, the sign $$\delta_{optimal} \in \{ -1, 1\}$$ of the shift that guarantees an optimal smoothing verifies : $$\delta_{optimal} = \underset{\delta_{optimal} \in \{ -1, 1\}}{argmax} \text{Price with a shift of } \delta * |\epsilon|$$.

With N the number of barrier observations, $$2^N$$ pre-evaluations enable to define the directions of the optimal smoothing $$\delta_{optimal} \in \{ -1, 1\}^N$$.

When N is large, it is not feasible to test 2^N combinations in a single evaluation.

There are several solutions to speed up the process.

You could divide the problem into M blocks and $$2^{N/M}$$ combinations per block are tested: * The directions of the smoothing of the last N/M autocall barriers are calibrated during the first calculation by setting to 0 the shift on all the previous barriers. * Then, you move to the next N/M previous barriers by taking into account the previously calculated values from the first calculation and setting to 0 the shift on all previous barriers. * You repeat this operation M times.

You could also get the directions of the smoothing independently from each other. It requires $$2*N$$ evaluations but the solution will not be systematically optimal: * You set all the other shifts to 0. * You test the two combinations: {-1,1} and keep the one for which the price is the most conservative. * You repeat this operation N times, meaning that it requires 2*N evaluations.

In practice, $$\delta(i)$$ depends greatly from the smoothing of the next autocall barrier, $$\delta(i+1)$$. It is therefore desirable to take it into account when evaluating $$\delta(i)$$ by modifying the previous algorithm as follows: * You set all the other shifts to 0 except the one from i+1. * You test the two combinations: {-1,1} and keep the one for which the price is the most conservative.

Note that this algorithm neglects the fact that the continuation value at observation date i depends on the smoothing on every observation dates after i (not only the next one). This solution is therefore suboptimal but faster as it requires $$2^1 + 2^2 \text{ * } (N-1)$$ combinations instead of $$2^N$$.

## 16.7 Multi-Asset: Worst-Of

The payoff is exactly the same except that we observe 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 Down-and-In Put.

## 16.8 Other variants

### 16.8.1 Autocall Twin-Win

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 if no knock-in event occured. As a seller, you must be careful to shift and smooth correctly around the barrier level as the discontinuity is twice the size of the discontinuity in a standard autocall.

### 16.8.2 Autocall with one star feature

In this variant, even if the worst-performing underlying closes below the Down-and-In Put barrier level, if at least one underlying closes at or above the one star barrier (usually its initial level), the product is redeemed at 100%.

While this one star feature is quite appreciated by the investors, it is not so much by the bank's risk departement. The reason behind this aversion is that the cega (sensitivity to correlation) can change sign because of this feature. Can you intuitively explain the impact of the star effect on the sensitivity to correlation?