Autocallables

Issuers that structure notes and the dealers on the other side use the coupon terms and the knock-in at maturity to build a note's profit and loss, then value and hedge it.

Early redemption and coupons decided on each observation date

An autocallable is a structured note that is redeemed automatically before maturity if the price of the underlying asset is at or above a set level on a predetermined observation date¹. On redemption, the investor receives the face value and the coupon, and the transaction ends. How these notes relate to exchangeable bonds (EB) is covered in Overview of Exotic Equity Products.

What the investor receives depends on three levels. In the final terms of Bank Vontobel AG (CH1512011946, a worst-of type that references three stocks; "the Vontobel final terms" below), all three are set as a percentage of the initial share price².

Some contracts start checking for early redemption only from a later observation date among the coupon observation dates (the Vontobel final terms are an example)². Some types also change the levels from one observation date to the next. In the example in the paper by Huang and Luo, the early redemption level rises step by step from 3,050 to 3,250 across the observation dates (the source does not state a unit)³. The worst-of type, which references several stocks and checks the worst one, also exists.

Coupon types

Some coupons are conditional: they are paid only when the share price on an observation date is at or above a level⁴. If the coupon level is set below the early redemption level, a coupon can be paid on an observation date when the share price does not reach the redemption level.

A variation on the conditional coupon is the memory coupon, which lets the investor collect coupons that were not paid earlier. How it pays is set by the terms of each product. Here, the example is the payment method in the Vontobel final terms. In these final terms, when the share price is at or above the level on an observation date, the note pays the coupon for that period together with the coupons that went unpaid since the last date on which a coupon was paid⁵.

$$C_t = A \times c \times (N_t + 1)$$

This is the formula in the Vontobel final terms. $A$ is the face value, $c$ is the coupon rate for one period, and $N_t$ is the number of observation dates before date $t$ on which no coupon was paid since the most recent coupon payment. In these final terms, the coupon from this formula is also paid on an early redemption date, in addition to the face value⁵. If the share price is never at or above the level on a later observation date, the unpaid coupons cannot be collected (a consequence of the formula).

Redemption at maturity and the knock-in put

If the note has not been redeemed early by maturity, the redemption amount depends on whether it has knocked in. Some types check for a knock-in using the share price during the term, and others use only the share price at maturity⁶⁷. This section covers only how the barrier affects the redemption amount at maturity.

$S_0$ is the initial share price and $S_T$ is the share price at maturity. For EB notes that check during the term, the redemption may be delivered in shares, and in value it equals the formula above⁶.

Only when the note has knocked in does the investor bear the loss from the share price at maturity falling below the initial level. This has the same shape as selling a put option with the initial share price as the strike that takes effect only after a knock-in, that is, a knock-in put⁶. For example, if the initial share price is 100 and it is 70 at maturity, the seller of a put is obliged to buy at 100 and loses 30. A knocked-in EB note is also redeemed at 70 against a face value of 100, so the loss is the same⁶.

In the type that checks only the share price at maturity, a small difference in the maturity share price, just below or just above the knock-in level, changes the redemption amount a great deal. The example below shows this.

Numerical example: two years, four dates

Consider a hypothetical example in which an autocallable linked to the shares of Company R has a face value of 1 million yen and a term of 2 years. There are four observation dates: 6 months, 1 year, 1.5 years, and 2 years (maturity). Company R's initial share price is 3,000 yen. The coupon has the memory feature, and the knock-in is checked only on the share price at maturity.

The example ignores the time value of money (discounting), the issuer's credit, fees, and tax. The issue price equals face value, and each amount is received on its observation date.

Figure description

The investor pays a face value of 1 million yen at purchase. After 6 months the share price is 1,900 yen, below the coupon level of 2,100 yen, so nothing is received. After 1 year the share price is 2,200 yen, above the coupon level, and the investor receives coupons of 60,000 yen, including the coupon from the first date. After 1.5 years the investor receives a coupon of 30,000 yen. At maturity, after 2 years, the share price is 2,800 yen and the note does not knock in, so the investor receives a coupon of 30,000 yen and the face value of 1 million yen, 1.03 million yen in all. The length of each arrow is a rough guide to the size of the amount.

In case C, without the memory feature the 30,000 yen for the 6-month date would not be received, and the total would be 1.09 million yen. Because the coupon has the memory feature, the total is 1.12 million yen. In case A the share price has risen 10%, but the investor receives only 1.03 million yen, the face value plus one coupon. In case D the investor receives 90,000 yen in three coupons, but the redemption amount at maturity after the knock-in is 550,000 yen, so the total is 360,000 yen below face value. In case E the share price has halved, and the redemption amount is also half of face value, 500,000 yen. In case F the investor never receives a coupon, but the maturity share price is above the knock-in level, so the face value is returned. If the maturity share price is exactly 1,800 yen, the redemption amount is 600,000 yen, and if it is even slightly above 1,800 yen, the amount is 1 million yen.

Valuation: discrete observation dates are hard to solve in closed form

Redemption timing and coupons depend on the share price on each observation date, so the price depends on how the share price moves along the way. An autocallable is a type of path-dependent option, and its valuation involves the underlying asset, interest rates, and volatility⁸.

In the framework of the Black-Scholes model, the value of a type that monitors the share price continuously has a simple closed-form formula. For the type with discrete observation dates, the formula becomes a multiple integral, and with more than 5 observation dates even integrating numerically becomes difficult⁹. In practice, the Monte Carlo method is widely used to value the discrete type. Its convergence is slow, so techniques such as variance reduction are needed. Another approach is to solve a partial differential equation with finite differences⁹. Because the payoff changes discontinuously with the share price, finite differences need measures such as smoothing the payoff to keep the accuracy⁹. See Monte Carlo Simulation for details.

Why the dealer on the other side finds hedging hard

Because the investor sells the knock-in put, the dealer on the other side is in effect holding that put. The general treatment of dealer hedging is in the section "Risks left with the dealer" of "Overview of Exotic Equity Products". What is specific to autocallables is that sensitivities move sharply near the observation dates.

In a numerical experiment on an autocallable referencing three stocks, delta and gamma change sharply near the observation dates, which makes it hard to keep delta and gamma neutral¹⁰. For an issuer that hedges dynamically by trading the underlying asset, profitability falls sharply when the market is unstable, when the payoff is complex, and when the investment period is long¹¹.

The shape of the investor's gains and losses

The investor's maximum receipt is the face value plus the coupons¹². Even if the share price rises, the investor receives only the face value and coupons at early redemption, and cannot earn more than the coupon from the rise⁶. This differs from holding the shares.

Losses come from the fall in the share price after a knock-in. The size of the loss equals the percentage by which the maturity share price is below the initial level, and if the share price goes to zero, the investor loses the entire principal invested even after receiving coupons¹². If the note does not knock in, the face value is returned whether or not coupons were received (case F). The investor also bears the issuer's credit risk¹³.

Even when the note has not knocked in, its price during the term can fall well below the issue price¹². The Financial Services Agency (FSA) points out that when an investor reinvests after early redemption, the purchase comes after the share price has risen, and the knock-in level also tends to be higher¹⁴.

Notes

1. The Options Industry Council, "Analysis on Structured Products and Listed Equity Options in Europe (Part 1)" (SRP, 2015), page 39, "Autocallable" in the Glossary of Terms. ↩

2. Bank Vontobel AG, "Final Terms Barrier Reverse Convertible with Conditional Coupon" (CH1512011946, 2026), page 2 (Barrier, Coupon Level, and Autocall Level) and page 3 (the difference between the observation dates). ↩ ↩²

3. Min Huang and Guo Luo, "A simple and efficient numerical method for pricing discretely monitored early-exercise options," pages 1–2 (section 1.1) and page 14 (Table 1 in section 7.1). ↩

4. Anil Sharma et al., "Hedging and Pricing Structured Products Featuring Multiple Underlying Assets" (ICAIF 2024), page 3 (section 3) and the Coupon Barrier in Table 1. ↩

5. The Vontobel final terms, page 2 (the memory effect) and page 3 (the Coupon Payment formula and Early Redemption). The symbols correspond to the Nominal Value and to N (the number of dates with no coupon paid) in the source. ↩ ↩²

6. Financial Services Agency, "投資信託等の販売会社による顧客本位の業務運営のモニタリング結果について" (Results of monitoring customer-oriented business conduct by distributors of investment trusts and other products, in Japanese) (June 2022), page 29, notes 26 and 27, and Chart 22. The redemption of EB notes that knocked in during the term, and how the loss corresponds to that of a put. ↩ ↩² ↩³ ↩⁴ ↩⁵

7. Bank Vontobel AG, "Final Terms Barrier Reverse Convertible with Conditional Coupon" (CH1512011946), page 2 (Barrier Monitoring) and page 3 (Redemption). An example of the type that checks only the share price at maturity. ↩

8. Anil Sharma et al., "Hedging and Pricing Structured Products Featuring Multiple Underlying Assets" (ICAIF 2024), page 3, section 3. ↩

9. Min Huang and Guo Luo, "A simple and efficient numerical method for pricing discretely monitored early-exercise options," page 2, section 1.1. A statement made within the Black-Scholes framework. ↩ ↩² ↩³

10. Anil Sharma et al., "Hedging and Pricing Structured Products Featuring Multiple Underlying Assets" (ICAIF 2024), page 3, section 3. A statement about a numerical experiment on an autocallable referencing three stocks. ↩

11. The Options Industry Council, "Analysis on Structured Products and Listed Equity Options in Europe (Part 1)" (SRP, 2015), page 22, "Dynamic hedging." ↩

12. Bank Vontobel AG, "Final Terms Barrier Reverse Convertible with Conditional Coupon" (CH1512011946), page 5, Prospects of Profit and Losses. ↩ ↩² ↩³

13. Bank Vontobel AG, "Final Terms Barrier Reverse Convertible with Conditional Coupon" (CH1512011946), pages 1 and 2, "The investors bear the Issuer's credit risk." ↩

14. Financial Services Agency, "投資信託等の販売会社による顧客本位の業務運営のモニタリング結果について" (Results of monitoring customer-oriented business conduct by distributors of investment trusts and other products, in Japanese) (June 2022), pages 29–30. ↩