Worst-of Options

Worst-of structures come up when they are embedded in structured notes sold to investors, and the dealer that structures them has to decide how to set correlation for valuation and hedging.

The lowest asset sets the payoff

The payoff of a worst-of call is based on the asset that rose the least by maturity among the reference assets¹. Write the initial value of asset $i$ as $S_i(0)$ and its value at maturity as $S_i(T)$. The reference level $m$ is the smallest of the ratios of the maturity value to the initial value.

$$m = \min_{1 \le i \le n} \frac{S_i(T)}{S_i(0)}$$

If the strike is set as a fraction $K$ of the initial values, the payoff of the call at maturity is $\max(m - K,\ 0)$², and the payoff of the put is $\max(K - m,\ 0)$³. For the call to pay anything, even the lowest asset must end above the strike. For the put to pay, it is enough that any one asset ends below the strike. Worst-of options are traded mainly over the counter, and their prices are not directly available⁴.

Consider a hypothetical example in which Company P buys a worst-of European call on asset 1 and asset 2. Both are currently at 100 yen, the strike is 100% of the initial values (100 yen), the maturity is 1 year, and the figures are per unit. The part of the lowest asset's value above 100 yen is the payoff.

Figure description

This is the payoff at maturity of the worst-of call (strike 100 yen) that Company P buys. The horizontal axis is the lower of the two assets' values at maturity, and the vertical axis is the payoff per unit, excluding the premium. If the lower value is 100 yen or less, the payoff is 0 yen. Above 100 yen, the payoff rises by 1 yen for each 1 yen the lowest value rises, reaching 20 yen at 120 yen and 40 yen at 140 yen. The value of the higher asset appears neither in the figure nor in the payoff.

Company P pays the premium at the start. In cases 1 and 4 the payoff is 20 yen, which is a gain if it exceeds the premium and a loss if it falls short. In case 2, asset 1 rose by 30 yen but asset 2 fell by 30 yen, so the payoff is 0 yen and the loss is the premium paid. In case 3 both assets fell, so the result is the same.

The call and put compared with single-asset options

At the same strike, the payoff of a worst-of call equals the smallest of the single-asset calls (ordinary calls that each refer to one asset only), and the payoff of a worst-of put equals the largest of the single-asset puts. This holds because the lowest asset's value is less than or equal to the value of every asset. Compare with the single-asset calls in cases 1 to 4 of the previous example.

The worst-of call never pays more than a single-asset call in any case, so its premium is also at or below that of the single-asset calls. The payoffs can be equal, as in case 3 where every payoff is 0 yen. The premiums are equal in the correlation-1 example later. In exchange, in case 2, where the asset 1 call would have paid 30 yen, the worst-of pays 0 yen.

The put is the reverse. In case 2 the lowest value is 70 yen, so the worst-of put pays 30 yen, which matches the larger of the asset 1 put (0 yen) and the asset 2 put (30 yen). Because the worst-of put never pays less than any single-asset put, its premium is also at or above that of the single-asset puts.

How correlation and the number of assets move the price

When correlation between the assets is low, there are more situations in which one asset is up while the other is down, and the lowest value tends to be lower. The prices in the table assume that volatility and correlation are constant and that each asset follows geometric Brownian motion. How the choice of correlation and volatility changes the price is covered in a later section. Consider a hypothetical example in which assets 1 and 2 are both currently at 100 yen with an annualized volatility of 20%, the strike is 100 yen, the maturity is 1 year, and interest rates and dividends are 0%⁵.

At a correlation coefficient of 1, the two assets move together, so the prices match those of a single-asset call and put (both 8.0 yen). In this example, the lower the correlation coefficient, the lower the call price and the higher the put price.

With more assets, the lowest value tends to be lower still. With a correlation coefficient of 0.5 for every pair and the other conditions unchanged, the prices are as follows⁵.

Adding an asset leaves the lowest value the same or lower, so the call payoff does not increase and the put payoff does not decrease. The values in the table agree with this direction.

Worst-of autocallables and knock-in

Worst-of structures are sold embedded in structured notes. In these notes, the payoff follows the worst performing of the reference assets, and the return of principal may be limited⁶. The mechanics of an autocallable and numerical examples are covered in Autocallables. Here we look at how making the autocallable test on the lowest asset changes the outcome.

Consider a hypothetical example in which Company P buys a 2-year worst-of autocallable note referencing asset 1 and asset 2, with a face value of 1 million yen. The observation dates are 1 year from now and at maturity, 2 years from now, and on both dates the test uses the lower of the two assets' levels relative to their initial values. The coupon is 30,000 yen per year, paid as a lump sum at redemption for the years held.

• On the 1-year and maturity observation dates, if the lower level is 100% or more, the note is redeemed and pays the face value plus the coupon.
• On the maturity observation date, if the lower level is below 100% but above 65%, the note pays the face value only (no coupon).
• On the maturity observation date, if the lower level is 65% or below, the note has knocked in, and it is redeemed at the face value times the lower level, with no coupon.

This example is simplified, and its terms differ from the example in the autocallables article. The coupon is paid in a lump sum at redemption, and the knock-in level is 65%.

With the same terms referencing asset 1 alone, cases B, C, and D would have been redeemed at 1 year. Tested on the lower level, asset 2 at 92% keeps all three from being redeemed at 1 year.

In case A both assets are at 100% or more, so the note is redeemed at 1 year. In case B the lower level at maturity is 98%, short of 100% but above 65%, so only the face value is paid. In case C, asset 1 has risen 30%, but asset 2 has fallen to 60% and the note knocks in, so principal falls short by 400,000 yen. The gain on the asset that rose does not enter the payment. In case D both assets are at 100% or more at maturity, so the face value and two years of coupon are paid.

One study includes an example of a coupon-paying autocallable that references the lowest of 3 assets and tests the level on quarterly observation dates⁷. The example here has 2 assets and annual observation.

With more reference assets, there are more situations in which at least one of them falls below the level. Some worst-of structured notes deliver the shares of the worst performing asset themselves when prices fall⁸. Sales practices and the risks of structured products in general are covered in Overview of Exotic Equity Products.

Why the valuation depends on the correlation assumption

One way to price a worst-of is to describe the movement of each asset's price as a correlated stochastic process and generate many paths with random numbers, using the Monte Carlo method⁹. The required inputs are the volatility of each asset and the correlation between assets. The price changes with the values used for these inputs.

In one paper's numerical experiment, two models matched to the same at-the-money index price gave different prices for a worst-of call. The two models were a model with constant correlation and a model in which the index level affects the movement of its constituents. The constant-correlation model gave higher values, especially in the money¹⁰.

Furthermore, the price of a worst-of changes even when the distributions of the individual assets and of the index (basket) are the same¹¹. For example, if two assets move from an initial 100 yen to 80 yen and 120 yen, the average of the two stays at 100 yen, but the worst-of is set by 80 yen. In a numerical experiment in the same paper, a 2-year put on the lowest of 50 assets gave two different prices depending only on how correlation was set¹². The prices of index options do not determine the value of a worst-of uniquely.

The dealer's hedging and correlation risk

The value of a worst-of depends on each asset's moves, each asset's volatility, and the correlation between assets, so the dealer on the other side carries risk to all three. In the example table in the previous section, a rise in correlation raises the value of the worst-of call and lowers the value of the put. In this example, both the seller of the call and the buyer of the put lose when correlation rises.

One paper states that around 2010, banks held a short correlation position as a result of selling products such as worst-of structures. As ways to offset part of it, the paper proposes correlation swaps and dispersion trading, which buys and sells options on an index and options on its constituents in opposite directions¹³. The correlation that affects the price of a worst-of is not determined by index option prices alone (previous section).

In autocallables, delta and gamma change sharply near the observation dates. One study shows that it is hard to keep delta and gamma neutral in a 3-asset autocallable¹⁴.

Notes

1. The Options Industry Council, "Analysis on Structured Products and Listed Equity Options in Europe (Part 1)" (SRP, 2015), the entry "Worst-of option" in the Glossary of Terms (PDF page 54). The example is a call on the FTSE 100 and the S&P 500. ↩

2. Benjamin Jourdain and Mohamed Sbai, "Coupling Index and stocks" (arXiv, 2009), "Application: Pricing of a worst-of option" (PDF pages 18–19), for the call formula. The symbols $m$ and $n$ in the text are a rewriting. ↩

3. Alex Langnau, "Introduction into “Local Correlation” Modelling" (arXiv, 2018 version), section 7 (PDF page 21), for the put formula. ↩

4. Maxime Malgrat, "Pricing of a “worst of” option using a Copula method" (Master's thesis, 2013), chapter 1, section D, "Worst Of options" (PDF page 20). Also called a call-on-min, traded mainly over the counter, with prices not directly available. ↩

5. These are not prices from a source but values computed for the conditions in the text by the Monte Carlo method. Each asset follows geometric Brownian motion with constant volatility and correlation. The run used 40,000,000 trials (including antithetic pairs with the signs reversed), rounded to one decimal place. The values for correlation 1 and for 1 asset are the call and put on a single asset, and they agree with the Black-Scholes value (about 7.97 yen). ↩ ↩²

6. FINRA, "Regulatory Notice 12-03" (January 2012), page 4, the "worst-of" example. The payoff depends on the worst performing of the reference indices, and the return of principal may be limited. ↩

7. Anil Sharma et al., "Hedging and Pricing Structured Products Featuring Multiple Underlying Assets" (ICAIF 2024), page 3, Table 1 and section 3. It references the lowest of 3 assets, observed quarterly, with a term of 4 years. ↩

8. The Options Industry Council, same report as above, "Worst of" under Convertible Structures (PDF page 41). A form in which, if any of the several reference assets is at or below the issue price, a set quantity of the worst performing asset is delivered in kind. ↩

9. Maxime Malgrat, same thesis as above, section 2, "Applications to pricing of options," and chapter 1, section D (PDF pages 19–20). Pricing by the Monte Carlo method, using correlated random numbers for the movement of each asset. Each asset's path is generated with constant volatility. ↩

10. Benjamin Jourdain and Mohamed Sbai, same paper as above, "Application: Pricing of a worst-of option" (PDF page 19). The result is that the constant-correlation model, with the correlation set to match the index at-the-money price, always gives a higher price than the paper's model, with a larger gap especially in the money. This is a result within the paper's numerical setup. ↩

11. Alex Langnau, same paper as above, section 7 (PDF page 21). A 2-asset basket in which one asset moves from 100 to 80 and the other from 100 to 120 (the basket is unchanged, while the option on the lowest value is affected). ↩

12. Alex Langnau, same paper as above, section 7 (PDF pages 21–22, Table 5). A put on the lowest ratio to initial value among the 50 stocks of EuroStoxx 50 (strike 0.6, with prices 0.306 and 0.463). Two cases in which the distributions of the individual stocks and of the index are the same and only the center of the correlation is changed. ↩

13. Antoine Jacquier and Saad Slaoui, "Variance dispersion and correlation swaps" (arXiv, 2010), abstract. Banks hold a short correlation position as a result of selling products such as worst-of structures, and correlation swaps and dispersion trading (opposite positions in index options and options on the constituents) are proposed as ways to offset it. ↩

14. Anil Sharma et al., same paper as above, page 3, end of section 2 and section 3. The explanation that delta and gamma change sharply near observation dates, making it hard to keep portfolios delta neutral and gamma neutral. ↩