Correlation: the hidden driver of multi-underlying structured products

This article examines the role of correlations and the implications of different modelling assumptions.

This article delves into nuances of correlation within multi-underlying structured products, offering key behavioral insights

In the progressively sophisticated world of structured products, where payoffs are engineered to align with investor needs across diverse market conditions, multi-underlying strategies play a pivotal role.

These structured products derive their value and performance not from isolated assets, but from the intricate interplay among them. Correlations determine the joint behavior of these assets and profoundly influence product pricing: a shock in correlation can substantially alter a product's value and, consequently, its risk profile.

This article delves into nuances of correlation within multi-underlying structured products, offering key behavioral insights. We explore not only the theoretical underpinnings but also real-world aspects illustrated through a concrete example.

Correlation concepts: a brief breakdown

Correlation measures the statistical - and not necessarily real - relationship between two variables. It indicates whether they tend to move together (positive correlation), move in opposite directions (negative correlation) or exhibit no discernible relationship (zero correlation).

Correlation is expressed as a value between −1 and +1
• +1: Perfect positive correlation (move in lockstep)
• −1: Perfect negative correlation (move inversely)
• 0: No correlation (no predictable relationship)

In the context of this article, we are interested in historical and forward correlations of stocks. And, no surprise, this is exactly the same discussion as for the volatilities to be used in a structured product.

Historical correlation

• Definition: correlation calculated using past returns (not price data) over a defined period and frequency.
• Use case: provides insights into the typical statistical relationship between stocks
• Limitations: past performance is not necessarily indicative of future results. Correlations can change over time due to shifts in market conditions and events at the individual corporate level.

Forward correlation

• Definition: an estimate of the correlation between two stocks over a future period—a prediction of how their relationship will evolve in the future.
• How it's calculated: often derived from related derivatives or statistical models projecting future price movements. Can also be inferred from implied correlations.
• Use case: necessary for pricing multi-underlying structured products and the related hedging strategies.
• Limitations: highly uncertain. Future performance is inherently unpredictable due to unpredictable changes in economic conditions and market sentiment, for example.

A key example of the use of correlations is the worst-of-stocks barrier reverse convertible structure, in which the relationship between the underlying stocks needs to be well understood.

Of course, the perspective is forward-looking: between today and the maturity of the product. Hence, the use of forward correlations.

Worst-of-3-Stocks Barrier Reverse Convertible (BRC)

Let's consider an example using constant forward correlations based on historical estimates under different assumptions..

A dynamic model—for example, one incorporating artificial intelligence—could also be used, although this is beyond the scope of this article.

Use case: three underlying stocks listed on the Swiss exchange: Nestlé (S1), Novartis (S2) and Roche (S3). That has been a pretty classic combinations for a long time.

One-year maturity with a coupon of 4.50% in CHF. Risk-free rate at 0.75%.

Barriers are all set at 60% of the initial price.Payout: at maturity, redemption depends on the underlyings' performance (slightly simplified).

1. Up-and-in (no barrier hit): If all underlyings' Observation Prices remain above their respective barriers throughout the observation period, you receive the nominal amount on the maturity date.

2. Barrier Hit (At least one observation price ≤ barrier)

All underlyings at or above initial spot price: If all underlyings' reference prices on the maturity date are ≥ their initial spot prices, you receive the nominal amount on the maturity date.

Worst-underlying delivery: If at least one underlying's reference price on the maturity date is < its initial spot price, you receive the underlying with the worst performance, delivered in quantity equal to that underlying's ratio.

Now, if we use a correlation matrix reflecting a two-year history at a weekly frequency, we get:

	S1	S2	S3
S1	1	0.34	0.38
S2	0.34	1	0.59
S3	0.38	0.59	1

_{Source: Evolids Finance}

The volatility of the stocks are (on the same basis): 21.38%, 20.23% and 25.37% respectively

And the theoretical price of structured product (our own Monte-Carlo model, using 20,000 simulations. One path is equal to one full joint trajectory of all three correlated stocks together): 100.921% (a barrier is hit in 7.06% of the paths, any stock on any day).

Using a correlation matrix reflecting a five-year history at a monthly frequency, we get:

	S1	S2	S3
S1	1	0.55	0.39
S2	0.55	1	0.49
S3	0.39	0.49	1

_{Source: Evolids Finance}

The volatility of the stocks are (on the same basis): 17.75%, 15.60% and 19.27% respectively.

And the theoretical price of structured product (our own Monte-Carlo model, using 20’000 simulations): 103.272% (a barrier is hit in 1.29% of the paths).

A last case with, a correlation matrix reflecting a one-year history at a daily frequency, we get:

	S1	S2	S3
S1	1	0.3	0.27
S2	0.3	1	0.57
S3	0.27	0.57	1

_{Source: Evolids Finance}

The volatility of the stocks are (on the same basis): 20.72%, 18.96% and 23.17% respectively.

And the theoretical price of structured product (our own Monte-Carlo model, using 20’000 simulations): 101.824% (a barrier is hit in 4.88% of the paths).

We can see that, the correlations (and the volatilities) vary depending on the look-back period and the frequency of the stock returns considered, even though these companies are all pretty mature: the past tells different stories but we can still use it as a basis for the forward correlations we are interested in. One can also note that within big pharma, the volatility can be quite different and the correlation, rather low.

Going back to the first case (two-year history, weekly frequency) and forcing materially higher correlations between the least volatile stock (Novartis) and the other two names, we would expect - under continuous barriers observation - that the barrier is breached less frequently, since the three stocks now move more closely together and are less likely to diverge enough for any single name to fall to its barrier independently.

A lower barrier-hit probability means the short down-and-in put option embedded in the BRC is worth less at inception; with the coupon unchanged (assumption), this translates into a higher price for the structured product - and a higher price, for a fixed coupon, mechanically reflects a lower total return to the investor.

The revised matrix and resulting price are:

	S1	S2	S3
S1	1	0.55	0.38
S2	0.55	1	0.75
S3	0.38	0.75	1

_{Source: Evolids Finance}

Theoretical price of the structured product (our own Monte Carlo model, 20,000 simulations): 101.139%, with the barrier hit in 6.52% of paths.

This confirms the expected relationship: the hit rate falls (from 7.06% to 6.52%) and the price rises (from 100.921% to 101.139%) relative to the base two-year weekly case.

Developing this type of structured product therefore hinges on estimating forward volatilities and correlations as accurately as possible. As shown above, these (two types of) inputs alone can move the theoretical price by more than two percentage points across plausible look-back windows - and in a live issuance, that translates directly into the coupon the issuer can offer: too conservative an estimate of volaltilities/correlations understates the embedded option value and leaves the investor under-compensated for the risk taken; too aggressive an estimate overstates it and exposes the issuer to mispricing the product.

Dividend yield matters as well: all three stocks carry high dividend yields, which lower the risk-neutral drift of each stock and, all else equal, increase the probability of a barrier breach, further influencing the coupon that can sustainably be proposed.

In our next article, still from our regular series on structured products and the Greeks, we will bridge the gap between traditional Strategic Asset Allocation and a more nuanced “Strategy Allocation” framework, demonstrating how (innovative) structured products are no longer just peripheral tools, but an essential component of modern portfolio design.

_{Image: MarekPhotodesign/Adobe Stock}

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This article is based on data and analysis provided by the SRP Greeks product. Find out more about SRP Greeks here

_{Disclaimer: This content is not intended as a solicitation or an offer; it is provided solely for informational purposes to professional investors. The information presented herein has been prepared with great care; however, errors may still occur.}