In the previous article, we considered the definition and calculation of sensitivities, or “Greeks”, for individual structured products. The main Greeks are named Delta, Gamma, Vega, Rho and Theta. Delta measures the sensitivity of the product price to the underlying asset, while Gamma represents the rate of change of Delta relative to the underlying price. The size of the Delta indicates the amount of stock, futures or other holdings required to serve as the primary hedge for the structured product position.

Gamma provides a measure of how quickly Delta changes, and therefore how frequently the corresponding hedge position will need to be adjusted to maintain a neutral position. A high Gamma value means that frequent rebalancing is necessary if the underlying moves significantly. High Gamma values are usually associated with direct and roughly “at-the-money” optionality that is highly sensitive to movements of the underlying asset.

The potential Gamma value is reduced when a product contains offsetting individual components or is deeply in or out of the money. In the case of out-of-the-money options, sensitivity approaches zero, while for in-the-money options, it becomes more linear to the underlying. Although these scenarios differ in terms of payoff (positive or negative), both exhibit lower option exposure, which is the key driver for hedging as captured by the Greeks.

Most traders would argue that the next most important Greek is Vega, which measures volatility sensitivity. Vega is closely tied to Gamma, as it can be considered “future” Gamma. Vega is highest for longer-dated products because changes in volatility can significantly affect future values of options.

Rho and Theta measure the rate of change in the product price with respect to interest rates and remaining time respectively. While it is important, Rho is usually easier to manage. Theta is primarily informational and does not represent any exposure to hedge.

** Calculating aggregate Greeks across products**

To measure the aggregate risk of a large portfolio of structured products, across an entire market or set of markets, it is necessary to combine the sensitivities of individual positions to calculate total risks.

The standard way for calculating the Greeks for a single option or structured product involves repeatedly using the pricing model, with the technique of “bumping” (i.e. changing product input parameters) to measure price changes.

A portfolio or collection of products will generally span many underlying assets, either through single-asset products with different underlyings or by considering the contribution of each underlying in basket or worst-of products.

The first obvious step in calculating aggregate Greeks for a portfolio is to find the overall Delta for each underlying. This requires identifying the distinct underlying assets across the portfolio and then summing up the Delta of each product multiplied by the notional size of each product.

This is a simple linear combination operation, as demonstrated in the example table below

Product | Notional (USD m) | Underlying Deltas | Currency Deltas (USD m) | ||||

Underlying 1 | Underlying 2 | Underlying 3 | Underlying 1 | Underlying 2 | Underlying 3 | ||

1 | 5 | 0.4 | 0.7 | 2 | 3.5 | ||

2 | 4 | 0.5 | 0.6 | 2 | 2.4 | ||

3 | 7 | 0.3 | 0.3 | 0.4 | 2.1 | 2.1 | 2.8 |

4 | 8 | 0.8 | 6.4 | ||||

5 | 12 | 0.7 | 8.4 | ||||

Total | 36 | 10.5 | 16 | 5.2 |

This table illustrates the Delta positions for 5 products across three underlyings. Product 1 has exposure to underlyings 1 and 2 (40% and 70% respectively), while the other products have exposure to one or more of these three underlyings in different combinations.

The currency Deltas columns are obtained by multiplying the unit Delta by the notional of each product. Since product 1 has a notional of USD 5 million, the currency Delta to underlyings 1 and 2 equate to USD 2 million and USD 3.5 million respectively.

The overall currency Deltas for the portfolio across all products are shown in the final row. These figures are important as they indicate the overall exposure to each underlying. For a trader managing these positions, they reveal the amount of each underlying that needs to be held, with underlying 2 having the highest requirement in this example.

**The importance of measuring Greeks for portfolios**

It is possible for Delta (as with any other Greek) to be negative for an individual product, which means the overall total may reflect some offsetting contributions. A trader managing the overall risk of the portfolio would analyse the breakdown this way, as it is important to isolate the exposure to each underlying.

This approach is suitable for an actual trading book or portfolio but can also be applied to all open products in a particular market or across markets. In this latter case, performing this calculation would give the overall exposure that the structured product market has created across the issuers involved in the market activity. This is important for understanding the size of the market and its hedging impact on different underlyings.

Aggregating other Greeks can be done similarly, calculating the combined Gamma position, volatility (Vega) and interest rate (Rho) exposures. These are all useful metrics for measuring the amount of market hedging activity that can be expected by assessing the size of Gamma, Vega and Rho across all products, underlyings and currencies.

For underlyings with significant issuance relative to the size of the underlying or its option liquidity, there will often be situations where the structured products market has a noticeable effect on underlying price movements and the observed implied volatility levels.

This information provides valuable insight into trading distortions and opportunities that may not be immediately obvious by simply tracking issuance trends in a more simplistic manner. Therefore, aggregate Greeks calculations serve some important purposes at both the portfolio or market level.

**Tags:**Valuations

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