Aggregating Option Deltas into Currency and Underlying Exposures

Summary

The document explains how to aggregate the deltas of individual options into a portfolio delta: multiply each option’s delta by its position quantity and sum across positions. The resulting exposure depends on the units and conventions used for each option, so a raw sum may not by itself communicate the book’s risk when contracts have different notionals, underlyings, contract sizes, or currencies.

One response recommends reporting delta in the currency used for book value and P&L. It interprets a dollar delta as a first-order sensitivity: a one-percent move in the underlying implies an approximate P&L change equal to one percent of that delta. Depending on the underlying, exposure may also be expressed in shares or futures contracts. The discussion provides a basic aggregation rule and reporting examples, but does not specify conversion conventions for multi-currency books, distinguish spot from futures delta, or cover nonlinear changes such as gamma. Despite the question’s wording, the answers address portfolio delta rather than gamma.

Key ideas

• Portfolio delta is the sum of each position’s delta multiplied by its quantity.
• Report delta in units that make the book’s first-order P&L sensitivity clear.
• Share-equivalent or futures-equivalent exposure can help describe delta for particular underlyings.
• The aggregation requires consistent units and conventions across instruments.
• Delta describes first-order sensitivity and does not capture gamma effects.

Tags

# How is delta reported for an options book?


# How is delta reported for an options book?












Say you have a collection of options, and a delta for each. How is this aggregated to a gamma for the entire portfolio of options? Each option will have its own notional, underlying, contract size, currency, etc.

Do you just average it with some weight? Do you scale the deltas so they correspond to the same "pct" increase in the underlyings, and then you just sum them?

## Answer by João (score 2)

https://quant.stackexchange.com/a/85518

You can calculate portfolio $\Pi$ delta $\Delta$ from the deltas of individual options

\begin{equation} \Delta = \sum_{i=1}^{n} w_i \, \Delta_i \end{equation}

$w$ being options quantity

## Answer by del (score 1)

https://quant.stackexchange.com/a/85516

in general you work out your delta in terms of dollars or whatever ccy your book value and p&l in expressed in, i.e your sp500 book with a delta of 10,000,000 means that, first order, if the sp500 goes up 1% you make 100,000.

depending on the underlying you will also show it in terms of nbShares for a single stock or nbFutures for an index.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.