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For FRM® Part I, portfolio delta, gamma and vega are the signed sums of the sensitivities of each option position. Multiply each per-option Greek by the number of options held, using a negative count for a short position, then add the contributions. A book with little net delta can still have material exposure to changes in the underlying price or implied volatility.
What each sum measures
Delta describes the local change in option value as the underlying price moves. Gamma describes how that delta changes when the underlying moves, while vega describes the local effect of a change in implied volatility. CFA Institute sets out these distinctions in its options strategies reading. GARP lists Valuation and Risk Models within its FRM Part I material.
Use the position count and sensitivity on the same unit basis. If the input is per option, count options; if it is per contract, count contracts. A contract multiplier must be applied when a per-option Greek is combined with a contract count. The sign of a short position belongs in the count, even when the option’s own delta is already negative.
A book whose delta cancels
Consider an invented book of calls on the same underlying. The figures below are illustrative per-option sensitivities. Assume each position count refers to one option unit, with no additional contract multiplier. Delta is value change per £1 underlying move; gamma is delta change per £1 underlying move; vega is value change per one percentage point rise in implied volatility.
| Long calls A | +10 | +0.60 | +0.020 | +0.12 |
| Short calls B | −15 | +0.40 | +0.030 | +0.20 |
For delta, the long calls contribute 10 times 0.60, or +6. The short calls contribute negative 15 times 0.40, or −6. The net delta is zero. Gamma is 10 times 0.020 less 15 times 0.030, giving −0.25. Vega is 10 times 0.12 less 15 times 0.20, giving −1.80.
The zero delta says the book has no first-order exposure to a very small underlying move at this starting point. It does not remove curvature: the negative gamma means delta becomes less favourable as the underlying moves away from that point. On the stated vega convention, a one percentage point rise in implied volatility would reduce the book’s value by approximately £1.80 if other inputs were held fixed. These are local approximations, not predictions of the full change in value.
Where sign and unit errors enter
- Short position sign. Entering the short calls as a positive 15 would turn all three of their contributions the wrong way.
- Per-option versus per-contract. A multiplier changes every aggregate Greek. Apply it once and consistently.
- Delta as a complete risk measure. A cancellation in delta does not cancel gamma or vega; calculate each sum separately.
For a broader place for this calculation in the syllabus, see the FRM programme overview. The study methodology can help turn the signed-count step into a repeatable calculation check.
Sources
- FRM® Study Material, Guide, Books, Practice Exams | GARP · garp.org · Retrieved
- Options Strategies | CFA Institute · cfainstitute.org · Retrieved
