Aggregating Greeks Across A Portfolio

A single option's Greeks are easy to reason about: this call has a delta of 0.60, meaning it moves like 60 shares of stock. A book with a thousand positions across fifty underlyings, a dozen expiries and a range of strikes looks impossible to reason about the same way — until you notice that every one of those Greeks is a derivative, and derivatives add. The book's total delta is simply the sum of every position's delta. No approximation, no simulation required for that step; it falls straight out of calculus.

A thousand small forces, one net push

Picture a tug-of-war with a thousand people pulling on the same rope, some pulling left, some pulling right, with different strength. You don't need to track each person individually to know which way the rope moves — you only need the net force, the sum of everyone's pull. Portfolio Greeks are that net force: each option is one person pulling on the rope in proportion to its delta (or gamma, or vega), and the book's total Greek is simply everyone's pull added together.

Why the sum just works

For a portfolio $\Pi$ made of positions $n_i$ contracts of instrument $i$, each with value $V_i$, the portfolio's value is $\Pi = \sum_i n_i V_i$. Because differentiation is linear, the sensitivity of $\Pi$ to any variable $x$ (stock price, volatility, time, rate) is

$$\frac{\partial \Pi}{\partial x} \;=\; \sum_i n_i \, \frac{\partial V_i}{\partial x}.$$

In plain English: the portfolio's delta (or gamma, vega, theta, rho) is just the position-size-weighted sum of every individual position's Greek — no interaction terms, no correction factor, as long as every Greek is measured with respect to the same underlying variable. The catch that trips people up is that "the same variable" has to mean the same thing across positions: delta with respect to Apple stock cannot be added to delta with respect to Microsoft stock and produce a meaningful single "delta," because a 1% move in one says nothing about a move in the other. Sums are only valid within a bucket that shares the same underlying risk factor.

Worked example 1: summing delta and gamma across three positions

A book holds: (a) long 20 calls on Stock X, delta $0.55$, gamma $0.04$ each; (b) short 15 puts on Stock X, delta $-0.30$ (a short put has positive delta contribution, so shorting it contributes $-1 \times (-0.30) = +0.30$ per contract), gamma $0.03$ each; (c) long 10 calls on Stock X, delta $0.80$, gamma $0.01$ each. All are 100-share contracts on the same stock.

$$\Delta_{\text{book}} = 20(0.55) + (-15)(-0.30) + 10(0.80) = 11.0 + 4.5 + 8.0 = 23.5 \text{ contracts} = 2{,}350 \text{ shares equivalent}.$$ $$\Gamma_{\text{book}} = 20(0.04) + (-15)(0.03) + 10(0.01) = 0.80 - 0.45 + 0.10 = 0.45 \text{ per share, per contract}.$$

To flatten delta, the desk sells 2,350 shares of Stock X. Gamma cannot be hedged with stock (stock has zero gamma); it needs offsetting options.

Worked example 2: what "small individually, large combined" looks like

Five positions each show a modest theta of -\40$/day on a risk screen — individually unremarkable. Summed:$5 \times (-$40) = -$200$/day. Over a single quiet month with no rebalancing (21 trading days), that book bleeds$21 \times $200 = $4{,}200$ purely to time decay, a number invisible if a risk manager only ever glances at each position's small theta in isolation rather than checking the portfolio total. This is precisely why desks watch the aggregated number, not the per-line one — no individual position looked alarming, but the book was never actually small.

The multi-leg strategy below is portfolio aggregation in miniature: each leg has its own payoff and its own Greeks, and what you see plotted is simply their sum, exactly the way the book's total delta and gamma above were sums of each position's numbers.

What this means in practice

Every trading desk's risk system computes book-level Greeks continuously and hedges those, not individual positions — a market maker with two offsetting option positions on the same stock does not hedge each separately, which would double the transaction costs for zero risk-management benefit. The aggregation only stays valid within a consistent risk bucket: total delta is meaningful per underlying, total vega should be split by expiry bucket (see Vega Bucketing And Term Structure Hedging), and gamma summed across strikes hides where the convexity actually sits.