Portfolio Greeks: measuring delta, theta and vega across a whole options book and setting a theta budget

What you will learn

• How to aggregate delta, theta and vega across positions on different underlyings.
• How beta-weighted delta expresses a whole book in units of one index.
• How to set a theta budget and a vega limit as percentages of the account.
• Which Greek is really driving your book and how to rebalance it.

Explanation

The single-option Greeks were introduced in the course Derivatives (lesson der07). A book is the sum of its parts, but only if every number is in the same unit.

Position Greeks are per-contract Greeks multiplied by quantity and by the contract multiplier (100 shares for most equity options; €5 per point for DAX options, €10 for EURO STOXX 50 options). Position theta of €18 means the position gains about €18 per calendar day if nothing else changes. Position vega of −€45 means a one-point rise in implied volatility across the position costs about €45.

Beta-weighted delta. Share deltas on different stocks cannot simply be added: 36 deltas on a €50 share are not the same risk as 36 deltas on a €200 share. The solution is to convert each delta into index-equivalent deltas:

beta-weighted delta = position delta × share price × beta ÷ index price

Beta measures how much the share tends to move when the index moves 1%. The result tells you how many index deltas your book behaves like; index positions are already in index units.

Theta budget. Theta is the income side of a short-premium book, but it is paid for with gamma and vega risk. A common rule of thumb is to keep daily theta between about 0.05% and 0.2% of the account value; more means you are selling too much premium. A vega limit works the same way: decide the maximum loss you accept for a given rise in implied volatility (for example 2% of the account for a 5-point rise) and divide to get the maximum negative vega.

Three reading rules:

• If beta-weighted delta is large relative to theta, your "income" book is really a directional bet.
• If vega is strongly negative while IV rank is low (lesson 2), you are short volatility at a poor price; reduce size or add long vega (calendars, lesson 5).
• Greeks are local estimates; recalculate daily and after every large move.

Worked example

Hypothetical example: account €20,000, index at 4,000 points, three positions. Greeks are per contract with multipliers applied. Fees and taxes are excluded.

Step 1, raw sums. Delta: 3 × 12 + 2 × 3 + 1 × (−2) = 36 + 6 − 2 = +40. Theta: 3 × 4 + 2 × 5 + 18 = 12 + 10 + 18 = +€40 per day. Vega: 3 × (−6) + 2 × 20 + (−45) = −18 + 40 − 45 = −€23 per volatility point.

Step 2, beta-weight the delta. A: 36 × 50 × 1.2 ÷ 4,000 = 0.54 index deltas. B: 6 × 200 × 0.8 ÷ 4,000 = 0.24. C: already −2 index deltas. Total = 0.54 + 0.24 − 2 = −1.22 index deltas. The raw +40 looked bullish; in index terms the book is slightly short. A 1% index move (40 points) changes it by about 1.22 × 40 = €49 before gamma.

Step 3, compare with the budget. Theta budget at 0.1% of €20,000 = €20 per day; actual theta is €40, double the budget. Vega limit: 2% of €20,000 = €400 for a 5-point rise, so vega must stay above −€80; actual −€23 is inside the limit. Conclusion: the book is collecting theta faster than the plan allows, which means more gamma risk than intended; reduce position C or A rather than adding a hedge.

Common mistake

Practice activity

Account €50,000. You trade only one iron condor type with theta +€18 per day and vega −€45 per condor. Your rules: daily theta budget 0.1% of the account, and a 5-point rise in implied volatility may cost at most 2% of the account. (a) How many condors does the theta budget allow? (b) How many does the vega limit allow? (c) Which limit binds, and how many condors may you hold?

Key takeaways

• Position Greeks = per-contract Greeks × quantity × multiplier; theta in € per day, vega in € per volatility point.
• Beta-weight deltas to one index before judging whether the book is long or short the market.
• Set a theta budget (a small percentage of the account per day) and a vega limit (maximum loss for a defined IV rise).
• Too much theta means too much gamma; reduce size rather than piling on hedges.

Next step

Lesson 12 looks at the practical side for traders in Germany: Eurex contract specifications, portfolio margin, access to US options under PRIIPs and the cost picture.