Position Greeks and the Book

Why Greeks add, and what the sums mean

Every lesson in this subject so far has described one structure. That is not how a desk is run, because positions do not stay separate: they are held at the same time, in the same account, on correlated instruments, and the risk that matters is the risk of the total. The reason the arithmetic is easy is that Greeks are **first-order sensitivities**, and derivatives of a sum are the sum of the derivatives — so the delta of a book is the contract-weighted sum of the deltas, and the same is true of gamma, vega and theta. On the worked book, five long calls at 0.52 delta and three short calls at −0.38 net to +260 − 114 = +146 shares before the puts, and the four long puts at −0.42 take it to −22. That is the first result and the most important one: three structures with 1,040 shares of gross exposure net to the equivalent of being **short 22 shares**, which is close enough to flat that a trader watching delta would conclude there was nothing to do. There is something to do. The gamma nets to +29.5 shares per dollar, the vega to +$69.50 a volatility point and the theta to −$19.20 a day, and none of those three cancel anywhere near as closely as delta did. What the sums describe is a **long-volatility book financed by time**: it is long gamma because all three legs are long options with more positive gamma in the long calls and puts than negative gamma in the short calls, long vega because the same is true of the volatility exposure, and short theta because every long option pays for its convexity with decay. That is one position, not three risks, and the scenarios make it concrete. On a $5 fall the book makes roughly 22 × 5 = $110 from delta plus ½ × 29.5 × 25 ≈ $369 from gamma — about $479 — and it makes that because both the long calls and the long puts gain delta as the stock moves toward or through their strikes. With the stock unchanged and volatility down three points the book loses 3 × 69.5 = $208.50, which is nearly eleven days of theta at the current rate. So the same book profits when the market moves and loses when it does not, and the honest question it raises is the one that belongs in a written policy rather than in a decision made under pressure: how many quiet days can the theta be financed, and at what point does a volatility decline turn a well-hedged book into a losing one? The book, added up — Long 5 × $100 calls (+260Δ) and short 3 × $105 calls (−114Δ): Net +146 shares before the puts ← · Long 4 × $95 puts (−168Δ): Book delta −22 shares — nearly flat · Gamma +29.5 per dollar: The book gets longer as the market moves, either way · Vega +$69.50 per volatility point: Long volatility, so a fall in implied volatility costs money · Theta −$19.20 a day: And that is the bill for the convexity Three positions that look separate are one long-volatility structure financed by time. Net delta hiding the risk is the reason the book, not the position, is the unit of risk.

Beta-weighting, and where a Greek estimate stops working

Two refinements are worth carrying out of the arithmetic. The first is **beta-weighting**: if the book holds options on several names, their deltas are not comparable until each is scaled by the position’s beta against a common index, because a 0.5 delta in a high-beta stock is more market exposure than the same delta in a utility. A book that looks market-neutral by raw delta can be short a lot of index beta, and the correction is a multiplication rather than a new concept. That matters most for the structure this book has: a set of long-premium positions on high-beta names is a larger long-volatility bet than the same deltas on defensives, and the beta-adjusted number is the one a limit is written against. The second refinement is the limit of the Greek estimates themselves, and it is the one that matters on the day it matters. Delta and gamma are local: they describe the book’s behaviour for small moves, and gamma’s quadratic term is the first correction rather than the answer. A 20% gap is not five times a 4% move, it is a different scenario entirely — every leg’s delta and gamma have changed, the short calls may be far in the money, and the volatility surface the vega was quoted off may have shifted shape rather than level. The way to price that is a full revaluation: what each leg would be worth at that price and volatility, given the time left. That is why scenario analysis sits beside the Greeks rather than being replaced by them, and why a risk policy names the shocks it will be tested against — a 5% fall, a 20% fall, a volatility spike, a session where the market gaps through several strikes — instead of relying on the day’s numbers to describe what could happen to them.

What the book costs to carry

The Greeks describe the risk of a book; the broker describes its capacity, and the two are not the same picture. Two books with identical net delta, gamma and vega can require wildly different amounts of equity to hold, because the requirement is computed from how the positions are built rather than from how they behave together. Reading the Greeks without reading the margin underneath them is reading half the document. Under the strategy-based rules that most accounts sit under, options are charged position by position. A long option costs its premium in full, since the most you can lose is what you paid. A cash-secured put is charged the whole strike, because the broker is reserving the cash to buy the shares. A naked short is charged a percentage of the underlying value plus the premium collected, with a per-share floor. And a spread is charged the width of the strikes minus any credit received — which is the defined risk, charged as though the offsetting long leg did nothing else. None of these numbers looks at the book’s net gamma or its net vega, and that is the whole point: the requirement is a property of the construction, not of the aggregate risk. Portfolio margin is the reverse, and this is exactly why the two regimes can disagree by a factor of three or four on the same positions. There, the requirement is the largest modelled loss across a scenario grid — which is the Greeks’ second-order behaviour written out explicitly and priced. A book that is genuinely hedged looks cheap under that model and expensive under the flat rules; a book that is diversified in name but short the same factor across every position gets no help at all, because the grid finds the scenario in which everything moves together. There is a second number that beginners conflate with the first. **Buying power** is not cash and it is not the premium collected. A short-premium book that has taken in twenty thousand dollars of credit has its buying power reduced by the requirement on those positions, not increased by the credit — so a book can be flush with premium received and simultaneously unable to open another position. Traders who measure their capacity by the cash balance discover this at the worst moment, when the position they want to add is the hedge for the position they already have. Concentration and expiry limits sit on top of margin and bite before it does. Brokers cap contracts per underlying, aggregate short exposure per expiry, and short positions in names they judge hard to borrow, and those caps are set by the broker’s own risk appetite rather than by any regulation. A book that grows by adding the same structure across more names will hit one of those walls long before it comes close to the account’s own tolerance, which is worth knowing in advance rather than at the click that is refused. The practical consequence is that managing a book has to include a capacity line, not just a risk line. A book sitting at eighty percent of its margin cannot take the position that appears next week, and a book whose requirement is risk-based will see that requirement rise on exactly the volatile days when the account is already losing — the same mechanism, one level up, that produced the margin calls in the earlier markets lesson. The book’s Greeks tell you what it does; its requirement tells you whether you are still allowed to hold it. • Strategy-based rules charge each construction separately and ignore the aggregate Greeks. • Portfolio margin charges the worst modelled scenario — the Greeks’ second order, priced. • Buying power falls with the requirement, not with the credit collected. • Concentration and expiry limits are the broker’s caps, and they arrive before your own. Check the requirement on the position *before* entering it, not after. The order ticket usually shows it, and a structure that cannot be financed is not a strategy — it is a plan to receive a margin call.

Sources

• Aggregating position Greeks and beta-weighting a bookTaleb, “Dynamic Hedging: Managing Vanilla and Exotic Options”
• Scenario analysis and the limits of Greek-based risk estimatesHull, “Options, Futures, and Other Derivatives”
• Why a gamma/theta trade is one position rather than two risksNatenberg, “Option Volatility and Pricing”