Dollar Greeks And Position Scaling

Two option positions both show "delta = 0.50" on a risk screen. One is 10 contracts on a $20 stock; the other is 10 contracts on a $2,000 stock. A trader who treats these as the same size of risk is off by a factor of 100 — the second position moves $100 for every dollar the first moves $1. Raw, per-share, per-contract Greeks are a unit of sensitivity, not a unit of money at risk, and comparing them across different stocks, strikes or expiries without converting first is comparing apples measured in centimetres to oranges measured in inches.

Converting apples and oranges to the same currency

A currency exchange booth doesn't ask you to compare "50 yen" to "50 euros" as if the numbers meant the same thing — it converts both to a common unit, say dollars, before anyone compares sizes. Dollar Greeks do exactly this for an options book: they convert every raw Greek, which is naturally quoted per share or per volatility point, into a dollar amount of exposure, so a risk manager can add, compare and rank positions across completely different underlyings on one screen.

The conversion

For delta, the dollar Greek multiplies the per-share sensitivity by the stock price, the number of shares per contract, and the number of contracts:

$$\$\Delta \;=\; \Delta \times S \times \text{multiplier} \times n_{\text{contracts}}.$$

In plain English: dollar delta answers "for a 1% move in the stock, how many dollars does this position gain or lose?" — because the dollar delta, $\Delta \times S$, times a $1\%$ move in $S$ gives $\Delta \times S \times 0.01$, and the convention of using $\Delta \times S$ directly (a "1-point" or "1%" dollar delta, depending on house convention) turns a unitless slope into a dollar figure comparable across every position in a book. Gamma gets the same treatment, usually scaled for a 1% move:

$$\$\Gamma \;=\; \Gamma \times S^2 \times 0.01 \times \text{multiplier} \times n_{\text{contracts}},$$

which answers "for a 1% move in the stock, how many dollars does my delta change by?" Vega and theta convert more simply, since they are already quoted per volatility point and per day respectively: dollar vega is $\text{vega} \times \text{multiplier} \times n_{\text{contracts}}$, and it directly means "dollars gained per 1-point rise in implied volatility."

Worked example 1: comparing two "delta = 0.50" positions

Position A: 10 call contracts (100 shares/contract) on Stock X at $20, delta $0.50$ each.

$$\$\Delta_A = 0.50 \times 20 \times 100 \times 10 = 10{,}000.$$

That is, $10,000.

Position B: 10 call contracts on Stock Y at $2,000, delta $0.50$ each.

$$\$\Delta_B = 0.50 \times 2{,}000 \times 100 \times 10 = 1{,}000{,}000.$$

That is, $1,000,000.

Both showed "delta 0.50" on a raw Greeks screen, but Position B moves 100 times more money for the same 1% stock move. A risk desk sizing limits by raw delta would have completely missed that Position B is the far larger risk.

Worked example 2: dollar gamma and a realistic hedge cost

A market maker is short 50 contracts (100 shares/contract) of a call on a $150 stock with gamma $0.03$ per share.

$$\$\Gamma = -0.03 \times 150^{2} \times 0.01 \times 100 \times 50 = -0.03 \times 22{,}500 \times 0.01 \times 5{,}000 = -33{,}750.$$

That is, -$33,750.

This means: for every 1% move in the stock ($1.50 here), the position's delta changes by roughly $33,750 worth of stock, which the desk must trade to stay hedged. If the stock is choppy and moves 1% up and back down five times in a day, the desk buys and sells roughly $33,750 of stock exposure five times over — the concrete, dollar-denominated cost of running a short-gamma book, not visible at all from the raw per-share gamma of $0.03$.

The payoff below is for a single contract, on a per-share basis — exactly the raw, unscaled number a Greeks screen shows. Multiply its slope by the stock price, the contract multiplier and the number of contracts, the way both worked examples above did, and you get the dollar figure that actually belongs on a risk report.

What this means in practice

Every risk system on a trading desk stores and reports dollar Greeks, not raw per-share Greeks, precisely because raw Greeks cannot be summed across different underlyings — you cannot add "0.50 of Stock X delta" to "0.30 of Stock Y delta" and get a meaningful number, but you can always add dollars. Aggregating Greeks across a portfolio is only possible once every position has been converted to the same dollar basis first.