3. The Greeks: A Language of Sensitivity

3. The Greeks: A Language of Sensitivity

Look at a car's dashboard. Not a single gauge shows you where you are — that is what the window is for. The gauges show something else: how fast your position is changing and what will happen to it a minute from now if you touch nothing.

An options desk is built the same way. The price of an option is "where we are now," and it goes stale every second. What professionals care about are the gauges: how the premium will react if the index moves a point, if a day goes by, if the market gets nervous. These sensitivities are called the greeks — after the letters of the Greek alphabet used to label them — and throughout the course we will call them by their international names: delta, gamma, theta, vega. For our course the greeks are no abstraction: two lessons from now it will turn out that they literally dictate trades to the largest participants in the market, and our indicators measure those trades.

Before the introductions, one shared property. Every greek answers the question "what changes if this one thing changes — all else being equal." In real life, all else is never equal: price, time, and volatility move at once, and the premium is the sum of all the effects together. The greeks are the decomposition of the move into axes.

Delta: the slope

Delta answers the first question: how far will the premium move if the underlying moves one point.

The curve on the chart is familiar from the last lesson — a call's premium against the asset's price. Delta is the slope of that curve at the point where the market stands now; three dashed tangents show it in three regions. Deep out of the money (the blue dot) the slope is nearly zero: delta 0.10 — the asset moves, the option barely stirs; a lottery ticket takes little interest in the news. At the money (yellow) the slope is 0.53: the premium catches roughly half of the move. Deep in the money (red) — 0.92: the option is already almost indistinguishable from the asset itself.

Delta has three readings, and we will need all three.

Sensitivity: delta 0.40 — the premium catches forty percent of the asset's move.

Position equivalent: a block of 1,000 calls with delta 0.40 reacts to the market like 400 units of the asset itself. This arithmetic looks like a bookkeeping detail — but in fact it is the formula market makers hedge by, and we will return to it a lesson from now as the master equation of the course.

Rough probability: delta is close to the option's odds of finishing in the money. Delta 0.10 — the market puts the odds at roughly one in ten. The approximation is imprecise, but for a quick read of the chain it does the job.

Calls have deltas from 0 to 1; puts, from 0 to −1: a put gains value when the asset falls.

Gamma: the curvature

Look at the delta chart once more. The slope is not constant: the premium curve is bent, and as the price rises, an out-of-the-money option gradually turns into an in-the-money one, its delta creeping from 0.10 toward 0.90. The speed of that transformation is the second greek.

Gamma — how much delta changes when the asset moves one point. If delta is speed, gamma is acceleration.

Here the axes no longer show the premium but delta itself against the asset's price. The blue curve is one month before expiration: delta flows from 0 to 1 smoothly, taking twenty points of road. The yellow one is five days out: the transition has compressed. The red one is the final hours of the option's life: delta leaps from zero to one within a couple of points. Gamma is the steepness of that transition, and it grows explosively toward expiration. The logic is simple: a month before the end, the question "in the money or not" will be decided a hundred more times — no need to hurry; in the final hour, every point of price settles the option's fate for good — and delta thrashes along with the price.

Remember the red curve. When we get to options that live a single day — and those are now the most traded options on the S&P 500 — their entire drama will turn out to be drawn right here: a market composed wholly of "final hours."

To the option buyer, gamma is a gift: his delta grows by itself when he is right and shrinks by itself when he is wrong. But to whoever is obliged to keep his total risk at zero, gamma is a source of endless forced scrambling. Who this unfortunate is, and why his scramble shows up on the futures chart — lesson 4.

Theta: the clock

Theta — how much premium an option loses per day that passes with the market motionless. This is the greek that "stole" the money on that Monday from the start of the last lesson: time value is a perishable asset.

Notice that the decay is not linear. Over the first half of its life, the option gives up noticeably less than half of its time value, while the final days burn at an accelerating rate — the closer the finale, the more each day of waiting costs. For the buyer, theta is rent paid for a chance. For the seller, it is income: like an insurance company, he pockets a slice of prepaid premium every quiet day. Just don't mistake theta for an annuity: it drips while the market is calm, and one sharp move hands back to the buyer, through delta and gamma, what theta had been collecting for weeks.

Vega: sensitivity to mood

Vega — how much the premium changes when implied volatility moves one percentage point. Recall the bells from lesson 2: the bell widens — every chance gets more expensive at once. Vega measures how sensitive a given option is to that.

The classic plot: the day before a major release, the index stands still while options get pricier — the market is inflating the bell in anticipation of the verdict. Next morning the news comes out, the uncertainty collapses, IV drops — and options get cheaper even if the market moved the "right" way. The buyer called the direction and still lost: vega ate more than delta earned. In the options market you can be right about the price and wrong about the uncertainty.

The second tier: names for later

Delta, as we saw, changes as the price moves — that is gamma. But delta has two more quiet enemies. It changes when volatility shifts — that effect is called vanna. And it changes from the mere passage of time — that is charm: look at the gamma chart once more — with every passing day the delta curve gets steeper, which means every option's delta is drifting somewhere even while the market stands frozen.

For now these are just two names, and it is too early to get properly acquainted: their power shows not in a single option but in the hands of someone holding them by the hundreds of thousands. Two of our indicators are named in their honor, and in lesson 11 it will become clear why.

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For most participants, the greeks are reference information: check, weigh the risk, decide. But there is a player in the market for whom the greeks are not a reference but an order. He does not choose his positions — other people's trades bring them to him. He is obliged to keep his risk at zero — and the greeks tell him how many futures to buy or sell right now, whether he likes it or not. Meet him in the next lesson: the market maker — the most predictable large player in financial markets and the main character of our entire course.