Options Greeks Explained: What Delta, Theta and Vega Measure

An option price moves for reasons other than the direction of the underlying, and the Greeks are the numbers that separate those reasons from one another. Delta, gamma, theta and vega each answer a single question about a single contract.

Each one is also a slope taken from a pricing model rather than a quantity measured in the market. That difference decides how much weight any of them can carry, and it is the part most explanations leave out.

What the Greeks Actually Measure

A Greek is a rate of change. It states how much a modelled option price moves when one input moves and every other input stays where it is. Delta tracks the underlying price, gamma tracks delta itself, theta tracks the passage of time, and vega tracks volatility.

The inputs are the part to keep hold of. An option price responds to the underlying, the strike, the time remaining, interest rates and volatility, and the page on what moves a forex option price covers those drivers and the contract types they belong to. This page stays with the sensitivities themselves.

Nothing in that list is observed. A pricing model takes the inputs, returns a theoretical price, and the Greeks are the partial derivatives of that price. Change the model or change one input and every Greek changes with it, without anything happening in the market at all.

The Options Industry Council, in its own material on the Greeks, describes them as guidance to expected behaviour rather than a guarantee of what a premium will do. That is the correct reading, and it applies to all of them equally.

There is a second condition inside the definition that is easy to read past. Each Greek holds every other input still while one input moves, and markets do not oblige. A sharp move in the underlying usually arrives with a change in expected volatility and one less day on the clock.

So a single Greek describes a move that did not happen in isolation. Reading them together is closer to what the position experienced, and reading one of them alone is where the estimate and the account statement start to separate.

Delta Is Not the Probability of Finishing in the Money

Delta is regularly presented as the chance an option expires in the money. A contract showing a delta of about half is described as a coin flip, and one showing a high delta as a near certainty. The shortcut is close enough to sound right and wrong in a way that matters.

In the Black and Scholes formulation the price of a European call contains two cumulative normal terms, written N(d1) and N(d2). The call delta is the first of them. The risk neutral probability that the contract finishes in the money is the second.

They are not the same number. The two arguments differ by the volatility of the underlying multiplied by the square root of the time remaining, so d1 always exceeds d2 whenever there is any volatility and any time left on the contract.

Because the cumulative normal function rises with its argument, N(d1) is always the larger of the two. Delta therefore overstates that probability, and it overstates it most where volatility is high and expiry is far away. The gap closes only as time runs out.

Two consequences follow. A delta read as a probability flatters long dated and volatile contracts, which are the ones where the reading is least reliable. And the second term, the one that answers the probability question, is the term almost no explanation names.

The same cumulative normal shape is also why a call delta sits between zero and one and a put delta between minus one and zero. Those bounds come from the function, not from a market convention.

The size of the gap is not fixed either. It widens with volatility and with time remaining, and it narrows as expiry approaches, which is why a short dated contract makes the shortcut look harmless. The same shortcut applied to a contract months out is describing a different quantity.

What to use instead depends on the platform. Some quote a probability of expiring in the money as its own column, and that column is the term the question asks for. Where no such column exists, treating delta as an upper bound on the probability is a fair reading, and treating it as the probability is not.

The Four Numbers You Will Actually Read

Rho, which tracks interest rates, appears on most screens and moves very little on short dated contracts. The four below are the ones that change a position from one session to the next, and each answers a different question.

Delta answers how much the price moves for a one unit move in the underlying. Gamma answers how quickly that first answer goes out of date, which is why a position can look hedged in the morning and unhedged by the afternoon without any new trade being placed.

Theta answers what the passage of time costs while nothing else changes. It is the reason a position can be right about direction and still lose, and the reason an option has a deadline in a way that a spot position does not. Dated products elsewhere have a similar clock, which the page on contract expiry sets out for CFDs.

Vega answers what a change in expected volatility does to the premium. It responds to the market view of future movement rather than to movement that has already happened, and the page on the implied volatility index explains where that expectation is read from.

Gamma and theta are best read as a pair rather than as two separate readings. A bought option carries positive gamma and negative theta together, so the holder gains responsiveness as the underlying moves and pays for it with every day that passes. A written option carries the mirror image of both.

That pairing is the whole trade being expressed. It also explains why a position can sit still on the screen and lose value: nothing moved, so the gamma side gave nothing back, while the theta side charged the day anyway.

One more distinction saves confusion later. The Greeks quoted on a chain belong to one contract, and a position holds several. Adding them across contracts gives the sensitivity of the position, which is the number that matters, and it is not what the chain is showing.

Why Two Platforms Show Different Greeks for the Same Contract

Pull up one contract on two platforms and the Greeks can disagree. Nothing is broken when that happens, and neither screen is quoting the market, because a Greek is not quoted by the market at all.

The first reason is the volatility input. A model needs a volatility figure to return a price, and the figure used is implied from option prices by the platform itself. Different source prices, different snapshot times and different interpolation across strikes all produce a different input, and every Greek shifts with it.

The second reason is the model. A European style contract and an American style contract are not priced by the same method, and platforms differ in how they handle early exercise and how they treat carry. Two correct implementations can return different slopes.

The third reason is presentation, and it is the one that misleads most often because it looks like disagreement about value. Theta may be shown per calendar day on one screen and per trading day on another. Vega may be scaled to one percentage point of volatility or to a different step.

Timing adds a fourth. A Greek is computed from a snapshot, and the snapshot on a quiet strike may be minutes older than the one beside it. Two screens refreshing on different schedules can therefore disagree about a contract that has not traded since either of them last looked.

The practical response is short. Read the specification of the platform before comparing any two numbers, and compare Greeks from the same source rather than across sources. A difference in convention and a difference in price look identical on the screen.

Where the Textbook Readings Stop Holding

Each Greek assumes small moves and orderly markets. Both assumptions weaken in the situations where a trader most wants an answer.

Close to expiry the numbers turn sharp. Gamma concentrates around the strike, so delta can travel most of its range within a small move in the underlying, and a position hedged on the last reading stops being hedged almost immediately.

Across a gap the readings describe a path that never happened. Delta and gamma are slopes measured at a point, and a market that reopens away from the previous close never passed through the prices in between. The estimate they give of the move is not the move the position took.

In a thin market the input itself becomes unreliable. Implied volatility is inferred from traded prices, and where the spread is wide or the strike rarely trades, the volatility figure carries that noise into every Greek computed from it.

There is also an assumption buried in the volatility input itself. A single volatility figure per contract implies the market expects the same movement whatever the strike, and traded prices say otherwise across the strikes of one expiry. Vega read from one figure inherits that simplification.

None of this makes the numbers useless. It sets the range they are useful over: small moves, liquid strikes and time still on the contract. Sizing decided by a Greek reading outside that range is sizing decided by an estimate the model does not stand behind, which is why hedging an existing position on delta alone tends to need constant attention.

Who Does Not Need These Numbers

A Greek exists only where an option contract exists. A spot forex position or a CFD on spot moves one for one with its underlying and has no gamma, no theta and no vega, because there is no premium and no expiry to decay toward.

Anyone trading only those instruments can leave the Greeks alone entirely. The costs on that side of the market are the spread, the overnight financing charge and the effect of leverage, and position sizing answers more there than any option sensitivity would.

The boundary is the contract, not the asset. A currency pair traded as spot carries no Greeks, and an option on the same pair carries all of them, so the instrument decides whether any of this applies rather than the market being traded.

The Greeks also do not settle whether a trade is worth taking. They describe how a price responds to inputs. Whether the premium is fair, and whether the position belongs in an account at all, are separate questions the model does not address.

Which Greek Answers Your Question

Choose by the question rather than by the list. Delta answers what a move in the underlying does now, and gamma answers how long that answer survives. Theta answers what waiting costs, and vega answers what a change in expected volatility costs.

If the question is the chance of finishing in the money, none of the four answers it. That figure is the second cumulative normal term in the pricing formula, and reading delta in its place gives an answer that is too high.