Option Greeks explained simply: what the measures show

The option Greeks are measures that show why the price of an option changes. Delta measures the response to the price, gamma the change in delta, theta the effect of time, vega the effect of expected fluctuation and rho the effect of the interest rate. These influences act at the same time, and none of the measures predicts whether the price will rise or fall. What calls, puts and the premium mean is explained in the article Options explained simply.

What do the option Greeks show?

The price of an option changes constantly. Several influences drive it, including the price of the underlying, the passage of time, expected fluctuation and the interest rate. The option Greeks measure how strongly each of these influences acts. Each measure looks at its influence in isolation, as if the others stayed the same. In reality, they all act at once.

A single example running through the article makes this tangible.

How does the premium respond to the price? Delta and gamma

Delta: the speed of the premium

Delta shows how much the premium is expected to change if the underlying rises by one unit. The OIC uses USD 1. A speedometer works as an image: it shows how fast the premium follows the price.

In the example, delta is 0.51. If the price rises from 100 to 101, the premium therefore rises by just over 0.5.

Delta is sometimes read as the probability that the option will be in the money at expiry, meaning that a call has value because the price is above the strike price. The OIC calls delta a proxy for this. It is not an exact probability.

Gamma: the acceleration

Gamma shows how quickly delta changes when the price moves. To stay with the car: the speedometer shows the speed, gamma the acceleration. In the example, gamma is 0.08. After the price rise of 1, delta rises from 0.51 to around 0.59.

According to the OIC, the simple calculation ‘delta times price movement’ is only a rough approximation. One reason is gamma: in the example, the premium does not rise by 0.51 but, calculated exactly, by 0.55, from 2.00 to 2.55.

What do time, fluctuation and the interest rate do? Theta, vega and rho

Theta: the melting ice cube

Theta shows how much value an option loses through time, calculated per calendar day. The image: an ice cube that melts even if nobody touches it. According to the OIC, time alone can reduce time value, even if the price stays the same. For buyers of calls and puts, theta is negative according to the OIC (in the model this holds without exception at an interest rate of 0%; at a higher interest rate, a put deep in the money can be an exception, own model calculation), and the loss in value accelerates as expiry draws closer. In the example, it is around −0.03 per day.

Vega: insurance before the weather turns

Vega shows how much the premium changes if expected fluctuation rises or falls by one percentage point. Travel insurance serves as an image: the more uncertain the weather, the more expensive the cover. In the example, vega is around 0.11. If expected fluctuation rises from 17.5% to 18.5%, the premium rises by around 0.11. How fluctuation determines option prices is described in the article on how volatility affects option pricing.

Rho: the interest rate in the background

According to the OIC, rho shows how sensitive the price of an option is to changes in interest rates. In its example, the OIC uses an interest rate change of one percentage point. According to the OIC, rho is positive for purchased calls and negative for purchased puts: rising interest rates make calls more expensive and puts cheaper. In the example, rho is around 0.04: if the interest rate rose from 0% to 1%, the premium would rise by around 0.04. The OIC writes that interest rate changes ‘impact longer-term options much more than near-term ones’. With the same assumptions but a term of one year, rho in the numerical example is around 0.47 (illustration).

What changes before expiry?

According to the OIC, gamma rises as expiry approaches, and the option responds more strongly to price movements. Vega falls, and the option responds less to expected fluctuation. For longer-dated options, the opposite applies. The numerical example (illustration) shows this pattern:

Own calculation using the assumptions of the numerical example (illustration), option at the money, price 100, rounded.

Why can an option lose value despite a rising price?

Because the influences act at the same time. The OIC points out that a price movement often goes hand in hand with a change in expected fluctuation. This works through vega and can offset the result from delta. The example calculates four steps, each recalculated with the model. The fall in fluctuation of 5 percentage points in one day is deliberately large so that the effect becomes visible. It is an illustration and says nothing about how expected fluctuation usually changes.

The price gain of around 0.55 is therefore used up completely: by the one day of time at around −0.03 and, above all, by the fall in expected fluctuation at around −0.55. The premium ends at 1.97, just below the starting value of 2.00. In the numerical example, with a fall of 3 rather than 5 percentage points, the premium would be around 2.19 (illustration).

Anyone who simply adds up delta, theta and vega individually and leaves out gamma estimates the total change in the scenario at about −0.06 to −0.09. Calculated exactly, it is −0.03. The measures are approximations and depend on one another.

What do the measures mean for buyers and sellers?

For the buyer and the seller of a call, the measures have opposite signs. For a sold call option, the OIC gives negative gamma, positive theta and negative vega. Delta and rho, derived by mirroring, are negative for the seller. For the purchased call from the example, delta, gamma, vega and rho are positive, and theta is negative.

Purchased call: own calculation of the numerical example, rho also according to the OIC. Sold call: gamma, theta and vega according to the OIC, delta and rho derived by mirroring.

Positive theta only means that the time value of the sold option decreases over time. It is not guaranteed income. With a sold call, a rising price makes the option more expensive and so works against the seller (negative delta). Negative gamma means that this disadvantage grows with further price movements. Negative vega means that rising expected fluctuation makes the sold option more expensive (derived from the signs). Someone who already owns the underlying and sells a call on it holds a covered call, as described in the article on the covered call strategy. The gain from a rising price is capped, and the underlying can lose value.

What risks and limits are there?

• Sellers: Someone who sells an option receives the premium but bears the risk. The OIC writes that the writer has ‘theoretically unlimited loss potential for a call option’ and ‘substantial loss potential for a put option’. Own derivation: with a sold put, the loss is mathematically limited to the strike price minus the premium, because a price cannot fall below zero. It can still be high. With a covered call (underlying held in the portfolio), the risk remains that the underlying loses value, and the gain is capped.
• Buyers: According to the OIC, the option no longer exists after expiry. If it is neither sold nor exercised, the entire premium can be lost.
• Snapshot: The measures are derived from pricing models. According to the OIC, they are dynamic, meaning they change with the price, time and fluctuation. A value from today does not apply unchanged tomorrow.
• Approximation: Delta is not an exact probability. The calculation ‘delta times price movement’ is a rough approximation.
• No forecast: The measures say nothing about whether the price will rise or fall.
• Limits of the calculation: The example assumes an interest rate of 0, no dividend, exercise only on the expiration date and constant fluctuation. Real options differ, for example due to costs, taxes, the bid-ask spread (the difference between the buying and selling price), interest rates and dividends.

Conclusion

The option Greeks break down the change in an option's price into influences: delta for the price, gamma for the change in delta, theta for time, vega for expected fluctuation, rho for the interest rate. In the numerical example (illustration), they all act at the same time, and despite a price rise the premium falls from 2.00 to 1.97. For buyers and sellers, the measures have opposite signs, and the OIC describes the loss potential when selling a call as theoretically unlimited; with a covered call (underlying held in the portfolio), the risk is instead the loss in value of the underlying, and the gain is capped. The values are calculated snapshots and not a forecast.