The Greeks

A trader who quotes and trades options looks at the delta of a position to decide the number of shares to trade as a hedge. A risk manager looks at gamma and vega to estimate the profit and loss when the underlying price or volatility moves.

What the Greeks measure

The Greeks measure how much the price of an option (the premium) changes when just one of the inputs that set it moves a little. They are used to grasp price moves in numbers, to decide how much to hedge, and to estimate the size of losses¹².

Consider a hypothetical example in which a European option, a call option (call), has an underlying asset price of 100, a strike price of 100, and a time to expiry of 1 year. The risk-free rate (a rate with no credit risk or similar risks) is 1% a year (continuously compounded), there are no dividends, and the volatility is 20%.

The Black-Scholes formula gives the theoretical price of an option from the underlying price, the strike price, the time to expiry, the interest rate, and the volatility. It puts the theoretical price of this call at 8.43. How the price is found is covered on the page The Black-Scholes Model. The delta of this call is 0.5596, and estimating from delta, the call price rises by about 0.56 when the underlying price rises by 1.

An option's price changes with the underlying price, the volatility, the time to expiry, the interest rate, and other inputs. Each of the Greeks corresponds to one of them, and the five main ones are as follows³. The basics of calls and puts are covered on the page Option Basics: Calls and Puts.

This page shows vega and rho per 1% (0.01) of volatility or interest rate, and theta per day (with 365 days in a year). Some sources give values per 1.00 (100%) or per year, so align the units before comparing with figures from other sources⁴. Values are rounded to four decimal places.

Deep diveDefinitions as partial derivatives and the formulas for a call

The Greeks are partial derivatives of the option price $C$ with respect to its inputs. Delta is $\partial C / \partial S$, gamma is $\partial^2 C / \partial S^2$, and vega is $\partial C / \partial \sigma$⁵. Rho is $\partial C / \partial r$⁶. Theta is the change as the time to expiry $T$ decreases, $-\partial C / \partial T$⁵.

For a call on an underlying with a dividend yield $q$, delta, gamma, vega, and theta are given by the following formulas⁵. $N(\cdot)$ is the cumulative distribution function of the standard normal distribution, $\phi(\cdot)$ is its density function, and $d_1$ and $d_2$ are the same as in the Black-Scholes formula.

$$d_1 = \frac{\ln(S/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}}$$ $$d_2 = d_1 - \sigma\sqrt{T}$$ $$\text{delta} = e^{-qT} N(d_1)$$ $$\text{gamma} = \frac{e^{-qT}\phi(d_1)}{\sigma S \sqrt{T}}$$ $$\text{vega} = e^{-qT} S \sqrt{T} \phi(d_1)$$ $$\text{theta} = -\frac{e^{-qT} S \phi(d_1) \sigma}{2\sqrt{T}} + q e^{-qT} S N(d_1) - r K e^{-rT} N(d_2)$$

In these formulas, vega is per 1.00 of volatility and theta is per year. To convert them to the units on this page, divide vega by 100 and theta by 365. For the example call, the formula's vega of 39.4479 becomes 0.3945, and the formula's theta of −4.4201 becomes −0.0121.

When the underlying price is high enough, $N(d_1)$ and $N(d_2)$ approach 1 and $S\phi(d_1)$ approaches 0. With no dividends, only the last term of the theta formula, $-rKe^{-rT}$, remains. Under the same terms as the example call (strike price 100, time to expiry 1 year, interest rate 1%), this is $-0.01 \times 100 \times e^{-0.01} \div 365 \approx -0.0027$ per day. This is why, in the section "How the Greeks change with the underlying price," a call's theta approaches the interest term rather than 0.

Signs for bought and sold calls and puts

The sign of each Greek depends on whether the option is a call or a put and whether it is bought or sold. The delta of a bought call is positive, between 0 and 1, and the delta of a bought put option (put) is negative, between −1 and 0³. For a sold option, the signs are the reverse of those for buying the same option³. The kinds of delta used for currency options are covered on the page FX Option Market Conventions. Setting the call from the example next to a put with the same terms gives the following values.

The call's delta minus the put's delta is 0.5596 − (−0.4404) = 1. For an underlying with no dividends, the deltas of a call and a put with the same strike price and expiry differ by 1⁷. This relationship follows from Put-Call Parity. For the same reason, gamma and vega take the same values for the call and the put⁸. Both are positive when bought, and when volatility rises, both the call price and the put price rise.

For a bought option, theta is negative in most cases, because an option's time value decreases as time passes⁹. In some cases, however, a bought option has a positive theta, such as a deep in-the-money put (one that would give a profit if exercised now) when interest rates are high¹⁰. In this example, rho is positive for the call and negative for the put: when the interest rate rises, the call price rises and the put price falls. For options with a short time to expiry, there is little need to take rho into account³. With the same terms but a time to expiry of 0.1 years, the call's rho is 0.0493, about one tenth of the 0.4753 for 1 year.

Worked example of estimating price changes

Multiplying a Greek by the size of the change in its input gives an estimate of the change in price. For the example call, the underlying price, volatility, time, and interest rate are moved one at a time, and the estimate from the Greeks is compared with the result of recalculating the price with the Black-Scholes formula. The price before any move is 8.4333¹¹.

In every row, the estimate and the actual change differ by 0.01 or less. In the underlying price row, the actual change of 0.5694 is 0.0098 larger than the estimate of 0.5596. This is because, as the underlying price moves from 100 to 101, delta itself also rises, by about the value of gamma (0.0197). If the move is twice as large, you double the estimate, but the larger the move, the larger the gap (see the section "When the estimates fall short").

How the Greeks change with the underlying price

The values of the Greeks change when the underlying price changes. With the same terms as the example call, changing only the underlying price gives the following values.

The delta of a bought call is close to 0 when the underlying price is well below the strike price and approaches 1 when it is well above, rising in an S shape in between³. Gamma and vega are large when the underlying price is near the strike price and approach 0 as it moves away³. The call's theta is also most negative near the strike price. When the underlying price is low enough, it approaches 0, but when the underlying price is high enough, it approaches the interest term rather than 0 (about −0.0027 per day in this example; the reason is in the deep dive "Definitions as partial derivatives and the formulas for a call")¹².

When the underlying price and the strike price are both 100, the call's delta is 0.5596, not 0.5. It is a little above 0.5 because of the interest rate and the volatility. As the time to expiry gets shorter, the S shape of delta becomes steeper near the strike price¹³. With the same terms but a time to expiry of 0.1 years, the delta is 0.0528 at an underlying price of 90 and 0.9400 at 110, closer to 0 and 1 than the 0.3532 and 0.7345 for 1 year.

The Greeks of a combination of trades

When you hold several options and the underlying together, the Greeks of the whole combination are the Greeks of each trade multiplied by its quantity and added up. For a combination that buys one unit each of the example call and put (buying a straddle), adding the values from the table in the section "Signs for bought and sold calls and puts" gives the following.

The combination's delta is small at 0.1192, so its price changes little when the underlying price moves a little. Its vega, on the other hand, is 0.7890, twice that of a single option: when volatility rises 1 percentage point, the combination's price rises by about 0.79, and when it falls 1 percentage point, the price falls by about 0.79. Its theta is −0.0215, so if one day passes with everything else unchanged, the price falls by about 0.02.

The price of a combination is the sum of the prices of its trades, so its change is also the sum of the changes of its trades, and the Greeks can be added up too¹⁴. Sold trades are added with the sign reversed.

The underlying itself gains 1 in value when the underlying price rises by 1, so one unit of the underlying has a delta of 1 and a gamma and vega of 0. If you buy one unit of the example call and sell 0.5596 units of the underlying, the combination's delta is 0.5596 − 0.5596 = 0, so when the underlying price moves a little, the changes in the values of the call and the underlying roughly cancel out. Hedging that brings delta close to 0 in this way is called delta hedging¹. The combination's gamma stays at the call's 0.0197, so as the underlying price moves, delta drifts away from 0. The larger the gamma, the faster delta drifts, and the more often the hedge amount must be adjusted¹⁵.

When the estimates fall short

Estimates from the Greeks are for small changes in the inputs. When the underlying price moves a lot, delta itself changes along the way, so an estimate from delta alone drifts away from the actual change. Adding the gamma term (gamma × change in underlying price × change in underlying price ÷ 2) to the delta estimate (delta × change in underlying price) brings it closer to the actual change². For the example call, compare a rise of 10 in the underlying price with a fall of 10.

Figure description

The estimate by delta alone falls further below the recalculated price the farther it is from the starting point, while the estimate by delta and gamma runs very close to the recalculated price. The figure draws the example call's price for underlying prices from 90 to 110 as three lines: the recalculated price, the estimate by delta alone, and the estimate by delta and gamma (the legend reads "Recalculated", "Delta only" and "Delta and gamma"). The three lines meet at 8.43 at an underlying price of 100. At 110, the recalculated price is 14.94, delta alone gives 14.03, and delta and gamma give 15.02. At 90, the recalculated price is 3.86, delta alone gives 2.84, and delta and gamma give 3.82. The delta-and-gamma line almost overlaps the recalculated line. The gaps are computed from unrounded values, so they can differ by 0.01 from the differences between the displayed values.

Delta alone underestimates the actual rise by about 0.9 when the price goes up, and overestimates the actual fall by about 1.0 when the price goes down. A bought call has a positive gamma, so when the underlying price rises, delta grows and the gain gets larger, and when it falls, delta shrinks and the loss gets smaller. Adding the gamma term narrows the gaps to 0.07 and 0.04.

This way of estimating gains and losses from delta and gamma is sometimes used in calculating historical value at risk (VaR), which estimates the size of losses on a portfolio that includes options².

A 2013 set of lecture notes by Haugh explains the Greeks on the basis of the Black-Scholes formula. The notes say that this model does not match how prices actually move in some respects¹⁶. For example, delta is calculated on the assumption that volatility does not change when the underlying price moves. If volatility moves along with the underlying price, the actual change in price also includes a part that comes through vega¹⁷.

Deep diveThe delta-gamma-vega approximation formula

When the underlying price moves by $\Delta S$ and the volatility by $\Delta \sigma$, a Taylor expansion approximates the change in price as follows².

$$\Delta C \approx \text{delta} \times \Delta S + \frac{1}{2} \times \text{gamma} \times (\Delta S)^2 + \text{vega} \times \Delta \sigma$$

The case $\Delta \sigma = 0$ is called the delta-gamma approximation. The "Delta and gamma" row of the table above uses this formula with $\Delta S = 10$ and $\Delta S = -10$. When you use vega in the units on this page (per 1%), enter $\Delta \sigma$ as a number of percentage points as well.

Notes

1. Japan Exchange Group, "デルタ（用語集）" (Glossary: delta, in Japanese). ↩ ↩²

2. Haugh, "Black-Scholes and the Volatility Surface," pages 11–12, "Delta-Gamma-Vega Approximations to Option Prices." ↩ ↩² ↩³ ↩⁴

3. Osaka Exchange, "理論" (Theory), Kitahama Investment School (in Japanese), the "リスク管理が最重要" (Risk management is most important) section (the explanations of delta, gamma, theta, vega, and rho, and the table of values for buying and selling calls and puts). ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

4. Vega and rho per 1% and theta per day are the units of this site's simulator. Osaka Exchange, "理論" (Theory), Kitahama Investment School, expresses rho for a 1% change in the short-term interest rate, and Japan Exchange Group, "セータ（用語集）" (Glossary: theta, in Japanese), explains theta for one day passing. Haugh, "Black-Scholes and the Volatility Surface" (pages 7–10), defines them as partial derivatives, with vega and rho per 1.00 (100%) and theta per year. ↩

5. Martin Haugh, "Black-Scholes and the Volatility Surface" (Columbia University, 2013), Section 4, "The Greeks" (pages 8–10). $d_1$ and $d_2$ are given after equation (13) on page 3. The source writes the cumulative distribution function of the standard normal distribution as $\Phi$. ↩ ↩² ↩³

6. Japan Exchange Group, "ロー（用語集）" (Glossary: rho, in Japanese). It gives rho as "the change in the option price ÷ the change in the short-term interest rate" (our translation). ↩

7. Haugh, "Black-Scholes and the Volatility Surface," page 8. When there is a dividend yield $q$, the put's delta is the call's delta minus $e^{-qT}$. ↩

8. Haugh, "Black-Scholes and the Volatility Surface," "Put-Call Parity" on page 7, and the explanations of gamma and vega on pages 8–9. ↩

9. Japan Exchange Group, "セータ（用語集）" (Glossary: theta, in Japanese). ↩

10. Haugh, "Black-Scholes and the Volatility Surface," page 11, "Options Can Have Positive Theta." ↩

11. Prices are calculated with this site's Black-Scholes calculation (src/lib/simulators/black-scholes.ts) and rounded to four decimal places. The example call's price of 8.43 is 8.4333 rounded to two decimal places. ↩

12. With this site's Black-Scholes calculation (src/lib/simulators/black-scholes.ts), under the same terms as the example call but with an underlying price of 200, theta is −0.0027. ↩

13. Haugh, "Black-Scholes and the Volatility Surface," page 8 (the description of Figure 2(b)). ↩

14. Japan Exchange Group, "デルタ（用語集）" (Glossary: delta, in Japanese), finds the delta of buying 10 units (1 unit = 1,000 shares) of a call option with a delta of 0.3 by multiplying the delta of one unit by the quantity: 0.3 × 10 × 1,000 = 3,000 (shares). ↩ ↩²

15. Japan Exchange Group, "ガンマ（用語集）" (Glossary: gamma, in Japanese). ↩

16. Haugh, "Black-Scholes and the Volatility Surface," page 7 (the start of the section "The Greeks"). ↩

17. Haugh, "Black-Scholes and the Volatility Surface," page 8 (the paragraph after the delta formula). ↩