Explaining the "Greeks"

Option traders look to make money by taking advantage of many different market forces; stock price changes, fluctuations in volatility, time to expiration etc. However, traders can also lose money if these factors move against their positions. So, traders need to understand all of these forces and be able to put a value on them to measure the risks/rewards.

Option Greeks are a group of calculations that help estimate the effect certain inputs have on the valuation of options. The Greek values most commonly referred to are Delta, Gamma, Vega and Theta. Other lesser known Greeks are Rho, Charm, Color, Speed and Weezu. Each of these Greeks help traders asses the risk of their option positions in order to place better trades, helping traders to answer questions such as:

  • How will the value of the option change as the stock price changes? (Delta)
  • What is the probability of the option expiring in-the-money/out-of-the-money? (Delta)
  • How fast will the option lose value as it approaches expiration? (Theta)
  • What effect will a change in the stock's volatility have on the option value? (Vega)

Here is a table that provides a brief definition of each option Greek:

Option Greek Definitions

GreekMeasures the
Delta Rate of change on the option price when the underlying price moves by 1 point.
Gamma Rate of change of the Delta when the underlying price moves by 1 point.
Theta Rate of change on the option price when 1 trading day passes.
Vega Rate of change on the option price when the volatility changes by 1 percentage point.
Rho Rate of change on the option price when interest rates change by 1 percentage point.
Charm Rate of change on the Delta when 1 trading day passes.
Color Rate of change on the Gamma when 1 trading day passes.
Speed Rate of change on the Gamma when the underlying price changes by 1 point.
Weezu Rate of change on the Vega when the volatility changes by 1 percentage point.

Calculations

The Option Greeks are outputs from an option pricing model. There is a fair amount of math involved but you don't have to know the formulas to be able to use the Greeks.

Most broker terminals that facilitate options trading will provide support for option Greeks as part of their offering. Interactive Brokers supports many option calculations that you can use in your trading screen. However, if you don't have a broker account or you would like to simulate various option scenarios to test the effect of different input parameters, go ahead and download my option spreadsheet. You can enter and play around with various payoff scenarios and test the effects each has on the outputs.

Here is what all the option Greeks would look like, calculated across 3 different expirations for ITM, ATM and OTM options. I used the Black and Scholes Model with a spot price of 1,000, volatility of 35% and interest rates of 2.7%.

Call Option Example:

StrikeDaysPriceDeltaGammaThetaVegaRhoCharmColorSpeedWeezu
900 49 116.53 0.819 0.00205 -0.397 0.963 0.945 0.00198 0.0000069 -0.0000167 0.01973
1000 49 52.82 0.537 0.0031 -0.558 1.456 0.652 -0.00038 0.0000324 -0.0000053 -0.00014
1100 49 18.66 0.257 0.00252 -0.441 1.182 0.322 -0.00279 0.0000117 0.0000103 0.01716
                       
900 140 147.57 0.74 0.0015 -0.297 2.01 2.278 0.00038 0.0000042 -0.0000059 0.01571
100 140 91.14 0.562 0.00182 -0.341 2.441 1.817 -0.00022 0.0000067 -0.0000031 -0.00066
1100 140 52.63 0.389 0.00177 -0.323 2.374 1.299 -0.00082 0.0000053 0.0000005 0.0096
                       
900 232 172.24 0.719 0.00121 -0.245 2.689 3.488 0.00013 0.0000024 -0.0000037 0.0134
1000 232 111.67 0.555 0.00139 -0.238 3.102 2.838 -0.0001 0.0000032 -0.0000022 -0.0017
1100 232 79.42 0.445 0.00142 -0.266 3.15 2.342 -0.00047 0.0000028 -0.0000007 0.00525

Put Option Example:

StrikeDaysPriceDeltaGammaThetaVegaRhoCharmColorSpeedWeezu
900 49 13.36 -0.181 0.00205 -0.33 0.963 -0.258 0.00198 0.0000069 -0.0000167 0.01973
1000 49 49.29 -0.463 0.0031 -0.484 1.456 -0.685 -0.00038 0.0000324 -0.0000053 -0.00014
1100 49 114.78 -0.743 0.00252 -0.359 1.182 -1.148 -0.00279 0.0000117 0.0000103 0.01716
                       
900 140 38.23 -0.26 0.0015 -0.229 2.01 -1.132 0.00038 0.0000042 -0.0000059 0.01571
100 140 80.77 -0.438 0.00182 -0.266 2.441 -1.972 -0.00022 0.0000067 -0.0000031 -0.00066
1100 140 141.22 -0.611 0.00177 -0.24 2.374 -2.869 -0.00082 0.0000053 0.0000005 0.0096
                       
900 232 56.64 -0.281 0.00121 -0.177 2.689 -2.115 0.00013 0.0000024 -0.0000037 0.0134
1000 232 106.96 -0.432 0.00139 -0.216 3.102 -3.388 -0.00016 0.0000032 -0.0000022 -0.0017
1100 232 160.34 -0.555 0.00142 -0.182 3.15 -4.507 -0.00047 0.0000028 -0.0000007 0.00525

Comments (16)

PeterMarch 5th, 2015 at 7:46pm

Hi Rajesh,

Yes, I've catered for dividends only via a Dividend Yield input. Does yours, instead, handle discrete dividends? If so, please feel free to send it to me - pop me an email from the contact page and I'll reply back.

For the Greeks, the best thing is to simulate the outcome by changing parameter that each Greek is forecasting. For example, if you were looking to validate Delta, note down the Theoretical Price and the Delta. Add the Delta to the Theoretical Price. Then, change the underlying price and verify that the new Theoretical has changed by the previous Theoretical plus Delta.

Know what I mean?

RajeshMarch 4th, 2015 at 5:13am

Excellent worksheet, truely the best so far I could find on web, to the point and very helpful.
Just one minor observation, your greeks probably doesn't consider dividend values, I have formulas with dividends as well if you want to update further..
Also like we can cross check call and put option price with put call parity, is there a way to cross check the greeks as well ?

Thanks in advance

PeterJune 10th, 2014 at 1:02am

Hi Sam,

That's a good suggestion for an article! I will work on that topic and place it under Options101.

Do you have any ideas for topics that you think would add value on this site?

sam June 8th, 2014 at 10:56pm

do you have anything on how to use a static delta and a dynamic delta to hedge a portfolio?

PeterMay 29th, 2014 at 6:49am

Hi James,

Yep, check out The Complete Guide to Option Pricing Formulas.

JamesMay 28th, 2014 at 11:00pm

Do you have any references that you can direct me to for pricing Asian options? In particular Asian options in relation to electricity markets. Fantastic site.

PoojaMarch 12th, 2012 at 7:08am

I am a student and this website is the best i have read so far.. thanks a ton

PeterJanuary 4th, 2011 at 3:47am

You would need to check the contract specifications with the exchange where the options are traded. However, generally speaking most stock/futures options are American style whereas index options are European style.

YEOJanuary 4th, 2011 at 12:54am

How do we know if the option trading is under 'AMERICAN OPTION' or 'EUROPEAN OPTION' ?

AftNovember 26th, 2010 at 2:24pm

Its a very good site, Especially for the newbees...Thnx..Aft

PeterAugust 8th, 2010 at 6:51am

Thanks Ritu!

RituAugust 8th, 2010 at 5:40am

Your's is the most exhaustive yet simple website in terms of information provided in layman terms.....

Trader TravisJanuary 22nd, 2010 at 11:49pm

Your site is very nice and you have done an excellent job of explaining the greeks without confusing folks. New traders need to read your way of explaining it before they are guided down the wrong path.

Trader Travis
http://www.learn-stock-options-trading.com/

Mani GhediaJune 23rd, 2009 at 11:25pm

I agree the greeks are super important. Option Volatility and Pricing is a fantastic book - worth reading. Excellent website nonetheless! Cheers

AdminFebruary 8th, 2009 at 3:38am

Thanks for the positive feedback and great suggestion! I will look to add greek explanations for all of the strategies soon.

pchaser87February 7th, 2009 at 11:08am

This is an excellent site, but I think you could expand a bit on the greeks. They are very important in options trading and you could easily go into more deatail (with practical examples?) without making it sound very complicated. Great work other than that!

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