Chapter 1-2 The Greeks

Option prices can change due to directional price shifts in the underlying asset, changes in the implied volatility, time decay, and even changes in interest rates.  Understanding and quantifying an option’s sensitivity to these various factors is not only helpful, it can be the difference between boom and bust.  The option “greeks” come from the pricing model (normally the Black-Scholes model) that gives us implied volatility and quantifies these factors.  Delta, Theta, and Vega are the greeks that most option buyers are most concerned with.


Delta is a valuable measure that can be used in evaluating buying and selling opportunities.  Delta measures an option’s sensitivity to changes in the underlying stock price.  It measures the expected price change of the option given a $1 change in the underlying.  Calls have positive Deltas and puts have negative Deltas.

For example, let’s say the Oracle (ORCL) Feb 22.5 calls have a Delta of .35.  If ORCL appreciates from $21.48 up to $22.48, the option should increase by $0.35.  The Delta also gives a measure of the probability that an option will expire in the money.  In the above example the 22.5 call has a 35 percent probability of expiring in the money (based on the assumptions of the Black-Scholes model).  However, we must note that this does not give us the probability that the stock price will be above the strike price any time during the options life, only at expiration.  Delta can also be used to evaluate alternatives when buying options.  At-the-money options have Deltas of roughly .50.  This is sensible, as statistically they have a 50 percent chance of going up or down.  Deep in-the-money options have very high Deltas that can be as high as 1.00.  This indicates that they will essentially trade dollar for dollar with the stock.  Some traders use these as stock substitutes, though there are clearly different risks involved.  Deep out-of-the-money options have very low Deltas and therefore change very little with a $1 move in the underlying asset.  Factoring in commissions and the bid/ask spread, low Delta options may not make a profit even despite large moves in the underlying.  Thus, we see that comparing the Delta to the options price across different strikes is one way of measuring the potential returns on a trade.  Option sellers also can use the Delta as a way to estimate the probability that they will be assigned.  Covered call writers usually do not want to be assigned; they use Delta to compare the probability of being assigned for execution with the potential return from selling the call.

Advanced traders often look to employ “Delta neutral” strategies by creating positions where the total Delta is close to zero.  The idea is that these positions should profit regardless of up or down moves in the underlying.  This approach has its own unique risks, however, and generally requires frequent adjustments to remain Delta-neutral.  To review, Delta represents an option’s sensitivity to changes in the underlying price.  The Delta tells us how much an options price will change with a $1 move in the underlying.  At-the-money options have a Delta of roughly .50 and therefore will change roughly $.50 for every $1 change, up or down, in the underlying stock.

The graph below illustrates the behavior of both call and put option deltas as they shift from being out-of-the-money (OTM) to at-the-money (ATM) and finally in-the-money (ITM).  From this we can see that calls and puts have opposite deltas – call options are positive and put options are negative.


This second graph (below) illustrates how the delta of both a call and put option changes as the underlying price changes.


Option Charm (Delta Decay)

Option charm, also known as delta decay, is the amount that delta changes as time progresses.  Charm is most helpful for traders who are looking to design strategies around delta (ex. delta neutral) as it can be used to better predict the overall performance of the strategy.  Furthermore, charm can be used to restructure a portfolio of options without affecting overall delta (whatever it may be) by substituting options with the same charm for one another.  A common example of this would be substituting an option on a stock with a dividend approaching for another stock without one as the payment of the dividend will affect the stock price and thus the options price.

The graph below illustrates the effect of option charm on the delta of an option.  The graph shows option delta across a series of strike prices calculated at 3 different points in time (2 days, 5 days, and 30 days).  From this you can see how the delta of an ITM call option approaches 1 as the option approaches expiration, as well as how the OTM call options approach 0 as the option nears its expiration date.



Theta represents an option’s sensitivity to time.  It is a direct measure of time decay, giving us the dollar decay per day.  This amount increases rapidly, at least in terms of a percentage of the value of the option, as the option approaches expiration.  The greatest risk for loss to time decay is in the last month of the options life.  The more Theta your position has, the more risk you have if the underlying price does not move in the direction that you want.  Option sellers use Theta to their advantage by collecting time decay every day.  The same is true of credit spreads, which are really selling strategies.  One use of Theta is through calendar spreads which involve buying a longer-dated option and selling a nearer-dated option.  This strategy takes advantage of the fact that options decay faster as they approach expiration.

The graph below depicts the effect on an OTM call option as it approaches maturity date.  Theta calculates the increment as each day passes.  Notice that in the last remaining days of an option’s life, it loses its value quickly.


Let’s use JDS Uniphase (JDSU) as an example to help connect the dots:

Going into earnings the implied volatility was highest for the May options, up near 64 percent.  Theta for the at-the-money calls was -.04 and for out-of-the-money calls was -.03.

  • June options had an implied volatility of 50 percent. Theta for ATM calls was -.02 and -.01 for OTM calls.
  • A calendar spread consisting of buying a June call and selling the May call would give you a positive Theta of +.02 whereas simply buying a May ATM call would give you a Theta of -.04.
  • A JDSU May ATM call spread against an OTM call (a vertical spread: buying ATM, selling OTM) gives you a Theta of -.01, still negative, but much reduced.


The Gamma metric is the sensitivity of the Delta to changes in price of the underlying asset.  Gamma measures the change in the Delta for a $1 change in the underlying.  This is really the rate of change of the options price and is most closely watched by those who sell options as the Gamma gives an indication of potential risk exposure if the stock price moves against the position.

The graph below shows Gamma vs. Underlying Stock Price for 3 different strike prices (45, 55, and 65).  As you can see Gamma increases as the option moves from being in-the-money reaching its peak when the option is at-the-money.  Then as the option moves out-of-the-money the Gamma then decreases.  Note that the Gamma value is the same for calls as for puts.  Thus, if you are long a call or a put, the gamma will be a positive number; and if you are short a call or a put, the gamma will be a negative number.



Vega is the option’s sensitivity to changes in implied volatility.  A rise in implied volatility results in a rise in option premiums and will increase the value of long calls and long puts.  Vega decreases as time to expiration decreases as implied volatility has a decreased effect.

The graph below plots the option Vega vs. Underlying Stock Price for 3 different strike prices (45, 55, and 65).  Notice that the behavior of an option Vega is similar to Gamma: increasing as the option moves from being in-the-money to at-the-money where it reaches its peak and then decreases as the option moves out-of-the-money.  Vega, like Gamma, is the same value for calls and puts.



Rho represents an option’s sensitivity to changes in interest rates.  An increase in interest rates decreases an options value because it costs more to carry the position.  Unlike the other option greeks, Rho is larger for options that are in the money and decreases steadily as the option moves out of the money.  Since most option traders tend to focus on options that are close to expiration and out of the money there is typically very little interest in this measurement.

The graph below plots the Rho of a call and a put option at 3 different points in time (2 days, 5 days, and 30 days), across a range of strike prices, with a stock price of $50.


Consider the following graphic that breaks down the option Greeks for Google (GOOG) puts and calls:


At the money options are highlighted in dark blue; at this point in time GOOG is trading at $612.  As you can see, Delta increases as the options get deeper in the money.  Theta decreases as you compare options with the same strike price from November to December.  Conversely, Vega decreases as expiration nears.  Finally, you can observe that Gamma decreases as you move in either direction away from the at the money strike price.

Using the Greeks to Buy a Call

Buying stock is a relatively easy process.  If you think it is going up, you buy it.  But when using options there are several additional layers of complexity and decisions to be made – What strike? Which expiration? Fortunately, we can use the Greeks to help us make these decisions.

First, we can look at the Delta.  The at-the-money call will have a Delta of .50.  This tells us two things- One, the option will increase (or decrease) by $.50 for every $1 move in the underlying stock.  If a stock is trading for $25 and the 25 strike call (Delta of .50) is trading for $2, then if the stock goes to $26, then the option should be worth roughly $2.50.  Out-of-the-money calls will have a Delta of less than .50 and in-the-money calls have a Delta greater than .50 and less than 1.  Two, the Delta tells the probability of expiring in the money.  A deep-in-the-money call will have a Delta close to 1.  This indicates that the probability that it will expire in the money is almost 100 percent, and that it will basically trade dollar for dollar with the stock.

Second, we can look at the Theta.  Theta is largest for the near-term options and increases exponentially as the call approaches expiration.  This works against us when buying short-dated options, and also leaves us the least amount of time for our position to work out.  Buying longer-term options – at least two to three months longer than we plan on holding the option – usually makes sense from this perspective.  We must balance this out with the Vega of the call.  The further in time you go out, the higher the Vega.  The practical import of this is that if you are buying options with higher implied volatility, often the case before earnings or when professional money managers are purchasing in big blocks, you have more exposure using those longer-dated options.  However, we are still left with the question of which option to buy.  The answer, as with most things, is which one will give you the most bang-for-the-buck.  For any given underlying asset look for the option with the lowest implied volatility.  This will have the lowest relative Theta and Vega exposure, and will provide the best return on investment.

The next step is to do a comparison of the Delta, Theta and Vega relative to the actual price of the options.  Deep-in-the-money calls have the highest Delta and lowest Theta and Vega, but they are probably not the best value when compared to the price of the option.  They will also be the most expensive. Basically, they will be the most capital intensive and represent the largest monetary risk.  On the other hand, far out-of-the-money options can also have low Vega and Theta.  They always have a low Delta, but again, those values may not be the best relative to the price of the option.  However, their probability of profit is very low.  “Near the money” options, two to three months out (depending on how long you want to hold the option) usually provide the best relative Delta, Theta, and Vega compared to the price of the option, in other words the most bang-for-the-buck.  Most option traders do not do this much analysis to just buy a call and it is for that exact reason that doing so can make you a more profitable trader.