In the options marketplace, the “Greeks” have zero to do with classic philosophers or toga parties (unless you’re trading from the fraternity house). For option traders, the Greeks are a series of handy variables that help explain the various factors driving movement in options prices (also known as “premiums”).
Many options traders mistakenly assume that price movement in the underlying stock or security is the only factor driving changes in the option’s price. In fact, it’s very possible to watch the option contract move up or down in value, while the underlying price stays still.
Mathematically speaking, the Greeks are all derived from an options pricing model. The most well known is Black-Scholes, but many variations are used. For equity options, it is most common to use some form of the Cox-Ross-Rubinstein model, which accounts for the possible early exercise of American-style options.
Each Greek isolates a variable that can drive options price movement, providing insight on how the option’s premium will be affected if that variable changes. This article will address each of the major Greeks: delta, gamma, theta, vega and rho, as well as dividends. We’ll explore each of these in terms of their importance, explaining how to interpret and use each of them in your trading decisions.
Getting a firm grasp on your Greeks will help you judge what option is the best to trade, based on your outlook for the underlying. If you don’t contend with the Greeks, though, you could be flying into your next option trade blind. Why not learn to speak a little Greek instead?
One of the biggest mistakes new options traders make is buying a call option in order to try and pick a winner. After all, buying calls maps to the pattern you’re used to following as an equity trader: buy low, sell high, in that order.
Options are trickier. Sometimes the underlying stock moves in the expected direction, but the option doesn’t, or even vice versa. Options with different strikes move differently when the underlying price moves up and down, and also as the option approaches expiration. Is there any mathematical way to estimate how much your option might move as the underlying moves?
The answer is delta – it provides part of the reason for how and why an option’s price moves the way it does. There are many different definitions of delta, but the explanation that follows is the primary one.
Delta is the amount a theoretical option’s price will change for a corresponding one-unit (point/dollar) change in the price of the underlying security – assuming, of course, all other variables are unchanged.
Keep in mind delta is determined by using a pricing model, hence the term “theoretical” in the definition above. So although this is how the marketplace expects the option’s price to change, there is no guarantee that this forecast will be correct. The same holds true for any other Greek parameter described below.
Imagine a stock is at $40 and we’re looking at the two-month, at-the-money (ATM) call with a strike price of 40 and a current price of $3. If the stock goes from $40 to $41 right now, so the only thing that changes is the stock’s price, how much would you expect the call option’s price to move?
The call price should increase by about 50 cents, to $3.50. How do you know? One way would be to look up the option’s delta, +0.50, which you can find in the Options Chains under the Quotes + Research tab.
Using the definition above, if the stock goes up $1, the call price should go up roughly by the amount of delta. Hence, it should go from $3.00 to about $3.50. This also works in reverse. If the stock went down by $1 instead, the call option should go down approximately by the amount of the delta, 0.50 or 50 cents, to a price of $2.50.
When looking up delta at TradeKing, you will notice that calls have a positive delta. One way to explain this is call prices tend to increase as the underlying increases. You may have also noticed that put deltas are negative. This is for a similar reason; puts typically increase in value as the stock decreases. These are the delta signs you’ll get when buying these options. However, when selling these options, delta signs reverse. Short call options will have a negative delta; short put options will have a positive delta.
Delta is dynamic
What if the stock moves from $41 to $42? Would the call option move another 50 cents, or more or less?
The answer is more than 50 cents. That’s because delta is dynamic: how close or how far away the stock is from the strike price determines how the option will react to stock price movement. Since movement from $41 to $42 would mean the call option was becoming more in-the-money (ITM), the delta will increase to reflect that. In this case, we estimate the delta to be about 0.60 or 60. Using the increased delta to calculate, the new price of the call option should be about $4.10 ($3.50 + $0.60).
At-the-money (ATM) call options, like our first example, usually have a delta around 0.50 or 50. (By the way, traders will often remove the decimal and just say “the delta is fifty”.) In-the-Money (ITM) call options usually have deltas between 0.50 and 1.00 (or 50 and 100), and out-of-the-money (OTM) call options usually have deltas between 0.50 and 0 (or 50 and 0), obviously never going below zero.
At-the-money put options have a delta around -0.50 or -50. Deltas of ITM puts typically range from -0.50 to -1.00 (-50 to -100). Put options that are OTM often have a delta ranging from 0 to -0.50 (0 to -50).
Using delta to predict price movement as expiration nears
Let’s take a look at a different example:
Many first-time options buyers will gravitate toward the 133 strike call, because it’s the cheapest. However, if you understand delta you’ll see why it’s so cheap.
It’s cheap because its delta is only 0.05 – that is, this call option wouldn’t move much for a one-point move in the underlying. Why is its delta so low? Because the strike is so far away from the underlying price, the odds of the underlying finishing above $133 in 11 days are very small.
If you were lucky enough to get a one-point movement up tomorrow in this stock, delta suggests the option price should move only about five cents. In the real world, though, it might not move at all. Sure, it will move according to delta, but the option also lost one day of time value, as measured by a Greek called “theta” (we’ll get to theta later on). When you factor in time decay, you’d be lucky if the option was still trading for 15 cents.
If a one- or two-point increase in the stock is expected before expiration, the 127 strike calls will most likely benefit you more than the cheap option (133 strike).
What’s the moral of the story? It’s crucial to understand how your option will move relative to the stock price. Without understanding delta, it’s hard to know which option will reward you the most if your forecast for the underlying security is correct.
Again, delta is dynamic: it changes not only as the underlying stock moves, but as expiration approaches. Gamma is the Greek that determines the amount of that movement.
Gamma is the amount a theoretical option’s delta will change for a corresponding one-unit (point) change in the price of the underlying security. In other words, if you look at delta as the “speed” of your option position, gamma is the “acceleration”. Gamma is positive when buying options and negative when selling them. Unlike delta, the sign is not affected when trading a call or put.
Just like when you buy a car, you may be attracted to more gamma/acceleration when buying options. If your option has a large gamma, its delta has the ability to approach one hundred (or 1.00) quickly, giving its price one-to-one movement with the stock. Option beginners usually see this as positive, but it can be a double-edged sword. When you have a large gamma, the delta can be affected very quickly, which means so will the option’s price. If the stock is moving in your favor, that’s great. If it’s doing an about-face and moving opposite to your prediction, changes in your option’s price may cause a lot of pain.
Gamma is highest for near-term ATM strikes, and slopes off toward ITM, OTM, and far-term strikes. This makes sense if you think it through: an option that’s ATM and close to expiration has a high likelihood to accelerate to the finish in either direction.
The graph below shows the gamma for a near-term option (15 days to expiration) with its underlying stock trading around $85. As you can see, gamma is clearly largest for the ATM strike price.
To better explain gamma, we need to revisit delta for a moment. Let’s go back to a previous example: an ATM call with a strike price of 40 and the stock also at $40. The new twist is there’s only one day remaining to expiration, instead of two months. Delta is still 50 because the option is exactly ATM. If the stock goes up, the call would be in-the-money; if it goes down it’d be out-of-the-money. In other words, you would have a 50/50 chance of the option finishing ITM on expiration. With this in mind, here’s an alternate definition and use for the term: delta is a guideline for giving odds of the option finishing ITM at expiration.
Now imagine the stock has moved up to $41, with one day remaining before expiration. The 40 strike call would already be in-the-money. What would the delta be now? Think about the second definition of delta. Being one point in-the-money with only one day remaining means the option has a higher likelihood of staying in-the-money. That likelihood translates into a much larger delta. In this case it may be close to 85.
What would gamma be then? Remember, gamma measures the acceleration factor of delta. When the stock was at $40, delta was 50. When the stock moved up one point to $41, delta increased to 85. The difference between the new delta and the old delta is gamma (85 – 50 = 35).
If we lengthen the time to expiration in this example, it would drastically change the way the option would act. Let’s now say the option has 60 days remaining until expiration, the stock is $41 and the call strike is still 40. What’s the probability of the option being ITM at expiration? It’s much lower because the stock has more time to move around between now and expiration. To reflect that idea, the delta on this option would be lower – probably around 60. The gamma was also much lower (around 10 or 0.10 when the stock was at $40).
If you want to get an idea of how much acceleration an option may have, you can look up gamma on TradeKing’s Call or Put Calc Chains before placing the trade. As with any Greek characteristic, there’s a tradeoff to consider. In this case, if you seek out options with high gamma for more acceleration, you’re also likely to get high theta (rate of time decay). This brings us to the next Greek.
Theta is the amount a theoretical option’s price will change for a corresponding one-unit (day) change in the number of days to expiration of the option contract.
Each moment that passes melts away some of the option’s value. Not only does the premium melt away, but it does so at an accelerated rate as expiration approaches. This is particularly true of at-the-money options. If the option is either very in- or out-of-the-money, its options tend to decay in a more linear fashion.
Because the price of the option erodes over time, theta takes the form of a negative number. However, its sign actually depends on what side of the trade you are on. Theta is enemy number one for the option buyer, and a friend to the option seller. Mathematically, this is represented by a negative number when buying options and a positive number when selling them. Theta may also be found on TradeKing’s Call and Put Calc Chain.
If we focus on at-the-money (ATM) options, there’s a quick and easy way to calculate and therefore estimate how fast an option’s time premium may decay. At-the-money options work best in this example because their prices only consist of time value, not intrinsic value (the value by which an option is in-the-money). This simplifies the calculations a bit.
At-the-money options move at the square root of time. This means if a one-month ATM option is trading for $1, then a two-month ATM option would be trading for 1 x sqrt of 2 or $1.41. A three-month ATM option would be trading for 1 x sqrt of 3 or $1.73.
If you work backwards and assume the underlying stock price and other variables have not changed, the three-month ATM option’s time value would lose 32 cents after one month passes. It’d lose another 41 cents after two months, and in the final month after three months have passed, the option would lose the entire dollar. It’s pretty obvious from this example that not only do options decay, but they decay at an accelerated rate as expiration approaches.
If we plot these points graphically you can see the accelerated curve of decay.
Since the time decay of ATM options accelerates as expiration nears, it makes sense that theta is a larger number for near-term options than for longer-term options. Consider XYZ trading at $100, the 100 call trading at $1.15 with an implied volatility of 20%, and seven days until expiration. The one-day theta for this option is -.085 or a negative 8.5 cents. If no other variables change except one day of time passing, this contract will trade for around $1.15 - .085, or $1.065.
What if this same contract had 180 days until expiration? The rate of theta here would be much slower than the seven-day option, about -.025 or negative 2.5 cents.
Time decay and volatility: an interesting relationship
If volatility increases, theta will become a larger negative number for both near- and longer-term options. As volatility decreases, theta usually becomes a smaller negative number.
Put in plainer terms, a higher-volatility option tends to lose more value due to time decay than a lower-volatility option. If you’re drawn to buying higher-volatility options for the action they bring, keep in mind that you’re also fighting time decay a bit harder with these contracts.
On the flipside, you may be more drawn to selling these high-volatility options because of the more rapid rate of decay they have over lower-volatility options. (Remember: time decay is the option seller’s friend and the option buyer’s foe.)
In either case, the rate of time decay is only one piece of the puzzle when analyzing opportunities. You’ll need to consider a variety of factors at once when deciding what, when and how to trade.
Considering time decay’s effect on your whole portfolio
It’s not only useful to look at theta on individual options; you should also consider net theta across your entire portfolio. If you are net long options in your account, your portfolio would likely have a negative theta. In other words, each day that passes your portfolio would suffer a bit from time decay. If you are net short options in your account, your portfolio would have positive theta, which means your account value may benefit with each day that passes.
TradeKing calculates both your individual positions and your net portfolio theta automatically. Simply login to your account and go to Accounts > Holdings - Options View.
At this moment in time, this sample account would theoretically lose $1,579.93 in one day from time decay alone. Keep in mind that many other factors beyond simple time decay would also affect your ultimate gains or losses: price swings on the underlying, changes in volatility, or a change in carry costs.
Vega is one of the most important Greeks, but it often doesn’t get the respect it deserves. Vega is the amount a theoretical option’s price will change for a corresponding one-unit (percentage-point) change in the implied volatility of the option contract. Simply stated, Vega is the Greek that follows implied volatility (IV) swings.
Don’t forget we’re talking about implied volatility (IV) here, not historical volatility. Implied volatility is calculated from the current price of the option using a pricing model (Black-Scholes, Cox-Ross-Rubinstein, etc.); it’s what the current market prices are implying future volatility for the stock to be.Just like the Greeks, implied volatility is determined by using a pricing model. Likewise, there is a marketplace expectation of how the option’s price might change due to this parameter. But as before, there is no guarantee that this forecast will be correct.
Historical volatility is calculated from actual past price movements in the underlying security. Historical volatility may also be referred to as stock volatility or statistical volatility. To look up an option’s vega, you’ll find it along with the other Greeks on TradeKing’s Call or Put Calc Chains.
Let’s consider an example: a 100 strike call option, with the stock trading at $100, 30 days to expiration, and implied volatility of 20%. The price of the option is $2.50 and the vega is equal to .115 or 11.5 cents. This means if nothing else in the marketplace changes except the option’s IV increases one percentage-point from 20% to 21%, this contract would trade for around $2.50 + .115, or $2.615. If volatility declines by one percentage-point (20% to 19%), you’d expect the option price to decline in the same fashion - by the amount of the vega. Which options tend to have large vega?
Vega is typically larger for options in the far-term, which have more time premium. It’s also usually larger for at-the-money (ATM) options versus in- or out-of-the-money contracts. Think of it in these terms: the further out you go in time, the larger the amount of time premium in an option’s price. More time until expiration means the contract is more susceptible to IV fluctuations.
Like gamma, vega is positive when you buy options and negative when you sell them. The sign is not affected whether trading a call or put.
Vega is also usually higher for option contracts that trade with higher implied volatilities, since higher volatility typically drives up the cost of the option. More expensive underlyings also translate to large vega. Options that meet either of these criteria are typically more sensitive to changes in implied volatility.
Vega: a tale of two stocks
To give you a feel for what this all means, let’s look at two very different (and fictitious) companies – High Flyer Tech, Ltd. and Stable Manufacturing Inc. High Flyer has all the characteristics mentioned above. It’s a higher-priced stock ($390), and has a higher implied volatility (33%) when compared to Stable ($75 and 17% respectively). To show you how vega levels are sensitive to all the factors mentioned above, we chose calls with expiration dates further in the future for High Flyer than for Stable (56 days versus 28 days until expiration).
First, let’s compare High Flyer’s ATM strike price, 390, to the in- and out-of-the-money strikes. As expected, vega is much larger for the 390 strike: 0.61 or 61 cents. This means if the implied volatility of this option moves one percentage-point up or down, the option value would either increase or decrease by 61 cents. It’s worth noting the vega for the out-of-the-money 440 strike is smaller, but it represents a larger percentage of the option’s premium. It is 44 cents of $5.50 (8.0%) compared to 61 cents of $22.10 (2.7%).
Now let’s turn to Stable’s calls with 28 days until expiration. Stable’s stock is trading at a much lower per-share price compared to High Flyer and with lower implied volatility. Here we’re looking at relatively nearer-term options. The vega for the ATM strike is .08 or 8 cents - much smaller than High Flyer’s ATM call at 61 cents.
At the same time, on a percentage basis, eight cents is still a major factor in the price; eight cents of $1.45 equals 5.5% of the option’s price. So if this contract’s IV moves just one percentage-point lower, this option will lose 5.5% of its value. Decreasing implied volatility is one of the most annoying occurrences for option buyers: sometimes you’re right about the direction, but you still lose on the trade because of a drop in IV, also known as an implied volatility crunch.
Rho is the amount a theoretical option’s price will change for a corresponding one-unit (percentage-point) change in the interest rate used to price the option contract. Typically the interest rate used here would be the risk-free rate of return. The rate associated with investing in Treasuries is traditionally defined by market experts as virtually risk-free. Rho is less important for those who trade nearer-term options, but may be useful for evaluating longer-term strategies.
Rho addresses part of the cost-to-carry issue: weighing the opportunity costs of tying up your cash in a long-term option versus other investments. (Whether or not the underlying stock pays a dividend can also impact cost-of-carry. Read below to learn more about dividends.)
Let’s consider an example: a 50 strike call option with the stock trading at $50. Let’s further assume we have 60 days to expiration, the annual risk-free interest rate is currently 5%, there are no dividends pending with this option, and implied volatility is currently 25%. The price of the call option is $2.25, and rho is .045 or 4.5 cents. This means if nothing else in the marketplace changes except the interest rate increases by one percentage-point, the call option would increase by the amount of the rho. In this case, the price of the call would increase in value by about five cents, to about $2.30 ($2.25 + $0.045 = $2.295). Interest rates don’t usually jump by a full percentage point at a time; a more likely occurrence is that they’d move a quarter-point (0.25%) or half-point (0.5%). For these fractional moves, multiply the rho by the amount of the interest rate change (either .25 or .50), and the result would be the theoretical impact on the option’s price (about one or two cents). If interest rates declined instead of increased, the call price would be expected to decrease by the amount of the rho multiplied by the change in the interest rate.
Here are a few takeaways about rho you may find useful:
Rho and vega react similarly when it comes to underlying price and time. Two factors that increase vega, or volatility exposure, are increasing time until expiration and higher underlying prices. These same factors also increase the option’s sensitivity to a change in interest rates. When compared to shorter-term options, longer-term contracts would usually have larger rho numbers, or higher sensitivity to interest rate shifts. Rho also tends to get larger the more expensive the underlying security gets.
Because rho relates to carry costs, calls usually have a positive rho value, while puts tend towards negative rho values. That is to say, an increase in interest rates would cause calls to become more expensive and puts to become less expensive. But keep in mind, this change in value has nothing to do with an investor’s outlook on the market. This change is only a result of how options pricing models work when they factor in a change in interest rates.
To bring this concept home, consider this example: let’s say you have $10,000 to invest. One choice is to invest it all in Treasury bonds, offering a theoretically risk-free rate of return. Another choice is to invest in $10,000 worth of stock – let’s say 100 shares of the same stock at $100 per share. And a third choice is to buy one ITM call option for $10 reflecting your interest to control 100 shares of stock. The call investment would be $10 x 100 = $1,000. This action would leave you with cash left over ($10,000 - $1,000 = $9,000).
If interest rates are high, the first alternative may be more attractive than the second because investing in Treasury bonds offers lower risk but still may offer a modest return when compared to investing in shares of stock. However, the third option can be a hybrid of the first two choices. This selection allows for the investor to participate in a market investment (call option) and also invest the left over money elsewhere (Treasury bonds). Although there is no guarantee that the call option (or the stock for that matter) would perform well, the return from the Treasury bond investment would boost the combined return when both investments are considered together.
The higher the interest rate, the more appealing the third choice becomes to investors. On the contrary, the lower the risk-free interest rate, the less attractive this combination investment turns out to be.
If you’re a fan of LEAPS options, rho can be a useful secondary or tertiary indicator to keep your eye on. LEAPS stands for Long Term Equity AnticiPation Securities; they’re basically extra-long-term option contracts. We’ve already established that longer-term options usually have larger rho numbers and are therefore more sensitive to interest rate shifts. TradeKing’s Call or Put Calc can help you predict how your LEAPS option may be affected by interest rate changes.
Dividends: as important as a Greek in impacting options prices
Dividends (either in cash or in shares) are paid by many companies to its shareholders, most often on a quarterly basis. Why should options traders care about dividends? Dividends are part of the options pricing model, and so that alone makes them relevant. But the real answer is that dividends can impact option price movement much like the Greeks do. If you want a better handle on why options prices change, you’ll need to watch your Greeks and any upcoming dividends on the underlying. Let’s go back to cost-to-carry for a moment. As stated earlier, rho helps an investor evaluate the opportunities and risks of interest rate changes on an option. The other factor impacting cost-to-carry is whether or not the stock pays a dividend. If an investor owns stock, carry costs can be determined using interest rates. However if the stock pays a dividend, the carry cost is reduced by the amount of the dividend the investor receives. Think of the interest cost as money going out, and the dividend as money coming in.
Because of this relationship, changes in dividends (increases, decreases, or the addition or elimination of them) will affect call and put prices. Since they somewhat counteract interest costs, they affect prices in the opposite manner as rho. Increasing dividends will reduce the price of calls and boost the price of puts. Decreasing dividends will have the reverse effect – call values go up and put values go down.
Just as the passage of time affects all the Greeks, it also affects dividends. The more time to expiration, the more quarterly dividends may occur in the same period. So any change in a dividend will have a greater effect on longer-term options than shorter-term contracts. Dividend amounts for any stock may be found under TradeKing’s Quotes + Research tab.
There are many factors that affect an option’s price, and the Greeks help us understand this process better. It’s possible for some Greeks to be working for your position while others are simultaneously working against it. If you understand how changing conditions can affect your options trades, you may be able to better position yourself accordingly.