Greeks: The what, why and how of options pricing

May 31, 2012 07:00 PM

Options can be frustrating for new traders used to the two-dimensional movement of underlying stock or futures prices. Options are three-dimensional and their movements are based on more factors than the simple strength or weakness of the underlying. 

Thankfully, guidance can be found in pricing models and the Greeks. “The Greeks give option traders a realistic expectation of how an option price will change if certain factors in the market change,” says Jim Bittman, senior instructor at the Options Institute at CBOE.

Simply stated, the Greeks are a group of mathematical models that each help to calculate the theoretical value of an option. While advanced models may involve many more variables, here we will examine the five that are most important for novice options traders.

∆ (Delta)

Delta tells you how far the value of an option is likely to move based on a $1 move in the price of the underlying asset. Delta is expressed as a range between zero and 1.00. The higher the delta, the more closely the option value will follow that of the underlying. Delta helps temper a trader’s expectations of when the underlying moves.

Alan Grigoletto, director of education at The Options Industry Council, explains delta to new traders by likening it to walking with a child. “Imagine that I’m holding the hand of a small boy who is three years old. For every step that I take, he correspondingly can only take one-third of my gait. He’s an out-of-the-money option and only has a delta of 33,” he says. “Now imagine the boy is 10 years old. He’s now an at-the-money option. Now, for every step that I take, he can cover half my gait and has a delta of 50. Then as he grows up and becomes stock, or adult-like, for every step that I take, he moves at the same rate. He’s now a deep-in-the-money option.”

Options with higher deltas tend to be more expensive, deep in the money and close to expiration. “You get more bang for your buck, but you pay more bucks,” Bittman says. “The art of the business is matching up the amount that you are willing to risk for the so-called delta involved.”

Traders who want to risk less capital might buy out-of-the-money calls, but they need to know that the option is not going to move as fast as the underlying stock, and that the likelihood of it expiring in-the-money is low. So, while you risk less money, the odds of success are much lower. 

Understanding delta is helpful for traders using options to hedge a position because it tells you how many options you need to buy to fully protect yourself against moves in the underlying. Note that one equity option controls 100 shares of the underlying stock, while one futures option controls one futures contract. 

Θ (Theta)

All options have two stores of value — intrinsic and time. While the intrinsic value of an option remains fairly constant throughout its life, it loses its time value the closer it gets to expiration. This is known as time decay, and is measured by theta (see “Wasting away,” below). 

Theta is not a constant, but accelerates on a concave or convex basis as the option approaches expiration. An at-the-money option’s value will accelerate to zero as it approaches expiration.

Grigoletto explains that theta tries to account for the unknowns of the future. “Anything can happen over the period of expiration. If you have a 30-day option, as you get closer to the last couple of days, there is less chance that you are going to get a dramatic move,” he says. “Still, there’s the opportunity that something dramatic can occur in those last few hours before expiration.”

Theta is particularly important for options with stocks or futures that are not moving much. “If you own an option, it tells you how much you will lose every day or week by owning the option if the underlying price is unchanged,” Bittman says. “If you are short the option, then theta tells you how much you’ll make if it remains unchanged.”

This is important because options that expire out of the-the-money are worthless. So, if a company’s shares are at $100, and you bought a two-month call at a $105 strike price for $1, you are not going to make any money unless the stock goes higher than $106 within the next 60 days. If the stock is at $105.30 with 10 days to expiration and the option is worth 50¢, you would still be losing money even though the stock moved in your direction. On the flip side, the option seller is sitting back and watching time decay working for him. 

Γ (Gamma)

At its simplest, gamma measures the change in delta. While delta tells how much an option’s value will change based on any move in the underlying asset, gamma measures the rate of that change. If a stock goes from $100 to $101, the delta may go from 50 to 56. Then as the stock goes from $101 to $102, delta may rise from 56 to 61. Gamma measures that change.

This often is referred to as the curvature of an option. “[Gamma is] the number of deltas that are gained or lost per a one-point move in the underlying security. It’s basically how long I am getting or how short I am getting for every one-point move in that underlying,” Grigoletto says.

Bittman advises that gamma tends to be more important to professional traders managing a portfolio of deltas. “Retail traders should be most concerned with delta and second with theta,” he says. “All the mathematical relationships hold and apply whether you are large or small, but typically the small guy is worried about making money and not about how fast his gamma is changing.” 


Vega, or kappa because vega is not really a Greek symbol, is used in options pricing models to measure the rate of change relative to a change in volatility. There are two types of volatility: Historical and implied. Historical volatility is a mathematical measurement of price movements through time and basically is a standard deviation. Implied volatility, the one used in the options pricing model, essentially is the measure of supply and demand for the options.

For most retail traders, vega is not a big factor except around times of increased volatility. “Volatility will change around earnings announcements if there is an unexpected announcement from the Federal Reserve, or an unanticipated economic report,” Bittman says. “For the typical retail trader, other than earnings periods, volatility of stocks tends not to change so dramatically that a 10- or 20-contract position will be impacted severely.”

Vega typically becomes more negligible the closer the option moves to expiration; longer-dated options feel a greater effect than shorter-dated ones. Grigoletto explains that this is because more volatility further out from expiration equals more possibilities before expiration.

Ρ (Rho)

In the current market environment, rho is not nearly as important as it has been in the past. Rho is the rate the model predicts an option price will change when the expected risk-free interest rate changes. 

With the Federal Reserve’s intention transmitted to keep interest rates low until late 2014, Grigoletto says rho is the least important of the five values at the moment. “You can look back to the 1980s where rho was very important because interest rates were hovering around 16%-17% and that affected the cost of carry. But that’s not the case today, so we’ll put that on the bottom of the most important for the current market environment.”

Ω (Omega)

Options are an investment tool that allows you to hedge your exposure, exploit changes in volatility and price, take advantage of where a market is not going as well as where it may go and, most importantly, allows you to define your risk clearly. Every trade is a balance of risk and reward, and options allow you to calibrate that risk/reward to your own specifications. Although option pricing can seem chaotic to the uninitiated, an understanding of the Greeks can bring order. “It’s not theoretical,” Bittman says. “It’s a mathematical relationship and this the way the option market works 95%-99% of the time.”

About the Author

Web Editor/Assistant Editor Michael McFarlin joined Futures in 2010, after graduating summa cum laude from Trinity International University, where he majored in English/Communication. With the launch of the new web platform, Michael serves as web editor for the site and will continue to work on the magazine, where he focuses on the Markets and Trading 101 features. He also served as a member of the Wisconsin National Guard from 2007 to 2010.