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Understanding the “Greeks,” Part 1: Delta

Understanding the “Greeks,” Part 1: Delta
March 07
00:35 2012

The analysis of options includes four important risk metrics, which are identified by the Greek letters delta, gamma, vega, and theta. (Based on that, it should not be difficult to divine why these four metrics are collectively nicknamed the “Greeks.”) Here we’ll examine what the first Greek, delta, measures. (This discussion will require an understanding of the fundamental workings of options, so if you do not have a good grasp of those concepts, please first review some of our material on options basics.)

It is important to recognize that all of the Greeks are derived from mathematical models. They are not “hard” numbers like an option’s price, bid/ask spread, volume, or open interest. For that reason, the Greek measures are only as good as the model that underlies them—something you should carefully consider before relying heavily on a particular set of Greeks to develop a trading strategy. This is also why you may see different conventions for representing the value of any given Greek.

Delta is a measure of how much impact a change in the price of the underlying stock will affect the price of the option. It is typically scored from 0 to 1 for call options and from 0 to ‒1 for puts. (For simplicity in this discussion, we’ll refer to deltas approaching 1 or ‒1 as “high” or “increasing” and those closer to 0 as “low” or “decreasing,” but please rest assured that we do recognize that mathematically speaking, ‒1 is in fact lower than 0.) For example, a delta of 0.47 means that a $1 change in the underlying share price should produce a $47 change in the option contract price ($0.47 x 100 shares). Since listed options represent 100-share blocks, some models use a value range of 0 to 100 or ‒100 instead to equate delta and price change.

In the simplest terms, delta increases the further an option gets into-the-money and decreases as it gets more and more out-of-the-money. For an option that is at- (or near-) the-money, delta typically hangs around 0.5 or ‒0.5. Here is a simple chart showing some values for a March call option on Amazon (AMZN) stock when shares were trading at $182.50 on February 20.

AMZN March Call
Strike Price Delta
195.00 0.23
190.00 0.34
185.00 0.47
180.00 0.60
175.00 0.72
170.00 0.80

As you can see, the delta for the 185 strike was very close to 0.50—only 0.03 removed. The stock price was only $2.50 away and furthermore had strong upward momentum, closing up $2.57 during the last trading session, which is why the delta for the 180 strike is a full 0.10 removed by comparison.

A mid-range delta value like this tells you that every dollar of movement in the price of the underlying stock will generate a change of roughly $0.47 in the price of the option contract. In essence, there is about a 50/50 chance that the option will end in-the-money. The further separated the stock price and the strike price are, the lower the chance that the stock price will cross that boundary. In the AMZN example here, the odds that the stock will fall below the 170 strike (thereby going out-of-the-money) are relatively low, as are the odds that it will climb all the way to the 195 strike by late March.

The sample values here are all near-the-money, however. Once delta reaches extremely high values (very near 1.0 or ‒1.0), the stock price and option price begin to move nearly in lockstep, changing almost one-for-one. This is because an option that is so deeply in-the-money is almost certain to remain there, and that predictability means the option’s value changes right along with the stock price. Conversely, at very low delta values, the option barely budges with changes in the stock price, simply because the odds of ever reaching that strike are minimal.

Delta is not a perfectly straightforward value, however. There are three primary factors that skew it.

First, as expiration approaches, delta tends to increase for options that are at- or near-the-money. Let’s consider our AMZN example again, adding data for the April contract.

AMZN March Call
Strike Price MAR Delta APR Delta
195.00 0.23 0.34
190.00 0.34 0.42
185.00 0.47 0.51
180.00 0.60 0.59
175.00 0.72 0.67
170.00 0.80 0.75

Notice that for those strike prices now in-the-money, delta is higher for the March contract than for April. Common sense will tell you that the odds that AMZN will fall to $170 by the April expiration are greater than for the March expiration, simply because there is more time for the price to move and for negative events to occur.

Second, delta is not constant. It varies continuously based on conditions in the market, and in fact gamma, which we address in Part 2 of this series, measures delta’s rate of change.

Finally, implied volatility (IV) affects delta. Volatility is merely how widely a stock’s price swings and how quickly it does so. Clearly, then, if delta considers that likelihood that an option will move out-of-the-money, high volatility increases the risk of that occurring.

Given the interplay of all these factors, it is easy to see why delta must be calculated using a mathematical model. It is also easy to see how models can differ, since one may choose to place greater or lesser emphasis on the contributions of certain factors—such as IV—to the delta value.

Delta is of particular importance when selling existing options (as distinct from writing option contracts). A more complex concept, called position delta, is employed when constructing option hedges, but that is beyond the scope of this discussion. Simple delta, however, will give you some indication as to what direction the market expects the underlying stock price to take.

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