# Understanding How Options Move With the Underlying Stock (Part 1)

02/03/2010 12:01 am EST

Focus: OPTIONS

Delta and gamma are two of the option “Greeks,” mathematical functions derived from an option-pricing model that help traders manage the risk of their positions. Delta tells you how much an option is like its underlying stock and how much its price should move with the stock.

Delta gives professional option traders an instant reference to hedge their positions so that they can trade “delta neutral,” which is simply their attempt to trade implied volatility and avoid the risks of directional movement in the stock. But delta is not static—it also changes as the underlying stock moves.

Thus, the way that delta changes is very important to option traders and must be monitored at all times, especially for large, complex positions. The primary tool to measure delta movement, and thus directional risk, is gamma.

Gamma is the sensitivity of an option’s delta to price changes in the underlying stock. While delta tells you how much an option price should change for every \$1 move in the underlying stock, gamma tells you how much the delta should change for that same move. In this way, as delta is the rate of change of the option price, gamma is the rate of change of the delta. Let’s look at an example of the mechanics of this, which you will be able to see on any quote chain, before we get into any more theory about what gamma is.

Let’s say we are looking at a quote chain for Amazon.com Inc. (NASDAQ: AMZN) options, with the stock currently trading around its previous day’s closing price of \$129.78. Today, if the AMZN February 130 at-the-money (ATM) call has a delta of 0.470, this means that for every one-dollar move up (or down) in the stock, the call option will gain (or lose) 47 cents.

If AMZN stock drops \$10, the option could lose as much as \$10 x .47, or \$4.70. But if the stock falls \$10 to around \$120, the 130 call won’t have a delta of 0.470 anymore, either. Actually, on the way down, the delta was also changing. So, all else being equal, the call option wouldn’t lose as much as \$4.70.

On the way down to \$120, and as the option price was losing value by virtue of that 0.470 delta, the delta was also changing, albeit by a much smaller magnitude. With the 130 call now about \$10 out of the money (OTM), the option may have lost over \$4.50 worth of value, and the delta was also reduced to, say, 0.170. This change in the delta means the option is now even less equivalent to the underlying stock, and has less probability of becoming more like the stock—about a 17% chance from this moment’s vantage point.

So, how did the option price change—but less than a 0.47 delta would have indicated it should—and how did the delta change, and what’s the relationship between the two?

Gamma is expressed in delta points gained or lost per one point change in the underlying. This “big” 0.30- point drop in the delta was brought about by its tiny gamma of 0.03, multiplied by the \$10 stock move. Gamma is always represented by a positive number, and it is a straightforward application to the delta—up moves in the stock are added to both call and put deltas, and down moves are subtracted from both.

Tomorrow in part 2, learning Greek “one letter at a time.”

By Kevin Cook, contributor, ONN.tv