Free Class
Options

# More Greeks

Russ Allen
Instructor

Last time I wrote about Delta, which is probably the best known of the Greeks. Wikipedia has a pretty good definition of these:

measure the sensitivity to change of the option price under a slight change of a single parameter while holding the other parameters fixed. Formally, they are partial derivatives of the option price with respect to the independent variables.”

In other words, each one of the Greeks tells us how much an option price should change when a specific thing happens.

The mathematical option pricing models, including Black Scholes and all the rest that have come after, assume that there are only four of these “independent variables” for any individual option, which has a given strike price and expiration date:

1. Underlying asset price
2. Time to expiration
3. Volatility
4. Interest rates

Each one of the Greeks tells us how much an option’s price should change when there is a small change in one of these four variables.

Here’s a quick rundown:

Delta: Sensitivity of option price to changes in underlying price. Shows expected option price change, in dollars per share, for the next one-point change in the underlying price.

Gamma: Rate of change of Delta with respect to underlying price change. Shows expected change in Delta, in percentage points, for the next one-point change in the underlying price. As you can see, both Gamma and Delta show aspects of the option price’s response to changes in the underlying’s price.

Theta: Sensitivity of option price to the passage of time. Shows expected option price change, in dollars per share, for the next one-day period of time.

Vega: Sensitivity of option price to changes in implied volatility. Shows expected option price change, in dollars per share, for the next one-percentage-point change in implied volatility.

(Interesting tidbit, or not:) Vega is not the name of any Greek letter. However, the glyph used is the Greek letter nu. Presumably the name vega was adopted because the Greek letter nu looks like the letter V, and vega was derived from vee by analogy with how beta eta, and theta are pronounced in English.

Rho: Sensitivity of option price to a change in interest rates. Shows expected option price change, in dollars per share, for the next one-percentage-point change in the risk-free interest rate.

Do we really have to care about these things? Can’t we just buy calls if we’re bullish and puts if we’re bearish?

If you’re looking for the easy-button solution, please do. After all, someone has to lose on options for the rest of us to make money. Best of luck to you.

But if you’re still reading, you already know the answer: yes, we do need to know these things. And they’re really not that hard to grasp in a usable way.

We already discussed Delta in last week’s article. It is the big one, the one that allows option market makers to figure out how to hedge their option positions. Without that, there would be no options market. Today let’s look at its cousin, Gamma.

Gamma is, you might say, the delta of the delta. It shows how fast Delta will change as the underlying changes. It is a positive number for long puts and for long calls, and therefore a negative number for short puts or short calls.

Very few retail traders create an option position based on Gamma. How, you may reasonably ask (as a student in an option class did just today) can Gamma be of any use to us? In fact it can, and here’s one big way.

We looked last time at creating a delta-neutral position. We can make one by buying and selling options whose total delta adds up to zero. Examples using only options would be (if we chose to do them in a neutral way) straddles, strangles, and condors. In fact we could make any option position delta-neutral by adding up the deltas of all our long and short options, and then buying or selling stock to balance out the net option delta. That’s what option market-makers do all the time. But the delta-neutral position gets “un-neutral” the second there is even a very small change in the underlying price. The deltas of puts and calls change as their “moneyness” changes, so the delta-neutral hedge needs constant adjustment (buying and selling of stock) to remain neutral. That’s fine for option market makers, who can afford to constantly fine-tune their positions. But what if we’d like to be neutral as to price and not have to make adjustments?

That can be done by creating a hedged position that is both delta-neutral and gamma-neutral.

First, by buying some options and selling others on the same underlying, we can create an option position that is gamma-neutral. This means that the total of the gammas of the long and short options is zero. For such a position, even pretty large changes in the underlying don’t change the delta. Incidentally, a gamma-neutral position also has very small or zero theta, meaning that there is next to no impact on it from the passage of time in its early days. The theta does increase eventually, but not much until expiration is pretty close.

Once a gamma-neutral position has been constructed, it can easily be made delta-neutral as well, by adding a position in the underlying stock. (Stock has no gamma, but it does have a delta of 1 per share. So adding stock to an option position changes its delta but not its gamma). The position is then a gamma-neutral and delta-neutral hedge. Being gamma-neutral, it is also (nearly) theta-neutral. So over wide ranges of both price and time, its value is not affected by movement of the underlying or by time decay. For many weeks or months, time and underlying price change will have very little effect. This is true whether the gamma-neutral position is a vertical spread, a calendar spread, a diagonal, or anything else, together with enough shares of stock to balance the deltas.

Huh? No impact from underlying price or from time? What’s left? What’s left is volatility. The option position still has a net Vega (positive or negative, depending on what options you used). The position will be affected by changes in Implied Volatility, and not much by anything else. If you want a pure bet on an increase or decrease in volatility, then a delta-neutral, gamma-neutral hedge will do the job.

The details of putting together such a position is beyond the scope of this article, but my point here is to answer the question “Who cares about gamma?” If what we want to do is trade volatility, then we do.