Gamma-Delta Neutral Option Spreads

Have you found strategies that make use of the decay of an option's theta that are attractive but you can't stand the associated risk? At the same time, conservative strategies such as covered-call writing or synthetic covered-call writing can be too restrictive. The gamma-delta neutral spread may be the best middle ground when searching for a way to exploit time decay while neutralizing the effect of price actions on your position's value. In this article, we'll introduce you to this strategy.

To understand the application of this strategy, knowledge of the basic Greek measures is essential. This means that the reader must also be familiar with options and their characteristics.

Theta is the decay rate in an option's value that can be attributed to the passage of one day's time. With this spread, we will exploit the decay of theta to our advantage to extract a profit from the position. Of course, many other spreads do this; but as you'll discover, by hedging the net gamma and net delta of our position, we can safely keep our position direction neutral.

For our purposes, we will use a ratio call write strategy as our core position. In these examples, we will buy options at a lower strike price than that at which they are sold. For example, if we buy the calls with a $30 strike price, we will sell the calls at a $35 strike price. We will perform a regular ratio call write strategy and adjust the ratio at which we buy and sell options to materially eliminate the net gamma of our position.

We know that in a ratio write options strategy, more options are written than are purchased. This means that some options are sold "naked." This is inherently risky. The risk here is that if the stock rallies enough, the position will lose money as a result of the unlimited exposure to the upside with the naked options. By reducing the net gamma to a value close to zero, we eliminate the risk that the delta will shift significantly (assuming only a very short time frame).

To effectively neutralize the gamma, we first need to find the ratio at which we will buy and write. Instead of going through a system of equation models to find the ratio, we can quickly figure out the gamma neutral ratio by doing the following:

1. Find the gamma of each option.

2. To find the number you will buy, take the gamma of the option you are selling, round it to three decimal places and multiply it by 100.

3. To find the number you will sell, take the gamma of the option you are buying, round it to three decimal places and multiply it by 100.

For example, if we have our $30 call with a gamma of 0.126 and our $35 call with a gamma of 0.095, we would buy 95 $30 calls and sell 126 $35 calls. Remember this is per share, and each option represents 100 shares.

• Buying 95 calls with a gamma of 0.126 is a gamma of 1,197 (9,500*0.126).
• Selling 126 calls with a gamma of -0.095 (negative because we're selling them) is a gamma of -1,197 [12,600*(-0.095)].

This adds up to a net gamma of 0. Because the gamma is usually not nicely rounded to three decimal places, your actual net gamma might vary by about 10 points around zero. But because we are dealing with such large numbers, these variations of actual net gamma are not material and will not affect a good spread.

Now that we have the gamma neutralized, we will need to make the net delta zero. If our $30 calls have a delta of 0.709 and our $35 calls have a delta of 0.418, we can calculate the following.

• 95 calls bought with a delta of 0.709 is 6,735.5.
• 126 calls sold with a delta of -0.418 (negative because we're selling them) is -5,266.8.

This results in a net delta of positive 1,468.7. To make this net delta very close to zero, we can short 1,469 shares of the underlying stock. This is because each share of stock has a delta of 1. This adds -1,469 to the delta, making it -0.3, very close to zero. Because you cannot short parts of a share, -0.3 is as close as we can get the net delta to zero. Again, like we stated in the gamma, because we are dealing with large numbers, this will not be materially large enough to affect the outcome of a good spread.

Now that we have our position effectively price neutral, let's examine its profitability. The $30 calls have a theta of -0.018 and the $35 calls have a theta of -0.027. This means:

• 95 calls bought with a theta of -0.018 is -171.
• 126 calls sold with a theta of 0.027 (positive because we're selling them) is 340.2.

This results in a net theta of 169.2. This can be interpreted as your position making $169.20 per day. Because option behavior isn't adjusted daily, you'll have to hold your position roughly a week before you'll be able to notice these changes and profit from them.

Without going through all the margin requirements and net debits and credits, the strategy we've detailed would require about $32,000 in capital to set up. If you held this position for five days, you could expect to make $846. This is 2.64% on top of the capital needed to set this up - a pretty good return for five days. In most real-life examples, you'll find a position that's been held for five days would yield about 0.5-0.7%. This may not seem like a lot until you annualize 0.5% in five days - this represents a 36.5% return per year.

A few risks are associated with this strategy. First, you'll need low commissions to make a profit. This is why it is important to have a very low commission broker. Very large price moves can also throw this out of whack. If held for a week, a required adjustment to the ratio and the delta hedge is not probable; if held for a longer time, the price of the stock will have more time to move in one direction.

Changes in implied volatility, which are not hedged here, can result in dramatic changes in the position's value. Although we have eliminated the relative day-to-day price movements, we are faced with another risk: an increased exposure to changes in implied volatility. Over the short time horizon of a week, changes in volatility should play a small role in your overall position. 

The risk of ratio writes can be brought down by mathematically hedging certain characteristics of the options, along with adjusting our position in the underlying common stock. By doing this, we can profit from the theta decay in the written options.