To understand Gamma risk just think about the following: What is the worst case scenario in a long gamma non-directional trade? The answer is simple, the worst case scenario is when the underlying doesn't move at all. The reason is because the options used will always suffer time decay, so you are losing money consistently when the underlying doesn't move. Conversely to make money you need the underlying to move as much as possible in any direction. We don't care about the direction of the move but we care about the magnitude of the move. To care about the magnitude of the move is another way of saying that we want the realized volatility of the underlying to be higher than the implied volatility of the options we paid for.

Now, Vega risk is more interesting. Option prices are very fickle, and they depend mainly on the expectations of future realized volatility, for some structural reasons most options (Index and equities) tend to come down in price when the underlying moves up, in other words during upside moves the implied volatility of the options goes down, the opposite is also true, during downside moves, implied volatility tends to rise. So even though you have a non directional trade, you are exposed to directional risk indirectly through Vega because in an upside move your options will tend to lose value (at least that component of value tied to implied volatility) and in a downside move your options will tend to gain value.

Contrary to popular belief, the most basic form of a non directional trade requires only one option. The idea is to be exposed to Gamma risk as much as possible and also to eliminate delta risk from the position. If you recall from the first section of this book, we could approximate the change of value in a call option as:

 

Of course this is a very crude model but I want to use it to illustrate how to setup a non directional trade with just one option. Please note the places where dS appears, dS is the change of value of the underlying at a given time, it could be positive if the underlying goes up, or negative if the underlying goes down. Notice that dS appears in two places, in one place it is squared and multiplies Gamma, and in the other one it just multiplies delta. Let's then do some trick here, let's look at the equation but this time let's remove the delta term first:

dC = 0.5*G*dS^2+Theta*time

The new equation is very interesting, as the payoff of the trade depends on Gamma (G), Theta and the square of dS which is the change in value of the underlying. The cool thing about the square of dS ( dS^2) is that, as all squares, it is always positive! So here you have it, it doesn't matter if the underlying goes up or down (positive or negative) our trade always makes money (well, there is that pesky Theta that is trying to take our money away). So by removing the delta component we managed to make our position non directional, or in more formal words we managed to create a delta neutral position. I can hear some people here now saying, wait a second, one thing is to remove one term from the equation in a book, but how can you do that in real life with real options? The answer is just ahead.

In order to implement a non directional trade with a single option, we need to remove delta from the equation, the act of doing that is called delta hedging. In its most crude form a delta hedge is as simple as to buy or sell a number of shares of the underlying that is equal to the delta of the option. So, the first thing we need to do once we choose a particular option is to find its delta and then we can hedge by using the following rules:

Of course there are a number of deep issues lurking here, the most critical of all is that there is no agreement as to what the correct delta value of an option is. Just recall that delta depends on the model used and also the future realized volatility used. However, for the purposes of creating a non directional trade we can skip over those issues for now and focus on the execution of the trade. For any practical purposes the delta used is the one that a vanilla BSM or American model produces when using the implied volatility of the option. Worst case scenario use the delta that your trading software is showing for the option in question.

Once we have an option and the corresponding delta hedge, the PnL of both positions look like this:

The blue line represents the profit or loss of our option, and the red line is the hedge we took. The total profit of our position is the difference between the blue line and the red line. That is, we make money on the regions where the blue line is above the red line, and along the same lines we lose money when the blue line is below the red line. As you can see we make money most of the time no matter what direction the underlying moves, however something that is critical here is that the graph is only capturing gamma risk. Please remember that our option is losing money every second that passes, so we are in a race against time. We want the underlying to move very fast, at least faster than the losses accumulated due to time decay.

Before even entering into a long gamma non directional trade we need to figure out a quick way to compute the minimum move that the underlying needs to realize for us to break even in a certain period of time, most of the time we are interested in the minimum daily move so we can decide if the trade is worth taking or not. In the case of a trade that is made of an option and a delta hedge, the quickest way of estimating the minimum move is by using the implied volatility of the option (the same one that was used to compute the delta hedge) and applying the following equation:

 

The value obtained will be a good estimate of how much the daily move of the underlying has to be for the trade to break even at least. Note how the minimum move heavily depends on the implied volatility, so opening non directional long gamma trades when IV is very high is inefficient as the minimum daily move will be very big.