## A brief history of GammaOptions are very fascinating instruments. They have an asymmetrical profit and loss profile. In other words, when you buy options, your losses are capped but your profits are not. That sounds fantastic, and in fact, if that were the only property of options, they would be the perfect vehicle for speculation. However, they also suffer from many sources of risk that we will address later on in this book, so they are not as perfect or as simple to trade as it seems. Beyond the asymmetry of the profit and loss profile, options have an even more interesting property: they are non-linear. This property makes them very powerful in terms of the kind of leverage they deliver. Sadly, the non-linear nature of options is not obvious to the novice trader, and in fact most of the information on the open internet and in most introductory books leads us to believe otherwise. Consider the following profit and loss profile, courtesy of Investopedia: By looking at it, you would think that options are linear. For instance you can see that the profits of that call option are rising in a linear fashion as the price of the underlying also rises. However, what the graph doesn't make clear is that this particular profit and loss profile only applies after the option expires. In fact, most of the websites and introductory literature on options that are out there only focus on what happens after options expire. Of course, after the option dies, so to speak, it behaves in a linear fashion. However, because we want to profit from our options before they expire, we need to start thinking about them in a completely different way. In this book, we will only think about options during the period of time when they are still "alive". In fact, all strategies and example trades in this book are designed to take profit or loss before the option expiration date arrives. With that being said, let's now consider the following graph:In the above graph, the non-linear nature of options is much clearer because we are looking at the profit and loss profile before expiration. The profit and loss profile is still asymmetrical, but now you can see that the profit section of the graph grows exponentially. That is, it is actually accelerating with each movement of the underlying. Also notice that the opposite is true: the loss portion of the graph shows deceleration; the losses become smaller as the underlying moves against us. This non-linear nature of the profit and loss profile is called optionality, and it is what makes options great when compared with linear derivatives that track the same underlying. Many people out there also refer to optionality as convexity, so if you hear any of the two terms, just recognize that they are the same and that they both refer to the non-linear nature of options. For those with a keen eye, you'll notice that the profit and loss profile displayed is for just one single option. You need to multiply it by 100 to obtain the real profit and loss when using a standard option contract. Also, from this point on and for simplicity, we will call the profit and loss profile a PnL profile.Now, I just mentioned that there is an advantage to having optionality/convexity, and that is that our profits will be accelerated while our losses will behave exactly in the opposite way. The following graph shows more clearly the advantage of having a convex position versus a linear position. To produce the graph, I used the same SPX 1960 Call with 3 days to expiration and the SPX sitting initially at 1936. The linear instruments used are an equivalent amount of ES-mini futures contracts, which are linear for the purposes of this example. Also, although there is a method to determine the equivalent position in ES futures, we will bypass the detailed discussion for now, but be assured this will be revisited in greater detail in later sections of the book. Finally, just in case it is not clear from the titles of the examples, the PnL profile displayed is what one would get if the SPX were to take the values in the x-axis during that particular time (3 days before expiration). Even though this is just an example, we can already see the superiority of optionality versus a linear instrument. In the above graph, we can clearly see that in the profit section, our convex profits just keep getting bigger and bigger when compared with the equivalent futures position. At the same time, we can see that our losses are substantially smaller when compared with the losses that a futures contract might incur if the underlying moves against us. I'm sure that by just comparing the two instruments side by side in the same graph, many of you are already devising clever ways of exploiting the different nature of the instruments used. Believe me, that kind of relative value trade will be pervasive across this book. If it is not obvious to you yet, don't worry. We will work on these kinds of trades in the later sections of the book. So at this point, we are already convinced that options are non-linear, but what does this have to do with gamma? The answer: everything. Gamma is nothing more and nothing less than a quantification of the convexity of the price of an option with respect to the price of the underlying. In other words, if gamma were zero, then we would get something like the red line in the previous graph - just a linear instrument. Any time that we see a convex PnL, that means that we are dealing with gamma, and by extension of this argument, because all options are convex (they have optionality), then all options have a non-zero gamma value. Now you can see why every time we trade options, we are in fact trading gamma.## The delta connectionBy this point, we know that gamma is an option parameter that tells us how convex the price of the option will be when the underlying moves. A high value of gamma means high convexity, while a low value of gamma means lower convexity (more linear-like behavior). Implicit in all of this discussion is that the price of an option is connected to the price of the underlying, which is natural, as options are a specific type of derivative, and in all derivatives, their prices are connected to the price of the underlying. So the connection is not a surprise. The interesting question then becomes: how exactly does gamma make the price of the option behave in a non-linear way? To answer that, let's go back to the previous graph. In particular, let's focus on the red line which is basically just a straight line. In a linear derivative (like ES mini futures), price always behaves in a linear fashion. Now, we remember from high school math that the general equation for a straight line is: y= b*x+a (b is called the slope) (a is called the y intercept) We also remember that "b" is the slope of the line, and "a" is called the "y" intercept. To be fair, I'm using the naming convention of the financial and statistical literature; back in high school, we called the slope "m" and the intercept "b". But beyond the names that we use, the important thing to remember here is that in a straight line, the slope is what controls how steep the line will be. Now, if we imagine for a moment that "y" is the price of our derivative, and that "x" is the price of the underlying, then the slope "b" of that equation is the parameter that connects the price of the underlying with the price of the derivative. This is a very important parameter for any derivative (including options). It's so important, in fact, that it has its own name. Instead of calling it "b", the name given to the parameter is delta. So now we see that delta is what connects prices in the two worlds (derivatives and underlying). If we assume that delta is a parameter that never changes, then the price of our derivative will look like the red line in the previous graph - just a linear instrument. So what do you think is required to get the blue line, the convex one? Some of you might have guessed it already. In order to obtain convexity, we need a delta that changes. More concretely, we need delta to change somehow as the underlying moves. We know now that because options are convex, it means that their delta is variable. Here, my friends, is where gamma enters into the picture. Gamma in a more formal way is a measure of how much delta changes when the underlying moves. When gamma is high, then the changes in delta are more dramatic, which is what is happening to the blue line in the profit area of the graph. You'll notice that it gets steeper as the SPX moves up. Because it is getting steeper, it means that delta is also getting bigger in those sections. Also notice that in the loss section of the profile graph, the blue line becomes shallower. The slope is more subdued, which means that gamma is affecting delta in a negative way because delta is being reduced in that section as the underlying moves against us. At this point we can say that in any option, there are at least a couple of connections going on: - Delta connects the price of the option with the price of the underlying.
- Gamma connects changes in delta with changes in the price of the underlying.
The connection between the underlying price, delta, and gamma is of a mathematical nature. For those of you that have some background in calculus, we can say that delta is the first derivative of the option price with respect to the underlying price. Along the same lines, gamma is the second derivative of option price with respect to the underlying price. For everyone else that didn't take or doesn't remember calculus that well, we can use a metaphor borrowed from physics: - Delta can be thought of as speed, or how fast or how slow the price of an option changes when the price of the underlying changes.
- Gamma can be thought of then as the acceleration of the option price. For instruments with zero gamma, there is no acceleration and their prices are linear. It is because of this acceleration that we get convexity in the price of an option (the blue line in our PnL profile).
So here you have it, my readers. The first two parameters that describe an option: delta, and gamma. Now you know that they are intimately connected, that they can be thought of as speed and acceleration of price, and also that gamma alone is what confers the convexity to our option prices, which in turn means that gamma is what makes an option an option. ## Putting it all to good useSo far this looks like a good theoretical discussion about the nature of option prices. Even though we know that options have at least two parameters so far (delta and gamma), we still don't know exactly how they look like. More importantly, we don't yet know where we get them from. The first issue can be tackled right away; delta and gamma are just numbers. For instance, here is a real example from a real option. This is the SPX 1935 Call expiring on 2016-2-05. The information was computed on 2016-01-29 at 4:15PM. Price: 19.85 Delta: 0.5054844 Gamma: 0.008015714Delta is a positive number, which is expected because call options always have positive delta. And just in case you are wondering, put options always have negative delta. More importantly what the delta number means is that by every unit change in the SPX value, the price of the option will change by 0.505488. So for example, at that time the equivalent SPX value was around 1936. If it goes up to 1937, our option will now be worth: 19.85+0.505 The sign is important. If SPX goes down by one point to 1935, then our option will be worth: 19.85-0.505 However, as we saw before, options are actually convex instruments, and delta changes every time the underlying moves. In fact, for very small changes in the underlying price, delta changes proportionally to the amount of gamma the option has. In this case, delta will change in a proportion to 0.0080 every time the SPX moves. So when SPX moves up, the price of the option will actually be higher than the one computed by just using delta. Here it is easier to see that gamma accelerates the price when the underlying moves in our favor because it keeps making delta bigger and bigger. Also notice that if the underlying moves against us, delta becomes smaller because it decreases in proportion to gamma, so as the move against us progresses, delta becomes smaller and smaller, resulting in our losses being capped. The exact nature of delta and gamma and how we can extract them from option prices will be left for a later section in this book. However, even if we don't yet know how to compute them, it is useful to know certain properties they have and how to exploit them in our favor during trading. ## A detailed look into deltaWith options, we have many choices. We can choose the expiration date, the type of option (calls or puts), and also the strike that we want. For each one of those options, we can get a particular delta value, so it can be helpful to visualize how delta changes, at least based on the strikes that we pick. The following graph plots the delta value for SPX call options expiring on 2016-2-05. The range that is plotted is all options between the 1835 and 2035 strikes. I picked that range because it covers about 100 points up and down from where the value of SPX was when the computation was run: 1936. The red line represents the SPX value at the moment of the computation, in this case 1936. Because we are using call options for this example, any strikes to the right of the red line are strikes for out of the money options (OTM), and all the strikes to the left of the red line are strikes for in the money options (ITM). The graph is interesting because you can conclude a few things about delta right away: - Delta for OTM options is lower than delta for ITM options.
- Deep OTM options have very low values of delta (almost zero), so that means that their price changes very little when the underlying goes up.
- Deep ITM options have very high values of delta (almost one), so they basically track the price of the underlying almost 1 to 1.
- Delta for the strike that is near the SPX value (the at the money strike) is about 0.5.
- The range for delta seems to be between 0.0 and 1.0.
- Because the underlying is moving up, our strike that was very close to the at the money value will now start to be in the money.
- Because options that are deeper and deeper ITM have higher delta than OTM or near ATM options, we know that the delta of our option will increase.
- A rough estimate of what the final delta value will be can be computed just by using the delta of an option that is already 20 points ITM (in this case the 1915 strike). Doing that yields a delta of 0.65.
So a 20-point move up will move delta from 0.505 to 0.65. Clearly, an acceleration of price is happening under that scenario. Of course, the real final delta won't be 0.65, as delta in reality depends on many other things, but this could be a good estimate to help us pick a good strike to initiate a position. A similar exercise can be done by creating a scenario where SPX drops by 20 points instead. In that case, we can say that the delta of our option will be reduced, and that should be similar to the delta of an existing option that is already 20 points OTM (in this case the 1955 strike). If we do that, we can see that the delta for that option is 0.34, a lower value, as we expected. From delta alone, we can derive a couple rules of thumb if we want to take full advantage of optionality: - If we are playing a big move in the underlying, it is better to pick an option with enough room in delta. In other words, don't pick ITM options with high values of delta, as you can see that it tappers off really quick as the option keeps moving more and more ITM. It is better to pick OTM options where delta can grow as the favorable move in our direction continues.
- Now, don't get crazy with the previous rule. Picking options that are too far OTM means we are picking options with very low values of delta, so they won't gain that much in absolute value. However, they could be interesting to play in relative value terms.
- Picking deep ITM options to play a directional move provides very little optionality in the profit section. However, it still provides nice optionality in the loss section. In other words, losses will be softened a bit.
It can be helpful to think about delta of the options as a dollar value. It is the value that will be added to the price of our option when the underlying moves $1. Once we figure out that delta can be seen as money, it can be easy to construct a graph that plots delta but as a percentage of the option cost. Please take a look at this: This is exactly the same delta graph but expressed as a percentage of the cost of our options. Now it is clearer why it is a better idea to pick out of the money options when playing a directional move. You can see that they produce the best bang for our buck in terms of initial delta. The in the money strikes are not that efficient, as the increase in value due to delta is dwarfed by the initial cost of the position (remember that ITM options are substantially more expensive than OTM ones). However, please don't go crazy picking options with 0.00001 delta. In fact, please wait before designing a real trade based on delta information alone. I mentioned that options are very complex, and there are many more parameters involved in making a trade decision, gamma being perhaps the most important one. ## A closer look into GammaWe also get an interesting profile for gamma when we consider different strikes in a particular expiration date. Please take a look at the following graph: The profile for gamma is very interesting. We can see that as we move away from the at the money strike, gamma starts to drop heavily. It also gets its maximum value around the ATM strike - but not quite there. The maximum value of gamma is actually on a strike slightly higher than the ATM one. Now, if we are buying options for convexity, then it is clear that we should not stray too far from the ATM strike if we want to get maximum convexity in dollar terms. Also, notice how this graph once again favors OTM strikes over all other ones when playing optionality. To see that, let's just repeat the mental exercise of the underlying moving 20 points up. If we start with ITM options, then it is clear that as the underlying moves up, gamma gets reduced all the time; that is, convexity starts to disappear. However, if our initial position is an OTM call, you can see that when the underlying moves up, gamma starts going up, then maxing, then it starts to drop again. So from the gamma profile alone, we could conclude than an OTM option is a better play in terms of convexity, because if the underlying moves in the direction we want, gamma will be higher (up to certain point), and our gains will be accelerated even more. It can also be very helpful to express gamma as a percentage of the option cost. When we do that we obtain the following graph: The relative gamma graph (relative to the option price) also shows why OTM options are favored when one wants to play convexity in a particular direction. It also reinforces what the delta graph was telling us: basically, that deep in the money options are not really plays in convexity at all. Instead, they are pretty much linear plays more along the lines of buying equivalent linear instruments (shares, futures). The only advantage that deep ITM options have over the alternatives is that if the underlying moves against us, the loss will be with convexity, which will make it softer so to speak. In other words, our losses will be slower than losses with the equivalent instruments. Gamma has another interesting property: it gets bigger as the expiration time gets closer. What this means is that the gamma for options that expire far in the future will be lower than the gamma for the same options that expire in the near term. The following graph illustrates the point very well: The above graph is the gamma value for an SPX 1945 Call option expiring at different expiration dates, starting with February 5th, 2016 (7 days from the point where these computations were run originally), and extending to April 15th, 2016. The graph is very telling; we can see without a doubt that gamma becomes very small the farther out we go in expiration dates. What this means is that if we are interested in trades with lots of convexity, we need to pick options with shorter expiration time frames. However, before jumping in and start trading convexity in OTM options right away, wait a little longer until at least this section is completed. There are more sources of risk, and in particular one that is ever-present with options, and which drastically changes our outlook when designing trades that take advantage of convexity. ## Gamma and TimeSo far we have discovered several very interesting things about options. We know they have delta and gamma parameters, and that they are convex (non-linear). We also know which strikes and expirations are more convex than others, so perhaps we think we can now adjust our trade design to take advantage of the convexity that they offer. However, there is much more to options that just those two parameters, and more importantly, there is a price we need to pay if we want to get convexity (there is no such thing as a free lunch). Given how wonderful this instrument is (thanks to convexity), there should be no surprise that options have a drawback. The universe seems to be very good at extracting a price when something seems to be too good to be true. In the case of options, the price that we pay is called time decay: options lose value every second that passes. This loss of value due to time is called Theta, and it is an option parameter that is usually expressed in dollar value of the loss per unit of time. Surprisingly, the unit of time in most options software is one year, so be careful if you see theta values that seem to be higher than the cost of your option (that is usually an indication that the theta value you are seeing is in an annual basis). In many retail platforms, theta is expressed in dollars per day, so they will give you a better idea of how much money we are losing each day by holding our option. This parameter is usually displayed with a negative sign, so it is more obvious that we are losing money every day.There is one particular thing about theta that is not very clear from the open literature on the web nor from certain option books, and that is that theta and gamma are intrinsically linked. If you try to get rid of one of the parameters, you will get rid of the other, too. In other words, if you manage to make the theta of your position zero, then that means that gamma for your position is zero too. There is no way around this; if you want optionality, you must pay time decay. As you might expect, because theta and gamma are connected, it is just natural that theta behaves in a similar way to gamma. And indeed, it does vary per strike, and also with time to expiration, as seen in the graph below: In absolute dollars, theta is greater around the at the money strike, and it becomes smaller as we move away from it on both sides. It could be tempting to conclude that theta is very small for OTM strikes, and indeed it is, at least in absolute dollars. However, remember that OTM options are also very cheap, so it is better to plot a relative graph of theta with respect to the cost of the option. When we do that, we discover something very interesting: When theta is expressed as a percentage, the picture changes dramatically. Deep out of the money options are affected greatly by time decay by very big values. You can see strikes with more than 50% value loss per day. So here is the first red flag when buying deep OTM options to play convexity: Yes, they have lots of gamma, and better relative delta, too. However, they lose value very rapidly compared with near ATM options. Theta also has an interesting behavior with time to expiration. The farther we are from expiration date, the lower the time decay we will get. Please take a look at this graph: This is the same SPX 1945 Call option that we have been using to compute gamma per expiration. We can see how theta is substantially lower for the April 15th, 2016 expiration (less than 40 cents per day loss) compared with theta for the February 5th, 2016 expiration. From this graph, we can also see that the last 3 weeks of life of an option bring a very accelerated time decay compared with the rest. The behavior is very similar to the opposite of what happens with gamma, which increases dramatically during the last 3 weeks of life of the option. ## The War against TimeIntroducing the concept of time decay for options produces an interesting war: the war between time and price. The question in the minds of most readers could be the following: Given that theta increases dramatically the closer to expiration the option is, and also in percentage terms as we move more and more out of the money in strikes, then, is there a way to pick the optimal strike/expiration to play a particular move? The answer is: of course! And we will revise it in more detail when talking about long gamma strategies later on. In the meantime, we can focus on a more basic question: What is the minimum amount that the underlying has to move in order to cancel the effects of time decay in a particular time period? We know that our option will gain in value if the underlying moves in the direction we want, but at the same time, it loses value as time passes. Ideally, we want to find what is the sweet spot: the minimum amount of the move that will keep us break even after some time. The simplest way to attack this problem is to consider the change in the price of our option due to a move in the underlying. We could use a very simplistic model of the price change using delta, gamma, and theta like this: dC = 0.5G*dS^2+D*dS+Theta*time
So to answer the question then, we need to find a value of dS that makes the whole equation zero (that is because we don't want our option to change value at all). Because theta is expressed in dollars per day, time is expressed in days, too. Solving an equation like the one we just used is very simple. It is basically a quadratic equation, and from high school math we know a general formula to solve it. Please remember that we are using a very basic equation, and that we are assuming that delta, gamma and theta never change. However, in real life, they change all the time. For this reason, the answer that we find using this simple method will undervalue the minimum move, but it can serve as a good starting point in the meantime. The next graph is the solution of that equation for all the strikes that we have been using so far, with SPX options expiring on 2016-02-05 that are 7 days away from expiration from the moment the computations are being executed. The graph is showing us the move in SPX points that must happen within one day so we break even. Using our crude estimator, we can see that the ATM strike requires that SPX moves at least 2.5 points within the next day in order for us to break even. The farther we go in terms of out of the money strikes, the higher the move needs to be. For instance, for the 2000 strike, we need a move of about 5 points to remain break even. Remember that our estimator always undervalues the move, so the real move has to be even bigger. Also, the reason the move needs to be so big is that we are using options that expire in the very short term (7 days). In order to give a sense of the level of underestimation of our crude model, I'm including the required move obtained by using a professional options pricing model and using numerical methods to solve for it: As you can see, the required move to break even after one day is actually higher than the one computed by our basic model, and it drives home the point that picking options that are too close to expiration requires bigger moves in the underlying for us to actually make money with them. The percentage error between our simple model and the full model varies per strike and also per expiration. The farther out the expiration of the option is, the smaller the error in our estimate. For options that expire very soon, you should add a 15% to 20% extra padding to the minimum move, and that will give you a more realistic idea of the magnitude of the move. For options with 30 or more days on them, you could use around 4% padding for the same move. Here is the same graph, but this time picking options that expire more than 30 days ahead. The idea is to see how dramatically it affects the break even move in the underlying: Using the March 4 calls, which expire about 37 days away, seems much better. In order to break even using ATM options, we only need SPX to move 1.1 points the next day, and in fact, using strikes bigger than 2000 doesn't require such a big move either, about 1.8 points. These values are valid for a move for the next day. Remember that as time passes, theta decay becomes greater, so the move will need to be bigger. But at least this gives us the general idea that "buying time" with options can actually work in our favor. I just want to caution not to go crazy here and go too far in expiration, because as we move farther away, our options become very sensitive to a different kind of risk that we have not discussed yet. In particular, in Index options, calls are very tricky because they tend to be affected negatively by massive upside moves. I will expand on this in the second section, but in the meantime, we can already see that the war against time can be won, but that we need to be careful in the strike and expiration we pick so we can be on the winning side. ## The end is not nearI'm sure that with the material presented so far, some readers are itching to start trading this newfound understanding of convexity and time decay. But how exactly should this be applied to real trading? Before answering this in detail, I must stress that options are very complex, and that even though we now have an expanded Greek vocabulary with the concepts of delta, gamma, and theta, there are still some unidentified risks that affect options that we need to take into account before we can start designing realistic trades. That being said, with the knowledge gained so far, we can start putting into place the basic rules of a trading system that seeks to exploit convexity in certain ways. I'm sure that each of you has in your mind a few of these rules. Perhaps some of you have even more rules than the ones that I'm writing about here. But just in case, here is a high-level summary of what we now know about those 3 parameters. - If we don't really care about optionality/convexity, and the only thing we want is a leveraged position with options, then buying deep in the money options does the trick nicely. By doing that we are forfeiting convexity and at the same time we are reducing the penalty associated with time decay.
- If our trading thesis calls for a violent move in the underlying in a short time frame, then we can choose OTM options with short expirations. The best strike should be far enough so we can get the max gamma peak before it starts fading again, but not too far, so time decay doesn't eat into our profits too much.
- If the timeframe of the move is not well established, then we need to minimize time decay. However, we still need to find a good balance with gamma so we can start making money as soon as the move starts. You don't want an option with such a low gamma that the gains can't overcome time decay.
- Don't get too crazy with short dated deep OTM options. The low gamma coupled with the high time decay will conspire against us, and the likelihood of making money with them is very low.
- And finally, even though it hasn't been mentioned explicitly here, convexity can also be sold. Of course, when selling convexity, pretty much all the rules are reversed and the design of successful trades becomes way more complex. We devote an entire section to short gamma trades for precisely that reason.
## Some parting words for the AdventurerI hope you are enjoying this book so far. After this section, I'm sure you have better conceptual tools to attack certain kinds of trades with options. However, the adventure is just starting. What follows after this introductory section is the real meat of the whole book. Take the time to process the information so far; it is the foundation of all the things to come. If you have any questions, please write to me, and I'll try to answer as much as I can. Even though it looks like we can do some real-life trading now, it is wiser to wait at least until the end of the second section, which deals with one of the most important aspects about options pricing. Again, thank you for taking the time to read this far, and I hope you don't get discouraged from continuing the adventure. If you feel ready, please jump to The Mysterious Smile of Volatility section now for more interesting discoveries. |