When we trade options, what we want to obtain is optionality. Even though it sounds familiar, what follows is a more detailed explanation of what optionality is.

### Optionality and leverage

Many people all over the market trade options because of the leverage they provide. However, the leverage obtained from options is very different to any other form of leverage out there. In fact, it has its own name: Optionality. I want to advance the notion that options are not really leveraged instruments (of course they are not) but instead provide the buyer with optionality. Understanding this concept provides key insights into the risks and advantages that option trading provides with respect to truly leveraged instruments like futures or CFD’s.

### Derivatives

Before moving into the meat of the argument, I want to spend some time talking about derivatives in general. A derivative is a financial instrument that trades like any other instrument but its value is actually tied to the value of another financial instrument. Shares are not derivatives as their price is set purely by market forces i.e supply and demand. Options, futures, forwards, CFD’s are common types of derivatives. Even though they trade in some type of market their value is not completely decided by market forces, instead a big component of the value is related to the value of the underlying they are tracking. Consider for a moment that derivatives are not constrained by supply (an infinite number of them can be created if anyone so wishes) so the usual market forces are not at play here.

### Delta

Given that derivatives are tied to an underlying it could be useful to have some kind of parameter that connects the value of the derivative and the value of the underlying. This parameter in fact exists and because of tradition is called delta. Delta is a parameter that quantifies the change of value of a derivative when compared with a change of value in the underlying. For instance, if our portfolio consists of 1 share of Apple, then we have a portfolio with 1 delta. If Apple goes up by $1 our portfolio also goes up by the same amount. In fact many derivatives out there like futures and CFD’s have a delta of 1 (or very close to 1), that is, they move in almost perfect synchrony with the underlying.

The question here is: if someone is offering me a derivative with delta=1 why would I buy it? I mean I could just go and buy the underlying and would get the exact same PnL without the hassle of using the derivative at all, plus, I could also get the dividends. The answer to why derivatives are so popular is leverage. Speculators buy futures and CFD’s not because there is anything magical about them (given that they have a delta of 1) but because those are highly leveraged instruments. They are leveraged because you are not actually buying anything, you are just allocating some kind of performance bond (margin) that is just a fraction of the total cost of the underlying you would be forced to acquire. A good speculator “buys” the derivative, and closes the position way before any settlement date collecting the profits in the meant time. Because margin requirements are a fraction of the cost of an equivalent underlying position the speculator realizes outsized gains and in the same token outsized losses too.

This is something very important to understand here, in delta1 derivatives (not a typo) there is leverage to both sides, that is, profit and losses are both leveraged which means you can be wiped out completely (or acquire a debt) if the position moves against you rather quickly. Just recall all the traders that were caught in the whole swiss franc fiasco last year, they were using leveraged positions.

So using futures or CFD’s is potentially dangerous and carries the risk of total account wipe out (or worse yet, acquiring a big debt with your broker). I know this could be news to some people here, but leveraged instruments carry great risks in general.

### Optionality

Options are very different to anything else out there, they are very complex instruments and the relationship to the underlying is not as simple as the one that futures and CFD’s have. In fact the delta of an option depends on many things and changes all the time. Because of this, the leverage that options provide is actually asymmetrical and varies over time. Options are very interesting because they provide an asymmetrical payoff, not only are profits unconstrained but losses are limited to the amount of money paid for the options (no risk of account wipe out).

Another characteristic of the asymmetrical payoff of an option is that is nonlinear, which is expected because delta changes all the time. This feature, this convexity of the PnL is what is called optionality. Not only are you making money if the underlying moves in your direction but the amount of profit that you make is higher and higher as the underlying moves (it is not a constant profit). With losses the opposite happens, they become smaller and smaller as the underlying moves against you. Of course if options had only this feature they would be the perfect instrument for trading (more on that later) in practice options are subject to many other sources of risk than futures or CFDs so it is trickier to make money with them. But understanding those risks and understanding optionality is fundamental to making money consistently with options.

### Gamma

Finally we arrive to the heart of this paper. We know now that there is a parameter called delta that connects the change of value of the derivative with the change of value of the underlying so we suspect that there could be another parameter that quantifies the convexity that options have, a parameter that tell us in fact what level of optionality we are buying and that is also connected to the value of the underlying somehow. Such parameter indeed exists and is called Gamma (for historical reasons too). What gamma is telling us is how much optionality we are buying when we buy an option. One way to visualize gamma is that it represents the amount that delta changes when the underlying changes however that doesn’t show us in what sense gamma represents convexity/optionality. A better way to visualize this is the following approximation to the price of an option (in this case a call option) valid only if we think delta and gamma are always constant and also if we ignore all other sources of risk for the option (basically unrealistic but helps me drive the point):

C = (gamma*S^2)/2 + delta*S+C0

As you can see gamma is a factor of the nonlinear part of the equation (In this case is the square of the underlying price). From high school math you can remember that the shape of those types of equations is a parabola so what we notice right here is that the price of an option is actually parabolic with respect to the price of the underlying and that gamma controls how strong the parabolic movement in price is. That my friends is what optionality is, gamma is optionality. If gamma were zero then an option would be a very boring derivative that only depends on delta (like a future contract). A parabolic movement means acceleration on the way up, and deceleration on the way down (which matches the asymmetric payoff of an option). Gamma in other words is what makes an option an option.

*Please remember that it was just a toy model for the price of an option, completely useless in real life but it captures the essence of the nature of gamma and delta.

### Buying options means buying Gamma

Now that we know that gamma is optionality then we realize that when we buy options what we are really buying is gamma, and of course we want to get the best bang for our buck. In other words we want to buy the most gamma possible for the less possible amount of money, if we do that we are in the right path of buying options for profit.

Because options come in all sorts of strikes and expirations I want to provide just a few facts about how gamma changes with respect to both. In general terms, the options with higher gamma are those that are at the money or very near to the at the money strike. So the farther away from ATM you go the lower the gamma you get. That is why deep OTM options are not the best play possible all of the time ( a few exceptions are deep OTM SPX calls, but that is a different topic). Also deep ITM options have very little gamma so when you buy them you are basically forfeiting any optionality and paying top bucks for mostly a delta1 play (which is fine too if that is what you want).

Also, in terms of time, gamma is higher the closer you get to expiration. What that means is that the same strike will have higher gamma for options that expire sooner, and very low gamma for far away expirations.

So if you are playing options because you are anticipating a big move in the underlying for some reason then you want options that expire soon (of course you don’t want them to expire before the anticipated move) and also you want them near the ATM strike. if you go too far in terms of time and strike you are not really using much optionality. A compromise has to be made as ATM or near ATM options are very expensive, a rough method is to pick options with at least 20% of delta or more (33% or more would be my preference). Also if you expect an imminent move please use the closest expiration possible or if you are not very sure of the time frame use the closest expiration that will keep your thesis valid (for instance if the big move doesn’t occur in the next week then the thesis is wrong).

### Two sides of one coin

There is a drawback to being long gamma (there is no such thing as a free lunch in this universe). As I mentioned before the price of an option depends on other sources of risk, one of them is basically time. The price of an option decreases with the pass of time, this is what people call time decay which is a well known fact of options pricing. What is not so well known is that time decay and gamma are actually connected. If gamma were zero there would not be any time decay due to optionality (only the usual interest and dividend related decays). This has practical implications because it means that the higher the gamma of our position then also the higher the time decay we experience. There is no way around it, if you want optionality you have to pay the price of time decay so we also need to include this side effect in the design of our trade.