Trading options may seem complicated, but there are tools available that can simplify the task. For example, modern hardware can take care of the fairly complex mathematics required to calculate the fair value of an option. To trade options successfully, investors must have a thorough understanding of the potential profit and risk for any trade they are considering. For this, the main tool option traders use is called a risk graph.

The risk graph, often called a "profit/loss diagram," provides an easy way to understand the effect of what may happen to an option or any complex option position in the future. Risk graphs allow you to see on a single picture your maximum profit potential as well as the areas of greatest risk. The ability to read and understand risk graphs is a critical skill for anyone who wants to trade options.

## Creating a Two-Dimensional Risk Graph

Let's begin by showing how to create a simple risk graph of a long position in the underlying—say 100 shares of stock priced at $50 a share. With this position, you would make $100 of profit for every one-dollar increase in the price of the stock over and above your cost basis. For every one-dollar drop below your cost, you would lose $100. This risk/reward profile is easy to show in a table:

To display this profile visually, you simply take the numbers from the table and plot them in the graph. The horizontal axis (the x-axis) represents the stock prices, labeled in ascending order. The vertical axis (the y-axis) represents the possible profit (and loss) figures for this position. Here is the two-dimensional picture that is produced:

To read the chart, you just look at any stock price along the horizontal axis, say $55, and then move straight up until you hit the blue profit/loss line. In this case, the point lines up with $500 on the vertical axis to the left, displaying that at a stock price of $55 you would have a profit of $500.

The risk graph allows you to grasp a lot of information by looking at a simple picture. For example, we know at a glance that the break-even point is at $50—the point where the profit/loss line crosses zero. The picture also demonstrates immediately that as the stock price moves down, your losses get larger and larger until the stock price hits zero, where you would lose all your money. On the upside, as the stock price goes up, your profit continues to increase with theoretically unlimited profit potential.

## Options and Time-Based Risk

Creating a risk graph for option trades includes all the same principles we just covered. The vertical axis is profit/loss, while the horizontal axis shows the prices of the underlying stock. You simply need to calculate the profit or loss at each price, place the appropriate point in the graph, and then draw a line to connect the dots.

Unfortunately, when analyzing options, it is only that simple if you are entering an option position on the day the option(s) expire, when determining your potential profit or loss is just a matter of comparing the strike price of the option(s) to the stock price. But at any other time between the date of entering the position and expiration day, there are factors other than the price of the stock that can have a big effect on the value of an option.

One crucial factor is time. In the stock example above, it makes no difference whether the stock goes up to $55 tomorrow or a year from now—regardless of time, your profit would be $500. But an option is a wasting asset. For every day that passes, an option is worth a little less (all else being equal). That means the element of time makes the risk graph for any option position much more complex.

On a two-dimensional graph displaying an option position, there are normally several different lines, each representing the performance of your position at different projected dates. Here is the risk graph for a simple option position, a long call, to show how it differs from the risk graph we drew for the stock.

Purchasing this February 50 call on ABC Corp. gives you the right but not the obligation to buy the underlying stock at a price of $50 by Feb. 19, the expiration date, which we'll say is 60 days from now. The call option allows you to control the same 100 shares for substantially less than the cost to purchase the stock outright. In this case, you pay $2.30 per share for that right. So no matter how far the stock price falls the maximum potential loss is just $230.

This graph, with three different lines, shows the profit/loss on three different dates. The solid line shows the profit/loss for this position at expiration, 60 days from now (T+60). The dashed line in the middle shows the probable profit/loss for the position in 30 days (T+30), halfway between today and expiration. The dotted line at the top shows the probable profit or loss of the position today (T+0).

Notice the effect of time on the position. As time passes, the value of the option slowly decays. Notice also that this effect is not linear. When there is still plenty of time until expiration, only a little bit is lost each day due to the effect of time decay. As you get closer to expiration, this effect begins to accelerate (but at a different rate for each price).

Let's take a closer look at this time decay. Say the stock price remains at $50 for the next 60 days. When you first purchase the option, you start out even (at the zero line with neither a profit nor a loss). After 30 days, halfway to the expiration day, you have a loss of $55. On expiration day, if the stock is still at $50, the option is worthless and you lose the entire $230. Observe the acceleration of time decay: You lose $55 during the first 30 days but $175 in the following 30 days. Together the multiple lines demonstrate this accelerating time decay graphically.

## Options and Volatility Risk

For any other day between now and expiration, we can only project a probable, or theoretical, price for an option. This projection is based on the combined factors of not only stock price and time to expiration, but also volatility. And the difference between the cost basis on the option and that theoretical price is the possible profit or loss. Keep firmly in mind that the profit or loss displayed in the risk graph of an option position is based on theoretical prices and thus on the inputs being used.

When assessing the risk of an option trade, many traders, particularly those who are just beginning to trade options, tend to focus almost exclusively on the price of the underlying stock and the time left in an option. But anyone trading options should also always be aware of the current volatility situation before entering any trade. To gauge whether an option is currently cheap or expensive, look at its current implied volatility relative to both historical readings and your expectations for future implied volatility.

When we demonstrate how to display the effect of time in the previous example, we assume that the current level of implied volatility would not change into the future. While this may be a reasonable assumption for some stocks, ignoring the possibility that volatility levels may change can cause you to seriously underestimate the risk involved in a potential trade. But how can you add a fourth dimension to a two-dimensional graph?

The short answer is that you can't. There are ways to create more complex graphs with three or more axes, but two-dimensional graphs have many advantages, not least of which is that they are easy to remember and visualize later. So it makes sense to stick with the traditional two-dimensional graph, and there are two ways to do so while handling the problem of adding a fourth dimension.

The easiest way is simply to input a single number for what you expect volatility to be in the future, and then look at what would happen to the position if that change in implied volatility does occur. This solution gives you more flexibility, but the resulting graph would only be as accurate as your guess for future volatility. If implied volatility turns out to be quite different than your initial guess, the projected profit or loss for the position would also be off substantially.

## Volatility and Time Synergy

The other drawback to estimating and inputting a value is that volatility is still held at a constant level. It is better to be able to see how incremental changes in volatility affect the position. That is, we need a graphical representation of a position's sensitivity to changes in volatility, similar to the graph displaying the effect of time on an option's value. To do this we use the same trick we used before—keep one of the variables constant, in this case, time rather than volatility.

So far we have used simple strategies to illustrate risk graphs, but now let's look at the more complicated long straddle, which involves buying a call and a put both in the same stock, and both with the same strike and expiration month. This option strategy has the advantage, at least for our purpose here, of being very sensitive to changes in volatility.

Again, say the expiration is 60 days from now. This is a picture of what the trade will look like exactly 30 days from now, halfway between today and the February expiration date. Each line shows the trade at a different level of implied volatility, and there's an increase of 2.5% in volatility between each line. The solid line is the profit/loss for this position at V+0, or at no change from the current level of volatility. The next lineup shows the probable profit/loss that would occur if implied volatility increased 2.5% within 30 days from now.

This method demonstrates the isolated effect of changes in implied volatility. As volatility increases, your profit increases (or, depending on the stock price, your loss lessens). The reverse of this is also true. Any decrease in implied volatility hurts this position and reduces possible profit—these effects on performance should be understood by the option trader before entering the position.

We mentioned earlier that to display the effect of volatility changes, we would need to hold time constant. But while the above profit/loss graph shows what the trade looks like only on a specific day, the effect of time is not completely stripped out. Notice that at a stock price of $50, the V+0 line shows there will be a loss of $150. That loss (for the long call and put combined) is solely due to 30 days of time decay.

As you gain experience and get a better feel for how options behave, it will also become easier to envision what a volatility risk graph would look like before and after the particular date being graphed.

## The Bottom Line

It is unlikely you would be able to predict off the top of your head what an option trade is likely to do. Even if you knew a trader bought 15 of the February 50 calls at $2.70 and sold 10 of the January 55 calls for $1.20, it would be difficult to project profit and losses. Visualizing how the trade is affected by changes in time, volatility, and the stock price is even harder.

But that's what risk graphs are for. They let you isolate the probable behavior of any option position, no matter how complex, to a single picture that is easy to remember. Later, even if a picture of the graph is not right in front of you, just seeing a current quote for the underlying stock will allow you to have a good idea of how well a trade is doing. That is why understanding how to use profit/loss diagrams is an indispensable skill for every options trader.