**Risk Graphs - Visualizing Your Profit Potential**

Jim Graham

Trading options may seem complicated, but there are tools available that can simply the task enormously. For example, calculating what the fair value of any option should be requires fairly complex mathematics, but a computer and the right software will take care of most of that work nowadays. But 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 that option traders use is called a risk graph.

The risk graph, often called a profit/loss diagram, gives you an easy way to understand what is likely to happen to an option, or any complex option position, in the future. Risk graphs allow you to instantly see your maximum profit potential, as well as the areas of greatest risk, by looking at a single picture. That makes the ability to read and understand risk graphs a critical skill for anyone that wants to trade options.

**Creating a Two Dimensional Risk Graph**

We will start at the beginning by showing how to create a risk graph using a position in the underlying – buying 100 shares of stock at $50 a share – as an example. 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.

With the stock price at: Your profit (loss) will be:

$62.50 $1,250

$60.00 $1,000

$57.50 $750

$55.00 $500

$52.50 $250

$50.00 $0 (the break even point)

$47.50 -$250

$45.00 -$500

$42.50 -$750

$40.00 -$1,000

$37.50 -$1,250

*page 2*

Since you paid $50 for the stock, the table shows that you will have a loss of $1,000 if the stock drops to $40 ($50 - $40 = -$10 x 100 shares = -$1,000). If the stock instead goes up to $57.50 you will have a profit of $750 to show for your position (all these figures ignore commissions and other trading costs). When the stock is trading for $50 a share you do not have either a profit or a loss – you are just breaking even.

This is easy to show on a two-dimensional graph. To create a profit/loss diagram for this position you simply take the numbers from the table and plot them in the graph. The horizontal axis (the x-axis) should be labeled with various stock prices in ascending order. The vertical axis (the y-axis) is labeled with the possible profit (and loss) figures for this position. That produces the following two-dimensional picture:

To read the chart you just look at any stock price along the horizontal axis, such as $55. Move straight up until you hit the blue profit/loss line. Then see where that point lines up with on the vertical axis to the left, $500 in this case. So at a stock price of $55 you would have a profit of $500, exactly what we calculated in the previous table.

The advantage of risk graphs is that they allow you to quickly 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 gives you an immediate understanding of both the risk and possible reward. You can see that as the stock price moves down, your losses get larger and larger, at least down to the point where the stock price hits zero and you lose all your money. On the upside, you can see that as the stock price goes up your profit will continue to increase, with a theoretically unlimited profit potential.

*page 3*

**Options and the Third Dimension - Time**

Creating a risk graph for option trades uses all the same principles we just covered. The vertical axis is profit/loss, while the horizontal axis shows prices of the underlying stock. You simply need to calculate what the profit or loss is at each price of the underlying stock, 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 on expiration day. If you are considering an option position on the day the option(s) expire, you can just compare the strike price of the option(s) to the stock price to see what your profit or loss will be. But at any other time between now 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, it made no difference if the stock went up to $55 tomorrow or a year from now. Your profit would be $500. But an option is a wasting asset. For every day that passes, all else being equal, an option is worth a little less. That means time is a critical element when evaluating the probable profit or loss of an option, and it makes the risk graph for any option position that much more complex.

In order to display an option position on a two-dimensional graph there will normally be several different lines shown, with each line representing the performance of your position at different projected dates. Let’s look at the risk graph for a simple option position – long one call contract – to show how it differs from the stock risk graph.

Purchasing the January 50 call on ABC Corp gives you the right, but not the obligation, to buy the underlying stock at a price of $50 by January 17, 2009 (the expiration date). The call option allows you to control the same 100 shares for substantially less than it cost to purchase the stock outright. In this case the trader paid $6.60 per share for that right. So no matter how far the stock price falls the maximum potential loss is just $660.

Notice that there are three different lines, with the line legend on the right showing you how many days out each line represents. The solid line shows the profit/loss for this position at expiration, 256 days from now. The dashed line in the middle shows the probable profit/loss for the position in 128 days, 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 that time has on this 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.

To see what this means, look at what would happen if the stock price were to remain at $50 for the next 256 days. When you first purchase the option, you start out even (at the zero line with neither a profit nor a loss). After 128 days, halfway to the expiration day, you would have a loss of $186. On expiration day, with the stock still at $50, the option is worthless and you lose the entire $660. So you lost $186 during the first 128 days due to time decay, but lost $474 in the second 128 days. By having multiple lines that represent different dates in the future you are able to see this effect graphically.

*page 4*

**Adding a Fourth Dimension – Volatility**

We mentioned earlier that calculating the probable profit or loss for options is easy only at expiration, when the only variable you need to calculate the value of an option is the stock price. For any other day between now and expiration we can only project a probable, or theoretical price, for an option based on the combined factors of stock price, volatility, and time to expiration. We then compare the cost basis on the option to that theoretical price to determine the probable profit or loss. The fact that the profit or loss displayed in the risk graph of any option position is based on theoretical prices, and thus dependent on the inputs being used, should always be kept firmly in mind.

Many traders, particularly those just beginning to trade options, tend to focus almost exclusively on the price of the underlying stock and the time left in an option when assessing the risk of an option trade. But changes in volatility also have a large effect on the value of an option. Anyone that is trading options should always be aware of the current volatility situation before entering any trade. To gauge whether an option is currently cheap or expensive, option traders look at its current implied volatility relative to both historical readings and their expectations for implied volatility in the future.

When we demonstrated how to display the effect of time in the previous example, it was assumed that the current level of implied volatility would persist without any 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. Naturally 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 it they are easy to remember and visualize later. So the preferred solution is to stick with the traditional two-dimensional graph.

There are actually two possible ways to handle this problem. The easiest way is to simply 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 also has some serious drawbacks.

The biggest problem inherent in this method is that you need to choose what number to input for future volatility. 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.

The other drawback to this method is that volatility is still held at a constant level. It would be better to be able to see how incremental changes in volatility would affect the position. So we would prefer to see a graphical representation of a position’s sensitivity to changes in volatility, similar to what we have when displaying the effect time has on an option’s value. To do this we use the same trick we used before – keeping one of the variables constant, in this case time rather than volatility.

We have used simple strategies to illustrate how to use risk graphs so far, but now we will get more complicated and look at a long straddle position. A long straddle involves buying both a call and a put in the same stock, with the same strike, and in the same expiration month. This option strategy also has the advantage, at least for our purpose here, of being very sensitive to changes in volatility.

This is a picture of what the trade will look like exactly 128 days from now on September 11, 2008, halfway between today and the January 17, 2009 expiration day. Each line shows the trade at a different level of implied volatility, with an increment of 2.5% between each line.

The solid line is the profit/loss (at various prices of the underlying stock) for this position at V+0, which means with no change from the current level of volatility. The next line up shows the probable profit/loss if implied volatility increases 2.5% by that date. Each additional line shows the trade at progressively higher levels of volatility, in increments of 2.5%. The line legend on the right indicates the exact increase each line represents.

The advantage to this method is that it allows you to see how changes in implied volatility will affect this position. As volatility increases, so does your profit (or at least your loss is smaller, depending what the price of the stock is). The converse of this is also true. Any decrease in implied volatility would hurt this position and reduce your possible profit.

This is a perfect illustration of why option traders need to understand how changes from the current level of volatility can affect the performance of a trade. As you gain experience and get a better feel for how option behave, it will also become easier to extrapolate in your mind what this risk graph would look like before and after this particular date.

*page 5*

**Conclusion**

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 January 50 calls at $3.00 and sold 15 January 55 calls for $1.25, it would be difficult to visualize what is likely to happen at various prices of the underlying stock. Visualizing how the trade is affected by changes in time and volatility is even harder.

But don’t worry, most people cannot do that. That's what the risk graphs are for. They let you reduce 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 a trade is doing. That is why understanding how to use profit/loss diagrams is an indispensable skill for every options trader.