# Probability-weighted graphs

From the explanation of profit/loss graphs you should understand how Options Laboratory chooses the x-axis price range for its predictions and graphs. In the profit/loss graph you can find the predicted payoff of a strategy at any specific stock price, on a date you choose by setting the projection date slider.

However, when that day arrives the stock won't have "any" price - it will have some specific price, or anyway trade within some (usually much smaller) price range during that day. It makes sense that prices near the current price are more likely than extreme increases or decreases; so, what can we say about the average or expected payoff for a strategy? Options Laboratory answers this question in a way that is consistent with option price theory and principles of investment and finance.

To understand what the program does, set up a graph as follows.

• From the Offline menu, choose Chain for textbook examples. This sets up a hypothetical stock AHSO priced at 101, and generates a simulated option chain.
• Go to the Trades window, which will now be empty except for a transaction quoting AHSO at 101. Double-click the blank line directly beneath that Quote transaction. This pops up a window where you can enter a transaction.
• In the popup dialog, click the Underlying radio button. Select a Buy transaction and 100 units (shares). Click the Add to trades button to store this purchase transaction.
• In the Graphs window, click the Probability weighted checkbox. You should get this graph:

As usual, there's a magenta dashed line at the current stock price, which is surrounded by ±1.0, 2.0 and 3.0-standard deviation bars. The program predicts that with this volatility, there is a 66% chance the stock price will end between 68 and 150 five years from now (9 Sept. 2002).

In similar fashion, the program can estimate the probability that the stock price will be between any two given prices on the projection date. It refines the analysis by dividing the possible price range into one-quarter standard deviation brackets from the low end of the predicted range, to the high end. These small ranges are called "buckets". The axis covers 6.0 standard deviations total (±3.0 standard deviations), so there are 24 buckets that together cover the entire x-axis range. Imagine 25 magenta vertical lines instead of just the six shown above.

Each bucket has a midpoint price, since we know its lower and upper price boundaries. If a bucket spans the range from \$79 to \$83, for example, its midpoint price is \$81. We can determine from the profit/loss calculation what the investment payoff  would be at each bucket's midpoint price.

We can also estimate the probability that the stock price will fall somewhere within that bucket. So we can estimate: there's a 4.1% chance the stock will be in this bucket; and if it is, the stock will be worth about \$81 a share, or \$8100 for the 100 shares we purchased. So the contribution of this bucket to the expected value of the stock over the entire range of possibilities is 4.1% * (8100-10,100) = -82 or a loss of \$82.  We place a point at (81,-82).

This procedure is performed for all 24 buckets and the points are connected with a line, resulting in the graph above. The sum of all these points, which is the area under the curve, is the probability-weighted expected payoff of our 100 shares of AHSO 5 years from now. In this example it is \$574.44.

## What does it mean?

This is not a prediction that the stock will end up at specific price returning \$574.44 after 5 years. The prediction is a statistical one. Roughly speaking it means: if you could make hundreds of similar trades, on different stocks with the same price volatility, entering each trade at a price of 101, then the average payoff or future market value on all these trades would be as indicated. But for any particular trade, anything could happen - governed only by a random, log-normal probability distribution.

(The log-normal distribution is a sort of bell-curve distribution not on the stock price itself, but on daily price changes. This results in the skewed curve illustrated above.)

Why does the program always predict an expectation of profit for owning stock? Because the underlying financial theory assumes that nobody would invest at all without at least an expectation of profit. The expected profit is related to the risk-free interest rate and the stock's price volatility. In fact, you can drag the volatility slider left and right to see the effect on this curve. The expected profit rises for higher volatility, modeling the reality that investors expect higher average returns for assuming more risk.

In these pictures, the exact y-value points are not numerically meaningful because they depend on the number of buckets chosen. With 48 buckets, each one would span roughly 2% of the stock price range, so there would be twice as many points, each half as high. What is meaningful is the shape of the curve, in particular the area under it. The expected payoff is the total area under the curve - the sum of all the buckets' payoffs. In our example, the curve has a lot more positive area (above y=0) than negative area. That's good. It means you're more likely to make money.

We can also derive the probability of making a profit from this analysis; it is just the proportion of total area that is in the positive region (where y is greater than zero). If we call the negative area N and the positive area P, then the probability of profit is  P / (N + P).

## How should you use this information?

### Keep yourself honest

One important aspect of the probability-weighted payoff curve is that it guides you away from a particularly common kind of judgmental error. Many people tend to think about call options like this: "This option is pretty cheap - a lot cheaper than buying the stock. I think this stock is going to go up, and if it goes up a couple of bucks I'll make a killing."

Unfortunately, there is an enormous body of evidence showing conclusively that most of the time, a stock is about as likely to go down in any given short time period, as it is to go up. Typically the probabilities are something like 51% up vs. 49% down. In fact, all of option pricing theory is built upon exactly this observation.

Of course, there are situations when the price movements are probably biased in one direction or another. But these are unusual conditions, and identifying them takes experience and judgment. You should always consider the possibility that the stock will behave in the more usual way, and the probability weighted graph shows you this at a glance. It takes all the possibilities into account in a rigorous, mathematically sound way.

### Compare alternative strategies or trades

Is a trade you're considering actually a good idea? This question depends on many variables, such as the strike prices and expirations of the options involved; the prices of those options; the stock's volatility. In fact, the option chain always offers many possibilities. Should you buy a call striking at 75, or 80? Is it better to write covered calls, or sell put options (which behave almost exactly the same way)? Which options should you choose for a particular combination strategy?

Every option strategy is different; but they can all be compared meaningfully in probability weighted terms. Even strategies with completely different structures can be compared this way. You're always comparing "apples to apples" when you compare the probability weighted statistics.

### Deciding when to get out

Suppose you successfully make some trades to set up a position you wanted. A little time passes, and the stock moves so that you have a significant profit. Should you hang on, or bail out? You can answer the question this way: if the expected profit for continuing to hold the position is about the same as your current profit, you should take your money and run. It may turn out, after the fact, that you'd have been better off to hold; but on average, over many trades, these analyses really do work.

Likewise, you might open a position and then have the stock move against you. Should you take your losses now, or hold on? The answer often depends on the nature of the strategy. Certain strategies can only become profitable after some time passes; initially, they're sure losers. A weighted analysis at different dates - drag the projection date slider back and forth - will give you an idea whether the strategy still has promise, and how much promise.