# Probability Lab

#### Probability Lab provides a functional way to think about options without the complex mathematics.

##### This page introduces the following concepts:

### PROBABILITY DISTRIBUTION (PD)

The initial concept to understand is the probability distribution (PD), which is a fancy way of saying that all possible future outcomes have a chance of coming true. The PD lets us know exactly what the chances are for specific outcomes. For example:

What is the likelihood that the daily high temperature in Hong Kong will be between 21.00 and 22.00 Celsius on November 22 next year?

We can take the temperature readings for November 22 for the last hundred years. Draw a horizontal line and mark it with 16 to 30 degrees and count the amount of readings that fall into each one degree interval. The amount of readings in each interval is the % possibility that the temperature will be within that interval on November 22, assuming that the future will be similar to the past. It works out that way because we took 100 readings. Otherwise you must multiply by 100 and divide by the number of data points to get the percentages. In order to accomplish greater accuracy we would require more points, so we could use data for November 20 through 24. Let us draw a horizontal line spanning each one degree segment at the height corresponding to the number of data points in that segment. If we use data from November 20 through 24 we would acquire more data and increased accuracy but it would be required to multiply by 100 and divide by 500.

These horizontal lines form a graph of our PD. They reveal the percentage possibility that the temperature will be in any one interval. If we want to know the likelihood that the temperature will be below a certain level, we must add up all the possibilities in the segments below that level. In the same way add up all the possibilities above the level to know the possibility of a higher temperature.

Respectively, the graph shows the possibility for the temperature to be between 21 and 22 degrees Celsius is 15% and the possibility that it will be anywhere under 22 degrees is 2+5+6+15=28% and over 22 degrees is 100-28=72%. Please note that the sum of the possibilities in all segments must add up to 1.00, i.e. there is a 100% chance that there will be some temperature in Hong Kong on that date. If we had more data we could make our PD more precise by making the intervals narrower , and as we narrowed the intervals the horizontal lines would shrivel to points forming a smooth bell shaped curve.

### STOCK PRICES

Just the same way as future temperature ranges can be selected possibilities, so can ranges of future stock commodities or prices or currencies. There is one vital difference however. While temperature seems to follow the same pattern year after year, that is not true for stock prices which are more determined by fundamental aspects and human judgement.

So the answer to the question, “What is the possibility that the price of ABC will be between 21.00 and 22.00 on November 22?” has to be more of an informed guess than the temperature in Hong Kong.

The information we have to work with is the current stock price, how it has moved in the past and fundamental data concerning the prospects of the corporation, the economy, the industry, international trade, currency and political considerations and so on, that may affect people’s thinking regarding the stock price.

Anticipating the future stock price is an unreliable process. Anticipating the PD of future stock prices seems to grant more flexibility, or at least we become more aware of the possible nature of the process. The more information and understanding we have the more likely we are to get it right.

### OPTIONS AND HOW THEIR PRICES INDICATE A PD

The prices of put down and call options on a stock are decided by the PD but the compelling fact is that we can hack the process. Specifically, given the prices of options, a PD indicated by those prices can easily be acquired. It is not required that you know how you can skip to the next section, but if you would like to know then here is one procedure that any high school student should be able to follow.

Suppose that stock XYZ is trading around $500 per share. What is the percentage possibility that the price will be between 510 and 515 at the time the option runs out about a month from now? Suppose that 510 call trades at $6.45 and the 515 call trades at $4.40. You can buy the 510 call and sell the 515 call and pay $2.05.

- If at termination time the stock is under 510, you lose $2.05
- If between 510 and 515, your profit is the average of your loss at 510 of $2.05 and your profit at 515 of $2.95 or $0.45
- If it is over 515, you make $2.95.

Further suppose that we formerly estimated that the possibility for the stock to be under 510 is 56% or 0.56.*

As long as options are “fairly” priced i.e. there is no gain or loss that can be made if the market’s PD is precise, then 0.56*-2.05+X*0.45+Y*2.95=0 where X=the possibility that the stock will be between 510 and 515 and Y=the possibility that it will be above 515.

Since all probable prices that are materializing have a possibility of 100%, then 0.56+X+Y=100 gives us 0.06 for X and 0.38 for Y.

* To estimate an entire PD you need to begin at the lowest strike and you need to take a guess as to the possibility below that price. That will be a low number, so that you will not make too great an error.

If you’ve read this far then you will also be intrigued to know how you can acquire the price of any call or put from the PD.

For a call you can obtain the stock price in the middle of each section above the strike price, subtract it from the strike and multiply by the possibility. For the last section, between zero and the lowest strike I would use ⅔ of the minimal strike and guess the possibility. Again, add all the results together to obtain the price of the put.

For puts you can take the stock price in the middle of each interval below the strike, subtract it from the strike and multiply by the probability. For the last segment, between zero and the lowest strike I would use 2/3 of the lowest strike and guess the probability. Again, add all the results together to get the price of the put.

Some may say that these are all very careless estimations. Yes, that is the nature of price speculation; they are careless and there is no point in pretending otherwise. Everybody is guessing. No one knows. Computer nerds with complex models appear unversed to be doing very accurate estimations, but the fact is that nobody knows the possibilities and your educated guess based on your understanding of the situation might be superior than theirs based on statistics of past history.

Note that we are disregarding interest effects in the discussion. We are also rearranging for the fact that options may be utilized early which makes them more relevant. When estimating the entire PD, this extra value needs to be accounted for but it is only compelling for deep-in-the-money options. By using calls to estimate the PD for high prices and using puts to estimate the PD for low prices, you can avoid the issue.

### THE PD AS INDICATED BY THE MARKET and YOUR OPINION

Considering that puts and calls on most stocks are traded in the option markets, we can estimate the PD for those stocks as suggested by the predominant option prices. I call this the “market’s PD,” as it is arrived at by the consent of option buyers and sellers, even if many may be up to date with the implications.

The highest point on the graph of the markets assumed PD curve tends to be close to the present stock price plus interest minus dividends, and as you go in either direction from there the possibilities decrease, first slowly, then more swiftly and then slowly again, approaching but never quite reaching zero. The forward price is the expected price at expiry as suggested by the possibility circulation.

Click the image above to view a larger version

The curve is virtually symmetrical except that somewhat higher prices have higher possibility than somewhat lower ones and much higher prices have lesser possibility than near zero ones. That’s due to prices tending to fall quicker than they rise and all corporations have some chance of some cataclysmic event occurring to them.

In the Probability Lab you can see the PD we estimate using option prices currently prevailing in the market for any stock or commodity on which options are listed. All you are required to do is enter the symbol.

The PD graph advances as option bids and offers change at the exchanges. You can now grab the horizontal bar in any interval and move it up or down if you believe that the price ending up in that interval has a higher or lower possibility than the consensus guest as expressed by the market. You will notice that as soon as you move any of the bars, all the other bars will move at the same time, with more distant bars moving in the opposite direction as all the possibilities must add up to 1.00. Also notice that the market’s PD remains on the display in blue while yours is red and the reset button will wipe out all your fiddle.

The market tends to expect that all PDs are close to the statistical average of previous outcomes unless an actual corporate action, such as a merger or acquisition, is in the works. If you keep track of the market or the particulars of specific stocks, commodities or industries, you may disagree with that. From time to time you may have a different opinion of the probability of certain events and therefore how prices may advance. This tool provides you with the facility to illustrate, to graphically express that opinion and to trade on that opinion. If you do not have an opinion of the PD as being different than the markets then you should not make a trade due to any trade having a zero expected profit (less transaction costs) under the market’s PD. The sum of each probable outcome (gain or loss in each interval) multiplied by its associated possibility is the statistically Expected Profit and under the market’s PD, it equals zero for any trade. You can select any actual trade and estimate the expected profit to prove that to yourself. Thus, any given moment you make a trade with an expectation of gain, you are taking a bet that the market’s PD is incorrect and yours is correct. This is true whether you are aware of it or not so you might be aware of what you are doing and sharpen your skills using this tool.

### THE BEST TRADES AND THEIR POSSIBLE OUTCOMES

Please feel free and play with the PD by dragging the distribution bars below. We display combination trades that presumably have favorable consequences under your PD. You can state if you would like to see the “optimal trades’ that are a combination of up to two, three or four option legs. We will show you three leading combination trades along with the consistent expected profit, Sharpe ratio, net debit or credit, percentage probability of profit, maximum loss and associated possibilities for each trade, given your PD, and margin necessity.

The best trades are the ones that have the highest Sharpe ratio of expected profit to variability of end result. Please note that the expected gain is defined as the sum of the gain or loss when multiplied by the associated possibility, as determined by you, across all prices. On the bottom graph you will see your predicted gain or loss that would result from the trade and the associated possibility, corresponding to each price point.

The interactive graph below is an indecent simulation of our real-time Probability Lab application that is available to our clients. Comparably, the “best trades” are displayed for illustrative purposes only. Unlike in the actual application, they are not optimized for your distribution.

When you favour a trade in our trading application, you might increase the quantity and submit the order.

In consecutive releases of this tool we’ll address buy writes, rebalancing for delta, multi-expiration combination trades, rolling forward of expiring positions and further clarifications or the probability lab.

Please play around with this interactive tool. As you do so, your understanding of options pricing and your so-called “feel for the options market” will intensify.