OPTIONS AND HOW THEIR PRICES IMPLY A PD
The prices of put and call options on a stock are determined by the PD but the interesting fact is that we can reverse engineer the process. Namely, given the prices of options, a PD implied by those prices can easily be derived. It is not necessary that you know how and you can skip to the next section, but if you would like to know then here is one method that any high school student should be able to follow.
Assume that AAPL is trading around $500 per share. What is the percentage probability that the price will be between 510 and 515 at the time the option expires about a month from now? Assume the 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 expiration time the stock is under 510, you lose $2.05
- If it is between 510 and 515, your gain is the average of your loss at 510 of $2.05 and your gain at 515 of $2.95 or $0.45
- If it is above 515, you make $2.95
Further assume that we previously calculated that the probability for the stock to be below 510 is 56% or 0.56.*
Provided that options are "fairly" priced, i.e. there is no profit or loss that can be made if the market's PD is correct, then 0.56*-2.05+X*0.45+Y*2.95=0 where X=the probability that the stock will be between 510 and 515 and Y= the probability that it will be above 515.
Since all possible prices occurring have a probability of 100%, then 0.56+X+Y=1.00 gives us 0.06 for X and 0.38 for Y.
*To calculate an entire PD you need to start at the lowest strike and you need to take a guess as to the probability below that price. That will be a small number, so that you will not make too great an error.
If you've read this far then you will also be interested to know how you can derive the price of any call or put from the PD.
For a call you can take the stock price in the middle of each segment above the strike price, subtract the strike price and multiply the result by the probability of the price ending up in that segment. For the tail end you need to take a guess at the small probability and use a price about 20% higher than the high strike. Summing all the results gives you the call price.
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 sloppy approximations. Yes, that is the nature of predicting prices; they are sloppy and there is no point in pretending otherwise. Everybody is guessing. Nobody knows. Computer geeks with complex models appear to the uninitiated to be doing very precise calculations, but the fact is that nobody knows the probabilities and your educated guess based on your understanding of the situation may be better than theirs based on statistics of past history.
Note that we are ignoring interest effects in this discussion, but with current interest rates, that is a small effect. We are also adjusting for the fact that options may be exercised early which makes them more valuable. When calculating the whole PD, this extra value needs to be accounted for but it is only significant for deep-in-the-money options. By using calls to calculate the PD for high prices and using puts to calculate the PD for low prices, you can avoid the issue.