Scrolling through my Quora feed the other day, I found someone asking for an explanation of the Black-Scholes Model. Everyone thinks that option pricing is difficult and arcane, but if you read the brief article below, you will have a better intuitive grasp of how option pricing works than 99% of option traders.
The following overview is from chapter two of my book, “The Framework Investing: Applying Value Investing to the World of Options” (McGraw-Hill, 2014).
Imagine you and your spouse or significant other have reservations at a nice restaurant. The reservation time is coming up quickly and you are still at home. The restaurant is extremely hard to get reservations for and if you are not there at your reservation time, your seats are given to someone else. Now, let’s assume that in the midst of the relationship stress you are likely feeling at the moment, you decide to lighten the mood by betting with your spouse or significant other as to whether you will be able to make it to the restaurant in time for your seating.
If you were a statistician attempting to lighten the mood of the evening, before you placed your bet, you would be attempting to factor in the following variables to figure out how likely or unlikely you would be to make it on time.
- How long do you have until your reservation time
- How far away is the restaurant
- How many stop signs / stop lights are there and how heavy is traffic
- What is the speed limit on the streets
- Does your car have enough gasoline to get to the restaurant
Let’s say your reservation time is for 6pm and now it is 5.35pm. You realize you will not be able to calculate an exact arrival time because there are some unknown factors—especially how heavy traffic is and how often you’ll have to stop at stop lights. Instead of trying to pick a point estimate of your arrival time, you decide to calculate the upper and lower bounds of a range of time over which you may arrive.
After assessing the input factors, let’s say your estimated arrival time range looks something like this:
In other words, you think that your best chance of arrival is the 15-minute range between 5:50pm and 6.05pm. If traffic is light, you’ll make it toward the beginning of that interval; if it is heavy, you’ll make it toward the end of that interval or may not make it at all. How willing would you be to bet on making it in time? How much would be a fair amount to bet?
This example illustrates precisely the process on which the Black-Scholes Model (“BSM”) and all other statistically-based option pricing formulas work.
The BSM has a fixed number of inputs regarding the underlying asset and the contract itself. Inputting these variables into the BSM generates a range of likely future values for the price of the underlying security and for the statistical probability of the security reaching each price.
The statistical probability of the security reaching a certain price (that certain price being a strike price at which we are interested in buying or selling an option) is directly tied to the value of the option.”
The math behind the BSM looks intimidating, but in fact, it is a pretty simple model of stock price movement based on what many economists considered the most valid theory of markets at the time the BSM was developed – the Efficient Market Hypothesis. (In my book, I walk through each of the assumptions that ties back to the example of the dinner reservation).
The model predicts a statistically most likely price, then builds a range around that most likely price based on an expectation of how much the stock’s price is likely to fluctuate in the future.
The price range projection made by the BSM theoretically looks like this:
The dotted line in the image above is the most likely price (the chance that the price will be above or below this line is 50/50) and the cone-shaped curve shows the range that the stock will probably vary (about a 2-in-3 chance).
An option is just a financial instrument that allows an investor access to a “range of exposure”. Here is a graphical representation of an investment in the purchase of a call option:
If the stock moves into that range of exposure, the investor stands to realize a gain.
The BSM simply overlays the pricing model (the cone-shaped diagram) on the range of exposure. The more likely the stock is to move into the range of exposure, the more expensive the option.
This call option (an At-the-Money “ATM” option):
Will be more expensive than this call option (an Out-of-the-Money “OTM” option):
Simply because the stock price is more likely to move into the ATM range of exposure.
The movement of stock prices that underlies the BSM are very simple. The concept of options as ranges of exposure is very simple. Overlaying the former on the latter is really simple. It follows that option pricing is really simple!
One of the most wonderful things about the BSM is that when you understand how to read a pricing screen, you can tell exactly what the market’s expectations for the future price of a stock are. Even if you never buy or sell an option, this knowledge is valuable.