Pricing an Option Part 1

Today, I'm going to talk about how to price options. I'm not going to do a deep dive but I will look at the different variables that go into pricing an option. In the past few posts, I mentioned we pay 2 or 3 for the option to buy/sell, how do we know what a fair price is?

Pricing an object, like a cup or a book, is relatively simple. There is the cost of creating the object (whether it is manufacture costs or labor costs) and a margin so the creator and seller can make a small profit. So, if it costs me $5 worth of materials to make a single shirt and it took me 3 hours of laboring (say I value my labor at $10/hour), then the total cost of creating a shirt is $35. I tack on some margin (say $7) and sell the shirt at $42. (This is oversimplified but the point is, it is easier to price a concrete object as opposed to a random event).

Before we get into pricing an option, we have to talk about how to fairly price something when the outcome is unknown (like an option, which, in essence, is a bet). We need to understand the concept of expected value.

What is expected value? The expected value is a probability-weighted average of all possible values. More intuitively, it is the long-run average value of repetitions of the experiment.

Let me share with you a simple interview questions often asked by trading firms to see if you understand the concept of expected value:

You and I are going to play a game of chance. You get to roll a fair six-sided die. Whatever result you get, I will give you that amount in dollars. For instance, you roll a 5, I will give you $5. How much are you willing to pay each time to play this game?

The real question that is being asked is: "What is the expected value of a fair die roll?" Or, in other words, if you were to roll this die infinitely many times, what would the average roll be?

Well, you have 1/6 chance of rolling each number, so the expected value is 3.5, as shown by the equation below:

1-pricing

It is worth noting that the expected value doesn't necessarily have to be one of the outcomes.

So, let's get back to the original question: "How much are you willing to pay each time to play this game?" Well, we now know that on average, the player will be earning $3.50 on each roll. So if you pay anything below $3.50 (say $3.49), you would eventually come out on top. The fair price of the game would be $3.50.

Let's look at another (slightly more complicated) interview question:

You and I are going to play another game. Much like the last game, you get to roll a die and you will get paid the amount you rolled. If you like your roll, that will be the payout, but if you don't like your first roll, you get to roll once more and have to keep the result. What is the fair price of this game?

So, we already know that the expected value of a die roll is 3.5, which means you would keep the first roll if it rolled 4, 5, or 6 (50% chance of happening and each roll (4, 5, 6) would have 1/3 chance of happening). In the other 50% (where you roll a 1, 2, or 3), you would roll the die again (which we already know has an expected value of 3.5). So, we can write this equation:

2-pricing

to arrive at the answer of 4.25