When I was young, the pricing of stock options and other derivatives seemed like an obscure black art. Then one day Don Brown showed me a simple example that made everything crystal clear. Today I’ll share an even simpler version of Don’s example.
Imagine a stock that sells for $10 today. A year from now it will be worth either $20 or $5. (Yes, I know that real-world stocks have a wider range of possible future prices. That’s why I called this a simple example.) What would you pay for an option that allows you to buy the stock next year at today’s $10 price?
You might think you’d need a whole lot more information to answer that question. You might expect, for example, that the answer depends on the probability that the stock price will go up to $20 rather than down to $5. You might expect the answer to depend on how much traders are willing to pay for a given dollop of risk-avoidance.
But the amazing fact is that none of that matters. The only extra bit of information you need is the interest rate.
Let’s assume, for example, that the interest rate happens to be 25%. (Yes, I know that’s unrealistic.)
Now let’s price the option. The key is to focus on my imaginary cousin Jeter, who never buys stock options. Jeter happened to wake up with $12 in his pocket today. Then he went out, borrowed $8, and used his $20 to buy 2 shares of stock.
A year from now, one of two things will happen. Either Jeter will get lucky, sell his 2 shares for $40, use $10 to repay his debt ($8 plus $2 interest), and pocket $30. Or he’ll get unlucky, sell his 2 shares for $10, use that $10 to repay his debt, and pocket $0.
Edit: Thanks to those who caught the typos in the above paragraph. I think they’re fixed now.
I, on the other hand, bought 3 stock options today. A year from now, one of two things will happen. Either I will get lucky and use my 3 options to buy 3 shares of $20 stock at a price of $10 each, pocketing a $30 profit. Or I will get unlucky and the stock price will plumment, in which case I will throw my option away and pocket $0.
In other words, Jeter and I are guaranteed exactly the same outcome next year. Either the stock price goes up, and we each pocket $30, or it goes down and we each pocket $0. In that strong sense, Jeter’s strategy and mine are perfectly interchangeable.
Jeter’s strategy costs him $12 out of pocket. Therefore my strategy must also cost $12 out of pocket — otherwise, nobody would ever pursue the pricier strategy. Since buying 3 stock options costs $12, the price of a single option must be $4. Problem solved.
Of course the real world presents far trickier scenarios, with stocks that can go up or down by any amount at any instant, and options that can be exercised at the time of your choice. But the trick for pricing these options is always the same: First invent an imaginary cousin Jeter who buys and sells only assets whose prices are already known. Devise a strategy for Jeter that mimics the value of the option under every conceivable circumstance. Devising that strategy can be a difficult technical exercise, but it’s not impossible. Now figure out what that strategy costs, and you’ve got the price of your option.
There are practical pitfalls galore. For starters, how do you account for the cost of trading, or for the fact that some of us have access to better interest rates than others? That’s why there’s still work to be done in this area. But ultimately, there’s just one big idea, and if you grasp this simple example, you’ve mastered it.