Calculus is our co-pilot

Updated: Mar 19, 2021

We love math because it predicts the future. It's beautiful stuff. We use our statistician/psychometrician/physics/patent law background to analyze quantitative methods for modelling options pricing.

We wrote an article on Seeking Alpha about how we pick some of our options based on calculus and gambling. Yep, calculus and gambling. We know, we know... calm down. It's very exciting. We thought you might find it interesting. Check it out:

We found that basic principles of gambling provide a good framework for thinking about the options market and options prices. When considering whether to make a bet, which is what options trading is, let's start at first principles. For example, someone offers to flip a coin with you. They'll pay you $1 if it's heads and you pay $2 if it's tails. Do you play? Good grief, NO. You would never play that game. Why? Because your EXPECTED VALUE is negative. Expected Value in gambling means "how much would I win on average if I played, like, a gazillion times." If you played this coin flip game a gazillion times, your average return, or Expected Value, is easy to calculate. It's simply the chance of winning times your win, minus the chance of losing times the loss.

50% * $1 - 50% * $2 = -$0.50

So, if you played this game 10,000 times, odds are you would lose about $5000.

If you played the game with even money where you win $1 for heads but lose $1 for tails, then your Expected Value is, of course, zero. In the long run, after a gazillion flips, you generally break even. (For those of you who are thinking about root expansion of random walks, we can talk about that later).

This principle can be expanded beyond coins. Say you have a six-sided die. You win $6 if it lands on 1 and you lose $1 if it lands on 2, 3, 4, 5, or 6. Do you play? You only win one out of six times, so it's not so clear. The Expected Value tells you if it's a good game for you:

1/6 * $6 - 5/6 * $1 = $0.1666.... Your Expected Value is positive, so you play this game all day long. If you play this game 10,000 times, you can expect to win about $1,600. Give or take.

The Expected Value can be done for more complex situations. What happens when you have a 20 sided die and you're offered this game: you win $8 for a 13 or an 11, $2 for any even number, and you lose $4 for all the rest. It's getting messy, right, but you know if you sat down you could figure this out. The answer is: you play. The Expected Value is $0.20. On average you win $0.20 per game.

Ok ok ok, so what's this got to do with options? Lots. We can make a probability model that says "here is the chance the option price hits $x by expiration." And we know what the gain/loss will be for an option at expiration, given the price: it's simple the stock price minus the strike price. So we can calculate the Expected Value by just adding up all the prices times their probabilities... -- wait. What's that you say?

"But guys, there's basically an infinite number of prices."

Oh yeah, we say. If only we had a tool to add up an infinite number of things...?


Check out the blog post for more!

Oh, and if you read this far, bless you. You rock.

Stay tuned and happy trading. Thanks for joining the NGT family!

Marc and Laura

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