Options Primer Part 1: Background
This is part one of a five-part series on options. We’re building toward understanding how options are priced and how to trade them. But first, two pieces of background: what stocks are, and what derivatives are.
Not financial derivatives - calculus derivatives. Rates of change. The options world borrowed the word “derivative” from math because options prices are derived from stock prices. To understand how that derivation works, you need to understand what a derivative measures.
Skip ahead: If you already know what stocks are and you’re comfortable with calculus (slopes of curves, rates of change), skip to Part 2.
What’s a stock?
A stock is a piece of ownership in a company.
WidgetCo has 1 million shares outstanding. If you own 1,000 shares, you own 0.1% of WidgetCo. If WidgetCo is worth $100 million total, your 0.1% stake is worth $100,000. That’s $100 per share.
The stock price is just: (what the market thinks the company is worth) / (number of shares).
When people say “WidgetCo is at $100,” they mean one share costs $100. If the company does well - makes money, grows, announces good news - people want to own it. Demand goes up, price goes up. If the company does poorly, people sell, price goes down.
A stock is a slice of a company. The price moves based on what people think that slice is worth.
The math we need
Options are priced using rates of change. How fast does the option price change when the stock price moves? How fast does it decay as time passes? These are the “Greeks” you might have heard of - delta, gamma, theta, vega. They’re all measuring rates of change.
To understand them, we need to understand one concept from calculus: the derivative. Not the financial kind - the mathematical kind.
Slope of a line
Start simple. A straight line on a graph.
The slope measures steepness: rise over run. How much the line goes up (or down) for each unit it moves across.
Drag the two points below. Watch the slope update.
A steep line has a large slope. A flat line has slope near zero. A line going downward has negative slope.
For a straight line, the slope is the same everywhere. Pick any two points, calculate rise/run, you get the same number.
Curves don’t have constant slope
Now consider a curve. The parabola y = x², for instance.
This curve bends. It’s steep in some places, flat in others. At x = -2, it’s diving down. At x = 0, it’s flat - the bottom of the U. At x = 2, it’s climbing up.
There isn’t one slope. The steepness changes depending on where you are.
So we ask a different question: what’s the slope at a specific point?
Zoom in until it looks straight
Here’s the key insight. If you zoom in far enough on any smooth curve, it starts to look like a straight line.
That local straightness is what we measure. The line that just barely touches the curve at one point - the tangent line - has a slope. That slope is the derivative at that point.
Drag the point along the curve. Watch the tangent line rotate. The slope display shows the derivative at each position.
At x = 0, the tangent is flat. Slope is 0. At x = 1, the tangent tilts up. Slope is 2. At x = -1, the tangent tilts down. Slope is -2.
The derivative of y = x² is 2x. Plug in any x value, get the slope at that point.
That’s a derivative
The derivative at a point = the slope of the curve at that point.
Different notation, same idea:
- f’(x) - “f prime of x”
- dy/dx - “the derivative of y with respect to x”
- The slope of the tangent line
The derivative tells you: how fast is y changing when x changes? If the derivative is large and positive, y is climbing steeply. If it’s small and negative, y is drifting slowly downward. If it’s zero, y is momentarily flat.
The derivative as its own curve
The derivative at each point is just a number. If we plot all those numbers, we get a new curve: the derivative function.
Below, the top graph shows y = x³ with its tangent line (red). The bottom graph shows f’(x) = 3x² - the derivative curve. The orange line is the tangent to that curve - the second derivative.
Drag either point. Three things update together:
- Red line: slope of the cubic (first derivative, f’(x) = 3x²)
- Green dot: your position on the derivative curve
- Orange line: slope of the derivative curve (second derivative, f’’(x) = 6x)
At x = 0, the red tangent is flat (slope = 0) and the orange line is also flat - the derivative curve has a minimum there. Move right: the orange line tilts up, meaning the slope is increasing. Move left: the orange line tilts down, meaning the slope is increasing in the other direction (becoming less negative).
The second derivative tells you how the slope is changing. In options, delta (first derivative) measures price sensitivity. Gamma (second derivative) measures how fast that sensitivity changes - crucial for understanding risk.
Coming up
Part 2 introduces options through a company called WidgetCo. You’ll buy calls and puts in a trading simulator and watch your positions gain and lose value.
Part 3 covers volatility - the uncertainty that determines what options are worth.
Part 4 formalizes the Greeks. They’re derivatives applied to option prices. Delta measures sensitivity to stock price. Gamma measures how delta changes. Theta measures sensitivity to time. Vega measures sensitivity to volatility.
Part 5: you stop buying options and start selling them. You become the insurance company.
The math you just learned - slopes, rates of change, derivatives - that’s the foundation. Everything else builds on it.