This part covers the mechanics of selling and why, mathematically, selling options tends to be profitable.
Options are priced using implied volatility - the market’s estimate of future stock movement. Here’s the key fact: IV is systematically higher than realized volatility.
Why? Options are insurance. Buyers pay a premium for protection against large moves. Sellers demand compensation for taking on tail risk. That compensation - the gap between implied and realized volatility - is the volatility risk premium.
Empirically, the VIX (S&P 500 implied volatility) exceeds subsequent realized volatility about 85% of the time. The average spread is about four volatility points.
This means selling options has positive expected value. The risk premium compensates sellers for absorbing losses during market crashes, earnings disasters, and black swan events.
The math: collect small premiums frequently, pay large claims rarely. On average, the premiums exceed the claims.
This gets called “picking up pennies in front of a steamroller.” The metaphor overstates it - you’re not picking up pennies, you’re running an insurance business. Insurance companies are profitable. They just occasionally pay out large claims.
But the metaphor captures something real: the P&L distribution is asymmetric. Months of steady gains, then one bad week erases them. You can be “right” 90% of the time and still lose money if the 10% hits hard enough.
A cash-secured put (CSP): sell a put, hold cash to buy shares if assigned.
WidgetCo at $50. Sell a $45 put, 30 DTE, collect $1.50 ($150 per contract).
The obligation: buy 100 shares at $45 if assigned. Need $4,500 cash to cover.
Drag the stock price. Above $45: keep $150. At $43.50: breakeven. Below: losing money.
Maximum gain: $150. Maximum loss: $4,350 (stock to zero, minus premium).
Stock drops to $44 at expiration. The put is exercised. You now own 100 shares at $45 cost basis. Time to sell calls.
A covered call (CC): own shares, sell a call against them.
You own 100 shares at $45. Stock at $44. Sell a $50 call, 30 DTE, collect $0.80 ($80).
The obligation: sell shares at $50 if called away.
Drag the stock price. Below $50: keep shares and premium. Above $50: called away, miss upside beyond strike.
Stock drifts to $47. Call expires worthless. Collect $80, still own shares.
What if you just kept selling options no matter what happens?
Sell a put. If it expires worthless, keep the premium, sell another. If you get assigned, you now own shares - so sell calls against them. If the call expires worthless, sell another. If you get called away, you’re back to cash - sell puts again.
That’s the wheel. CSP → assigned → CC → called away → repeat.
The 45-day expiration comes from Part 4: theta accelerates near expiration, but so does gamma. TastyTrade’s research found 45 DTE balances premium collection against the risk of sudden moves.
Try it. Sell puts until assigned, then sell calls until called away.
Run the wheel long enough and you’ll get called away right before a rally. Stock gaps up, your shares are gone at the strike price, and you watch the stock climb without you.
That’s the trade-off. You collected premium along the way. But when the stock doubles, you don’t double with it.
The wheel works when realized volatility stays below implied - when the market overestimates how much stocks will move. Historically, that’s most of the time. The strategy fits investors who want income over growth, can tolerate holding individual stocks through drawdowns, and have the capital to secure puts.
Retirees sometimes run wheels on blue chips for income. It’s active management with real risks, but the math favors sellers. If that sounds like work, JPMorgan’s JEPI and JEPQ run covered call strategies on the S&P 500 and Nasdaq respectively. You get the volatility risk premium without the position management.
This series was about the math. Options are derivatives - their prices derive from stock prices, time, and volatility through functions we can write down. The Greeks measure sensitivity to each input. Buyers pay for movement. Sellers collect premium for absorbing risk.
Whether you trade options is a different question. But now you know what you’re looking at.
This is Part 5 of a 5-part series on options. Part 4 covers the Greeks. Part 1 is where it started.
]]>It depends. On where the stock started. On how much time is left. On whether volatility spiked with the drop. On how far away your strike is. The answer is a function of all these variables at once.
The Greeks are tools for navigating this. But to understand what they’re actually measuring, we need to talk about the surface your option lives on.
An option’s price depends on several inputs: stock price, time to expiration, implied volatility. Strike is fixed when you buy. That leaves three moving variables - which means the full picture is four-dimensional (three inputs plus price). There’s no way to draw that.
But we can take lower dimensional slices. Hold IV constant and look at how price changes with stock and time. That’s a surface:
Drag to rotate. This is a $50 call with IV fixed at 45% - one slice of the larger space.
Hold time constant too, and we get a 2d curve:
Same option, but now we’re only watching how price changes with stock. We’ll use both views throughout.
Back to the surface: see those colored dots with arrows? Each dot is an option at a different position. The arrows show which way you’re exposed - how the option responds when things change, or when your option moves across the surface.
Those arrows are the Greeks. They work in the full space too, pointing in directions we can’t draw.
Delta measures how much your option moves when the stock moves $1. It’s the most immediate question: if WidgetCo goes up $1 tomorrow, how much do I make?
You buy a $50 call, 30 days to expiration. It costs $2.85. Delta is 0.52.
If WidgetCo goes from $50 to $51, your call gains roughly $0.52. If WidgetCo drops from $50 to $49, your call loses roughly $0.52.
Drag the stock price. Watch how the blue tangent line (delta) changes slope as you move.
Delta ranges from 0 to 1 for calls. Deep out of the money (stock way below strike), delta is near 0. At the money (stock at strike), delta is around 0.5. Deep in the money (stock way above strike), delta approaches 1 - moving almost 1:1 with the stock.
For puts, delta is negative - ranging from -1 to 0. A put with delta -0.5 gains $0.50 when the stock drops $1.
Another way to think about it: delta is how much stock you effectively own. A 0.52 delta call on 100 shares behaves like owning 52 shares. You paid $285, not $5,000 - the option amplifies your capital while exposing you to 52 shares worth of movement.
Every day, your option loses value. That’s theta.
Your $50 call has theta of -0.08. You’re losing $0.08 per share per day - $8 per contract - just by holding it.
Why? An option is a bet that something will happen before a deadline. The value of that bet depends on how much time is left for it to play out. With 30 days, anything could happen. With 1 day, much less likely.
Drag the time slider. Watch how the red tangent line (theta) steepens as expiration approaches.
Theta accelerates near expiration. With 30 days left, you lose $8/day. With 7 days left, you might lose $15/day. At 30 days, losing one day costs you 1/30th of your remaining time. At 7 days, losing one day costs you 1/7th. Same absolute time, bigger percentage hit.
Here’s what theta feels like in practice: You buy a call. The stock does nothing for a week. You check your position. It’s down 5%.
Delta isn’t constant. It changes as the stock moves.
That’s gamma - how fast delta changes per $1 stock move.
Your $50 call has delta 0.52 and gamma 0.04.
If WidgetCo goes from $50 to $51:
If it continues to $52:
Move the stock price. Watch how the purple tangent line (gamma) shows whether delta is accelerating or decelerating.
Why care? Gamma tells you how unstable your exposure is. High gamma means your delta can flip quickly. At $50, your call has delta 0.52 - you’re moderately exposed. But gamma is high. If the stock moves $5, your delta might be 0.75 or 0.25 depending on direction.
Low gamma means stability. Deep ITM, delta is 0.95 and gamma is tiny. The stock can move $5 and delta barely changes - you’re locked in at near-1:1 exposure.
Vega measures how much your option moves when IV changes by 1 percentage point.
Why does IV matter? Remember: IV is the market’s estimate of how much the stock might move. Higher IV means bigger expected swings. Options are bets on movement - if bigger moves are expected, your option is worth more. When IV drops, the market is saying “actually, calmer than we thought” - and your option loses value because the big move you were betting on now seems less likely.
Your $50 call has vega of 0.08. If IV goes from 45% to 46%, your call gains $0.08 per share - $8 per contract.
Drag the IV slider. Watch how the green tangent line (vega) shows sensitivity to volatility changes.
This is what makes earnings plays tricky. Before earnings, IV is high - uncertainty is priced in. After earnings, uncertainty resolves, IV collapses. You can be right about direction and still lose because IV dropped faster than the stock moved.
Those arrows from the first surface:
Now you can read them.
Each point has two arrows: blue (delta) points along stock price, red dashed (theta) points toward expiration.
Calm waters (top left) - ATM, 45 days out. Both arrows moderate length. You have time, you have exposure.
The cliff (bottom left) - ATM, 5 days out. The red theta arrow is longer - time decay is accelerating. The blue delta arrow is similar - stock sensitivity hasn’t changed much.
Deep ITM (right side) - $12 above strike, 30 days out. The blue delta arrow is long - this option moves nearly 1:1 with stock.
The Greeks don’t operate in isolation. On any given day, all of them are working simultaneously.
Drag the point around. Watch delta and theta change as you move across the surface. Near expiration (the cliff), theta accelerates. Deep in the money, delta approaches 1. At the money with time left, you’re in the sweet spot - moderate exposure to everything.
| Greek | Measures | Range |
|---|---|---|
| Delta | Stock price sensitivity | 0 to 1 (calls), -1 to 0 (puts) |
| Gamma | Delta’s rate of change | Always positive |
| Theta | Time decay per day | Negative for long options |
| Vega | IV sensitivity | Always positive |
When you’re long options: delta is your directional bet, gamma means big moves help you, theta works against you, vega helps on IV spikes.
When you’re short options, the signs flip. Theta works for you. Vega works against you. That’s Part 5.
The Greeks are local. They tell you what happens for small moves right now. They don’t tell you which direction the stock moves, what IV will do, or how big the move will be.
You can have perfect Greek management and still lose money because the stock went the wrong way.
Part 5: you stop buying options and start selling them.
This is Part 4 of a 5-part series on options. Part 3 covers volatility. Part 5 covers selling options.
]]>Real markets are more complicated. WidgetCo is about to have a bad quarter.
Volatility measures how much a stock moves. But there are two different measurements, and understanding the gap between them is half of options trading.
Historical volatility (HV) looks backward. Take the last 30 days of prices. Calculate how much the stock moved day-to-day. Annualize it. That’s HV.
WidgetCo’s HV is about 22%. Over the past month, the stock has been moving at a 22% annualized rate. Not wild, not dead. Normal industrial company stuff.
Implied volatility (IV) looks forward. Or rather, it’s what the market implies about the future by how it prices options.
To understand IV, we need to talk about pricing models.
Physicists have a joke about simplifying a cow to a sphere to make the math tractable. The spherical cow isn’t accurate, but it’s useful.
Black-Scholes is the spherical cow of options pricing. It assumes stock prices move smoothly in a random walk, volatility stays constant, you can trade continuously with no fees, and a bunch of other things that aren’t true. The model won a Nobel Prize anyway, because it’s useful.
More sophisticated models exist: binomial trees that let prices jump discretely, Monte Carlo simulations that handle path-dependent options, stochastic volatility models that let volatility itself fluctuate. Those are beyond scope here.
For this series, we’re trying to learn to drive, not build the engine. I implemented Black-Scholes for the interactive widgets on this page. It’s a black box: stock price, strike, time to expiration, interest rate, volatility in. Option price out. Good enough to illustrate the concepts.
The trick: we already know the option’s price - it’s whatever the market is trading at. Stock price, strike, expiration, interest rate - all known. Only one unknown: volatility.
So, given all these caveats, if we run my flawed black box in reverse, we get a value for volatility. That’s the (according-to-my-flawed-algorithm) implied volatility - what the market implies about future movement by how it prices options today.
Log returns are how HV and IV are actually expressed mathematically. Understanding them helped the concept click for me.
Why log returns instead of regular percent changes?
Simple return: stock goes from $100 to $110, that’s +10%. Goes from $110 to $100, that’s -9.1%. You’re back where you started, but +10% and -9.1% don’t cancel.
Log return: \(\ln(110/100) = +9.53%\). \(\ln(100/110) = -9.53%\). They cancel exactly. Log returns are symmetric and additive - they compound correctly over time. This is why volatility is measured in log space and why pricing models assume log-normal price distributions.
The formula for historical volatility:
$$\sigma = \sqrt{\frac{252}{n} \sum_{i=1}^{n} \left( \ln \frac{P_i}{P_{i-1}} \right)^2}$$
Take each day’s log return, square it, average them, take the square root. The 252 is trading days per year - it annualizes the daily volatility.
What does this number actually mean? An IV of 20% says the market expects the stock’s annual log returns to have a standard deviation of 20%. If returns are roughly normal:
For daily movement, divide by \(\sqrt{252} \approx 16\). A 20% annual volatility means roughly 1.25% expected daily movement. WidgetCo at 22% HV moves about 1.4% on a typical day.
When you see “IV: 25%” on Yahoo Finance or Robinhood, that’s this number - annualized standard deviation of log returns.
WidgetCo’s IV is 25%. The market expects slightly more movement than has been happening. Maybe the new PE ownership adds uncertainty.
When IV > HV, options are “expensive” - you’re paying for more movement than has occurred.
When IV < HV, options are “cheap” - you’re paying for less movement than has occurred.
Neither tells you which way the stock moves.
Let’s see what happens when volatility changes.
It’s Q3 2020. WidgetCo’s new manufacturing line has been running for six months. The PE firm pushed an aggressive timeline - 18 months to double capacity. Quality control got squeezed.
News breaks on a Tuesday morning: WidgetCo recalling the Widget Pro line. Stress fractures. Twelve injuries reported, no deaths. Lawsuits pending.
Stock opens at $102. By lunch, $88. By close, $79. Over the next week, it bottoms at $65.
Before the recall: IV at 25%. During the crash: spikes to 65%. A week later: settles around 55%.
The stock dropped 36%. IV nearly tripled.
Uncertainty. Will there be more recalls? How big are the lawsuits? Could the company go bankrupt? Nobody knows.
Options are insurance. That uncertainty gets priced into them the same way hurricane season gets priced into Florida homeowners policies.
Say you held a $100 call from Part 2. You bought it for $4.20 when the stock was at $102.
By close, the stock is at $79. Your $100 call is now out of the money - if expiration were today, it would be worthless. But expiration is 30 days away.
Drag the slider to watch IV rise as the stock falls. Two lines: orange shows actual value (IV spiking), gray shows hypothetical value (IV constant at 25%).
At $79, the stock needs to rally 27% to reach your $100 strike. With 25% IV, the market expects about 25% movement per year. A 27% move in 30 days? The math says nearly impossible. Your call is worthless.
But IV isn’t 25% anymore. It’s 65%. The market is saying: we have no idea what happens next. Lawsuits, more recalls, bankruptcy, or maybe a surprise recovery. With that much uncertainty, a 27% swing isn’t crazy. Your call is worth $0.84.
That’s what IV is. It’s the market’s estimate of how much the stock might move. Higher IV means bigger expected swings.
Q1 2021. WidgetCo stabilizes around $52. The lawsuits are pending. Management is “restructuring.” Everyone’s waiting for the next earnings call.
IV spikes to 85% in the days before earnings. Why? Because nobody knows what will be announced. Revenue up or down? More recalls? Bankruptcy? A surprise turnaround? The stock could go anywhere. That uncertainty is what IV measures.
You buy a $55 call for $3.20. The option is expensive because the market expects big moves.
Earnings drop. Stock goes from $52 to $56. You were right about direction.
Post-earnings: Your $55 call is worth $2.10. You lost $1.10.
IV collapsed from 85% to 40%. The earnings announcement resolved the uncertainty. Now everyone knows: revenue down, restructuring plan in place, lawsuits ongoing. The range of possible futures just narrowed. The market no longer expects 85% annualized movement - more like 40%.
Your option was priced for a world of 85% volatility. Now it’s repriced for 40%. The stock moved $4 in your favor, but the expected-movement component of your option’s value shrank faster than the stock-moved-toward-strike component grew.
IV isn’t one number. Every option has its own IV, depending on the strike price.
Drag the slider to see IV at different strikes.
The curve isn’t flat - it’s a smile. Out-of-the-money puts (left side) have higher IV than at-the-money options (center). That’s skew.
Think about what a $35 put actually insures against. WidgetCo at $52 falling to $35 - that’s a 33% crash. In a world where that happens, it’s making headlines. The company is in crisis. Volatility spikes. We watched exactly this during the recall.
If you’re selling that insurance, you price it accordingly. Extreme scenarios cost more. When we run Black-Scholes backward on these pricier contracts, we get higher implied volatility.
2021-2023 in the simulator. WidgetCo stuck between $45-60. Three CEOs. Constant uncertainty.
No cooperative mode.
A few things to notice as you trade:
IV stays elevated. Options are perpetually expensive. The premium you pay reflects real uncertainty about this company’s future.
Every earnings is an IV collapse. Before earnings: IV spikes. After earnings: IV collapses. Regardless of which way the stock moves.
News gaps aren’t tradeable after the fact. New CEO announced: stock gaps 15% overnight. If you weren’t positioned before, you missed it. After the gap, IV collapses - the news is priced in.
When you buy an option, you’re not just betting on direction.
The option’s value depends on IV. If you buy when IV is high and it drops, you lose money even if the stock moves your way. If you buy when IV is low and it spikes, you make money even if the stock barely moves.
Part 2 mentioned that delta isn’t the whole story. Now you see why. The stock can go up and your call can go down - if IV falls faster than the stock rises.
Part 4 formalizes all of this. Delta measures your exposure to stock movement. Vega measures your exposure to IV. Theta measures what time costs you. They’re all partial derivatives of the option price. But you can’t understand what they’re measuring until you understand volatility.
This is Part 3 of a 5-part series on options. Part 2 covers what options are. Part 4 covers the Greeks.
]]>Here’s how Wikipedia defines it:
In finance, an option is a contract which conveys to its owner, the holder, the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified date.
One phrase at a time:
“The right, but not the obligation” - You can do something, but you don’t have to. If exercising the option would lose you money, you just… don’t. You walk away.
“To buy or sell” - There are two flavors. The right to buy is called a call. The right to sell is called a put.
“At a specified strike price” - The price is locked in when you buy the option. Doesn’t matter what happens to the market afterward.
“On or before a specified date” - Options expire. After that date, they’re worthless. (We’ll use “American-style” options, which can be exercised anytime before expiration.)
The catch? You pay for this right upfront. That payment is called the premium. If you never use the option, you lose the premium.
We need a stock to trade. Enter WidgetCo (ticker: WGT).
(“Widget” is a placeholder name economists use for a generic product. WidgetCo is our fictional company - a bland, stable industrial manufacturer. Think of it as “Generic Corp.”)
WidgetCo was founded in 1952 by Earl Widget in Akron, Ohio. Three generations of the Widget family ran it until 2019, when the last heir sold to a private equity consortium. Revenue: $2.3 billion. Employees: 12,000.
Stock’s at $100.
A call is a contract. It says: I have the right to buy this stock at a fixed price, no matter what happens.
WidgetCo is at $100. You buy a $100 call expiring in 30 days. It costs you $4.
That $100 is the strike price. It’s locked in. Doesn’t matter what happens next.
Say WidgetCo announces a breakthrough widget. Stock jumps to $150. Your call says: “I can buy at $100.” So you do. You buy at $100, sell at $150, pocket $50 per share. Minus the $4 premium, that’s $46 profit per share.
Or say WidgetCo’s factory burns down. Stock drops to $60. Your call still says “I can buy at $100” - but why would you? You wouldn’t. The contract expires unused. You’re out the $4 you paid, nothing more.
That $4 is the premium. It’s what you pay for the right. If you never use the right, you lose the premium. If you use it, your profit is whatever you made minus the premium.
A note on pricing: Options are quoted per share, but each contract controls 100 shares. So a “$4 option” actually costs $400 to buy. When you exercised at $100 and sold at $150, you made $50 per share - but that’s $5,000 total (100 × $50), minus the $400 you paid, for $4,600 profit.
This series quotes prices per share because that’s how you’ll see them displayed. Just remember: multiply by 100 for the actual cash changing hands.
Drag the slider to see how your call’s value at expiration depends on the stock price:
Below $100: flat at -$4. You lost the premium, nothing more. Above $100: slopes upward. You’re making money.
A put is the mirror image. It says: I have the right to sell this stock at a fixed price, no matter what happens.
WidgetCo is at $100. You buy a $100 put expiring in 30 days. Costs $3.50.
Say WidgetCo’s CEO gets arrested for fraud. Stock crashes to $50. Your put says: “I can sell at $100.” So you buy shares at $50 on the open market, then immediately sell them at $100 using your put. That’s $50 profit, minus the $3.50 premium = $46.50.
Or say WidgetCo announces record earnings. Stock jumps to $130. Your put says “I can sell at $100” - but the market will pay $130. Why sell at $100? You wouldn’t. Contract expires unused. You’re out $3.50.
Above $100: flat at -$3.50. Below $100: slopes upward.
Time to play. The simulator below lets you buy calls and puts on WidgetCo, then fast-forward time to see what happens.
The stock is being nice to you right now. Cooperative mode: it tends to drift in whatever direction makes your positions profitable. This is practice.
Try this:
Now try a put:
Play around. Get a feel for it.
As you played, you might have noticed: the stock moved $8, but your option didn’t move $8. It moved less.
That’s delta - the ratio of option movement to stock movement.
Your call has delta of about +0.5. That means for every $1 the stock moves up, your call gains about $0.50. Your put has delta of about -0.5 - it gains $0.50 for every $1 the stock moves down.
We’ll formalize delta in Part 4. For now, just notice: options amplify your exposure (you paid $4 to control $100 of stock) but they don’t move 1:1.
When the stock is at $100 and your strike is $100:
But watch what happens as the stock moves.
Stock rises to $110:
Stock falls to $90:
This is the second derivative from Part 1. Delta itself has a rate of change. That’s called gamma. We’ll cover it in Part 4.
If options were perfectly priced, directional betting would break even. The $4 premium already reflects the market’s probability estimate. Over enough trades, wins and losses balance out.
So what are you paying for?
Exposure to volatility - whether the stock moves more or less than the market expects.
The option expires eventually. Every day that passes, it loses value. The right to buy at $100 is worth something today. Worth less tomorrow. Worth nothing after expiration.
And if the stock might move a lot, options are worth more. Dead calm? Worth less.
When you buy options, you’re betting the stock moves more than expected. When you sell them (Part 5), you’re betting it moves less.
Part 3 covers volatility - the key input to option pricing.
This is Part 2 of a 5-part series on options. Part 1 covers background. Part 3 covers volatility.
]]>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.
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.
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.
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.
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?
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.
The derivative at a point = the slope of the curve at that point.
Different notation, same idea:
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 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:
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.
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.
I use LLMs to help write this. If something reads like an AI wrote it, that’s a failure. Let me know: journal@ehrlich.dev.
First up: people sometimes ask about options trading when it comes up that I do it for fun. The answer takes a while, so I wrote it down. Five parts, starting with what a derivative is.
]]>