Options Primer Part 3: Volatility
The trading simulator in Part 2 was rigged. The stock drifted whichever way made your positions profitable.
Real markets are more complicated. WidgetCo is about to have a bad quarter.
Two kinds of volatility
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.
The spherical cow
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
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:
- ~68% of the time, the stock ends the year within ±20% of where it started
- ~95% of the time, within ±40%
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.
Back to WidgetCo
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.
The recall
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.
Your call during the crash
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.
Volatility collapse
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.
The volatility smile
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.
The dark years
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.
What this means
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.