On Monday, I noticed that Robinhood ($HOOD) vol screened cheap in the Trade Ideas tool. But that tool uses 30-day constant maturity IV. Since HOOD earnings was just about 30 days out on Monday, the interpolation gave the earnings vol no weight. The pre-earnings vol is in the low 50s, which is, indeed at the bottom of the range for HOOD implied vol.

I looked at the vol that includes earnings.

HOOD reports earnings on February 10th. The February 13th expiry is currently priced at 64% ATM implied volatility.

At first glance, 64% might seem elevated but let’s decompose what the market is actually pricing. When a known event, like earnings, falls within an option’s expiry, the market assigns extra volatility to that expiration. But how much of that IV comes from the event itself versus normal trading day volatility?

I’m gonna lay out the numbers and then get to the process.

• Expected earnings move or straddle: 8.55%

• Event volatility (earnings day): 10.72% single-day vol (169.8% annualized)

Why?

The ATF straddle approximation tells us that a straddle ~ .8 x vol

Well, if we assume the earnings straddle is 8.55% then we just divide that by .80 (or multiply by 1.25 which is the arithmetic burned into trader brain) to get 10.69%

• Trading day volatility (pre-earnings): 54% annualized = 3.41% per day

Where do these numbers come from?

Let’s start with the earnings straddle…why 8.55%?

Here’s a handy secret. A good first guess what the market’s estimate for an earnings moves is the mean move size of the last 4 or even 8 earnings.

I just asked Gemini.

It’s a good first guess but then you run that number through our Event Volatility Extractor:

Once you’ve extracted the lump of variance that comes from an 8.55% move on a single day, the remaining variance until expiry is then divided over the remaining days. That’s what that calculator does. It tells you that the ex-earnings implied vol is 54% IF you accept that the earnings move is 8.55%

Since the IVs that precede the Feb 13th expiry are in the low-50s then the term structure ex-earnings is smooth and sensible. If it wasn’t, then we know the market is pricing a very different move size for earnings.

We are just slicing a pizza pie. The whole pizza is the total variance until Feb 13th, currently encompassed by 64% IV. The bigger you make the earnings slice, the smaller the remaining slices (regular trading days) have to be. If the extracted trading day vol turned out to be much lower than 54%, then the market must be expecting a bigger earnings move to account for the difference. Conversely, if it extracted to 62%, the market is pricing a smaller earnings move than 8.55%. The smooth term structure tells us 8.55% slices the pie correctly—each regular day gets roughly the same-sized piece. The Feb 13 expiry sits naturally in line with surrounding expirations.

But…

• If you think that’s too high for earnings, you could sell the Feb 13th expiry and buy the expiry preceding it. If you think it’s too low, you could do the opposite.
• If you think it’s a fair price, then you can simply judge the implied trading day vol on its own merit — 54%.

[Our tools will programmatically do this so that we can then use the ex-earnings vols in our standard Trade Ideas cross-section algo. Until then, we are adding a filter that allows you to exclude names with earnings upcoming from the cross-section sorter.]

So is HOOD vol cheap?

The Trade Ideas algo thinks it’s relatively cheap. Relative depends on your universe. Based on the universe I calibrated on (over 100 liquid ETFs and stocks) it screens cheap.

But an obvious follow-up question is…does it look absolutely cheap compared to its own history?

The answer is ‘“yea”. It’s not screaming cheap, but it’s on the cheaper side.

[This is where it helps to have context. Like if you follow the stock closely and have any feels on it then knowing the options are a bit cheap can inspire some trade structures that get you more juice for your knowledge.]

A few views into its history:

The current IV curve is lower than median realized vols,and a bit higher than current realized vols. BUT…current realized vols are also less than 25th percentile. They only need to sneeze up to median levels for these options to price much higher (especially if they maintain the same VRP ratio which is totally reasonable).

If you prefer time series, the current 30-day IV is sitting near the 1-year low for 1-month implied vol (red line).

Recapping some of the more challenging points:

• The entire “cheap vol” thesis depends on whether the 8.55% expected move is reasonable.
• While 8.55% matches HOOD’s historical average, that’s not how we finalized the number we should use. It’s a starting point that we then test to see if that move size would produce a smooth or humped term structure. If it causes the term structure to jump higher than we are using too small of an estimate, if it causes it to invert sharply, then we are using too high an earnings estimate. If your head hurts, you’re doing this right. Maybe 8.75% or 8.35% makes the term structure a touch smoother but you can use the calculator to see how much little adjustments like that flow through to an implied trading day vol. It has a bigger impact than you might think…changing the earnings day straddle by .25% can move the trading vol by a .5 to 1 point. This is below the threshold anyone except high volume vol traders and market-makers should care about.
• The embedded risk you take when “renting the straddle”: the implied earnings move compresses as you approach February 10th – perhaps because the market decides HOOD’s earnings will be less volatile than historical patterns suggest – then your “cheap” pre-earnings vol becomes less cheap. You’d be holding a position where the event vol component is shrinking, pulling down the value of your straddle beyond normal theta decay. You’re not just betting on realized vol exceeding 54%. You’re also betting that the market continues to price in an ~8.55% earnings move.

Key Takeaway

Decomposing event volatility matters for cross-asset comparison and relative value analysis. A 64% implied volatility might look high in isolation, but after extracting a 170% event vol component (calibrated to produce a smooth term structure), you’re left with 54% trading day vol – which can then be evaluated against your regular toolkit.

 

_______________________________________________________________________________

Appendix: Recipe for Cross-Sectional Analysis With Earnings Names

I used Claude to encapsulate and synthesize a recipe. You can decide how it did:

One of the most powerful applications of event extraction is enabling apples-to-apples comparison across tickers – even when some have earnings and others don’t.

The Problem

Standard cross-sectional vol analysis breaks down when comparing:

• AAPL at 35% IV (no events)

• NVDA at 48% IV (earnings in 30 days)

Which is really “cheaper”? You can’t tell without extracting the event component.

The Recipe

Step 1: Identify Events in Your Universe

For each ticker in your analysis:

• Check earnings calendar (next 30 days typically)

• Note FOMC weeks for macro-sensitive names

• Flag other known catalysts (FDA decisions, etc.)

Step 2: Extract Base Vols Using Term Structure Smoothness

For each ticker with events:

a) Pull the full term structure of ATM IVs

b) Use the Event Volatility Extractor with different move size assumptions

c) The “right” move is the one that produces a smooth, non-humpy base vol term structure

This is the key insight from the NVDA example: too high an earnings move creates an unnatural dip after earnings; too low creates a spike. The correct assumption produces a smooth power law curve.

Step 3: Record Your Assumptions

For each extraction, document: Ticker, Earnings date, Assumed move size (%), Term structure fit quality (R²), Your confidence level (tight/loose)

Step 4: Run Cross-Sectional Analysis on Clean Vols

Now compare:

• AAPL: 35% IV (no adjustment needed)

• NVDA: 44% base vol (extracted from 48% dirty vol with 6.5% earnings move)

Calculate percentile rankings using the clean vols for all four dimensions: IV percentile (using base vols), RV percentile, VRP (base IV – RV), and Term structure steepness (using base vol term structure).

Step 5: Understand What You’re Betting On

When you identify NVDA as “cheap” after extraction, you’re making TWO assumptions: (1) Base vol of 44% is cheap relative to history, and (2) Market will continue pricing ~6.5% earnings move (your assumption holds).

The Cross-Sectional Edge

By extracting events, you accomplish two things:

1. Expand your opportunity set: Instead of excluding 30-40% of your universe during earnings season, you can analyze everyone on equal footing

2. Identify hidden opportunities: Sometimes the “expensive looking” ticker with earnings is actually cheap on a base vol basis, or vice versa

The market often prices earnings mechanically (historical average moves), but base vol can be at extremes. Finding names where base vol is at the 5th percentile but dirty vol looks “normal” because of earnings—that’s where edge lives.

On the last day of November, Kevin bought a bunch of cheap SPY options about 10 minutes to the close and scored. Trades:

Looking at this prompted me to write this post which I’ve had on my mind for a long time: how to think about 0DTEs (or from the bulk of my historical experience — options on the last trading day).

The moontower.ai uses a “volatility lens” for discernment in the option market. But we don’t have a suite of tools for analyzing 0DTE. If we did, we would use a different approach than we do for options broadly. (I’m nothing if not opinionated about how to think about options and being opinionated is part of what you pay for.)

I’ll give you a hint. To think about 0DTEs properly, you must think about time. Any consideration about “vol” can easily be swamped by what you assume about time.

I’ve written about time in options before:

• understanding variance time
• If you annualize volatility with 252 days can you use that number in a 365-day option model?
• weekend theta
• the dirties are down the cleans are up

[The closest hint as to what we’re going to build on was a birdie asked how to model a 1-day option]

But these articles do not address intraday time decay. Option theta is large on the last trading day, while vega is small. 0DTE option pricing is far more sensitive to “How much time remains until expiration?” than notions of volatility.

But the question of how much time remains until expiry is not so simple. Without a concept for how much time remains, we can’t appreciate whether Kevin’s trade was a lucky outcome or strong ex-ante decision.

We’ll unpeel the problem, and in doing so, you’ll get a new view into 0DTE prices.

The most effective way to do this will be to build from a naive model of time passage to a more realistic one to see how it influences option values.

Note: This topic just got way more timely (pun most definitely intended) in light of the Nasdaq’s SEC bid to increase trading hours to 23 hours per day.

Setting the scene

• We are looking at options on our stylized friend, the $100 stock with a 16% vol (corresponding to an expected ~1% daily standard deviation).
• The options are American-style and expire Friday at 4pm.

It’s currently Thursday at 4pm. There is 1 DTE.

Let’s establish a few starting measures. The calculations used in the tables that follow will use the same process.

What’s the $100 straddle worth?

Using our handy approximation:

straddle = .8Sσ√T
straddle = .8 x 100 x .16 x √(1/365)
straddle = $.67

What’s the vega of the straddle?

Straddle vega is defined by change in straddle price per 1 point change in vol. We just rearranged the formula:

vega = straddle/σ = .8S√T / 100
vega = .8 x 100 x √(1/365) / 100
vega = $.042

What’s the 30-minute theta of the straddle?

This is where we need to think differently. If the stock goes nowhere in the next 24 hours the straddle goes to zero. It decays 67cents. But this is far too blunt of a measure if we are trying to think about pricing an option intraday. It’s not illuminating nor useful for our aperture. So we sprinkle in judgment. We’re going to compute a 30-minute theta. There is nothing special about 30 minutes, but you’ll see that just selecting a shorter theta window allows us to reason about time’s relationship to the price of the straddle, and we already hinted that this is the most important driver of price.

We don’t need Black-Scholes. We can compute the theta numerically by pricing the straddle in 30 minutes and taking the difference. If 1 DTE corresponds to 24 hours, then 23.5 hours corresponds to 23.5/24 or .979 DTE

straddle in 30 minutes = .8 x 100 x .16 x √(.979/365)
straddle in 30 minutes = $.663

30-minute Theta = straddle now - straddle in 30 minutes
30-minute Theta = $.67 - $.663  $.007
30-minute Theta = $.007

The naive and invisible assumption

To say that 24 hours equates to 1 DTE and 23.5 hours equates to .979 DTE assumes that “volatility time” passes at the same rate as “wall time” (it’s called “wall time” because clocks are on the wall).

But volatility passes unevenly. Sometimes in bursts. Think of earnings or Fed announcements. The straddle decays instantly after the news is out. Mechanically, traders “crush the vol” in their model to simulate this, but traders on Friday also do things like keep their vols the same and “roll their dates” forward. These are pragmatic kluges to imperfect models to account for the fact that vol does not pass uniformly with time.

[That concept is covered thoroughly on an interday basis in the articles I link to above, but we are zooming in to intraday in this post.]

Our calculations assumed time passes uniformly. Let’s extrapolate the calculations to see what that looks like:

Observations from the uniform time passage assumption

• Theta increases as we approach expiry making the straddle decay faster towards the end of the day
• The straddle’s sensitivity to vol (vega) becomes smaller than 30-minute theta by about noon New York time.

Implication

The value of the straddle is quite sensitive to how much time remains. As we get closer to expiry, it’s clear that even a difference of 10 minutes in your assumptions of how much vol time remains is worth several volatility points.

If market participants understand that time does not pass uniformly, their opinions about the cheapness and expensiveness of the straddles will vary at any singular point of time even if they all agree that the straddle was worth $.67 with 1 DTE!

Differences of opinion are the basis of trading. If you measure time naively, you are a sitting duck for someone who models it better and can buy/sell from you because you are mispricing the “rent” for the next X hours. Of course, this is all masked by 0DTEs by nature having noisy outcomes, but I assure you this is a casino market-makers like being the croupier in.

Towards better time assumptions

From non-linear to the “U-shape”

It is widely understood that market volumes are not uniform throughout the day. The opening and closing 30-minute periods punch above their weight out of a 6.5-hour trading day. VWAP algorithms, which target the day’s “volume-weighted average price”, send child orders in proportion to the day’s volume signature rather than slicing the order evenly throughout the day.

It turns out that this volume profile is strongly linked to the intraday volatility profile. That’s what we care about for pricing options. While unexpected news can change the value of assets without any trading occurring (a gap can be a large update in bids and offers with minimal volume — think of trading halts), trading itself is a source of volatility.

Here is just one of many papers that show not only volume profiles but how intraday volatility follows the same pattern. The authors summarize volume and volatility trends from 6 years of SPY tick data:

Notice the bottom of the U-shape showing the midday lull in volume and volatility.

I made a table characterizing the mean volume curve by hour. It’s not directly from the paper, but derived by eyeballing, but it’s perfectly adequate for our purposes.

When I was a market-maker, our option models “decayed” the day in a similar pattern. There are 13 half-hour periods, but by 10am we believe more than 1/13 of the “vol time” had elapsed, yet from noon to 12:30 pm, the straddle barely decays. Our Option City streaming software let you specify a curve for how much time remained in the day for any hour.

Addressing the “overnight”

Another improvement to our assumptions is to acknowledge that the overnight period from 4 pm yesterday until 9:30am today, despite encompassing 17.5 out of 24 hours, does NOT represent nearly 2/3 of the market risk or volatility.

For exposition, we will show 2 different adjustments.

1) “Overnight has no volatility” assumption

This is naive in the opposite direction from the original calculations, which assumed that an overnight hour and a market hour were equal. In this adjustment, the straddle doesn’t start decaying until the market opens. The full decay occurs in just 6.5 hours.

2) “Overnight has 30% of the 24-hour volatility” assumption

We prorate the passage of time so that overnight DTE sums to .30 and the remaining .70 DTE is encompassed by trading hours.

Demonstration

This table assumes the U-shaped profile for how much volume has elapsed as a stand-in for how much time remains. The volume profile:

We construct pricing for both types of overnight assumptions.

Visually:

In the naive 0% overnight schedule, the straddle doesn’t start decaying until the open and the decay is steeper intraday since there’s only 6.5 hours to erode the entire straddle.

Let’s zoom in on the straddle trajectories because this highlights just how different one’s valuations can be as soon as the clock starts ticking and traders’ assumptions of DTE start diverging:

The straddles start and end in the same place but the interim is where the buys and sells happen.

Kevin’s 683 SPY call with 10 minutes until expiration

We’re going to roll with the compromise decay schedule since it’s the most realistic of the 3 choices:

U-shaped decay profile assuming overnight is 30% of the DTE

From the table, we see the last half-hour represents .102/365 DTE.

How about the last 10 minutes?

We could divide .102 by 3, but realistically (and the paper confirms this), the last 15 minutes contain even more “volatility time” than the second-to-last 15 minutes. Still, let’s be conservative and just divide by 3 since Kevin is buying.

DTE = .102/3
DTE = .034 

When Kevin bought the 683 call for $.02 he said SPY was $682.05 bid. Just to be thorough, let’s estimate the 682 straddle

SPY 10-minute straddle = .8Sσ√T
straddle = .8 x 682 x .16 x √(.034/365)
straddle = $.84

We are going to compute the call using a Black-Scholes calculator, but I like to inject homework questions when there’s an opportunity for estimation practice:

💡Estimate the 683-call based on the straddle price without an option calculator

I just used my Black-Scholes function in Excel with these inputs:

Stock price = 682.05
Strike price = 683
IV = .16
DTE = .034/365
RFR = 0

683 call = $.105

Those calls are realistically worth about a dime and Kevin lifted them for $.02!

 

Post-closes and contrary exercise

For Amercian-style exercise, expiration isn’t really 4pm because you can abandon an option that was in-the-money at 4pm or contrary exercise an option that was out-of-the-money.

This has a value. If suddenly there was a bomb dropped in the Middle East and USO expired just below the strike, you could exercise the calls to get long or abandon the slightly ITM puts. Similarly, if you were short the calls, you should expect to get assigned. If you were short the puts, you should expect not to get long at the strike as you will not be assigned.

Let’s say bearish news came out after the close while the futures are still trading. The futures tank. You can beta-weight all your prices from the close to construct a theoretical price for each name. Then make your contrary exercise and abandon decisions. You are effectively shorting the market at the closing price and then you buy the same dollar notional or beta-weighted notional in futures contracts to lock in a differential. You’ll have basis risk since your share positions won’t be in exact proportion with SP500 weights, but this will be small relative to your theoretical profit.

Likewise, if you are short options, you should expect to accumulate deltas in the wrong way, so you’d need to estimate how many wrong-way deltas you’ll acquire and hedge those with futures. While this will help have a neutral delta for the next day, you will have locked in a theoretical loss. Which makes sense — you were short options that turned out to have value after the close, so the 4pm mark did not reflect your actual p/l nor risk.

This matter of valuing options after the close is not academic. The procedure described here was a regular part of our expiration workflow and checks. This goes beyond equities too. In commodity options, there are “look-alike” European cash-settled versions of the American-style options. Since you can contrary an American style if the name was near pinning there would be an active market in the EOO or “exchange of option” which was the price for the euro-american “switch”. The American trades premium because you get another couple hours to look at the market so the switch was a referendum on the post-close straddle.

This was very common in natural gas options 15-20 years ago. EOOs didn’t trade electronically, so you basically had a mental scroll of where switches would tend to trade on past expiries to have an idea of what the post-close straddle could be worth. In practice, market-makers would always have the pins on the same way so they all needed to do the same risk-reducing trade, causing the switch to find a risk premium where a contra was content to either open or add more.

[I believe in 2018, the NYMEX changed American option specs so the options were auto-exercised at expiry. Probably to appease option shorts who didn’t like position surprises when they got their contrary assignment notices later in the evening.]

If you are long an equity option and your clearing firm doesn’t require you to decide to exercise/abandon until an hour after the close, (clearing firms have their own unique “cutoffs” and with a phone call you might be able to massage that) then you basically get a free look at the futures for the strikes near the expiry price.

Let’s say that hour could be worth as much as the mid-day 1-hour lull from 12:30-1:30pm or about .06 DTE. How would that have affected Kevin’s 683-call?

.06 + .034 (the DTE for the last 10 minutes of the day) = .094 DTE

The 683 call with .094/365 DTE is worth $.32 via B-S calculator or 3x what it was worth if you thought the post-close had no option value. In practice, the post-close is likely worth very little most of the time, and quite a bit in the event that news hits between 4 and 5pm.

I started researching/writing this post about a month ago. It took a strange arc. It began with me wondering about “up vol” vs “down vol” or how vol acts differently in rallies vs selloffs. Then it ran straight into a topic I read about this summer (the title is a clue). It will awaken both seasoned and novice option traders with both inspiration and discomfort. Which is to say, I’m really happy I wrote it, but also feel like there’s a lot more to this than what I can cover today (and sparring with LLMs about it is definitely affirming this feeling).

Before we start unfolding, one more meta thought.

While working on this I benefited from a pedagogical technique that I didn’t plan, but believe you can engineer. I mentioned it in one of my “learning science” articles, myelination:

The “hypercorrection effect” is the phenomenon where you remember corrections to wrong answers better than when you give a correct answer off-the-bat when the question is difficult. Generating a prior makes you own a prediction. When it breaks, surprise becomes the teacher.

I’ll walk you through the same steps I took, which reinforced, even with all my years, just how nebulous the concept of volatility can be and how it touches trading and investing in practice.

A popular starting point: napkin math

Before pulling any data, I wanted to test my market intuition. I start with some guesses about how the S&P 500 behaves off the top of my head:

• S&P 500 volatility hovers around 16% annually. I heuristically think of this as some blend of “volatility when the market is up” and “volatility when the market is down”.
• 2/3 of the months are positive
• Risk reversals suggest upside vol is about 10% below some “base” vol
• Downside vol is about 30% above this “base” vol

If the full market vol is 16%, and I have asymmetric volatility in up/down months, what’s the “base” volatility?

Let x = base volatility
Up month vol = 0.9x (10% lower)
Down month vol = 1.3x (30% higher)

Full variance = 2/3 × (0.9x)² + 1/3 × (1.3x)² = 16²
Full variance = 2/3 × 0.81x² + 1/3 × 1.69x² = 256
Full variance = 0.540x² + 0.563x² = 1.103x² = 256

Therefore: x = √(256/1.103) = 15.23%

So my base vol would be about 15.23%, giving me:

• Up month vol: 0.9 × 15.23% = 13.71%
• Down month vol: 1.3 × 15.23% = 19.80%

For monthly returns, I figured the standard deviation would be roughly 16%/√12 = 4.62% per month.

As for expected returns, I guessed the market delivers about 80 basis points per month (~10% annually).

If 2/3 of the months are up and 1/3 are down, and the average is +0.8%, what are the typical up and down returns?

Let’s call up months +U% and down months -D%:

2/3 × U - 1/3 × D = 0.8
2U - D = 2.4

If monthly volatility is about 4.6%, what would typical up and down returns be?

Assuming monthly returns are normally distributed with a mean of 0.80% and standard deviation 4.62%, the probability of a positive return is 57% (leaving 43% negative).

Probability of market down
Z-score = (0 - .8)/4.62 = -.173
P(Z ≤ -0.173) ~ 0.431

[Wait a minute...for N(.8, 4.62) P≤0 ~43% but I assumed the probability of a negative month is only 1/3. This is a clue some of my estimates are wrong OR the distribution is not normal. We're going to bring the real data in soon and the appendix will expand the discussion. I won't bury the lede -- my estimate of p≤0 is correct! But I get some other estimates wrong and, well, the returns aren't normally distributed. We're going to make sense of all of this.]

Again, my unconditioned estimate of monthly return is .80%.

Now I want to estimate the monthly return given that the market is up. Let’s try translating to math language:

I want the return at the midpoint of the positive portion of the distribution.

That’s at the 43.1% + 56.9%/2 = 71.5% cumulative probability point.

P(Z ≤ X) ~ .715
Solve for X using Excel:

NORM.INV(0.715,0.8,4.62) = 3.42

For a N(0.8%, 4.62%) distribution, the 71.5th percentile gives us +3.42%.

If 2/3 of the months are up, and the expected return in an up month is+3.42% but the overall mean is 0.8%, the down months must average -4.44% to balance the equation above.

Validate: 2/3(3.42%) – 1/3(4.44%) = 2.28% – 1.48% = 0.80%.

Reality Check

Time to test these intuitions against actual data. I pulled daily S&P 500 returns from January 2016 through October 2025—nearly a decade covering COVID, Fed policy shifts, and retail investing mania.

Market batting average:

• Up months: 81 out of 118 (68.6%) ✅ Pretty close to my 2/3 guess!

Returns:

• Average up month: +3.43% ✅ I estimated 3.42% —boom!
• Average down month: -3.93% ❌ I estimated 4.44%.
• Overall monthly average: 1.13% ❌Higher than my 80bps estimate

Volatility:

• Full sample annual vol: 18.23% ❌Higher than my 16% guess.
• Mean vol in up months: 12.47% ✅ I estimated 13.71%— so-so.
• Mean vol in down months: 20.43% ✅ I estimated 19.80%—not bad!

All of these were calculated from daily returns, whether it was the full sample or if they were then grouped into months.

That’s weird…

This is where things got interesting. My intuitions were pretty decent about up and down vol. I decided to check if the weighted average of monthly volatilities would recover the full sample volatility:

Weighted variance = 0.686 × (12.47%)² + 0.314 × (20.43%)²
                  = 0.686 × 0.01556 + 0.314 × 0.04175
                  = 0.02377

Weighted vol = √0.02377 = 15.42%

Wait. The full sample vol using daily returns is 18.23%, but the weighted average of monthly vols is only 15.42%.

That’s an 18% gap in volatility, which is large, if we consider typical vol risk premiums of ~10% just to give a sense of proportion.

In variance terms:

• Full sample: 332.33 basis points (ie .1823²)
• Weighted average: 237.70 basis points (ie .1542²)

Missing: 94.74 basis points

Where did ~30% of the variance go?

Let’s take a detour before we go into the arithmetic.

The Coastline Paradox

I’ve been reading Geoffrey West’s book “Scale” and this anomaly reminded me of the coastline paradox—the closer you look at a coastline, the longer it becomes. These excerpts tell the story of Lewis Richardson’s discovery in the early 1950s when he discovered that various maps indicated different lengths for coastlines:

Richardson found that when he carried out this standard iterative procedure using calipers on detailed maps, this simply wasn’t the case. In fact, he discovered that the finer the resolution, and therefore the greater the expected accuracy, the longer the border got, rather than converging to some specific value!

This was a profound observation because it violated basic assumptions about measurement, which we hold to be objective to some underlying reality. But Richardson’s discovery is intuitive once you think about it:

Unlike your living room, most borders and coastlines are not straight lines. Rather, they are squiggly meandering lines… If you lay a straight ruler of length 100 miles between two points on a coastline or border… then you will obviously miss all of the many meanderings and wiggles in between. Unlike lengths of living rooms, the lengths of borders and coastlines continually get longer rather than converging to some fixed number, violating the basic laws of measurement that had implicitly been presumed for several thousand years.

When you use a finer resolution (shorter ruler), you capture more of these wiggles, leading to a longer measured length.

This gets better. (Also, you should read this friggin’ book!)

The increase follows a pattern:

When he plotted the length of various borders and coastlines versus the resolution used to make the measurements on a logarithmic scale, it revealed a straight line indicative of the power law scaling.

The practical implication:

The take-home message is clear. In general, it is meaningless to quote the value of a measured length without stating the scale of the resolution used to make it.

Risk exhibits the same property. It depends on the resolution at which you measure it and forms the link to the question: where did those 95 bps of variance go?

While I’ve pointed this out before in these articles:

Volatility Depends On The Resolution

Risk Depends On The Resolution

…I didn’t drill down to the mathematical decomposition for why this is true. We will do that in a moment but in words:

When we calculate monthly volatilities and average them, we’re essentially “sampling” risk at a monthly resolution. But when we calculate volatility from all daily returns, we’re capturing additional variation that exists between months—variation that gets smoothed away in monthly aggregation.

Understanding What Is Masked With A Test Score Analogy

Let’s illustrate with a tangible example. Imagine three classes taking the same test:

Class A (Morning class): Scores: 75, 80, 85 (mean = 80)
Class B (Afternoon class): Scores: 65, 70, 75 (mean = 70)
Class C (Evening class): Scores: 85, 90, 95 (mean = 90)

If we calculate the variance two ways:

Method 1: Pool all scores together
All scores: 75, 80, 85, 65, 70, 75, 85, 90, 95

• Mean = 80
• Variance = 83.3 (average of squared deviations)

Method 2: Average the within-class variances

• Class A variance = 16.7 (sum of squared deviations is 50, then divide by 3 samples)
• Class B variance = 16.7
• Class C variance = 16.7

Average variance = 16.7

The gap: 83.3 – 16.7 = 66.7

This missing 66.7 is the variance that comes from classes having different average scores (80, 70, 90).

The Law of Total Variance captures this precisely:

Total Variance = E[Var(Score|Class)] + Var(E[Score|Class])
      83.3     =        16.7         +        66.7

Circling back to our example:

• The “Full Sample Volatility” (18.23%) or 332 bps is the Total Variance
• The “Weighted Average Volatility” (15.42%) or 238 represents only the first term: the Within-Group Variance
• The “Missing Gap” (95 basis points) is the second term: the Variance of the Means

Intuitively:

The market doesn’t just wiggle around a static zero line every month. Some months the whole market shifts up (+3.43%), and some months it shifts down (-3.93%). If you only look at volatility within the month, you ignore the risk of the market shifting levels entirely. Simply averaging monthly volatilities ignores this “Between-Month” risk.

Bonus Reason Why Averaging Volatilities Misleads: Jensen’s Inequality

There’s another subtle effect at play: Jensen’s Inequality. This mathematical principle states that for a convex function (like squaring for variance), the average of the function is not equal to the function of the average.

💡See Jensen’s Inequality As An Intuition Tool

In this context:

• Variance is proportional to volatility squared (convex function)
• The average of squared volatilities ≠ the square of averaged volatilities

First of all, in our data, each month has a different number of trading days (19-23). When we calculated monthly volatilities, we essentially gave equal weight to each month regardless of how many observations it contained.

But even in months with equal days, averaging volatility is dangerous

The March 2020 Example:

• March 2020: 22 trading days, 91.53% annualized volatility
• October 2017: 22 trading days, 5.01% annualized volatility

In our “average of monthly vols” calculation, these months contribute equally. But their contribution to the full sample variance is vastly different:

March 2020’s contribution = (91.53%)² × 22/2473 = 74.54 basis points of variance 
October 2017’s contribution = (5.01%)² × 22/2473 = 0.22 basis points of variance

March 2020 contributes 334 times more to total variance despite being weighted equally in the monthly average!

Practical Implications

For Option Traders
The difference between realized vol at different sampling frequencies directly impacts estimates of volatility. The shorter the sampling period the higher the volatility on average. When computing realized vols based on tick data, a method sometimes known as “integrated vol”, there is a minimum sampling frequency that, if you dip below, causes the vol to explode because it is simply capturing “bid-ask bounce”. The minimum threshold can vary by asset, so by using a volatility signature plot (a plot of vol vs sampling frequency) you can see where this threshold lives.

Conversely, it’s reasonable to expect that estimating long-term vols by sqrt(time) scaling from shorter dated vols may overshoot. See the appendix on the discussion of power law scaling in the context of the coastline paradox, keeping in mind that term structure scaling takes a power law shape, but the exponent needn’t be 1/2.

[Even if you conclude that upward sloping term structures are unjustified or at least reflecting a risk premium, do you understand why it’s weakly, if at all, arbitrageable? I think this would make a good interview question for an option trader to demonstrate how they think about risk-taking and capital (and business generally). I’ll withhold my answer because I like the question too much.]

For Portfolio Construction
When combining assets with different measurement frequencies (daily equities, monthly real estate, quarterly private equity), be aware that risk measured at different resolutions isn’t directly comparable. This is not a perfectly overlapping reformulation of the “volatility laundering” criticism of slow-to-mark assets.

Conclusion: Respecting the Fractal Nature of Risk

This little jaunt from napkin math to data analysis shows how risk, like coastlines, is fractal. The closer you look, the more you find.

When reconstructing measures of risk from lower resolution assumptions that were quite strong, I found gaps which point to my oft-repeated:

Risk depends on the resolution at which you measure it.

The resolution at which you measure risk affects three things:

1. Aggregation effects: Higher frequency captures more granular variation
2. Weighting effects: Different time periods get different implicit weights which can be decomposed by the Law of Total Variance
3. Jensen effects: The non-linearity of variance creates gaps when averaging

The market’s full 18.23% volatility tells one story. The 15.42% average of monthly volatilities tells another.

Technical Note: This analysis used realized volatility calculated as √(Σ(X²)/n) × √252, treating daily returns as having zero mean. This approach, common in high-frequency finance, effectively assumes the drift is negligible compared to volatility at daily frequencies—a reasonable assumption given that daily expected returns are typically 0.04% while daily standard deviation is over 1%.

Appendix — Various Topics

🌙The Variance Decomposition

When measuring at daily resolution across all data:

Var(returns) = E[X²] - E[X]²

When measuring at monthly resolution, then averaging:

E[Var(returns|month)] = E[E[X²|month] - E[X|month]²]

The difference between these is:

Var(returns) - E[Var(returns|month)] = Var(E[X|month])

which implies The Law of Total Variance.

The law states that the total variance of a dataset can be broken into two parts:

1. The average of the variances within each group (Within-Group Variance)
2. The variance of the means of the groups (Between-Group Variance)

Var(X) = E[Var(X|Group)] + Var(E[X|Group])

🌙The Napkin Math Validation

The algebra used to solve for the “base volatility” x is known as a mixture model:

Total Variance = (Prob_up × Var_up) + (Prob_down × Var_down)

It’s only valid if the means of the up/down months are close enough that the “Variance of Means” component is negligible for a rough guess.

🌙Skewness in monthly returns

Actual Monthly Statistics (S&P 500, Jan 2016 – Oct 2025)

• Mean: 1.13%
• Std Dev: 4.39%
• Median: 1.80% (notably higher than mean)
• Up months: 68.6% (81 out of 118)

If monthly returns were truly N(1.13%, 4.39%), we’d expect only 60.1% up months.

But we actually get 68.6%—an 8.5 percentage point gap. This gap, as well as the difference between mean and median demonstrate negative skew. The left tail is longer, meaning occasional large down moves.

It’s classic equity pattern: stairs up, elevator down. The bad months are worse than the good months are good, but the good months happen more often than a normal distribution predicts, even net of a positive mean return. Both the higher mean and the skewness.

If I ran through my same logic above using actual data:

Probability of market down
Z-score = (0 - 1.13)/4.39 = -.257
P(Z ≤ -0.257) ~ 0.399

I want to estimate the monthly return given that the market is up. Let’s try translating to math language:

I want the return at the midpoint of the positive portion of the distribution.

That’s at the 39.9% + 60.1%/2 = 70% cumulative probability point.

P(Z ≤ X) ~ .70
Solve for X using Excel:

NORM.INV(0.70,1.13,4.39) = 3.43

Market return given that it’s up: +3.43% (coincidentally matching reality)

We go back to this identity with the true mean and volatility to solve for the down move:

.601 × U - .399 × D = 1.13
.601*(3.43) -.399D = 1.13
D = -2.33

If the distribution was normal N(1.13%, 4.39%), we expect the down moves to be -2.33% on average with 40% down months, but the actual data shows the down moves occurred only 31.4% of the time, but were -3.93%!

🌙Coastlines and Power Laws

The generic power-law relationship:

y = A · xⁿ

Where n is the exponent that determines how drastically y responds to changes in x.

You can see the sensitivity by comparing different exponents:

• If n = 1/2:
  To double y, you must increase x by a factor of 4 (because 4^(1/2) = 2).
• If n = 1/4:
  To double y, you must increase x by a factor of 16 (because 16^(1/4) = 2).

West writes:

“To appreciate what these numbers mean in English, imagine increasing the resolution of the measurement by a factor of two; then, for instance, the measured length of the west coast of Britain would increase by about 25 percent and that of Norway by over 50 percent.”

In the British case, doubling the resolution increases the coastline by 1.25x, therefore, the exponent, n, must be ~ 1/3

2ⁿ = 1.25
n log 2 = log 1.25
n = log 1.25 / log 2 = .32

Yesterday in “trader” is a uselessly broad term, I boosted several of Euan Sinclair’s insights about option trading. I saved one for today because it’s an actionable trade that I agree with and I believe it exists because it’s unintuitive to everyone but volatility traders who are a tiny minority of the traders at the scene when the particular setup presents itself!

We are going to describe the trade and what its success depends on.

First, what did Euan say exactly?

🔎The Bubble Trade – Selling Puts on Meme Stocks

Euan argues that when stocks enter extreme bubbles (GME, AMC-type moves), selling puts captures massive premium with volatility correlated to price movement.

The opportunity here is to sell puts. I’m selling puts. I’m getting a huge premium for those cuz vol’s high. If the stock keeps going up, well, that’s fine, right? Those puts are going to expire worthless. But if the stock goes down, I’ve got a huge cushion there because vol’s going to come in.

Why Not Calls?

Once something’s in a bubble, it can continue to be in a bubble. Like it’s doing something stupid. And once someone’s stupid, their stupidity knows no bounds, right? Once it’s become unmoored from reality. So selling calls is insane.

The Edge – Volatility Correlation:

GME the vol when it started I don’t know probably like 50 or 60. When it went nuts? I don’t know what it was because my system topped out at 1000. So it was above a thousand right? So we know vol has gone up as the stock’s gone up. So what happens when the stock comes down? Vol comes down. If let’s say I sell the 40%. So I’m selling the 200 strike puts when GME is 500 and implied vol’s a thousand if it drops back to 200 that implied V is probably going to drop back to you know 200. It’s quite likely I’ll make money because the vega is made up for any delta effect.

I have written about this idea before in What Equity Option Traders Can Learn From Commodity Options.

In that post, I talk about…

1. Option market-maker @DeepDishEnjoyer calling attention to puts going UP in value as GME and its vol ripped higher:

This is quite odd from a first principles perspective. GME closed 17 handle on Friday. Today it meme squeezed up because of Roaring Kitty. A basic model is: it continues meme’ing – then these puts expire worthless or the meme ends and we go back to where we were at at Friday. But note that you could have sold these puts at 75 cents today even though they closed in the 50s on Friday!!!! They should be actually be worth *less* since there is no state of the world where downside vol increased.

That’s easily anywhere from 20-40 cents of EV on these puts. And indeed that’s where these puts landed now. So why does it happen? Well, market makers don’t pay a large amount of attention to the wings of their vol surface. ATM implied vol got correctly bid, but they moved the…rest of the surface in parallel EVEN THOUGH THAT MAKES NO SENSE IN A SCENARIO WHERE A STOCK MEME GAPPED UP. Again, vol follows fairly two discrete paths that are intimately tied to stock price – vol is high when the stock is memeing, vol necessarily dies down when it stops.

At the money implied vol should increase. But the strike vol of the 10 strike put should not be massively increasing as the probability of going *below* 10 has not increased today from yesterday, while the options market is implying it has.

2. My instinct to sell those downside puts via put spreads…but the market is quite good at pricing the vertical spread! Commodity markets in particular since they are deep and accustomed to pricing options in physical squeeze scenarios.

Euan harps on this class of trade where you sell puts in bubble or squeeze scenarios where the vol explodes. He covered it in Retail Option Trading, which came out a year ago. We discussed it with respect to DJT stock, arriving to the same conclusion that whether or not Trump won, those DJT puts would come in hard regardless of the stock’s direction.

(He did the trade, I balked. It was a nice winner for him even when the stock fell in the weeks after Trump’s victory was declared).

My desire to cap the risk and bet on the distribution ruins the trade because the blunt source of the trade’s p/l is vol coming in hundreds of points. Any version of this trade where you buy options is like choosing to drive on a tightrope instead of a wide avenue. Just take more risk to load the results fully to the edge, but decrease your size.

Listening to Euan explain this on the podcast got it rattling around my skull again. The intuition behind this trade runs deeper than “vol is high, so sell it” because the stock is going to make large moves. It’s about how the vol surface changes as the stock moves, but to appreciate how much wind you have at your back, we can show not only option values but also describe what’s happening along the path with Greeks.

Let’s consider the $10 strike put of a meme stock that has surged to bubble territory. Maybe it was $10 and shot to $50. Its vol explodes. Let’s look at a matrix of spot and implied vol pairs for the $10 strike with 30 DTE.

If the stock is $50, we’ll pretend the ATM vol is 500%

The downside strikes will trade at a discounted vol as the skew inverts. We’ll go with a 400% vol on the $100 strike, the cell with the red border.

As both the stock and vol fall back to earth, those puts don’t perform and this is assuming there is still 30 DTE. There’s no time decay embedded, just stock and vol changes. If the stock and vol both suddenly halved, the puts still LOSE half their value. It’s very difficult to win on being long those puts if the vol stabilizes regardless of where the stock goes. I mean you almost need them to go in the money to win.

Oh and just to put ridiculous vols in context, this is a table of vols and what they imply for monthly standard deviation of returns. Just divide vol by √12:

It’s educational to see this through the lens of option Greeks.

The Mechanics of Vanna

Vanna is a higher-order sensitivity that answers “how does the delta of this option change with volatility”.

OTM calls have positive vanna because, as you raise volatility, the call delta increases. The vol and delta change in the same direction. ITM puts share the same vanna as their corresponding OTM calls because raising vols makes ITM put deltas less negative.

OTM puts have negative vanna because raising vols makes put deltas more negative (more likely to finish ITM). Their delta is moving opposite the sign of the vol change. ITM calls also share the negative vanna since raising the vol lowers the ITM call delta (less likely to finish ITM).

You can be long vanna by being long OTM calls or short OTM puts.

If you are long vanna, you get:

• longer delta as vol increases
• shorter delta as vol decreases

You can be short vanna by being short OTM calls or long OTM puts.

If you are short vanna, you get:

• longer delta as vol decreases
• shorter delta as vol increases

Let’s test our comprehension with the familiar —the SPX. Asset managers, at the margin, buy protection and overwrite calls against their long positions. Therefore, market-makers, on balance, tend to be long calls and short puts. In other words, they are long vanna.

We know the SPX exhibits inverse spot-vol correlation. As the index goes up, vols tend to fall. If market-makers are long vanna, their delta changes with the same sign as the vol. If vol falls, the market-makers get shorter. The calls they own provide “less length” and the puts they are short “become smaller” as vol falls. The lingo used to describe this position is that it “decays short”.

Think of it this way, if a market maker is long call/short put and has the delta hedged with short index futures, if all the options go to zero, thus not spitting off any more deltas, then the market-maker is just short futures. Therefore, the glide path of the portfolio as time passes or vol falls is “decaying short”.

If they are long vanna, and vol increases as the market falls, then their delta changes with the sign in vol. Vol goes up, they get longer!

Notice what is happening.

As the SPX goes up, the market makers are getting longer gamma because the index is going towards their longs. The gamma effect makes them longer, but if the vol is falling, the vanna effect makes them shorter. The vanna and gamma effects on delta are directionally offsetting although the gamma effect is usually larger.

On the downside, the market maker gets shorter gamma as the index falls to their shorts but since vol is increasing the delta is also increasing due to vanna. The sign of the vol change and delta change is the same. But this is bad for the market maker! In a falling market, both the vanna and gamma effects conspire to make them longer delta. Meanwhile, in the rising market, the gamma benefits are offset by the vanna.

Let’s stop for a moment to make something clear. This is all description. Knowing this is not an edge any more than knowing that moving air is called wind. The question of strategy comes down to price. The steepness of the skew is either the cost or compensation for your vanna, depending on whether the spot/vol correlation works for or against your position.

In the SPX you collect vol points in the differential between the OTM call you own and the OTM put you sold. It boils down to:

Did you collect enough vol points to compensate you for the fact that you will NEED TO SELL MORE shares (vs a constant vol world, ie no skew) to hedge when the market falls toward your short gamma region and you will NOT GET TO SELL as many shares when the market goes up (vs a constant vol world)?

If you are short vanna in SPX (you get shorter delta as vol increases) you pay for the privilege in vol points.

In markets with positive spot-vol correlation—think squeezing commodities or meme stocks in full mania—you pay to be long vanna. If you get longer delta when vol increases, this is aligned with the positive spot-vol environment, amplifying your gamma as the market rallies towards your long calls.

But let’s examine the downside. After all, the thrust of this post is what happens to the puts in meme stocks as the market falls.

If you are short puts in a falling market, you are getting shorter gamma. This makes you longer delta on every downtick. Not desirable. But you are long vanna. As vol falls, your delta gets shorter. Long vanna means the change in your delta follows the same sign as the vol change.

In the SPX upside situation, the market maker was long vanna, so their long deltas shrank while vol was falling, dampening gamma’s delta lengthening effect. Here, it’s the downside move that coincides with vol falling. The long vanna effect on delta directionally offsets the short gamma effect on delta.

That these puts don’t perform for longs (and pay off the put sellers) is the vanna effect winning. Consider the delta of those options we looked at earlier for different pairs of vols and stock price.

If the IV falls from 450% to 250%, the $10 put has the same delta despite the stock being 40% lower. Again, we assumed no passing of time. If time passed, those puts would be even “further OTM” in delta or standard deviation terms.

💡The Volga Asterisk

There’s another Greek at work: volga or “vol gamma”. When you’re short OTM options, you’re short volga, which means as vol falls your vega is getting smaller. Maybe you make $1 on the first 10-point decrease in vol, but only 50 cents on the next 10 points. I cover volga in more depth here: Finding Vol Convexity

💡2 Vannas?

I never looked at an option’s vanna in a pricing model or the vanna of my position. I’m using it here to name effects that option traders know from experience, even though older option traders probably don’t say “vanna”. Vanna actually has 2 definitions. The one we are using here is the change in delta per change in vol. In practice, I’d say this cashed out as “these options have more/less gamma than what the model said” but it would have been higher resolution to just attribute vanna.

The second definition of vanna is change in vega for a change in underlying. I wouldn’t track this number explicitly, but this is something you must be keenly aware of. You position for it on purpose. Your risk shocks, a matrix of greeks for various combinations of spot and vol, show your vega under different market assumptions, thus capturing this definition of vanna even without naming it vanna.

In my lingo, “owning the skew” means owning the premium region of the surface where the market expects vol to increase if the spot heads there. That would be the downside in SPX or the upside in my cotton story. If cotton rips higher, I get longer vega because my OTM calls become closer to ATM. You pay up in vol points to “own the skew”. Those vol points are the price for having the vega winds at your back. Like any price, it can be too low or too high.

I really don’t care for skew as the basis for a vol trade. I’ve talked about how the skew is pretty good at knowing where the bodies are. I can tell you from my cotton and nat gas days that I was quite contrarian on the topic. I would hold my nose and buy expensive skew if I had a strong conviction on the directional outcome. Owning the skew was an insurance policy in case I was wrong about a directional trade I had unusually strong conviction in (like owning expensive calls to create an ITM synthetic put position to bet on a sell-off).

Strike Vol Dynamics

I’ve been throwing around vague statements like “vol coming in” or “vol going up” with respect to spot changes, but you should be asking, “Kris is this err, Vol, in the room with you right now?”

Well, no. There is no such thing as Vol. We have many numbers known as strike vols that when pushed through a model with other assumptions, generate a contract price. Those prices are the only thing our boss, P&L, cares about. The strike vols give us a ruler to compare and normalize. The Greeks, in turn, depend on them. This allow us to understand our risk and make sense of how these option prices respond to all the ways these contracts are battered by market circumstances.

If the vol on those OTM puts is low enough relative to what will actually happen, the strike vol won’t decline as much as you need it to. If the meme stock gaps to $0, the strike vol doesn’t even matter. The realized move drives the entire outcome and the puts go to $10 of intrinsic.

On the other hand, if the stock is squeezed and hard-to-borrow, and starts falling due to the supply of lendable shares loosening, this will lower the cost to be short and reduce the value of puts relative to calls! In other words, the very thing you might be scared of, the stock falling once the squeeze ends, might coincide with put prices weakening!

What is the story of the edge for selling puts on a meme/squeeze names?

In Laws of Trading, quant trader and Jane Street alum Agustin Lebron emphasizes something that many might not expect of quants — a belief that an edge should be easily explained by a qualitative story.

I actually think this particular trade is emblematic of a trade whose edge makes perfect sense. It sits there because it’s so unintuitive.

Let me get this straight. We acknowledge that a bubble or squeeze is happening, that the price of the asset is going to fall, and that the right trade is to…sell puts?!

Well, yea. Welcome to markets.

This is one of those trades that exhibits the “curse of knowledge”. You and I understand how vol surfaces work. We understand that when the market moves in the way people expect, that it is a stabilizing move. It is a move in a direction where people are more comfortable or at least less wary of selling options. In the SPX, that’s to the upside where the world is happy to clip profits, business as usual.

In squeezes and memes, stability is lower. That’s where the world makes sense again. Implied vol will come in just as it went out on the way up. When GME ripped higher nobody knew what was going on, but they knew GME at $60 is not its new “home”.

The public correctly understands that put options allow you to bet on the stock going lower. They don’t understand that the main input into its price is volatility so that they can be directionally right and still lose. They are non-economic with respect to what the contract is worth because their scrutiny stops at “you said puts go up when stocks go down”. Plus, when a stock turns into a football the gambler sees a window for a wildly assymetric payoff. They want upside or downside lotto tickets. They don’t want to sell options even if it’s the edgey side. All this commotion for bounded upside? Huh? To quote Dave Mustaine, “It’s still we the people, right?”

What could make the edge in this trade disappear?

Simple.

The risk-taking capacity of the vol-aware traders overwhelming the public’s demand to make the obvious but ill-structured bet on the stock going lower.

But keep in mind, the demand benefits from the trade being both:

a) obvious (the stock is gonna go back down)

and

b) most easily expressed by buying puts (anyone who thinks to outright short the shares is on display like a brontosaurus in a natural history museum somewhere)

For the opportunity to die, either:

a) retail stops trading, in which case, how did the meme stock take off in the first place

or

b) the situations become so common and retail gets burned so frequently that they finally realize that there is such a thing as a positive delta put.

That said, I literally just explained how this all works and I’m still not holding my breath.

 

Related:

How Options Confuse Directional Traders

Last week, in American options are not vanilla, we covered not only the concept of early exercise for American options, but rules for “optimal” early exercise.

If you want this broken down in video form, I direct you to Sheldon Natenberg’s explainer in CBOE’s educational series:

📽️ Early Exercise of American Options (CBOE, video lesson)

Today we’ll not only get into the common model used to price American-style options (you can use them for European-style as well, while Black-Scholes only works for European), but you can get hands-on to see how they work.

Just to tie a bow on last week’s post and not give you a false impression that early-exercise rules are dry calculations, here’s a shower thought I had laid out in a progression:

1. American reversal/conversion values are lower than European R/C because of early exercise. Basically, the expected value of T is smaller than T itself. “DTE” in the world of American-style options is not deterministic.
2. The spread between the Euro vs American R/C is a function of interest rate volatility. But I’ve never seen the spread directly modeled because the R in option models is taken as constant and “managed” at the level of portfolio Greeks [and general judgement].
3. I heard a few years ago a big MM took a hickey on early exercise mispricing during the 2022 rate hikes. That feels like a clue. I’m guessing they overvalued calls/undervalued puts because their R/Cs turned out to be too high. Amercian R/C values turned out to be much lower than the Euro values as they were assigned. In other words, the spread, which represents the value of the option to exercise early, was greater than expected.
4. I don’t know the details of what happened at that MM but I’m just guessing. If anyone wants to enlighten me you know how to reach me. I’m purely curious.
5. If this didn’t make sense, but you want it to, you like to be nerdsniped, which I appreciate. But this is definitely not something to be practically concerned about.

Before we go on to the tree models, how’s this for an oblique, albeit grim, option play via Darkfire Capital LLC:

The survivor option:

Ok, here’s your free money of the day tweet – on your deathbed, instruct the trustee of your trust to buy as many brokered CD’s with the lowest coupon/longest maturities possible.

Once the death certificate is issued, forward it to broker and have them exercise the survivor’s option – bang, that CD priced at 88 is now par.

Laugh heartily from your coffin.

 

Tree models

Natenburg tells us that tree models are easier to grasp than Black–Scholes and can price both European and American options. He explains that the Cox–Ross–Rubinstein (CRR) binomial model remains one of the most popular implementations of trees to this day.

They work by pricing options just before expiry then working backward to today. At each node you ask: exercise now or wait?

Another SIG bootcamp exercise was to build these in Excel from scratch.

I used an LLM to help me code up both a tutorial and simulator so you can learn this stuff without signing a non-compete 🙂

Step 1: Build the Stock Price Tree (forward in time)

💡CRR Parameters — where they come from

I’ve written a step-by-step explainer of the risk-neutral probability formula if you want to build up from intuition to math:

📐The General Formula to Back Out The Risk-Neutral Probability (moontower)

Step 2: Backward induction (the magic)

For European options, you skip the max with intrinsic (no early exercise), using only the Hold value.

A word on convergence

The binomial tree is a discrete way to approximate continuous price movements. As you increase the number of steps:

• Each time slice gets smaller.
• The tree gains more branches, resembling a smooth diffusion.
• The option price converges toward the “true” theoretical value.

Claude’s “rule of thumb” shows diminishing returns since you’re doing roughly 10× more computation for that extra 0.9% improvement:

• 100 steps ≈ 99% accuracy
• 1,000 steps ≈ 99.9% accuracy

Get your hands dirty

🌲Moontower Cox-Rubinstein Binomial Tree Lab: A self-explanatory demo

The green nodes represent early-exercise candidate conditions. A nice way to explore the tool is to see where the clustering occurs based on the inputs to build your understanding of what makes an option more or less likely to be exercised early.

 

🖥️Black-Scholes and Cox-Rubinstein side-by-side calculator

 

I’ve always found it amusing that the most commonly traded options, American-style equity options contracts, are considered “vanilla”. Because they can be exercised early, their valuation is an instance of a famously difficult problem — explore/exploit. The only reason I might refer to them as “vanilla” is not for them being simple, but simply common.

The last time you bought a TV you encountered this problem — do I keep researching or pull the trigger? From shopping or channel-surfing to giant commitments like who to settle down with or businesses to pursue, the problem sits at the heart of decision-making across all domains though it tends to be more formally considered in fields like computer science, finance, game theory, and operations engineering.

💡See my notes from Brian Christian talking about Algorithms to Live By to see how explore/exploit is seen in everything from “bandit” problems to child development and learning strategies.

The Black-Scholes formula is elegant. It’s a closed-form equation that you can implement in a common financial calculator. As trainees, we programmed it into our bootcamp standard-issue 12C:

But Black-Scholes doesn’t work for American-style options.

💡If you need accessible, non-formal refreshers on Black-Scholes, see:

• The Intuition Behind The Black Scholes Equation
• A Visual Appreciation For Black-Scholes Delta
• Slides to Moontower’s The Equation That’s a Strategy

The equation is a factory — it takes in raw material and squeezes out a hot dog on the other side. But it has no visibility on the path between input and output. But that path is key to the core question:

What is the optimal stopping time of an American-style option?

When should we exercise the right to “stop” the option early?

While American options let you exercise anytime before expiration, you usually shouldn’t. The value of optionality (your right to wait) is typically greater than the small benefit of early exercise.

This discussion is not only useful but fun since we invoke microeconomics in real-life.

It starts with 2 key questions.

WHY would you exercise early?

• Puts: to collect interest sooner on the short stock position.
• Calls: to capture a dividend

WHEN is it worth it to exercise early?

We check 2 tests in sequence:

a) Is the benefit worth more than the optionality you give up?
Compare the gain from early exercise to the value of the out-of-the-money option at the same strike.

Examples:

• Put: If a stock is $100 and you own the 120-put, you can exercise the right to sell/short the stock at $120. Is the interest you earn on the $120 until expiry worth more than the 120-call, which you are effectively selling at 0?
• Call: Let’s say this $100 stock pays a $1 dividend and you own the 80-call. Is the 80-put, which you are giving up, worth less than the dividend?

The cost of the exercise (the time value of the option) vs its benefit (interest or dividend) is just the first step. But now you need to zero in on the when.

This is where we have to go to “marginal” cost/benefit. In other words…

b) Is one day’s cost of waiting (theta) greater than one day’s benefit (interest or dividend)?

When daily theta decay exceeds daily interest/dividend gain

Let’s get more concrete.

Stepping through the tests

We start with assumptions.

✔️Stock price = $100
✔️DTE = 60
✔️RFR (the rate you earn on cash in your account) = 5%
✔️Implied volatility = 20%
✔️ No dividends

We will analyze the 105 strike put.

I used a Black-Scholes European style calculator to compute the option values. You’re supposed to use an American calculator, but since I’m trying to explain the exercise rules, that would mask some exposition. Pointing out the European calculator’s mistakes will be better for learning.

Ok, we start with this table of our option values for each day until expiry.

Column explanations:

• Put theo: Black-Scholes value for the 105 put given our assumptions
• Call theo: Black-Scholes value for the 105 call given our assumptions
• Total interest: interest you’d collect until expiry if you exercised the right to sell shares at $105. Computed as 105e^(.05 * DTE/365) – 105
• theta from the option model
• 1 day’s interest computed as 105e^(.05 * 1/365) – 105
• Test 1: value of the call – total interest. This is a blunt total comparison of the put I own’s time value (represented by the call) vs the interest I’m forgoing by not exercising
• Test 2: 1 day interest that I forgo vs 1 day optionality represented by theta. This represents the marginal comparison of the interest vs optionality for 1 day.

At 12 DTE, the European model is telling us that the 105 put is worth LESS than its intrinsic value of $5. That’s a clue!

That’s the point at which the time value of the put (ie represented by the call on the same strike…remember put/call parity means the call is “in” the put) is LESS than the interest you’d earn if you exercised early.

💡The European put can and will trade under intrinsic. The American-style option should not trade less than $5 because if it did, you would simply buy the put, buy the stock, exercise immediately and have a risk-less profit of the amount it traded under intrinsic. So if for some reason you could buy the American-style 105 put for $4.92, you’d buy the stock for $100, then exercise the put which allows you to sell the stock at $105. Between the stock and put you’ve laid out $104.92 but your proceeds from the sale are $105. You pocket $.08 with no risk.

At 12 DTE, the 105 put has passed Test 1 (interest > option value). Visually, you can see this on the upper chart. But look at the lower chart…

The lower chart is the visual of Test 2 (1-day interest exceeding 1-day optionality).

5 DTE is the point of optimal early exercise.

Let’s do this again a bit faster with the 108 strike.

Since this put is deeper ITM, the corresponding 108 call is headed to zero earlier and the interest one earns on $108 is a bit more than the interest on $105 so without working through any math we expect to exercise the 108 put sooner than 5 DTE.

For the 108 strike, Test 1 (total interest exceeds the call value) is satisfied with 55 days until expiry, but the optimal exercise point isn’t until 20 DTE.

Interest “rent” is constant; theta is NOT

The daily interest benefit is linear. It’s basically total interest divided by DTE. But you may have noticed the theta values are curved. It’s intuitive that theta for an ATM option increases in an accelerating way until expiry. After all, with 1 DTE, the ATM option is entirely extrinsic value and you know in 24 hours the time value goes to zero.

For the 108 call, it has no value several days before expiry so there’s nothing to decay. The option went through the steepest part of its depreciation days earlier. Test 2 depends on when 1d interest and 1 theta “cross”.

For reference, this is a visual of theta vs DTE for strikes of various moneyness. Remember, the stock is fixed at $100. The closer the option is to ATM, the later it experiences its steepest decay. With a week to go, the far OTM 109 call has no decay. It’s already worthless.

Optimal exercise of American calls

You exercise calls early to capture a dividend. You must be the shareholder of record on the “record date” to be entitled to the dividend. When the stock goes “ex-dividend” it means any holders of the stock are NOT entitled to the dividend.

Owning a call option does not give you rights to the dividend since you are not a shareholder. That’s why the cost-of-carry component of option pricing discounts calls by the amount of the dividend.

When a stock goes “ex”, meaning the dividend has been paid out, the shares fall by the amount of the dividend which makes sense — the balance sheet has shed X dollars per share of cash.

The owner of the call will experience the drop in share price without any dividend receipts to make up for it.

💡If a $100 stock pays a $1 dividend and the shares open at $100 your brokerage or data vendor will say the stock is up $1 on the day. Unchanged would mean the stock should open at $99. If you own a dividend-paying stock it’s not “extra” return. The company just chiseled off a piece of its value and gave it to you in cash. It’s economically a wash. If they didn’t pay you the cash the company would retain it, the enterprise value would be unchanged and your return is the same. Your cash flow is different but of course you could have sold 1% of your shares to the same effect. In fact, that’s more tax-efficient. Of course, these are all first-order mechanical considerations. The properties of companies that pay or don’t pay dividends is a separate point of debate. If you do not believe stocks fall by the amount of the dividend, meet me at the corner of Trinity and Rector. I’ll be in a trenchcoat with a suitcase of Euro-style call options to sell you on a lovely selection of fat dividend yield aristocrats.

The optimal exercise of ITM American calls is easier. Test 1 is simple. Is the dividend I’m receiving greater than the value of the OTM put I’m giving up? [Again the put value tells me the time value or optionality of the ITM call I actually own]

What about the optimal timing of the exercise?

The marginal thinking represented by Test 2 is straightforward. The benefit of exercising the call early is discrete — it’s a dividend on a specified date. If the dividend is worth more to me than the time value of the call, I shouldn’t give up the time value until the last moment I have to capture the benefit. I should exercise on the day before the stock goes ex-dividend so I’m the shareholder of record.

Real-world considerations

• Early exercise decisions are directly dependent on interest rates (for puts), dividend amounts (for calls) and volatility (which influences the optionality you are surrendering when you exercise the option). Just think of the benefit you receive vs what you are giving up and what influences those quantities.
• Stock settles T+1. If you exercise a put on Thursday, your short share proceeds from the stock hit your account Friday, meaning you collect interest over the weekend. When I started in trading, stock settlement was T+3, so Tuesdays were “put day”. That’s the day you’d exercise to capture interest over the weekend. From 2017 to 2024, “put day” was Wednesday as the standard settlement was T+2. Microstructure nerds might be aware of a famous pick-off trade in the early aughts where a SIG alum bought shares from a NYSE specialist requesting T+1 settlement knowing that the company was going to pay a giant special dividend the next day. This ended up being very expensive to the seller. And eventually, to the buyer as this maneuver landed them in court. The option world is littered with dividend shenanigans. The range of ethical codes is wide and can certainly extend to “a moral obligation to relieve dumber people of their money” or “legal fees are part of my expected value calculation”. Having spent time in the trading world, I’m not surprised to when I notice these familiar moralities in tech, but a distinction in trading is pro vs pro violence was ok, ripping off customers was killing the golden goose.
• Sometimes companies announce a large dividend suddenly that the exchange will treat as special. Strike prices will be revised lower to account for the special dividend keeping the economic impact on options unchanged. That said, incremental changes in dividend policy are risks to option holders. Increased dividends lowers calls/raises puts all else equal.

We have an option calculator that allows you to compare the “early exercise premium” of American to European options:

https://www.moontower.ai/tools-and-games/option-pricing-calculator

🔗This is fully explained in Using Log Returns And Volatility To Normalize Strike Distances

The computation of this distance, besides being dependent on an estimate of volatility which we can borrow from the options market, assumes the asset is lognormally distributed. If you believe, as the options market certainly will if you look not at the at-the-money vol, but the far out-of-the-money call vols, that there is more positive skew than a lognormal distribution then our 3.6% estimate is too low.

But that logic is moving us in the right direction. We want to take the intel embedded in the options market when considering the price in the prediction market. We expect the liquid options market with much more volume and money behind it, to be the best guess as to the “fair price” of a proposition. If there’s an edge, it will be in the mispriced prediction market.

A prediction market bet can take a binary flavor. For example, “Probability that BTC settles above X by some date”

It can take a “one touch” flavor. “BTC to touch but not necessarily settle above X by some date”

Of course, “touch” is more likey than “settle” because “touch” encompasses all the times BTC settles above X, but also includes all the cases where it breaches X and falls back below X by expiry.

We can get information about the price of both binary and one-touch scenarios from the option market.