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.

Last week, in part 1, we backfilled prerequisite knowledge:

1. Distance in return space: equal percentage moves aren’t equal in compounded or log space
2. Vol bonus vs vol tax: trend and chop change the distribution of a levered asset
3. Derivatives-on-derivatives: options on the underlying ETF are inputs into pricing options on the levered ETF.
4. Vega convexity: OTM options have “vol gamma” or volga which makes their sensitivity or vol changes vary depending on the IV level. This is not true for at-the-money (technically at-the-forward strikes) options.

Each of these ideas plays a role in the question that brings us here today.

How do we price options on a 2x levered ETF?

Besides being a practical question, especially with the explosion of levered ETFs hitting the market, this exercise in pricing options on levered ETFs offers something more valuable: it’s pedagogically isomorphic to the fundamental problem of pricing options in the first place.

Sorry for the big words: pedagogically isomorphic. But as I was working on this 2-parter, it conjured that familiar feeling that every derivatives trader would recognize — feeling your way through a problem that has an uneasy tension:

It feels like this should be a solved problem with an agreed-upon solution, but as you mentally step through scenarios, the path dependence suggests that it cannot be a solved problem.

To say that the pricing options on a levered ETF is isomorphic to pricing options in general is a recognition that the same essential structure repeats itself. In both cases, the solution depends critically on your assumptions, and the practitioner must understand how those assumptions bias the model’s output. The model doesn’t deliver a “true” price that you can mechanically apply. Instead, it invites discretion in interpreting its output, demanding that you recognize where your assumptions are doing the heavy lifting.

The bad news is that you won’t walk away from this post with an Answer with a capital A. The good news is there’s room for disagreement if assumptions vary which means there might be opportunity to trade.

Either way, I think the bridge between the concepts we backfilled earlier and how they affect option pricing on a levered ETF will not only be enlightening but tighten your grip on option concepts that feel slippery in application. You’re not just learning a technique; you’re developing judgment about when and how models guide decisions versus when they merely formalize your beliefs.

Starting Point: Double the Vol

We’ll start with the obvious observation: a 2x levered ETF has roughly twice the daily volatility of the unlevered one. That’s our baseline.

Imagine both ETFs trading at $100.

The unlevered ETF has an implied vol of 20% across all strikes.

A reasonable starting model is to double the volatility at each equivalent strike from the underlying.

But what counts as equivalent?

(In our example, we are saying all the strikes have the same vol, but in the real world where strike vols vary, this question of equivalent strike is critical.)

Percent vs Log Moneyness

If the unlevered ETF has a 90 strike (10% OTM), what’s the corresponding strike on the 2x ETF?

Under percent moneyness, the answer is 80. The formula for this:

Equivalent Strike For Percent Moneyness = 2K-S

For example: 90*2 – 100 = 80

A 10% move in the unlevered ETF means a 20% move in the 2x ETF.

But in log-space, the math gives you an 81 strike. We start with the definition of logdistance from Using Log Returns And Volatility To Normalize Strike Distances:

Equivalent Strike For Log Moneyness = K² / S

Both make sense—just not at the same time.

If the unlevered ETF gaps down 10% then the equivalent strike to the 90 strike will be the 80 strike for the levered ETF with 80 and 90 being the new ATM strikes, respectively.

But if the unlevered ETF meanders lower, accumulating a total return of -10% then the log return will have been slightly less. (Just think of how a 10% annual return must mean less than a 2.5% return compounded quarterly to get to the same place).

The point is that “equivalent strike” is an assumption, not a fact. Each definition carries a bias about how returns behave and we don’t know what held until after the fact.

For the sake of heuristics:

• Percent moneyness assumes fatter tails and linear compounding of shocks.
• Log moneyness assumes smoother diffusion and constant proportional risk.

This table shows the equivalent strike and option values for the percent and log-monyeness assumptions:

Take note of the highlighted strikes which show how the difference in equivalent strikes vary as we get further from the spot price. Also note that distance when measured in standard devs for equivalent strike is the same for logmoneyness. This is a tautology as we defined distance in logspace or Black-Scholes terms.

💡The forward price

Another wrinkle in specifying the equivalent strike is the forward price. If the 2x ETF is hard-to-borrow but the regular ETF is not, the ratio of forward prices will be different than the ratio of the spot prices. The equivalent strike by any method will end up accounting for that if you replace S with F in the formulas. But if you have a pair vol trade on between the unlevered and levered ETF you are also exposed to any changes in the funding differentials between the 2 names because it will be inherited by the option prices.

Observing the effect on OTM calls vs OTM puts

If we price the 2x ETF’s options using log-equivalent strikes and double vol, the ATM options will indeed be about twice as expensive as their unlevered counterparts.

OTM strikes won’t scale that cleanly.

OTM calls on the levered ETF will be worth more than 2x at the equivalent strike while OTM puts will be worth less than 2x.

You can likely explain this in several ways. For our purposes, which ultimately is about pricing options, our going to attribute this phenomenon to volga aka “vol gamma.”

In Finding Vol Convexity, you see a chart of vega at different vol levels. You can infer that OTM call volgas are higher than OTM put vegas as they are “picking up more vega” as vol increases.

💡A non-visual explanation

We can reason our way to the same conclusion by recalling arbitrage bounds. Upside call values are arbitrage bounded by the stock price which itself is unbounded. Put max values on the other hand are bounded by the strike price. Upside options have more to gain from vol going up.

The Problem with Doubing Vol: Amplifying Skew

While doubling vol doubles the value of ATM options, it more than doubles OTM options because vol convexity makes them increasingly sensitive as IV rises.

Recall this chart from part 1:

What’s the problem? Doubling vol, especially on already skewed puts, risks modeling put spreads too cheap.

To keep vertical spreads from becoming unrealistically cheap, the levered ETF’s vol surface likely has flatter skew than the underlying’s—it passes through less of the amplification than a pure 2x model would predict.

🧪An untested hypothesis

As an example of something I might try to throw against the wall:

Perhaps the lower strike put shouldn’t trade below the price that keeps an equidistant, vega-neutral put spread identical between the levered and unlevered ETFs.Sometimes model-building is oberving the market prices, trying to fit the behavior, and then trying to reverse-engineer what rule, in words, the streamer’s model seems to be projecting.

The Problem with Return Assumptions: Path Dependence and Time Resolution

In Risk Depends on the Resolution, I show that volatility scales not only with sampling period* but with time.

💡The coastline paradox

This is not just a property of vol. The length of a coastline is not a given. It depends on the length of ruler you use to measure it.

A 6σ daily move is possible. (It happened on Oct 10th intraday in IWM)

A 6σ annual move is a fantasy.

As explained earlier, the “correct” equivalent strike depends on how you assume vol scales with time. Black-Scholes’ lognormality says variance is linear with respect to time (or volatility is proportional to root time). It’s just an assumption.

Realistically, equivalent-strike mapping probably has a curved relationship with time, with the growth process looking more lognormal as you go out in maturity.

Reconciling the Trade-offs

Skew amplification

We have competing effects:

Fat tails, especially in near-dated or same-day maturities, demand higher vols, but that same adjustment can make longer-dated put spreads too cheap.

The skew might vary in the unlevered ETF to account for the time-dependence of distributions, but the effect is amplified when we double the vols. The pass-through values risk being excessive.

My guess:

Levered ETFs should show larger vol premiums than 2x for the far OTM put tails (1-5d deltas) in near-dated expiries but smaller than 2x differences in vols at equivalent strikes for meatier strikes (say 5-30d).

🧪Testable expression of this idea: Is the 45/25/5 delta put fly empirically expensive if you used 2x vol at equivalent strikes?

Browsing option surfaces, skewed options at equivalent strikes on SSO and NVDL (2x versions of SPX or NVDA) trade at lower vols than 2x the underlying ETF vols just based on 1-5 month expiries.

Market widths mask what’s going on in the far tails of the 2x ETFs or in some cases the strike ranges don’t go far enough.

Path dependence and modeling the stock process

You can always try pricing options on the 2x by Monte Carlo simulation based on a tree of the underlying.

You still need to specify a diffusion process. Just a thought on that:

My former colleague at SIG, quant Vivek Rao, has a repo that simulates terminal prices (not log prices) under a Student-t process and makes the case that this conforms better to the vol smiles we actually see in markets. This might be especially relevant for nearer-dated distributions, which look more normal than lognormal.

Wrapping up

No matter how you price options on a 2x ETF there will be a bias. Even the simple question of which strike on the levered ETF maps to the unlevered ETF depends on assumptions of the stock diffusion process.

If you use percent difference (ie 2K-S) you will structurally underprice the 144% call versus some using log difference (K² / S) to find the equivalent strike. The path (trend vs chop) will ultimately be the judge of who was right.

I suspect this is how I got smoked to a short SCO (inverse levered oil) call back in 2014 after OPEC lifted production quotas to bury the US shale competition. Oil went down relentlessly in a crescendo of fundamental realignment and self-reinforcing gamma flows as oil liquidity providers are structurally short puts. A seemingly far OTM SCO call turned out to be “much closer” because of the path USO took.

(Think of it this way: if USO goes down 10% 2 days in a row, the stock is down cumulatively 19%. The inverse levered ETF is up 44% not 38%).

These modeling problems are endemic to option pricing, generally not just levered ETFs. If you price options by sampling vol daily instead of weekly in a name that trends, you’ll sell the option too cheap. The problem is that you extrapolated vol via root time but that understates vol for a trending asset.

[See Adjusting volatility for autocorrelation. If only we knew ahead of time if an asset would trend or mean-revert!]

Pricing options forces you to stare straight at your assumptions. The path, the volatility resolution, the return process—all of it shapes how “equivalence” behaves.

The point isn’t to fix the bias but to know which way it leans. This awareness is the space between what’s priced in and how it differs from your analysis. It’s the foundation of disagreement and ultimately trades.

“They” say human labor will be irrelevant by 2027. By then, any business you can think will be solved by capital (electricity and tokens) before you brush your teeth in the morning.

You either get rich in the next year or join the permanent underclass.

So we aren’t shocked that the hottest fads in investing is pure return fuel:

• double, triple, even 5x levered ETFs
• options (record volumes, with nearly 2/3 of SPX options in 0DTEs!)

You’re not gonna break outta that underclass clipping 10% per year amirite?

I’m not here to judge whatever you think you need to do. The Moontower SOP, as always, is:

I give you tools, you use them as you see fit.

We need to talk about the collision of these trends:

options on levered ETFs

We can hack together a pricing framework. I say “hack” because we use a few building block concepts, combinations of arithmetic and logic, to construct a model. The process fuses several ideas I present often, reinforcing your comprehension of the basics, while alerting you to common misunderstandings.

We’ll do this in 2 parts.

Today, we cover four essential concepts:

1. Distance in return space

2. “Vol bonus”

3. Derivatives on derivatives

4. Option vega and convexity

Next week, we arrange the basics into a model for finding a fair price for an option on a levered ETF.

Essential Concepts

1) Distance

• A $100 stock that goes up 10% 2 consecutive days is now $121.
• A $100 stock that goes down 10% 2 consecutive days is $81.

In return space, $21 on the upside is the “equivalent distance” to $19 on the downside.

2) Vol “bonus”

I’ve written as much as can be written about vol drag/volatility “tax”/volatility “drain”, whatever you want to call it. It’s the “chop”. Up 10%, down 10% yields a cumulative return of -1%.

Trending leads to a vol “bonus”.

Imagine a 2x levered ETF on the $100 stock from earlier. Let’s say the ETF also starts at $100.

◾Reference stock goes up 10% 2 days in a row..

$100 —> $110 —> $121 or a 21% cumulative return

🟩2x levered ETF goes from $100 —> $120 —> $144 or a return of 44%

If you sized your ETF position half as large to have the equivalent risk or beta exposure you will have outperformed an equivalent risk holding over 2 days, although indifferent after just 1 day.

The vol bonus comes from trending.

If you step through a random walk where a stock can either go up or down X% you find that in most paths, the ones where we chop or recombine frequently (the paths fenced in red), your compounded return is less than if you simply added up all the daily returns and applied them to $100. Be careful: volatility doesn’t change your expectancy, only the distribution of return.

The infrequent, large vol bonuses offset the frequent vol taxes to keep the expectancy the same, even though you “usually” suffer a vol tax.

3) Derivatives on derivatives

A levered ETF is a derivative with a fair value. At any given moment, its NAV should be equal to:

NAV at the start of the day * (1 + reference asset’s return * leverage factor).

This is approximate since you should also deduct one day’s worth of expense ratio and the fund’s trading costs divided by the share count, however these values will have negligible impacts on daily fair value calcs for most users. The point is that a levered ETF is a derivative just like all ETF values are derived from an underlying basket, future, or security.

We want to price options on the levered ETFs which means we need a volatility surface. The reference asset’s option chain will provide a consensus vol surface. We pull a page straight from the arbitrage trader’s handbook — use a liquid market to price a closely related market after adjusting for the differences. This will provide a fair value of the levered ETFs options relative to the reference asset’s option prices.

4) Option values, vega, and curvature

We will cover one last “basic” before trying to reason our way towards a model for pricing options on levered ETFs.

Consider an at-the-money option.

💡Technically I should say at-the-forward but we can assume a RFR of 0% in which case ATM = ATF

If we double the vol, what happens to the option price?

It doubles.

The approximation for an ATF call:

ATF call ~ .4 * S * σ * √T

where:

S = stock price
σ = implied volatility annualized
T = fraction of a year until expiry

If c doubles, the call doubles. It has a linear dependence on vol.

💡Visual derivation of the ATF option approximation

Look at the approximation again.

What’s the vega of the ATF call?

Before you go searching for a Black-Scholes calculator, just recall that vega is the change in option value for a 1 point change in vol.

If σ increases by 1 point, the ATF call increases by .4 * S * √T

If we are pricing a 1-year ATF call on a $100 stock, if vol increases by 1 point, the ATF call goes up by $.40

Therefore, by definition:

ATF vega ~ .4 * S * √T

💡Example: Consider a 1-year ATF call on a $100 stock. If vol increases by 1 point, the ATF call goes up by $.40

Notice that the vega itself has no dependence on the volatility.

This is only true for ATF options!

Out-of-the-money option vega DOES depend on the vol level. It’s not hard to understand this intuitively.

Consider the 60 strike put on a 16% vol, $100 stock expiring in 3 months. 16 vol is roughly SP500 vol. The 3-month, 40% OTM put is probably worthless.

💡16% is a 1-year standard dev. 16%/√4 = 8% quarterly standard deviation. That put is 5-standard deviations OTM. Save me the Taleb-stanning that option is worthless.

A worthless option has no vega. If I raise the vol from 16% to 17% that put is still worthless. It has no sensitivity to the vol.

But that can’t be true for all levels of vol. If we 10x the vol to 160% then that put is now only 1/2 standard deviation OTM. It’s most definitely NOT worthless.

💡Re-framing volatility as time: Increasing the vol by a factor of 10 is algebraically equivalent to increasing time-to-expiry by √100. Even if we stick with 16% vol, a 40% OTM put on the SP500 with 100 years til expiry is clearly valuable

So we know that the 60 put acted like it had 0 vega when vol went from 16% to 17% but when vol 10x we expect the option to have a non-negligible valuation. Somewhere along the line, this option “picked up vega” or sensitivity to volatility.

While the ATF option is a purely linear dependence on implied vol, the OTM option’s dependence ranged from 0 to some positive number. This is the source of vol convexity.

💡See Finding Vol Convexity

A far OTM option might have no vega. But as implied vol increases, that option’s strike “becomes closer” to the ATM strike. I mean if vol is 200% there’s really very little difference between the 70 strike and the 60 strike in terms of standard deviation.

As vol increases, OTM option vega increases. If you keep jacking up the IV, the vega eventually peaks. The maximum vega an option can have is the ATF vega, which has no dependence on vol level at all.

I computed European-style option values using Black-Scholes for a range of strikes for 3-month options on a $100 stock using 20% vol at each strike.

Then I doubled the vol to 40% on each strike and computed the difference in option values:

The ATM options have the most vega and will be the most sensitive to vol so naturally they go up the most.

But remember, at 20% vol some of those OTM options would be close to worthless but when vol doubles, an option like the 80 put went from $.04 to $1.16! On the chart, it gained $1.12.

If vol doubles instantly, your ROI on an ATM option is 100% but doubling the vol on an OTM option leads to comical ROIs:

At the extremes, you get divide by zero errors as the return is infinite on a previously worthless option.

💡Clarification: I’m measuring changes in the OTM option on each strike. In other words, the “extrinsic” option. So for the 90 strike we are using the put, but the 110 strike, the call. The price changes will be the same for the call or put on the same strike, but the ROI due to a vol change shouldn’t be muddied by the instrinsic value which is constant on our examples, so we use the OTM option.

We’ve covered the essentials, next week we’ll answer:

• How do we translate the reference asset’s vol surface to the levered ETF?
• What does this mean for strike selection?

A traditional way to think of a stock price is the expected value of its future prices weighted by their probability and discounted to the present. Ignoring the cost of money, in a binary world a $100 stock could be fairly priced if it was 50/50 to be worth $200 or $0. It is also fairly priced if there’s a 20% chance of it going to $500 and an 80% chance of $0.

There’s this vocal VC named Keith Rabois who aggressively cheerleads his companies. I don’t know the guy. His online persona exudes many standard deviations of F-U confidence. Sounds par for the uber-rich these days, but he’s extra fun because he’s pugnacious. And got baited into a silly pride bet.

Here’s a tweet from investor @compound248:

We wouldn’t talk about this in the Masochists section because this is fairly basic financial reasoning. The type that really needs to obvious to everyone in a society is flirting with a simulcrum of the movie Rat Race. But it’s appropriate to spell out the opportunity here in gambling terms.

Keith is offering an even-money bet, his $100k to Compound248’s $100k on the stock multiplying by 31.5

If you think in odds:

Keith is offering even money on a 30.5-1 odds proposition

That might be more clear when you think of Keith buying the stock. If he buys $OPEN he risks losing the stock price or 1 bet and if the stock goes up 31.5x he wins 30.5 (because the 1 bet or amount of cash he spent for the stock is not part of his win or return).

It’s similar to how a stock 2x’ing is 100% return, or 10x’ing is a 900% return. A stock that 10x’s paid 9-1. You risked S, you won 9S if S is the stock price.

Normally when you buy a stock, you get paid dollar for dollar as it moves times the number of shares you have. If the stock doubles you make S in profits which is how much you risked when you bought it. You are paid in proportion to the move.

Keith needs a heroic move to simply get paid even money. His proposition is an arbitrageable violation of how return works. I don’t know anything about Compound248’s outlook on $OPEN by him taking the bet. He could be bullish or bearish. When you hear the proposition, your mind shouldn’t go to “Is $OPEN a good or bad investment?” because what you should do doesn’t depend on this assessment.

Keith’s offer is free money regardless of your outlook.

He’s laying 30.5-1 odds where the max loss is $100k.

So you solve for “How much OPEN do I need to buy to make a $100k profit if it pays 30.5-1?”

It’s simply:

1/odds * bet size

1/30.5 * $100k = $3,280.21

I need to buy $3,280.21 worth of shares. Since the stock was $8.48 that’s just about 387 shares.

If the stock goes to 0, you lose the $3,280.21, but Keith hopefully pays $100k. If the stock does go up 31.5x, you break even.

You could also structure the hedge so that at a stock price of 0, you break even. You buy $100,000/$8.48 or about 11,792 shares. If the stock hits Keith’s bogey you get paid 30.5 on your $100k and you happily peel off 1 bet size to him as a tip. Any share quantity you buy between 387 and 11,792 is a guaranteed win.

An amusing post-script to this story is HF manager Benn Eifert requesting $10mm of action on this proposition. Of course, Keith said no — he’s confident not stupid. Keith said he did the bet with Compound248 just to shut the “troll” up or something.

I don’t understand how rewarding a troll with the easiest money I’ve ever seen is anything but encouraging future trolls, but maybe this is why Keith is rich and I’m writing on the weekend.

This is a fun one.

A good friend and mentor from the pit sent me this (lightly edited and hyperlinked):

I was catching up on moontowers and reading your cotton story. Made me think of one that you can use if you ever cover interest rates.

I started in trading with silver options. I was quite meek as I had never clerked. I was just backed and thrown in the pit at age 22. Anyway, a couple of years in, silver had rallied from around $4 to $7. The whole pit was short long dated $4.50 calls to a spec who was holding the long. These things were easily exercisable as interest rates were high. Great short to have.

Broker [badge redacted] (you may remember him) offered the synthetic put at zero. No one knew what he was doing. I bid 2 ticks under and he said sold.

So I said “1000”. I think the biggest trade in that pit was like 200.

Anyway I took 1000, exercised them and made 10 grand before commissions. The carry was probably north of 50 ticks and he increased his shorts by around 800 as all the other locals got hit with my exercises. The other locals were pissed.

So I came in through a broker and bid 2 ticks under the next day. Locals hit me. I got lucky because I was clearing [redacted prime broker] and they neglected to put in my exercise. They delayed it by mistake (and gave me the interest as I recall.)

Anyway the locals thought they found someone who would hold these things and just wanted a synthetic put for a credit. I kept doing the trade for a week.

[Redacted broker] got it right by the second day. I think I made another 5 or 10 grand before the locals figured it out and stopped doing it.

Anyway. I hope if you relay this no former silver option traders subscribe. They still don’t know it was me!

 

There’s a lot going on here!

Option pricing mechanics, pit dynamics, deduction.

Let’s start with the option pricing.

I don’t want to deny you the opportunity to figure that part out for yourself.

In the following list, I’ll start with important clarifications but as the list unfolds the material is more of a hint than just reference information.

Like a quiz show, buzz when you understand why “locals” [ie the other traders] were willing to” sell 2 cents under” and why my friend was willing to buy that level.

Clarification and hints…

• 1 silver option references 1 silver future. This is typical in commodities whereas equity traders are used to an option having a multiplier of 100
• 1 silver future references 5,000 troy ounces of silver. So if silver is $10 an oz, it’s a $50,000 contract
• The minimum increment, or “tick”, in silver options is .001 or 1/10th of penny. Since it references a 5,000 oz contract, a penny is worth $50 and a tick is worth $5.
• This story happened many years ago. Look at those silver prices: “Silver had rallied from around $4 to $7. The whole pit was short long dated $4.50 calls to a spec who was holding the long.”
• Silver options are American-style meaning you can exercise them anytime.
• “Random assignment”: when an option is exercised, any contract in that series held short is equally likely to be assigned regardless of the clearing firm or account. See rules.
• When a broker quotes a “synthetic put” (and yes synthetics are not just identities but directly traded orders!) the convention is to bid or offer as a “differential to intrinsic value”. When my friend bid “2 ticks under”, it’s understood that he’s willing to pay 2 ticks under “intrinsic value”. If the futures are $7 and the broker sells the $4.50 call 2 ticks under than the trade package is:
  • broker sells the $4.50 call at $2.498 to my friend
  • brokers buys the future for $7.00 from my friend
  • Make sure you understand this: to buy a synthetic put it to buy call and sell the future “1-to-1” (meaning for every option you buy, you sell 1 future.) Another way to express that is you hedge on a 100 delta. A synthetic put is assumed by everyone to be a simultaneous package of “long call, short future” in the same way as a straddle is call + put on the same strike or strangle is call + put on different strikes. “Synthetic put” is a real tradeable thing not just the name for an option identity.
• While futures are subject to initial and maintenance margin, option premiums are settled in full. In other words, the premium is not itself marginable. That is normal but worth stating because there are some option markets where the premium is marginable (ie WTI options on ICE)

At this point, you should be able to understand my friend’s side of the trade. If you don’t want to bother, stick around, I’ll spell it out soon enough but also that part should feel really obvious to anyone that’s ever owned an American option.

The harder question is:

Why were the broker and the other locals willing to sell him the synthetic put 2 ticks under?

Your last hint is a line already tucked into the story:

The whole pit was short long dated $4.50 calls to a spec who was holding the long. These things were easily exercisable as interest rates were high. Great short to have.

This is fun stuff. Let’s unpack it.

Why did my friend buy the synthetic put 2 ticks under?

It’s an American-style option. He buys the $4.50 call for $2.498 and sells the future at $7.00.

He exercises the call immediately, effectively paying $6.998 closing out the future he sold at $7.00. He makes 2 ticks or $10 actual dollars x 1,000 contracts. $10k profit (before exchange fees which probably claimed ~ 25% of that). Call it $7,500 for a minute’s work and no risk.

How could my friend’s buy be so good if he also says:

The whole pit was short long dated $4.50 calls to a spec who was holding the long. These things were easily exercisable as interest rates were high. Great short to have.

The key:

The carry was probably north of 50 ticks

50 ticks = 5 cents

In other words, the interest on $2.50 of intrinsic option premium was 5 cents or about 2%

💡While a market-maker will be hedged with futures, you only need to post margin to maintain that leg. You can satisfy the collateral requirement with T-Bills, so you really are capturing the the interest on the short option premium without having it offset by hedge funding which is close to zero.

💡In the story, there’s 5 cents of carry. If we knew the DTE, we could back out the interest rate prevailing back then. If we knew the interest rate, we could back out the DTE. But we have neither. We just have the recollection that the call had about 50 ticks of carry.

Think of the typical local’s position:

The whole pit was short long dated $4.50 calls to a spec who was holding the long.

The locals are hedged. They are short the deep in-the-money calls and long futures against them on a 100 delta. In other words, they are short synthetic puts. The puts on that strike are worthless however if the spec never exercises the calls, the locals will get about 50 ticks of interest. It’s like selling a worthless put for a nickel.

💡If you’re so Taleb-pilled you can’t imagine a put being worthless, just pretend you could buy the actual $4.50 put for a tick or the $4.75 put for 3 ticks. There’s such a thing as arbitrage bounds. Options 101 stuff.

Amateurs will say things like “that’s worthless” because they believe a price “can’t get there” but when I say it’s worthless I mean by arbitrage. Like the value of the put on that strike is dominated in such a way that if you could sell it at a positive value there’s free money on the board. An option trader thinks in a matrix of arbitrage relationships when they examine a chain.

You can see why the locals were willing to sell the put 2 ticks under. They didn’t think they would get assigned! They assumed they were going to collect 50 ticks on a riskless position.

The mistake was in thinking my friend wouldn’t exercise the calls. Perhaps they thought he needed to cover risk and if they were all short the calls he was just closing. But that doesn’t make sense to me since he was a “meek” trader before then and certainly wouldn’t have been short 1000 calls.

That’s the most puzzling part to me. I understand why they might believe a customer like the spec would not exercise the call optimally, but when a market-maker buys them you need to update. “This guy who trades options all day is buying an American-style option below intrinsic, I wonder what he’s going to do?”

I don’t want to be too snarky because “street smart” is probably one of the most salient features of a pit trader. I’ll assume I’m missing a detail that makes their decision justifiable.

 

The rude awakening

“The other locals got hit with my exercises. The other locals were pissed.”

This is the random assignment. When my friend exercised his calls the free interest gravy train slowed down as suddenly some of the locals got assigned which closed their positions (the short call goes away and the long future they were hedged with is liquidated to the exerciser).

💡In random assignment, you “expect” to be assigned on your pro-rata portion of the OI held by all shorts. How many you actually get assigned on can vary from this theoretical expectation because the sequential 1 by 1 assignment process is memoryless just like dealing from a deck of cards with replacement each time. Just because you got tagged on the last one doesn’t change your odds of getting tagged on the next one.

Let’s keep unpacking the story.

Poisoning the well

My friend masked his behavior to blend in with customers and used a broker to “bid 2 ticks under” the next day. The locals didn’t think it was a market-maker going through a broker. The locals were reasonable in not suspecting a local to be hiding themselves behind a broker order for 2 reasons:

1. Paying commission which would eat into the slim margin even further.
2. This behavior is known as “poisoning the well”. This type of pro vs pro crime was considered bad form. Just like in poker how the pros try to avoid each other and just eat the fish. Of course from the outside, the norm is considered anti-competitive.

Let’s address both of these.

Regarding #1:

My friend may have worked out a deal with the broker. After all, the broker stands to make a thousand bucks or so for no real effort and if the order is contingent on a reduced commission rate they’ll probably go for it.

[I used to do this all the time. Wet the brokers’ beak. We like to talk about nerd stuff and math here, but trading is a business like any other business. Giving out orders is currency. Traders who pay lots of commission magically get the first call. Whether it’s commodities or equities brokers can solicit the other side rather than bring an order to the pit and instead just cross it without worrying about the order being broken up (ie bettered) they can double-bill. Brokers are fiduciaries and their business is competitive, get too many bad fills and you risk losing a customer, but there’s a lot of leeway in discretion. I’m using a broad brush, the rules and details vary across asset classes, but the mechanics rhyme everywhere.]

Regarding #2:

Prisoner’s dilemma.

How much is it worth to defect?

Will we return to stasis and you got away with one or did you really poison the well?

Will anyone find out?

Does any of this change if you don’t really fit in with the club and can’t hope to be part of it?

And then there’s pure disagreeability mixed with self-confidence. I was at Max’s soccer game last weekend and the ref in a moment of deprecating humor said “They told me when I was a kid you need to either be smart or likeable. And I’m definitely not smart”. I don’t think this particular buddy ever cared about going along to get along. But he’s also super-smart.

I’ll add my own perspective to the game theory stuff. I did nothing but watch markets get tighter, more competitive and more ruthless over time. The edge you never would have settled for becomes something you’d kick grandma down the stairs for before you know it. Someone is going to defect.*

This is chained to another observation. In a landscape where there’s excess profit (ie too much reward per unit if risk) there is always someone sandbaggin’. They aren’t showing how smart or fast they are because they will harvest at this level before it tightens and not have to tip off to others what’s possible. I know this firsthand because of a friend that runs a brilliant trading operation in a niche space in which he never shows the market how fast he really is. Imagine my lack of surprise when I read about this “don’t show” tactic from HFT-er Liquidity Goblin’s Let’s Pretend We Have An Edge (paywalled).

A fortunate mistake

I got lucky because I was clearing [redacted prime broker] and they neglected to put in my exercise. They delayed it by mistake (and gave me the interest as I recall.)

Anyway the locals thought they found someone who would hold these things and just wanted a synthetic put for a credit. I kept doing the trade for a week.

[Redacted broker] got it right by the second day. I think I made another 5 or 10 grand before the locals figured it out and stopped doing it.

My friend was able to milk the ruse for a bit longer because of an error! The prime broker failed to process the exercise which reassured the locals briefly that they were not getting picked off. Sounds like it gave him at least another day to buy more synthetic puts for a credit before the locals got wise to the game.

Knowing is half the battle

Let’s put a bow on this like an 80s cartoon ending lesson.

Being short those synthetics to a single counterparty who doesn’t realize the calls should be exercised is good fortune. But once someone wants to buy those deep ITM calls opening, you know that they know what they’re doing.

I’ll tell you from personal experience that anytime someone wants to trade a deep option or a reversal/conversion where there is little or no open interest your guard goes up. You check your funding assumptions. You think harder about what your option model is saying about the value of an American vs European style rev/con. The difference between the 2 represents the value of the early exercise option for that strike. Those are modeled via trees simulations and difficult to debug as opposed to closed form equations for European Black-Scholes.

Only the paranoid survive.

---

 

*I refer you to this excerpt from A Former Market Maker’s Perception of PFOF:

SIG wasn’t know as the “evil empire” on the Amex just because of the black jackets we wore. They understood the risk-reward was completely outsized to what it should be 25 years ago. They were amongst the first to tighten markets to steal market share. They accepted slightly worse risk-reward per trade but for way more absolute dollars. They then used the cash to scale more broadly. This allowed them to “get a look on everything”. Which means you can price and hedge even tighter. Which means you can re-invest at a yet faster rate. Now you are blowing away less coordinated competitors who were quite content to earn their hundreds of percent a year and retire early once the markets got too tight for them to compete.

SIG was playing the long game. The parallels to big tech write themselves. A few firms who bet big on the right markets start printing cash. This kicks off the flywheel:

Provide better product –> increase market share –> harvest proprietary data. Circle back to start.

The lead over your competitors compounds. Competitors die off. They call you a monopoly.

Recently, I’ve been writing a lot about option funding — implied rates, reversal/conversions, and financing stock positions, and even a touch on early exercise. The posts from earliest to most recent:

• the easiest win in options is for stock traders
• implying the cost of carry in options
• what hides in the option chain
• the art of paranoia

You can’t really overstate how important this is, especially to pros.

On the same theme, Stanford lecturer and HF manager Kevin Mak published an outstanding and detailed post:

Holding a high cost to borrow stock? Here’s how to collect the borrow fees using the options market (link)

I want to quote a few key parts:

I cannot stress enough that if you want to be competitive in capital markets, you cannot afford to make these types of mistakes and forgo this return.

Kevin is quite blunt:

If you think this is too complicated to follow, to be blunt, you likely do not have the mental fortitude to have alpha in markets (you may still make money via luck). If you CAN figure this out, but couldn’t be bothered to spend time on it, that’s probably fine, but I suggest staying away from holding a stock with a high cost to borrow. If you insist on holding high cost to borrow stocks, and “not worry” about collecting the free borrow/lending fees, you really should be playing triple-zero roulette, or splitting every pair of 10’s at Caesar’s Palace instead because that would have a lower edge, and be way more fun.

I suggest that by using options to refinance your position, you “inherit the market maker’s funding rates” which are almost certainly better than yours, whether you are long or short.

This is Kevin’s way of saying this which might land better for some (emphasis mine):

If this is arbitrage, it shouldn’t exist right? This unique situation is you “arbitraging” your own holdings. You’re basically holding an inefficient asset since you can’t lend it (or collect the lending fees) so this lets you own it in a more efficient manner. Nobody is going to compete away this arbitrage because nobody can access your holdings except you. In fact, it’s the arbitrage happening in the open markets which is pushing the value of synthetic long to be equal to the long stock (and collect borrow) position.

Kevin wisely tucks all the brain damage into the endnotes to not ruin the flow and central message of the post. I love his intro to them:

This is a beginner treatise on a synthetic long position and covers ~98% of what you need to know about it. The last 2% I could write 25,000 words about and not be finished.

In the spirit of the endnotes, I want to share a recent real-world example of a scenario that resides in that annoying “2%”

Alex, a mutual of mine on X, asked the following:

I bought some deep itm $AIV calls, I think Jan exp, a couple weeks ago because I thought they were cheap vs my financing cost. Not much extrinsic value… Underlying had silly spike after hours, up then down.

Was the right way to trade this, to short the underlying after hours? I thought about exercising to sell, but didn’t want to burn a few months of time value for nothing, even if I didn’t really pay for it.

I’m going to step through the conversation, injecting meta-commentary to bring it down a level.

Me: Your trade expression was because of financing but you are asking about what you should have done with your delta. I’m not being pushy, I’m just trying to help you answer your own question.

There are 2 things I can see so far:

• Alex is conflating his strategic financing decision with tactical delta management. I do the Socrates thing as a habit when learners entangle multiple issues into a single question. If I just jump ahead and answer each part of the implied questions, I rob Alex of a valuable opportunity — to debug his own thinking by breaking the big question into its components. I notice many people getting option or trading concepts rolled into a ball of yarn, which tells me they don’t understand each string as well as they should. When you’re forced to decompose your own question, the deficiency is self-apparent.
• He’s showing good instincts about not wanting to exercise the calls because he’d be “burning a few months of time value.”

💡Option Pricing Clarifications

1. The first [but not final] check on whether you should exercise a call is if the value of the OTM option on the strike is worth more or less than the cost of carry.
   1. For an ITM call, is the dividend you are exercising for worth more than the put on the same strike?
   2. For an ITM put, is the interest on the stock short if you exercise greater than the call on the same strike?
2. Alex said he thought about “exercising to sell,” but to be clear, you can’t exercise the calls after-hours and have it settle the next day. There is a cut-off for exercise notices. He would have needed to short the stock after hours, then exercise the next day, meaning he would have to carry a short for 1 day. This tactic would have flattened his delta, but he would have to incur any financing ramifications for the 1-day short.

This simple scenario highlights a risk to using heavily-discounted synthetic futures to express a long:

If you plan on liquidating the length before expiration, you are exposing yourself to the change in funding that you otherwise locked in.

In this case, Alex is exposed to the borrow he may have to pay on short shares if he chooses that route to cut length. One day’s borrow, even if steep, is unlikely to make or break the whole trade, but the bigger issue is the “locate”. There might not be any shares to borrow from your broker. If Alex was long the stock instead of the synthetic stock, he could just sell it in the after-market.

If Alex chooses to cut length by selling the ITM call, it may be for the same discount that attracted him to buying them in the first place. It’s like buying a home for a discount because it has lots of street noise. It’s a problem you can’t outrun because you will be on the losing end of the discount when you sell the house.*

Stay groovy

☮️

 

*Tangential, but I see people misunderstand this in the private school vs public school debate. If you pay up to live in a good school district, you are exposed to 2 risks:

1. The carry cost on the premium. If you pay 20% more for a good school district, the funding cost on the extra ~20% of mortgage is the cost. If schools are the deciding factor between an $800k or $1mm house and mortgage rates are 5% then your pre-tax cost of the better schools is ~$10k per year. This is probably much less than private school tuition, especially if you have multiple kids.
2. The perception of school quality. The “school premium” can expand or contract over time. You can win or lose to this when you sell.

In general, the true cost of the premium you pay for good school districts is very small relative to private school tuition, making your VORP hurdle for private school choice high (although worth it for many people).