Giant swaths of the investment industry are doing nothing but selling soothing balms. Placebos. It is the vitamin industry at best and a penis-pill pop-up banner when it “democratizes private investment.”

But it could be no other way.

If you can statistically prove your strategy has alpha you also know exactly how to price it to take all the surplus. The market for making alpha is like any market — it equilibrates based on supply and demand. There’s more wealth out there in search of a return than the capacity to absorb which is why pod shop bosses feast.

I’m not knocking this. Their CAGR, ie returns net of vol drag, through insane market environments after fees are perfectly fine. In fact, it seems that the HF market is more efficient compared to the amount of over-earning for beta that went on in the early 2000s between the dot-com meltdown and the GFC. People that made several lifetimes of loot telling just-so stories to allocators who didn’t notice how Moneyball applied to their field.

The GFC disillusionment revealed many stories to be no more than fairy tales. There was an opening for a new story that perfectly complemented the spread of technological capacity and its rider, technical skill.

Evidence-based investing.

With large data sets and faster computers we could solve investing like a physics problem. Engineers aren’t fooled by steak dinners and silver-tongues. The softest stuff they read is Kahneman who holds the why for why their factors work. All of it has the sheen of the scientific method.

Except there’s one lingering inconvenience. It’s an inconvenience that’s obvious to gamblers. I think most investors can feel it in their bones. Not surprising, we are all natural gamblers to some extent (we eat hot dogs and let strangers drive us around).

The inconvenience is the uselessly long feedback loops. Which we are going to discuss. But it’s worth mentioning that the feedback loops double as a defense for asset managers. They get to say “this works over time and if it worked all the time it wouldn’t work”. That’s true but it doesn’t solve my problem — I STILL HAVE NO FEEDBACK — plus the defense IS convenient to the fee collector.

So we’re left with “keep buying my pills because they might work”. Good luck getting a refund if the fish oil doesn’t make you live longer. The whole arrangement is irreducibly uncomfortable. That’s why it’s called the Paradox of Provable Alpha (and why I need to one day finish writing moontowermoney).

All of these thoughts were stirred up again as I listened to Adam Butler on Excess Returns.

An early quote in the interview:

The size of these edges is so small relative to the noise we encounter daily — especially compared to the gyrations of the underlying indices — that it’s very difficult to make high-confidence, informed choices in advance. In other words, it’s hard to know which edges or strategies to allocate to in a portfolio with any certainty that they’ll outperform a random selection of other possible strategies over the next 10, 20, or 30 years of your investment horizon…your skill in selecting strategies in advance based on even very long histories of performance is pretty close to zero.

I should clarify — Adam distinguishes investment or factor style edges from trading or niche forms of investment that rely on some form of information advantages that have built over time. I think Adam would agree with me when I say trading is like any other business but for superficial reasons gets confused with investing.

I agree with his understanding of pod shops:

Pod shops are really looking for people that genuinely have alpha, so I think it’s useful to kind of distinguish between what we might call sort of systematic factor strategies and alpha.

Alpha comes from somebody who has very particular niche insight or information or experience within a fairly narrow domain of the market. So, for example, we have a client who allocates to a municipal bond manager. Now, this manager has a hard cap at about a billion dollars. The team that runs it spun out of what used to be the largest muni market-making desk — worked there for 20–30 years.

What did that give them? Well, it gave them access to knowledge of where all of the flows from muni bonds — all of the issuance from the muni bond sector — are coming from, the different state governments, who the decision-makers are there, how they can get inside information on what type of issuance is coming down the pipe. And then, being at the center of flows in the muni market — which is a very niche segment of the market — right?

I think that’s just one example, but there are many. For example, somebody who worked for 20 years in the electricity markets — and electricity is a very nuanced pricing market with a very small number of key players, and is largely driven by changes in regulations at the state level and the county level. So, having very specialized knowledge of that, from having worked and gained experience inside the sector, gives you a real edge.

Right, so these are the types of strategies and people that the pod shops are looking for right now. These typically tend to be fairly illiquid strategies — right? You can’t have Elliott Management, a $70 billion firm, running just a niche electricity strategy or a niche muni strategy. But the goal is to find hundreds of people who are all running these niche little strategies, that will all require liquidity to take advantage of opportunities at completely different times from one another, and putting them all together in a diversified basket.

Now, I’m sure there are also very scalable strategies in there as well, that maybe are running more liquid equity strategies or option strategies or whatever. What I fundamentally believe — and my insight from knowing people at those shops — is that the majority of the alpha that you can’t get anywhere else at scale comes from the assembly of many different, less liquid, small niche players that are all operating together in an ensemble.

That last sentence harkens right back to the idea of combining multiple strategies into a portfolio that has a higher Sharpe than any of the constituents by letting the zigs neutralize zags to shrink the denominator.

So I’m nodding along then the host, Jack drops a tight line:

I think Corey [Hoffstein] showed in Factor Fimbulwinter that the amount of time we would need to show that is longer than our investing lifetime.

So this does become about faith.

The verb “showed” following the subject “Corey” is a clue that I get to learn something cool today.

So I read the article referenced:

🔗Factor Fimbulwinter (8 min read)

I’m going to jump to the end because I think Corey stages why he takes the approach he does in the article (emphasis mine):

The question we must answer, then, is, “when does statistically significant apply and when does it not?” How can we use it as a justification in one place and completely ignore it in others?

Furthermore, if we are going to rely on hundreds of years of data to establish significance, how can we determine when something is “broken” if the statistical evidence does not support it?

Price-to-book may very well be broken. But that is not the point of this commentary. The point is simply that the same tools we use to establish and defend factors may prevent us from tearing them down.

Corey uses fire to fight fire.

This is where the learning begins. Let’s see what he does.

We ran the following experiment:

1. Take the full history for the factor and calculate prior estimates for mean annualized return and standard error of the mean.
2. De-mean the time-series.
3. Randomly select a 12-month chunk of returns from the time series and use the data to perform a Bayesian update to our mean annualized return.
4. Repeat step 3 until the annualized return is no longer statistically non-zero at a 99% confidence threshold.

For each factor, we ran this test 10,000 times, creating a distribution that tells us how many years into the future we would have to wait until we were certain, from a statistical perspective, that the factor is no longer significant.

Sixty-seven years.

Ok, I’m going to raise my hand in class.

I didn’t really understand the method.

Awesome I get to learn something new…which means you do too! This is pretty cool.

Luckily, there’s a tireless teacher known as ChatGPT. I wrangled with this professor at office hours until I was able to have it teach me in words I or a middle-schooler can understand.

🧪 How They “Repeat with New 12-Month Chunks” — Using a Coin Flip Example

We’re walking through the logic of how a statistical test updates with each new batch of data — using a coin flip as our analogy.

🎯 The Goal

We want to detect whether a strategy (like a stock factor or a biased coin) has stopped working — i.e., its returns have gone flat.

Think of it like this:

• A coin used to land heads 60% of the time.
• But now it’s just a fair coin (50/50), we just don’t know it yet.
• So we flip it 12 times (like one year of monthly returns), check what it shows, and keep flipping until we’re statistically convinced it’s no longer special.

🧪 The Setup

• The coin is now fair (true heads probability = 50%).
• We start with a belief: “maybe it’s still biased to 60%.”
• We flip the coin in 12-flip chunks, and after each chunk, we update our belief.
• This mimics what the researchers did — taking random 12-month samples from a flat return series and asking: “Does this still look like a working strategy?”

🔁 Step-by-Step Walkthrough (One Simulation)

Let’s pretend we have a long list of coin flips (each 1 = heads, 0 = tails), all drawn from a fair coin.

Here’s a fictional sequence:

[1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, ...]

We’ll take this in chunks of 12 flips, like 12 months of flat returns.

1️⃣ Year 1 — First 12 flips

[1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1]

• Number of heads: 6
• Sample mean: 6 / 12 = 0.50

We compare this to our hypothesis that the coin is biased to 60%:

“Is 50% close enough to 60% that we still believe the coin is special?”

Yes — this result is plausible from a 60% coin, so we keep going.

2️⃣ Year 2 — Next 12 flips

[1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0]

• Heads: 5
• Mean: 5 / 12 = 0.4167

Now combine both years:

• 24 total flips
• 11 total heads
• Cumulative mean: 11 / 24 = 0.458

Still not far enough from 0.60 to be statistically confident the coin isn’t biased.

3️⃣ Year 3 — Another 12 flips

[0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1]

• Heads: 6
• Cumulative now: 36 flips, 17 heads
• Cumulative mean: 17 / 36 = 0.472

Now let’s check this against our original belief (that the coin is 60%).

We run a z-test:

• Expected mean under H₀ (the hypothesis): 0.60
• Standard error: sqrt(0.6 * 0.4 / 36) ≈ 0.0816
• Z = (0.472 – 0.60) / 0.0816 ≈ -1.57

Since -1.57 is not extreme enough (we need z < -2.58 to reject at 99% confidence), we still can’t say the coin is fair.

🔁 Repeat the Process

Each year:

• Add 12 more flips
• Update the total number of heads and flips
• Recalculate the cumulative mean
• Re-test: “Are we now confident the coin isn’t biased?”

Eventually, the sample mean will drift far enough from 0.60 that the test crosses the 99% threshold. At that point, we’d say:

“I’m now 99% confident this coin is no longer special.”

💡 Why This Works

Even if the coin is fair, short sequences can look biased just by chance. You might see 8 heads out of 12 once in a while — that doesn’t mean the coin works.

So the researchers repeat this full process — from scratch — 10,000 times, each with a new random sequence of fair flips.

Then they record:

“How many years did it take before the test figured out the coin was dead?”

On average, the answer was:

67 years.

🧮 What This Means for Investing

If a strategy (or “factor”) stops working but still produces noisy returns — it might take decades before we’re statistically confident it no longer works.

The noise in the short term can mask the truth for a long time.

 

I enjoyed Corey’s technique because it gives you a sense of proportion between signal and noise. In so many domains where an assertion is made, that proportion is absent. I always think about how a CRO I worked with would reflexively try to put error bars on any metrics presented in a chart. It’s good epistemological hygiene. It automatically triggers awareness of base rates and outside views. It’s not a panacea for truth, but it rules out obvious bullshit —randomness sold to you as signal. That will save you time and money in life. It may not increase your top line but it will save you your bottom line.

Thursday’s post the dirties are down the cleans are up took the form of an extended interview question. If you are high-volume professional vol trader, the topic of vol time is fundamental but I don’t see much written about it. I hope my posts on it fill the gap.

For non-pros it probably best serves as a bicycle for the mind or a seed of inspiration but I wouldn’t stress over it. I suspect it does since tweets like this are popular even though I’m pretty sure the engagement on them isn’t coming from a bunch of practitioners in the middle of the trading day:

Most IVs you encounter use a 365-day model. To convert to a 251-day model (or any other tenor model) you multiply by the square root of the DTE ratio.

In the spirit of Thursday’s post and the tweet, I’ll pose 2 “interview-style” questions that can be answered in seconds. They require making a reasonable assumption. I’ll give the questions here, then I’ll post an assumption as a hint after the questions for those who need help forming one. The answers are at the end of the post. (Ignore cost of carry — also if you asked about that you’re way ahead of the game).

You do not need any calculators to answer these (just mental arithmetic).

❓#1: Volatility

It’s the close on Wed. Options expiring next Tuesday and next Friday have the same dirty vol (ie the same vol in your off-the-shelf 365d model). Does one of them have a higher clean vol? Explain. List any assumptions.

❓#2: Price of a straddle

It’s Friday close. Next Friday’s ATM straddle is $5. What price is the Friday ATM straddle expiring in 2 weeks to be the same clean vol? You do not need option calculator.

An assumption you could use to help:

We’ll use a calendar specification that states non-business days count as 50% of a business day for vacation time purposes.

If we continue to denominate our basic unit, a full trading day, as 1.0 and weekend days or holidays as .5 we get the following tenor:

251 x 1.0 + 114 * .5 = 308 day calendar.

---

Answers to the “interview” questions

#1: Tuesday has a higher clean vol than Friday.

Relative to a “dirty” year where each day is treated as equal vol, a “clean” year in which variances passes more slowly over a weekend, the Tuesday expiry has less time to expiry than Friday’s ratio of clean to dirty DTE.

If the model implies the same dirty vol for Tuesday and Friday, we can infer Tuesday must have the higher clean vol bc it has relatively less vol time vis a vis a dirty model.

#2: The second Friday straddle must be $7.07

The approximation for an ATF straddle is .8*S*σ*√t

Since there is no cost of carry we can assume ATF straddle = ATM straddle which is what the question asks about.

We don’t know S or σ but the question asserts the same dirty vol for both Fridays.

We know there’s twice as much time to expiry for the 2nd Friday and we know the earlier Friday straddle is $5 so the second Friday straddle can be computed as

$5 * √2 since there’s 2x as much time til expiry.

So the second Friday straddle = $7.07

💡This is one of those useful trader math ideas — for a given vol the price of the straddles only varies by square root of the ratio of DTE. One of those mental arithmetic things I found myself using constantly especially with short-dated options where you’re like “if the 1-week straddle is X, the 2 week is…”. This is also a clue to the degree to which I ditch the idea of “volatility” altogether in near-dated options and “think in straddles” and move sizes. This intuition is habitual but you can also see why it has theoretical support — vega p/l is a less of an influence on short-dated options. Results mostly come down to “how much did this thing move vs how it was priced”.

Returning to the question.

The $7.07 straddle is based on the same dirty vol.

But does that translate to the same clean vol the same as the first Friday straddle?

Yes — the ratio of dirty to clean DTE is the same for both expiries!

I just want to say that you people are sick. This is the most viral tweet I can remember sending in recent times.

Because I happened to be helping the 3rd grader with improper fractions recently I saw 1.25 as 5/4 which is immediately recognizable as a square root of 25/16.

Sprinkle in some trader math that condemns you to see the sqrt(251) as 16 and you get an even more compact version:

Step by step:

The real masochism in that thread happens further below…

People will message me with charts like this to ask my opinion of a pairs trade:

This is not a pair anyone sent me but the form is the same.

I guess-and-tested a few pairs whose daily return correlation is at least as high as some pairs I’ve been sent. In this case, I picked 2 homebuilders whose correlation over the last 500 calendar days turned out to be .90

[I used the hedge ratio tool from last Thursday to find this pair]

I respond to these messages with a few things that come to mind but I’ll flesh the responses out more in this post. Think of it as a map of data and interpretation pitfalls. There are bits in here for both novices and pros that’ll stimulate a “I never thought of that before”.

The charts above aren’t worthless but they’re very faint beeps on a metal detector during a beach stroll.

Let’s see why…

 

Thoughts as they come to me when I look at the chart…

👁️The first thing my eyeballs say to my brain when looking at a vol spread chart is “why not a ratio chart?”. Over the period, the mean vol spread of LEN to DHI is +.83 and the vol level is about 40%

If vols doubled and the vol spread had the same average I’d be surprised, so a vol ratio strips out the level-dependant assumption that is silently suggested in a spread chart.

The nature of the relationship is of practical importance because it affects how you weight a pair. Any weighting scheme has trade-offs and matching the trade-offs to the bet you are expressing is, well, advisable.

I won’t rehash weighting here but both of these links go into it:

• Weighting An Options Pair Trade (moontower)
• To dollar-weight or share-weight a dislocation in a highly correlated pair? (moontower notes from Financial Hacking)

In this case, the spread and ratio seem good enough proxies for one another. A scatterplot rather than a time series would provide a more granular look but given the 30d maturity and no earnings extractions the extra granularity would probably cause us to overemphasize extremes that would probably be smoothed by arduous surface cleaning.

🔁Possible spot/vol artifacts

If LEN price drops relative to DHI I expect its vol will expand more than DHI. To the extent that a widening vol spread simply reflects recent price action we may deem the vol aberration “more justified” or “not an opportunity”.

Whether a widening vol spread being explained by a sell-off is meaningful or not is unavoidably context-dependent. If LEN falls relatively after earnings, I’d expect its vol spread to narrow from a wider reading that we likely sampled before earnings. But this is just an example of “news flash: lazy low-resolution charts are too underpowered for a trading thesis”.

That said, I do believe the idea of spot/vol mattering in terms of how much weight we’d give an aberration even qualitatively is at least relevant enough to pivot data in a few ways to understand the shape (or lack thereof) of a relationship.

Let’s play “charts plus thoughts”:

• We see a hint of vol/spot effects in the downward-sloping trendline. As LEN price increases relative to DHI its vol premium falls. But the r-square is trash. Weak relationship.
• This isn’t saying that spot-vol relationship in the stocks vs their vols is weak just that the vol ratio to price ratio relationship is weak. Remember these stocks’ returns are .90 corr. When LEN goes down, it’s vol expands but DHI usually goes down when LEN goes down (and its vol will likely be expanding as well in that case).
• My background is in commodities where the spot/vol correlations in general are higher than they are in single stocks (but not as strong as equity indices which I’ll address a bit more below). Here’s just LEN IV vs spot

  

  The grey trend line shows a weak relationship between spot and vol in terms of slope and a correlation that would be around .55-.60…but then I drew those red dashed lines because the story is really:

  LEN is been like a 30% vol stock give or take 5 clicks in a price range of about $165 give or take 30% (ie an annual st dev). It sold off recently with the rest of the world and vol went up to 40% or about 1/3, as the stock stepped from one economic platform down to another.

  As a matter of generalizable perspective:

  All of this feels interesting from the altitude of an investor’s vantage point, but beneath the level of signal from a trading pov. The distinction is useful for any type of money decision — am I getting trader or investor resolution? Is the option trade I’m doing matched to a trader’s dashboard or an investor’s?

   

The vol ratio vs price ratio chart showed a weak relationship. Not unexpected. LEN vols and price ratios are correlated. So instead let’s scatterplot the 1-month change in vol spread (our first chart said it’s a reasonable stand-in for ratio) vs the 1-month relative performance of LEN vs DHI directly. This is not about levels but about motion.

Honey badger, I mean the vol spread, don’t care. For a pair of 30% IV names that are 90 corr, 5%-10% relative performance is significant (can you use the risk-remaining lesson in last Thursday’s post to show that? The learning sticks if you apply it.)

And yet the scatter doesn’t validate the IV spread caring that much.

Let’s do a final chart before chatting.

We saw above that there’s a faint vol ratio to spot ratio relationship in the pair. As LEN goes up relative to DHI its vol ratio declines a touch but there’s a useless amount of noise from our distance.

Straddles are another way to look at vols and spot prices in a combined way. The approximation for a straddle price depends on vol, spot price, and DTE. DTE is constant. If spot and vol were negatively correlated with a -1 beta you’d expect straddle prices to stay constant over many price levels. If vol has a less than a -1 beta to spot then as price increases straddles increase and vice versa. Over large moves I always expect this to be the case.

For smaller moves, the constant straddle might hold as it often does in something like crude oil, a market where traders have a history of speaking in terms of straddles (or “breakevens” in case you ever wondered about Ari’s twitter name) instead of vol. In that market, discussing vol without price level is like owning one shoe.

For single stocks, this effect is going to be less pronounced. I expect straddle prices to rise with price as the spot effect in the straddle formula overwhelms the declining vol. Voila…the LEN – DHI straddle spread vs price ratio. Upward sloping af.

[Additional thought on this chart:

It likes like there are 2 channels which lovingly accommodate the trendline’s horny advance. I didn’t bother, but when you see things like that, it’s usually a clue to color code your dots by date. I don’t mean to abuse the metaphor but there are probably a couple periods of note here.]

🚧Detour on spot-vol correlation in single stocks

The spot-vol correlation in single stocks will vary by sector. The correlation looks fairly small in these homebuilders. We saw that directly in the spot/vol charts, the change in vol ratio vs change in spot ratio, the change in the vol spread vs change in relative performance, and in the strong positive relationship between straddle spread and price ratio suggesting straddles don’t stay constant but are driven by price more than vol.

One reason I could imagine homebuilders in particular having a muted spot vol relationship is they benefit demand side from lower mortgage rates. I’ll put my finger in the air and say they act like a levered 60/40 portfolio. They are diversified if stocks and bonds are anti-correlated.

[In fact, one way to potentially think of their vols is possibly from the quanto lens that one might think of pricing EWJ vol by considering the individual vols of Nikkei vs Yen. The options on EWJ relative to the legs imply a correlation between Japanese equities and FX — which is substantial in an export-led economy.]

Even though spot/vol corrs vary by sector, overall the skew and spot/vol correlations will be smaller than equity indices. Consider 2 reasons that derive from logic not vol surface data which only reflect the logic:

✔️Stock returns are positively skewed

Most companies go bankrupt. A few drive the total return of the indices. Black-scholes’ positive skew lognormal distribution is not a horrible description of companies…pump up the vol and it says, the median return is negative but there’s a long right tail. See Is There Actually An Equity Premium Puzzle?

✔️Average single stock skew must be less than index skew

An equity index is a portfolio of stocks. The vol of a portfolio depends on the vol is positively related to the vol of its constituents and their cross-correlation. If the correlation falls, all else equal, the portfolio vol falls.

Index downside skew must account for both pathways of vols increasing:

1. The average stock in the basket’s vol increasing
2. The cross-correlation of the stocks in the basket increasing

Most stocks share a systemic, undiversifiable risk — the possibility of the economy as a whole slowing down. This shared risk gets priced in index skew as an extra kicker over single stock skew because as correlation goes up, index vol increases faster than the increase in single stock vols.

Recast as a trading statement:

If you could sell all the single stock skew at the same level you could buy the index skew you’d be getting “correlation going to 1” for free.

 

What the vol spread chart hides

In traditional investing, we look at charts that prices travelling from the lower left to the upper right. We can’t feel the drawdowns along the green path it draws. Our eyes focus on the endpoints. Looks easy. All the suffering that you are getting paid for is eradicated, nothing but an expectation of profit for showing up remains.

This turns out to be (has been?) a salutory illusion. The chart makes it easy to buy while hiding the source of edge — a time horizon to not care about the dips. The time series of a mean-reverting vol spread does the same thing.

Let’s break down its lies.

🔮Snooping ahead

The mean and standard deviation are known for the sample period only after the sample period is over. If we transport ourselves to a place on the graph where it’s “obvious” to sell we are doing so with the benefit of hindsight. At the time of the trade, we didn’t know we were “there” on the finished chart. This insidious self-deception happens easily when we look at charts, don’t bring it with you when you move towards actually trading.

🧩Implied vol is only part of the puzzle.

The actual p/l of an option trade done to capture a vol discrepancy is a combination of:

1. vega p/l or change in implied
2. your realized p/l or what I call the “tug of war between your gamma and theta”. For a 1-month option the tug-of-war is going to have a large influence on your results but the chart’s thesis was IV driven.

    

If the market were strong-form efficient all the money you’d make on mean reversion of the IV spread would be lost on your realized vol p/l if the IV spreads perfectly predicted subsequent stock movements! I’m not saying that they do, I am saying that any focus purely on the IV spread only studying half the p/l drivers.

Discrepancies result from flow. Updating fair value based on flow is an inexact science, but to assume that the full amount of discrepancy is edge is the same as saying the flow has zero information.

🍦Vanillas are the real exotics

Vanilla options are messy. How?

1. This time series is a constant maturity vol. To replicate it, you’d need to rebalance the options of the 2 expires bracketing 1-month on each trading day.
2. As the spot prices of the pair move around, your moneyness and greeks change knocking your initial weighting off balance. Contrast this with a variance swap where holding time constant, your gamma is constant over the range of strike prices. The pricing is exotic but the experience is familiar because it ties mechanically back to a familiar calculation — the standard deviation of close-to-close returns. Vanilla pricing is familiar but the experience is exotic.
3. You were trying to capture a vol discrepancy which means you probably want to delta-hedge to isolate the edge.

    

🩺Clinical vs practical

Maintaining a desired exposure in vanilla options requires rebalancing both the options and deltas. To do this faithfully every day as a taker would cost more than any perceived edge is paying.

This means in practice you must be willing to warehouse and tolerate noise. This means you must trade smaller for a given amount of edge if you want to maintain the same risk/reward of the clinically hedged expression.

In the real world, this type of trading is the domain of entities that can sell on the offer and buy on the bid, allowing them to leg the hard leg for edge (the liquidity of which will also be the limiting reagent on the trade).

If you are not a market-maker, trade the leg you have more conviction in or use the history to enter/exit the leg that fits into your overall portfolio more synergistically.

 

Last word

These charts should be thought of as indicators as opposed to backtests. The allure of these charts is they hint “backtest”. By peeling back their insidious, white lies, we can approach them with the right humility —they are clues. The opening statement in a trial, not the final verdict.

 

Stay groovy

☮️

I fired up Corey Hoffstein’s goated Flirting With Models podcast to hear Scott Phillips discuss “ugly” edges in crypto. This episode came highly recommended in my corner of twitter. It does not disappoint.

But I want to zoom in on one part. Scott says:

If you did what I do perfectly, you’d be up well over Sharpe two, and probably with size. But the way that I do it, our four-year Sharpe is 1.7 with retail-level costs. So cross-sectional momentum is a little bit better than that, but you don’t have the really nice positive skew of trend. And cross-sectional carry is about a Sharpe 1.7 as well, and slightly orthogonal. So you blend the three of them together, and then you’re at Sharpe two easily — and without even good execution…The math holds. Returns scale with the square root of independent bets.

Scott misspeaks here (easy to do in a conversation) but what Scott means is volatility scales with the square root of independent bets (returns scale linearly). This concept underpins one of my most important posts — Understanding Edge. It is the basis of all trading businesses without exception. It’s Day 1 learning at a prop shop. Munger has that “Take a simple idea and take it seriously” advice…This is THE idea Jeff Yass took seriously.

I’m repetitive on log and compounding math for 2 reasons that extend beyond the shock factor of the “lilypads in a pond” puzzle:

a) Investing is a serially repeated game so compound returns are our primary concern

b) Option theory sits atop logreturn math which is just continuous compounding (in fact e, the Euler constant of 2.718, is your growth of $1 if you continuously compound at 100% rate for 1 unit of time. See Using Log Returns And Volatility To Normalize Strike Distances)

We’ve confronted this math many times from different angles:

• The Volatility Drain
• Path: How Compounding Alters Return Distributions
• Examples Of Comparing Interest Rates With Different Compounding Intervals
• Well What Did You “Expect”?
• Geometric vs Arithmetic Mean In The Wild
• Understanding Log Returns
• I Felt Bad For Picking My 3rd Grader Off
• Getting Comfortable With Log Charts
• you can ONLY eat risk-adjusted returns
• Growth rate = 70% * (doublings/years)

Compounding is a multiplicative process that describes how returns scale across time. “Volatility drain” reminds us that volatility’s interaction with growth is embedded in that process.

We already understand how compounding scales across time as a function of volatility.

But volatility itself has scaling properties.

That’s what Scott means when he says (or meant to say) it grows by the square root of independent bets. Regular readers have seen me scale a daily return to an annual return by √251 or ~16 many times. But I don’t always remember to say that this assumes no correlation between daily vols (mapping to the “independent bets” language).

In multiple posts, we have seen that volatility is understated in the presence of autocorrelation:

• Thinking in N not T
• Volatility Depends On The Resolution
• the option market’s point spread (part 2)

Autocorrelation affects how we scale vol through time per asset.

This is only half the volatility scaling story. It’s one vector.

The second vector is how we scale vol across our portfolio.

This depends not on autocorrelation, but on pair-wise correlation.

Independent bets are simply bets that have no (ie zero) correlation with one another.

It is why you can blend several strategies and end up with a composite Sharpe greater than any of the components.

Let’s look closer.

Let’s consider an equal-weighted portfolio of ETFs:

• IBIT (BTC)
• GLD (gold)
• HYG (high yield credit)
• IEF (10-year notes)
• QQQ (Nasdaq stocks)

Method:

1. I used the hedge ratio tool to fetch the correlation of daily returns for each of the 10 possible pairs (5 choose 2) for the past 16 months.
2. I chose a reasonable vol for each name and then backed out an expected return by assuming the Sharpe Ratio of each asset is .40. I’m ignoring RFR in SR — I’m just using expected return / vol.
3. I convert the correlation table into a covariance table to do the matrix multiplication for computing portfolio vol. I explain the math in writing, video, and in a spreadsheet I’ve shared before:
   1. ✍🏽Computing portfolio vol (moontower)
   2. 🔻 Portfolio Vol Spreadsheet (download)
   3. 🎥Portfolio Vol Explainer (Moontower YouTube)

 

Here’s my work for the equal-weighted portfolio:

What to take note of:

• If we compute the portfolio vol in the “stressed” condition where all the pair-wise correlations are 1.0, then the portfolio vol is equal to the weighted average vol of the components. In this case, the portfolio vol is 20%. There’s no diversification benefit to the portfolio vol or SR!
• When we use the actual correlations the portfolio vol drops to 12.5%. Since the expected return is unchanged the SR improves from .4 to .64. Massive diversification benefit.
• The ratio of the portfolio variance to average weighted stock variance is the average weighted pair-wise correlation. The average weighted stock variance is the same as the portfolio variance if all the correlations are 1.0
  • For the option fans: this math is exactly how implied correlation is backed out of the index option market. The ratio of index variance to weighted average stock in the basket’s variance is the implied correlation! The cheaper index vols are relative to stock vols, the more implied diversification benefit there is in the index. This concept sits at the heart of dispersion trading.

A portfolio that equal weights IBIT and IEF sounds a bit comical. Let’s re-do the analysis with an inverse vol-weighted portfolio:

Now IEF has 10x the weight IBIT does. If you multiply each component’s weight by its vol you’ll notice they each have equal risk contribution.

The normal portfolio has a Sharpe of .68 but if all corrs go to 1.0 the SR drops to .40 and the portfolio vol goes from 6% to 10%.

Even if the math isn’t interesting, the principle should be seared into your risk brain. Adding zero or negative correlation assets to a portfolio can improve its risk/reward even if the asset’s individual SR is inferior to the average component. An asset’s stand-alone properties are not as important as how it contributes to the risk/reward of your portfolio just as how an asset’s arithmetic return does not mean much if don’t consider its vol. After all, we care about compounded returns.

These dual concepts, timer-series volatility (influenced by auto-correlation) and cross-sectional volatility (influenced by pair-wise correlation of components) give a fuller picture of how returns AND volatility accumulate through time. You care how both the numerator (return) and denominator (risk) scale.

Application to allocators

Professional allocators recognize that much of the low-hanging fruit of long-term results is sound portfolio construction. Basic hygiene in how you stack investments by understanding their properties in various states of the world is something you have more control over because volatility and correlation are more stable than expected return predictions. Simple optimizers are incredibly sensitive to expected returns — which is a bit self-defeating when you realize this is the metric we know the least about.

To add a bit of practical color. I have a good friend — a physics academic turned investor — who was CIO for the family office of one of the partners of [insert the best quant firm you’ve heard of and it’s probably the one]. He admit something he found embarrassing — a shockingly small number of ideas underpinned the family office’s approach and none of them is a secret. The scaling property of portfolios is one of them. If they sourced 16 investments that generated a high-quality portfolio, in order to double the shape they needed 4x as many additions to that portfolio of similar quality (which was already high) to double the portfolio Sharpe. Pretty much impossible when all the investments are LP stakes in quant funds (due diligence alone is too labor intensive, never mind finding that many great managers).

[I’ll get around to toying with this tool a bit more to tinker my way to seeing how the scaling works — for example going from a 3 stock to 12 stock portfolio where I add names with equal SR but different correlations. I’m down to the wire on writing this post as it is.]

Application to traders

The last part of this video shows how the interaction of return and volatility can inform how you think about sizing risk.

If you prefer text, the heart of the material is captured here:

Understanding Daily vs. Annual Sharpe — and Sizing Risk with Intent

Let’s start with something simple but important.

🗓 Daily SPX Returns

The S&P 500 has a daily mean drift of about 3 basis points, with a standard deviation of about 100 basis points in normal conditions. That gives you a:

Daily Sharpe ≈ 0.03

(This ignores the risk-free rate, which is roughly 1.5 bps/day.)

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📆 Annual SPX Returns

Now zoom out.

The S&P’s annual return is roughly 8%, with an annualized volatility of 16%. That gives:

Annual Sharpe = 8 / 16 = 0.50

Here’s the interesting part:

Annual Sharpe (~0.50) ≈ 16× Daily Sharpe (0.03)

Why? Because volatility annualizes by the square root of time:

√251 ≈ 15.8, or roughly 16

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🎯 Sharpe Ratio Benchmarks

Let’s anchor a few examples to make this more intuitive:

• SPX benchmark: 3 bps return / 100 bps vol → Sharpe 0.03
• Sharpe 16 strategy: 1% return/day, 1% vol/day → Sharpe 1.0 daily
• Sharpe 2 strategy (annual) =
  → Daily Sharpe ≈ 2 / 16 = 0.125

So:

A Sharpe 2 strategy means your daily standard deviation is ~8× your expected return.

That 8-to-1 ratio is a helpful sanity check.

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💰 Backing Into Risk Limits

Let’s say you want to size your book around a Sharpe 2 strategy. Here’s one framing:

• Daily P&L target: $1,000
• Daily volatility: $8,000
  → Daily Sharpe = 0.125
  → Annual Sharpe = 2.0

Now annualize:

• Yearly expectancy: $1,000 × 251 ≈ $251,000
• Annual volatility: $8,000 × √251 ≈ $126,744

What does this imply for annual ranges?

• A +1 SD year: $124,256 – $377,743
• A +2 SD year: ~0 – $500k
• You’re brushing against zero at 2 SDs

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📐 How to Use This for Book Sizing

You don’t just ask how much do I expect to make?

You ask:

• What kind of daily swings are consistent with my goals and risk tolerance?
• Can I size positions and set Greek tolerances so that my volatility lands near ~8× my expectancy (for a Sharpe 2 strategy)?
• Do I have enough emotional and capital runway to survive the downside scenarios?

Via prompting ChatGPT:

Helper Notes

I replaced my IBIT shares into long calls when BTC vol got crushed this week while selling my VIX futures which held up quite well on the equity rally, a possibility I suggested when I explained that the elevated but also flat term structure of VIX suggested vol was here to stay in 2025. From vol speed round:

As I was looking at IBIT I thought I’d share a habit I picked up from distant past of ETF arbitrage — estimating the prem/discount baked into the ETF price I’m about to trade:

1. Go to the IShares web page to grab 3 values (highlighted)

2. Enter them in my spreadsheet to compute the ratio of NAV to BTC price. The cells with the box around them are via the previous close values. I then apply the fair ratio to the live BTC price compute IBIT’s live premium or discount.

Net of last week’s April expiry the flurry of trading from the last few weeks has left me net long delta. That’s not accidental. I had plenty of room to be a buyer on balance in the turmoil since cutting lots of portfolio delta heading into the election.

[If interested…I rebalanced much of stock delta into t-bills bills then re-deployed into bonds, TIPs and silver in late January.]

Coming out of the April expiry last week, my position gained ES futures, VIX futures, IBIT, with my only outright option position being some May AAPL put spreads.

Back in November I published this video which show how I look at the risk of our household portfolio.

But in the flurry of trading while I’m mentally separating what I’m doing for edge and my total portfolio, I do have some delta bias under the hood to effectively dollar cost average since I de-risked in October.

But I want to keep close tabs on how much I’m adding which means summing positions in a vol or even better beta-aware fashion. NVDA is far more volatile than SPY so just as traders combine all their deltas into SPX or some other benchmark terms I like to also normalize back to SPY equivalent risk.

This is a doorway into hedge ratios and beta-weighting. Beta-weighting is not just vol weighting but includes correlation.

I’ve written extensively about how to do this in:

🔗From CAPM To Hedging (17 min read)

That post steps through derivations and the basics of correlation math but in my personal portfolio spreadsheet I needed to make a little widget where I could just put 2 symbols and get a hedge ratio.

We actually have a calculator in the app but you need to supply the inputs.

However I realized with Excel’s built-in =stockhistory I could not only get the inputs, but easily plot time series and scatterplots to understand the shape of the “idio” risk and variation in the beta over time.

🚀We are going to add this functionality to the moontower.ai app. We already have the data, just need to extend the UI.

In the recent month, I wanted to understand how many SPX delta equivalent I had in VIX futures. I used VXX as a proxy put it into Excel and voila got some idea of how much “less” long SPX I am with the VIX futures incorporated.

So….

I made a video walking through the Excel tool. It’s some of the most practical traders-use-this-everyday-type knowledge. We skip the derivations and jump to how do I actually use this data to size hedge ratios or estimate my book’s beta.

I hope you find it as useful as I think it is. (If not, I’ve got more work to do on the explanation side so let me know!)

Paid subs get the spreadsheet. If you have the stockhistory function in Excel this will work seamlessly for you. It’s addicting to toggle thru pairs so fair warning!

The spreadsheet link below has 2 bonus items not shown in the video:

1) A widget for spitting out the hedge ratio not just running data

2) A little position radar template on a separate tab to keep your gameplan organized

Before getting to the sheet, tomorrow night is Ricki Heicklen’s Trade Gala party in SF. I mean yesterday we learned that Jane Street made $20B in 2024 which doubled what they made in 2023 which shattered their record from the prior year. I think they are the most profitable firm in the world per employee (they have about 3k employees).

I think it was Morgan Housel who once wrote that very few orgs in the world could say their edge is just flat out “we’re smarter than you”. There’s usually some other sauce. He gave the example of RenTec. Jane Street is another that can boast the same. This is a cool opportunity to see how that crowd thinks.

This is a special moontower invite that Ricki sent me. She’s repeatedly acknowledged that anyone she meets that comes out of this community is a breath of fresh air which she poses as a compliment. I’m not one to insult a compliment but it’s a testament to y’all. I don’t control how you act.

Anyway the party starts at 8pm and runs thru 6am…well I’ll let you click on this yourself. This is actually a custom invite for Moontower readers:

Trade Gala Party

A quick description..Trade Gala is a limited-access costume party with markets, a puzzlehunt, and demo-ing a new trading game.

If you want to attend the trading BootCamp the rest of the weekend sign up below. It’s truly ridiculous. If you are a novice it will open your mind to what trading looks like thru the eyes of sharps. It is hard to come by this in the level of detail you’ll see here.

If you are a pro, well, you might want to know you might notice some things that you are up against. There’s a lot to learn for everyone.

trading.camp

It’s view only but you can save a copy for yourself and do what you like from there.

1. When you click the link it the link will open in your browser.
2. Download a Copy. (see pic)

A reminder in the spirit of being attuned to seemingly far-fetched risks:

If short selling were restricted in any way, the value of puts relative to calls on the same strike increases in a put-call parity framework.

Another way to say this is being long stock is more valuable since only long sellers can sell. If puts increase relative to calls on the same strike, as they do when borrow costs increase, that is like a synthetic future on the stock trading at a discount to the stock price.

Similarly, being short futures vs being long index or SPY expresses the same bet. If futures start trading at a sustained discount to the cash index value it could also reflect an implied probability of short-selling restrictions being imposed.

If futures are trading at full carry and you think such a ban is possible it’s an asymmetric bet to put on conversions (ie short futures, long stock or for options short combos long stock).

Broadly speaking, the derivatives market becomes the underlying market. When the underlying market becomes encumbered do to technical frictions, derivatives are no longer just “derivatives”. The arbitrage mechanism is severed. The derivatives market becomes the home for price discovery.

I’ve seen this happen throughout my career. Just a few examples in addition to hard or impossible to borrow/short situations:

1. HYG becoming a liquid referendum on illiquid high-yield bond market
2. Cotton option synthetics continuing to trade even when the underlying futures are locked limit up (my trading claim to fame is selling the all-time nominal high in cotton in late 2011…it was an option synthetic around $2.25)
3. When a stock is halted, the premium or discount to any ETFs holding that stock computed using the halted stock’s “last” price before the halt implies the price the stock will open when it resumes trading. These situations were common during the dot-com heydey when stocks would be halted intra-day on pending news announcements.

🔗Related

📽️Teaching Options Basics With Live Data (Moontower YouTube)

📔Synthetics: Alternate Realities (Market Jiujitisu)