appreciating diversification

This week, I hosted class #4 of the Investment Beginnings for local kids aged 12+.

The series’ materials are here:

https://notion.moontowermeta.com/investment-beginnings-course

This is the specific material for class #4:

I also created a web version of the game:

☀️🌧️Sun/Rain Game

While I’ve been doing the series for kids, I think a lot of adults could even benefit. The overall arc of the presentation:

  1. Last class’s game ended with a humbling but common result, hinting at a key pillar of investing.
  2. We use a few facts to dispel the recency bias that all investors carry with them.
  3. They learn what the fundamental nature of stocks predicts about their individual and group behavior.
  4. We widen the meaning of diversification beyond stocks, which was extremely easy to do in light of March 2026.
  5. We play a game that makes the implications for portfolios concrete.

While moontower readers span a wide range of investment experience (although overall quite interested in investing and money), here are a few ideas that I hope are presented in ways that might augment even your understanding or at least help you explain to learners in your life.

The most naive strategy is hard to beat

The kids spent Class 3 picking stocks based on a bunch of variables they could sift through, only for the equal-weight benchmark to beat everyone except the team that contrarily concentrated in the highest momentum company that is very much still an enigma to the market (TSLA).

The equal-weight strategy which I just called a monkey (although it’s not random, just dumb) beat 2/3 of the 15 individual stocks themselves.

The reason you shouldn’t be surprised that the naive strategy is hard to beat

Companies eventually die, but indexes shed them before they are in hospice.

Only 17% of the original S&P 500 companies from 1957 survived 50 years. The average company lifespan on the index was 33 years in 1964 — it’s now under 20. Kodak invented the digital camera in 1975 and buried it because of the innovator’s dilemma.

In a crash, stocks remember they’re all stocks.

Diversification works differently in good years than bad ones. In the class data, stocks spread widely in bull years. Then we looked at Jan 2022 to Jan 2023: 13 of 15 stocks fell together, the spread collapsed.

I didn’t want to lean into the word correlation, but I noticed a different way to convey the same idea. The inter-quartile range (IQR) of annual returns was smallest in the worst years. This chart is rich with insight. Notice the IQR’s visually but also how the equal-weight portfolio performed relative to the individual stock and median stock returns each year:

These observations are non-CAPM ways to arrive at the familiar language of diversifiable risk (company-specific stuff you can eliminate for free) and systematic risk (market-wide stuff you can’t diversify away but do get paid to carry). The crash revealed which was which.

If we zoom out from stocks alone, we see a race where the leaders change each year

The Novel Investor quilt shows 15 years of annual returns ranked best to worst across 9 asset classes. The diversified portfolio, that gray-ish bar, never wins a year nor comes in last. Note commodities, gold and BTC are absent from the series.

How do you think they would influence the gray portfolio?

The Sun/Rain Game

This leads to a game where we can build some intuition about the role of non-stock assets in a portfolio.

If you look at the sheet you can see how the kids actually did (I changed the kids names to letters):

The game’s punchline is that owning the anti-correlated asset despite it having a worse expected return than the “good” asset leads to a better long-term portfolio.

But this is so unintuitive that I got a student’s question wrong during the discussion!

I’ll explain the mistake here.

A student asked if we played the game for 100 years instead of just 20 years, if owning the good asset ONLY would have led to the best return. I initially said no, then corrected myself and said yes because it has the higher expected return.

But I was right the first time. The answer is definitely NO.

It comes down to the fact that the good asset has an expected arithmetic return of +5%, BUT it has a negative expected CAGR or geometric return.

The math:

The company is 50/50 to return+40% or -30% in any given year.

.5 x 40% + .5 x -30% = +5%

But over 2 years, you expect 1 up, 1 down. Compounding math:

1.4 x .7 = 98%

You expect to lose 2% over a 2-year sequence of about 1% per year.

Formally, we compute the expected CAGR by multiplying (note how the arithmetic or single period return is added):

1.4^(1/2) * .7^(1/2) – 1 = .9899 -1 = -1%

[The exponents represent the probability of each outcome. If there were 3 outcomes, you’d have 3 terms and the exponents sum to 1.]

In the long run, the good asset destroys value. So you do not want to concentrate in it despite its superior expected arithmetic return.

The CAGR is being killed by volatility drag, which is the asymmetry of the fact that if you lose 30% you need to return 42.9% to get back to even, but the “up” years only return 40%. You are falling behind over time.

The bad asset returns -10% half the time and +8% half the time. It’s a “worse” asset, but it’s less volatile. Taking this quality to its extreme, isn’t this what cash is?

In arithmetic terms, our average return if we allocate to each asset equally is +2% (50% x 5% + 50% x -1%). But that portfolio is less volatile because one stock zigs when the other zags. The diversification cuts the volatility MORE than it cuts the expected return, leading to a better risk/reward!

If we rebalance each year back to an equal-weight portfolio, we “pull” the expected CAGR closer to the expected arithmetic return. It’s the only way we can get close to eating those expected arithmetic returns. Otherwise, they don’t really exist for you over time.

This table is worth staring at:

Here’s a message one of the dads sent me after the class:

Measure Your Own Diversification

I made you a tool to compute your portfolio vol and see how much the cross-correlations between your holdings have been reducing total vol from the vol that the individual assets contain. You can tinker by adding ETFs of other asset classes to your equities (ie GLD or USO or TLT etc) to see how they affect the volatility.

If you just want inspiration for an idea, use the tool to compare the Mag 10 index (MGTN) realized volatility with the average realized volatility of its holdings. The index is conveniently equal-weighted, 10% in each name.

Two ways to try this on your own portfolio:

🌐To run in your browser

https://colab.research.google.com/github/Kris-SF/data-pipelines/blob/main/portfolio-vol/portfolio_analysis.ipynb

⚠️Just push through the warning it spits off

The output will includes metrics and charts:

 

🖥️To run locally

git clone <https://github.com/Kris-SF/data-pipelines.git>
cd data-pipelines/portfolio-vol
pip install -r requirements.txt
jupyter lab portfolio_analysis.ipynb

Either way, edit the WEIGHTS dict and the START / END dates, then Run All.

how a high implied vol can be cheap

EWY, the South Korea ETF, was an interesting source of disagreement in our Discord about whether the vol was expensive or not. This is the IV vs trailing RV:

Based on realized vol calcs using daily sampling, IV approaching 50% looks rich.

But EWY had been grinding up since the beginning of the year. (It tanked along with the dollar this week after the Iran strikes.)

 

It was up 25% in February alone.

If we annualize that to a vol:

25% * √12 = 87% vol

More than 2x the realized vol and significantly higher than the “rich”IV.

The posts below discuss this sampling issue from several angles.

  • Risk Depends On The Resolution | 4 min read
  • Volatility Depends On The Resolution | 5 min read
  • The Option Market’s Point Spread (Part 2) | 11 min read
  • Thinking In N not T | 6 min read
  • A Misconception About Harvesting Volatility | 3 min read
  • The Coastline Paradox in Financial Markets | 11 min read

There’s no single “realized” volatility. Every time you delta hedge you sample a unique volatility such that is possible for a long delta hedger and a short delta hedger to both make or both lose money depending on the timing and size of their hedges.

Because we are cursed with memories, every good trade we do, we wish we did bigger, and every bad one we wish we did none of. Our memories, combined with the noise inherent in delta hedging is a recipe for madness. That’s why all option traders are unpleasant and wish they had chosen a career where they can simply clip a fee from the collective net worth of society, which has been steadily levitating for the past generation, raising (good) but compressing (boring) the fortunes of the clever and the dimwitted alike.😉

Since realized volatility is sensitive to how we sample it, it’s worth looking a bit closer to how it accumulates. This exploration is likely to inspire your own research or even guide your thinking on how to get your head around return behavior that, despite being common and familiar, remains, as my kids say, confuzzling.

In this post:

  • The Trend Ratio — what the ratio of weekly-sampled to daily-sampled vol tells you about trending vs choppy regimes
  • The Variance Contribution Ratio — a single number that tells you whether a trend was a slow grind or a one-day event
  • Broad patterns across 35 liquid ETFs over a decade (~97K observations)
  • What TR implies for delta hedging — the tradeoff between rebalancing noise and sampling bias
  • What happens to forward vol after grinding trends, and what that means for pricing
  • A self-contained Jupyter notebook that fetches from yfinance and reproduces everything

 

the shape of volatility

EWY had a grinding rally. You can describe this as momentum, autocorrelation, trend. These are all ways to say the stock went on a quite a run. These descriptions mask something even more fundamental that we should make explicit. The notability of this run, even before describing its steady behavior, is that it was volatile.

Even if it’s 1% per day for 20 days this is volatile in the sense that the movement in the stock was unusual. We do not expect EWY to find itself over 20% away from where it was a month ago. Plain and simple. If we tallied all monthly returns, a move of that size would stand out as an outlier.

If a dog is wearing a dress, we would acknowledge that unusual observation before describing the color or material of the garment. Similarly, before describing the shape EWY’s move, we take it in, “That’s pretty remarkable.” You’d need to have a narrow definition of volatility, a definition that is divorced from an honest view of reality, to think otherwise.

It’s settled then, EWY was volatile. Great. Now we can think about the shape of the volatility. I’m going to introduce 2 measures that we can use in conjunction to classify volatile moves.

Trend Ratio

A common way to compute a realized vol for say 20 trading days is to average the sum of squared daily returns, take the square root, then annualize by √251. We’ll call this 20d RV sampled daily or 20d_RV for short.

Now compute the same realized vol but sample weekly instead of daily. The method is the same except for 2 variables:

  • 5-day returns instead of daily returns. Note that means only 4 data points, not 20.
  • Since you sampled every 5 days, you annualize by √251/20

We will call this 20d RV sampled weekly or 20d_RV_w

The ratio of weekly-to-daily vol captures how much “trend” was present relative to chop. We can call this Trend Ratio (TR).

TR = 20d_RV_w / 20d_RV

When TR > 1, the market has been trending. The point-to-point displacement exceeds what you’d expect from the daily noise. When TR < 1, daily returns have been partially canceling or mean-reverting within the window.

As of the last day of February 2026:

EWY

20d_RV_w = 49.9%

20d_RV = 40.6%

TR = 1.23

Variance Contribution Ratio

Imagine 2 stocks.

Stock A: Moves 1% every day. Its vol annualizes to 16% if you sample daily

Stock B: Moves .60% 19 days, and 3.6277% on 1 day. Its vol also annualizes to 16% sampled daily

Both A and B accumulated the same amount of variance, but for A, each day contributed 1/20 of the variance. Stock B’s most volatile day contributed 65.8% of the total variance!

💡Variance is the square of returns. We care about variance because realized p/l in options is proportional to variance. If you are short gamma, a 6% move costs you more than 2x a 3% move.

We will define a Variance Contribution Ratio (VCR) as the fraction of total variance explained by the single largest squared daily return. Hence, the VCR for a 20d window:

VCR20 = max(r²) / Σ(r²)

If all 20 days contributed equally to variance, VCR would be 1/20 = 5%.

Snooping ahead for a moment, the median VCR across 35 liquid ETFs for the past decade is about 25%. This means one day typically explains a quarter of the whole month’s variance. A major departure from the uniform case. The real world is lumpy.

 

Boiling vs jumpy frogs

A high TR reading tells you the market trended, but not necessarily how. By filtering TRs by VCR or vice versa, we can distinguish grinding or frog-boiling trends versus a trend characterized by larger jumps. From there, we can study subsequent realized volatility behavior.

I grabbed 10 years of daily return data for 35 ETFs spanning equities, fixed income, fx, and commodities from yfinance (~97,000 observations)

The details of all the calcs and code are in this notebook:

🔗https://github.com/Kris-SF/public_projects/blob/main/vol_ratio_vcr_study1.ipynb

Here’s a high-level summary:

Across all tickers, we can see that the median trend ratio is ~95%. In other words, volatility sampled weekly is about 5% less than if you sample daily. More frequent sampling over the same time window generally leads to higher vol computations, so this is not a surprising result.

If VRPs are typically 10-15%, then VRPs are about 1/2 to 1/3 larger if you sample weekly. An interesting observation for someone debating how often to hedge. The trade-off, of course, is noise. We can see the distribution of trend ratios in the blue histogram. Again, that’s across all tickers. For individual tickers, you can look up the standard deviation of the Trend Ratio. We will look at them graphically below in a bit. The distribution of TR appears well-balanced.

On the other hand, we can see that VCRs have a strong positive skew. The median VCR is ~25%, meaning it’s normal for 1 out of 20 days to comprise 25% of the total variance! It’s never the case that the distribution is truly uniform, but there’s about a 1 in 20 chance that a single day can comprise 50% of the variance. Remember, there are no single stocks in this universe, so earnings are not a factor. If interested, you could change the tickers in the notebook to study single stocks.

What’s normal at the ticker level?

Trend Ratios by ticker:

Commodities seem to exhibit more trending behavior than equities, but the overall feels compact with a range of TRs from .9 to 1

VCRs by ticker:

It seems like SLV and FXY have had about 10 to 20% higher VCRs than the typical name suggesting they are more prone to a single jumpy move in their return stream. Because we are looking at the median VCR I don’t think the recent SLV chaos is skewing the data. If I exclude SLV data from June 2025 until now, the median VCR only drops from 29.5% to 29.4%.

 

Classification

Split TR and VCR at their medians to get a blunt classification framework:

Summary:

Grinding Trend: 20,744 (21.3%)
Spike Trend : 21,605 (22.1%)
Choppy Grind : 28,046 (28.8%)
Spike Revert : 27,150 (27.8%)
TOTAL : 97,545

EWY’s move was textbook upper-left quadrant grinding trend. High TR, low VCR.

Let’s set VCR aside for a moment. It’s nice that the recent VCR confirms that the variance was not especially lumpy, but we can see that with our eyes. The question that prompted this whole post was whether the elevated TR, the fact that the less frequently sampled vol was much higher than the daily vol, meant anything for future volatility? Is the high IV actually expensive, or does the option’s market somehow balance both measures of realized vol?

Phrased generally:

Does the elevated TR tell you anything about subsequent realized vol?

For every observation, I computed both the current TR and VCR, then looked at what happened to daily realized vol over the next 20 trading days. To be clear, this is the window that is 20 days hence, so there are no overlapping days between the TR reading and the subsequent volatility.

I’m specifically interested if daily sampled vol exhibits any tendencies. I sorted all observations into TR quintiles and measured the median percent change (technically the log change) in RV20d from the current window to the next window.

The pattern is monotonic and the direction of change is what I’d expect.

In Q1 (lowest TR, most choppy) forward daily RV declines. To be fair, I had no expectation about whether it would increase or decline, merely that as we increase the TR, the subsequent RV would increase.

[To articulate the logic: there’s additional information in the less frequently sampled vol at the margin, perhaps uncovered by splitting the data into quintiles. We are looking for benefit in the margins as we accept that there is less total information than more frequently sampled vol. After all, daily vol sample would converge to a good estimate of an asset’s true vol faster than once a year observations. This is also why you would prefer daily data about a trading strategy versus monthly.]

As we ascend quintiles, Q5 (highest TR, most trending) precedes a median increase of +3.4% in RV20d.

The daily estimator was understating the expectation of the next period’s vol if we assume it would be unchanged. The next period, daily RV partially “catches” up.

3.4% isn’t a huge number, but it’s material. If you thought 50% vol is fair, now you might pad that to 51.7% but…it’s highly variable and positively skewed. The mean vol increase is 14.9%, which would mean raising your fair vol from 50% to 57.5%!

This is the histogram of the percent vol increase in the subsequent period for the 5th quintile of trend ratio:

Be careful, the standard deviation of that vol change is huge. This is all the quintiles:

 

That EWY elevated IV over daily-sampled RV starts making a lot more sense because its trend ratio of 1.23 is in its top quintile.

 

VCR adds independent information

High VCR predicts vol decline, holding TR constant. This is partly mechanical. To take an extreme example, when one day accounts for half your variance budget, vol drops when it rolls out of the next window. But it’s also real: spike regimes tend to cluster and then subside.

To examine how VCR may interact with TR, we construct a heatmap. Each cell shows the median percent change in daily RV from the current 20-day window to the next, broken out by TR (columns) and VCR (rows).

Reading left to right (TR axis): Higher TR predicts vol increase, and this holds within nearly every VCR row. Look at the 15-20 VCR row: it goes from roughly flat at low TR to +11% at high TR. The pattern repeats row by row.

Reading top to bottom (VCR axis): High VCR predicts vol decline across every TR bin. The bottom row (VCR > 50) is negative across the board, ranging from -30% to -3%.

We would find EWY in the upper right corner (high TR, low VCR) the grinding trend zone. Subsequent vol rises from +8 to +12%.

Recall from the four quadrants that grinding trend is the least common, showing up about 21% of the time. But this is still frequent enough that you can easily bid an IV equivalent to the trailing daily-sampled vol.

I just doubt that the market will give it to you. But at least you know to screen for this and at the very least not be tricked into selling an insufficiently high IV.

It’s trivial to compute a VCR as well, so you can add this filter as confirmation that the trend is boiling a frog not just a jump.

The Notebook

Again, I’ve open-sourced the full Jupyter notebook behind this analysis.

🔗https://github.com/Kris-SF/public_projects/blob/main/vol_ratio_vcr_study1.ipynb

It fetches data directly from Yahoo Finance, constructs all the variables from scratch, and reproduces every chart above. You can change the ticker universe, the window length, or the sampling frequency and re-run the whole thing.

Note the code computes TR and VCR using a zero-mean estimator for realized vol (dividing by N, not N-1). This is deliberate, we’re measuring total quadratic variation including drift so the zero-mean formulation is standard in the vol trading world

hard earned trading wisdom

Euan Sinclair needs no introduction from me.

I’ll cut straight to the gold.

He’s been a repeat guest on Erik’s Outlier Trading podcast a few times. His writing and interviews mince no words. Despite never sugar-coating the reality of trading, I found his most recent interview even bolder. It’s this witch’s brew of insight that is somehow both timeless and underreported.

I’ll start with an idea from the interview he did from late 2025 that always bears repeating before moving to the more recent chat.

The most common misconception about trading volatility

In Euan’s 2025 chat with Erik, he was asked “What is the most common misconception about trading volatility?”

He zoomed in on the mistaken logic that because volatility is mean-reverting, selling it when it’s high assures a profit since it always comes back down.

There are several facets to the mistake. One is with respect to how volatility can cluster based on a market regime.

Saying volatility’s mean reverting is true, but the means also change, you know? So, if you sold volatility in, I don’t know, March 2020, right? Volatility didn’t go back to 15 for about a year, right? Volatility had a new normal. So just because something’s mean reverting doesn’t mean you’ll make money because it comes back.

Another flaw is in understanding that you are exposed to both realized and implied vol.

The other thing that can be wrong is that you’re not directly trading volatility. You’re trading the spread between implied and realized. And that doesn’t have to be mean-reverting and it doesn’t have to be negatively correlated with the level of volatility either. So just because the implied V comes down, that’s not necessarily going to help you if the realized vol still is higher than the implied.

[Kris: I remember suffering through a short option position in nat gas in the expiry right before I got married. The V09 option cycle (expiry date was late Sep 2009). Implied vol got up to 110% but it realized more than implied for our entire holding period. We chopped ourselves up on the short gamma even though we had “positive vega p/l” on some of the marks.]

It’s easier to understand this when you scrap the concept of “vol” for a moment:

Forget about volatility, right? Volatility is just a way of turning option prices into another thing. It relies on a model. Forget about all that, right? Let’s say you’ve got a straddle and the straddle’s trading at five bucks and that straddle normally is trading at two bucks. So you’re like, “Oh, that’s really high. I’ll sell it.” And then as soon as you sell it, it moves by $10. You know, the stock moves by 10 bucks. You’ve just got hammered, right? But you know, the straddle drops down to a dollar. So you were right about the price of the straddle, but you weren’t really just trading that.

Finally the VIX chart illusion:

That’s one of the problems people get wrong is selling volatility spikes isn’t as appealing as it looks as when you look at the VIX graph over time and you’re like, “Oh, the VIX these spikes. It always comes down again.” Yeah, but you that’s not what you’re trading when you’re trading options.

See this post from the Liberation Day period if this is not clear: you can’t trade spot VIX.


The rest of these excerpts are from Euan’s 4/1/26 appearance on Outlier. Emphasis mine.

Where to start: known effects

Erik asked Euan where retail traders should start. Euan’s answer is a tour through what he actually believes works, why it works, and how to think about learning from past data without fooling yourself.

If you’re prepared to have low enough expectations that they’re realistic, and then really work at this, um, and it doesn’t have to be a full-time job, but you can’t just think this is my extra income. You know, this isn’t like driving for Uber.

Erik: Thinking a little more deeply then about the kind of things retail traders should look at — I went through an interesting exercise on my own, because people would ask me for ideas on places they might want to start looking, and I always struggled to give a good answer. There’s a million variables. But I’ve recently started directing people toward really well-known market effects — stuff you could go look at research papers on SSRN in mass, well-researched stuff. Something like time-series or cross-sectional momentum as a general market effect.

The reason I’ve been going that route is that there are so many things you have to do right as a retail trader. Even if you’re doing all the right things but it’s centered on an effect that isn’t really there but you think it is, that’s massively detrimental long term. Even though the returns might not be — as you talk about — stuff that would give you Lambo money this week, you’re at least building the infrastructure around something we know exists.

Are there effects or markets that you think are better suited for those first few repetitions for a trader?

Euan: It’s not so much about finding a market where you have an edge. It’s not like, “Oh, you’ve got to trade shitcoins,” or “You’ve got to trade options on pharmaceuticals.” The actual instrument and sector are of lesser importance than the concept of what makes this thing work.

There are a few things in finance that we know broadly are real. Momentum is one. The other one I’d say, if you’re just starting out and you want to set up what I’d call a real trading operation, would be carry.

If you really understand the concept of carry, particularly as it applies to futures basis trading — in theory, we know that a future is going to coalesce to the spot price at expiration. But that doesn’t tell you how it’s going to do it. It doesn’t say the future’s going to stay there and the spot’s going to go up toward it. It doesn’t say the other way around. But what actually happens most of the time is that the future moves toward the spot price. Naively, that’s exactly what you wouldn’t think happens — you’d think the future was an expectation of the future or whatever. But no. It doesn’t matter what you naively think. That’s a very strong effect.

If you know about that effect, you’ve now got lots of places you can look for ways to apply it. And if you understand the effect, you’ll know places that are better than others. You look for something with a big basis. You look for something with high volatility, because that also gives you more of a basis. And you want something to move a lot, because if nothing moves, you can’t make any money.

That leads you to something like the VIX. The VIX futures have a very high basis to the cash usually. If you look at the difference and you annualize it, if that entire basis is realized, that can be 100% a year. Are you going to get 100% a year? No, because lots of other things are going to happen, and you’re not going to realize all of that, and there’s volatility. But that’s now an effect where you can start saying, “Okay, now I know this. What can I do to harvest it?”

Then you start saying, “Well, clearly I want to be short the future if it’s above the cash. That’s risky. What do I do to hedge that risk until I’m happy?” And maybe you’re like, “Okay, I’ll do a future spread. I’ll be short the front one, long the back one. What ratio? What futures?” Eventually you’re going to come up with — and there are papers written on this exact effect, I haven’t just pulled this one out of nothing.

A lot of the stuff’s out there. People tend to have this idea that no one’s going to tell you anything that works, whereas literally there are thousands of pages written on stuff that works. The universe isn’t going to give you money the way you want it to necessarily, but this carry effect — once you understand it in the VIX, then you’re like, “Holy shit, this also works on a ton of other commodities. It works in bonds.” And then, “Okay, if it works in bonds, does it work differently in treasuries, credit, corporate bonds?” And the answer is yes. So now you’ve got spreads. So relative value is the next thing you start looking at.

Pick a bunch of carry situations, learn to put them on, learn to manage them. They largely take care of themselves, but you have to adjust, you have to understand the risks. It’s like when you fly a plane — you’re learning on a Cessna, learning slowly. You don’t just get into a MiG and blast off. But this is that Cessna. It’ll get you money, and it’s a real trading operation.

Then you move on to relative value. These spreads move around. Maybe I can scalp those spreads. How do they move around? What’s the range? And then, “Well, sometimes they don’t move around.” Okay, and that’s going to lead you to momentum. That’s your next one.

Everyone should read that book by Antti Ilmanen, Expected Returns. It breaks down — it’s about 400 pages long, that’s how detailed it is — and it talks about the volatility premium for about four pages. It goes into hundreds of things like this that work. Carry is a huge unifying feature. Relative value. Momentum. The variance premium and options — again, that’s there. I wouldn’t recommend people start with it. It’s kind of slippery and a good way to lose a ton of money unless you’ve got everything else sorted out. But it’s another one.

That’s where I’d start. I’d start with one thing, move to the next, keep adding. Once you got to three or four things, that’s probably all you can handle. You’re not going to be able to have 10 different strategies and keep things together. That becomes a major logistical operations issue.

Where to expect edge: the hard leg is where the money is

[Kris: My biz partner would say the “hard leg” is where the money is.]

Typically in the world, you get paid for doing something that makes the world somehow better. You provide a service, and typically there’s something unpleasant about that. Otherwise people will just do it themselves.

If you’re selling flood insurance, that’s a tough business because you make money, make money, make money, and then you get absolutely blitzed. That’s a tough thing to live through. Everyone thinks they can, but in reality it’s a lot harder than you think. We’re very bad at figuring out how we’ll feel when something bad happens. Similarly, the guy in the flood who’s cleaning out the sewers, he gets paid.

If you’ve got a clear idea of, say, the carry trade — why am I getting paid to do the carry trade? Largely because I’m providing a source of risk insurance for other people. I’m short those futures that are going to go massively up when the market — like a couple of weeks ago — when the VIX spikes from 18 to 30, those short front-month futures are going to hurt you way more than the long back months. And they have to. Because if you hedged that, then there’s not going to be any premium for you to take out of the trade. All of these things are risk premia. In order to get risk premia, you have to accept that risk. Over time you’ll be fine, but you’ve got that unpleasant nature of the payoff.

Another good rule of thumb: anything where you think this is a bad idea because the risk looks unpalatable — there’s probably edge in there somewhere. If you as a professional trader are like, “Yeah, this looks dangerous,” you’ve got to accept that most of the rest of the world also thinks it looks dangerous, and you get paid to manage that fear. You are the bomb disposal guy. That’s why you’re getting paid.

So if it’s like, “Ooh, there’s no way I’d sell options over the weekend, that’s dangerous” — all right, what side do you think the edge is on? The guy who is prepared to sell options over the weekend, or the guy who wants to buy options over the weekend?

Walking into danger — I’m not saying you always have to, and it’s a risk judgment; you don’t have to take every risk that’s out there. But if it makes you nervous, there’s probably edge in there somewhere. If you can get it to a point where you’re comfortable with it, or can diversify, manage, or hedge it, that’s a good place to start making money. That’s why you get paid for selling options. You don’t get paid for buying options.

Again, difficult thing to say, because for the last three or four years, retail made a ton of money buying 20-delta calls in the indices. By the way, historically, what’s the worst option to be long? Index 20-delta calls. If you think that’s your edge — no. Take that money and good for you. But that’s not the way the world has typically worked. It might work like that for the rest of my life, who knows. You’ve got three years where that worked really well, and 100 years where it didn’t. Just as long as you know that.

Look for any situation that legitimately makes you uncomfortable, and there’s probably something in there.

Learning from the past

Erik: Two follow-ons. I’ll give you both, and you can pick. The first is how to use past information to inform future projections or predictions. The second is pricing of convexity and some of the internals behind that. You pick whichever one sings to you.

Euan: I think the first one is probably more suited for a podcast format.

Basically, the only thing we have to predict the future is the past. And there are two things you can say: the future will be somewhat like the past, or it won’t be like the past. Of those, clearly the better one is to say it will be somewhat like the past. That just makes sense, and usually that’s the way it’s been.

The problem, particularly in finance, is it’s an adaptive system with people on the other side of, and everyone is going through this thought process to a certain degree as well. Everyone’s looking at the same information. Everyone’s saying, “This is the way it’s behaved.” Everyone’s trying to predict on the same thing. Largely coming to a spread, but broadly the same conclusions.

So using the past to predict the future is always going to be murky. And there’s an unfortunate tendency we have now. Like, 30 years ago, no one did backtesting, because you couldn’t. There was no data, no computers. Even before Excel — you’d have to write a program in Fortran. It was hard. No one did it.

Now, everyone does backtesting. The problem is that means most people do it wrong. They see backtesting as a way to find patterns, and they’ll test stuff, do cross-validation, test walk-forward, go out of sample, and they’ll find things. The problem is they look at so many things. Of course they’re going to find something. That’s the big problem with using the past — you think you’re doing statistics correctly, but really all you’re doing is looking at stuff over and over again until you find it.

You’ve also got to remember — and this is an Aaron Brown thing as well — as soon as you’ve looked at some data once, you’re done. That data is in sample forever. You can’t say, “Oh, I’ll try my trend-following system on the S&P. That didn’t work. I’ll try a mean-reversion system on the S&P.” The only reason you’re trying that mean-reversion system and think it might work is because your trend-following one didn’t. So you’re already overfitting. It’s not so much overfitting to the numbers. It’s applying information you’ve learned by looking at it already, even if you’ve looked at it for a different thing.

At this point, there is practically nothing I can do to study the VIX that’s going to tell me anything, because I’ve looked at the VIX for so long that I know what’s in there already. I’m not actually making any sort of new judgment. All I’m doing is applying what I know has worked in the past because I’ve looked at it so many times.

The way to address that is to start with something you believe first. Do you believe in carry? Yes. Do you believe in momentum? Yes. Cross-sectional momentum? Yes. Risk premia? Yes. And then you come up with an idea, and then you say, “Does this work in the past?” And you test it. If it works, great, you can make it better. If it doesn’t, you give up.

We see this with options particularly. People are like, “Well, I tried selling strangles on Monday and holding them all week and it didn’t work. But then I found if I sold straddles on Tuesday and got out on Thursday, it did work.” Both of those are driven by the same effect — they’re both variance premium plays. If one of them didn’t work and the other one did work, all it means is you got lucky picking the entry point for one and not the other. As soon as you’ve looked at that second one, you’ve just completely overfit the whole thing.

That’s one of the big problems people make by using past data. You use past data to come up with the overall belief. Carry means something. Credit spreads mean something. We’ve got, I don’t know, 5,000 years of stories about credit. Credit’s older than money. It’s a real thing.

Looking at numbers should be the last thing you do, and only to confirm something you’re already pretty sure works. My risk premia thing — I’m sure it works. I can test does it work better in a one-month option or a one-year option, but I should not be just blasting combinations in until I find something that is the best.

One of the most dangerous words that’s come out in quant finance is certainly “alpha drift.” This isn’t a retail problem either. I see this in quant firms all the time. They optimize and think they’re doing it out of sample, but they’re not, because they’ve already — this is the fourth model they’ve run on the same data because they keep looking at the S&P 500 or something.

Optimization is a horrific thing. It’s one of the worst things to ever hit the world of finance — that concept that you can make something better, or perfect, or optimal.

The other one, by the way, is theta. I can’t think of a thing that’s cost any more money than theta — this idea that options decay over time. That is literally not what the Black-Scholes equation is telling you. The Black-Scholes equation is literally telling you they don’t. If everyone knew an option was decaying over time, no one’s going to pay for it now — they’ll buy it tomorrow, it’ll be worth less. The number of people who’ve fallen into that “I’m harvesting theta” thing — and there’s plenty of influencer types who are peddling that story, typically also owning a brokerage at the same time.

The difficulty of learning from the past is that people think they can learn too much from the past. You have to discount everything you know based on how much you actually believe the thing. Show me an effect you found in the market — I don’t know what it is. A few years ago, “gold goes up on Fridays” was a big thing. Everyone looked at the data — look, gold went up on Fridays. Okay, why would that happen? “Well, it’s because fund managers are scared of risk and they go into gold on Fridays.” All right, find me one of those people. Not you, or some other person who buys gold on Fridays. I mean someone who runs an appreciable amount of money — doesn’t have to be Pimco, but people who run hundreds of millions or billions. Show me one of those people. No one’s ever managed to do that.

Does it go up on Fridays? Well, it looked like it did. T-stats, sure it did. Now how much do you actually trust that? On a scale of zero, which is candlestick charts, to 10, which is carry is a real thing — I’ll give gold on Fridays a three. Could it be true? Sure. But I don’t see a compelling reason. You look at enough things, you’re going to find that.

That’s the problem people get. They look at the t-stats and go, “Look, statistics, man.” All right — now how many other things have you looked at that didn’t work? Because you’ve got to include those.

The philosophical problem of what you can learn from testing

Starts with sound reasoning:

I’d like to start with the idea that people buy bonds at the end of the month because of window dressing and rebalancing of 60/40 funds. There’s an idea. If the stocks go up more than bonds in a given month, people rebalancing have to sell the stocks and buy the bonds. That should have an effect. That’s your hypothesis.

So let’s look for situations where that hypothesis has a market effect. Do you see it? Do you see it in situations where your hypothesis would tell you you should see it? Equally, do you not see it in situations where your hypothesis tells you you should not see it? If this also happens on the second week of the month, maybe it’s not what you think it is.

I’d always like to start with a reason and then test the reason.

…But there’s thousands of years of philosophers who pointed out all the contradictions in this stuff. Nothing in the world is out of sample. The only person in the world who’s out of sample is a baby who doesn’t know anything. That’s the great curse — the more you know. With me and the VIX, it’s not that I’m the greatest VIX person in the world or even close. I’ve just looked at it, I know it, I can’t be surprised by anything it does. That’s the downside of experience: nothing is out of sample…

The best you can do is find a situation that looks like it’s somewhat constant over time. If you look at the statistics of the VIX, the distribution is pretty constant. If you look at what it did in the ‘90s and in the 2000s, break it down in big blocks — it kind of looks the same. Whereas stocks tend not to. You can’t look at Tesla 10 years ago and Tesla two years ago. The business models were totally different, but in one case the stock was a dollar, and over the last two years it’s been between 200 and 400. These are different situations. Stocks are not stationary.

But really, one of the things you have to do is accept how little you know about the world. The good traders never say, “I’m right because of this, and this, and this.” The good traders are like, “I think I might be right because of this. But on the other hand, I could be wrong because of this and this and this.”

Aaron Brown, who’s probably thought more about this stuff than anyone I know, says that a bad trader is always saying “and furthermore,” and a good trader is saying “on the other hand.” A good trader is looking for holes in their argument. A bad trader is continually trying to find other reasons why they’re right.

David Hume — smart guy, all philosophers do is think about thinking — they were pointing out the problems of trying to learn from induction hundreds of years ago. There’s no answer. That’s the only information we have, but we can never really draw certain conclusions from it. So I always apply a big whacking discount to everything. And it’s not statistics. It’s a meta level above that. It’s like belief. I’m not just saying there’s a statistical rule for how you judge if this is good or bad. In fact, literally there isn’t. There’s an actual degree of belief that is independent from what the numbers are telling you.

Vibes vs quant is a false dichotomy — there’s vibes in everything

[The general construction of his portfolio isn’t] based on statistics. It’s based entirely on my degree of belief. Like that carry trade we talked about — how much do I believe that? A lot. Given what I know about its volatility — and I mean “things move around” volatility, not standard deviation of returns — given I know how much that is, given I know how scary it can be, but given I know the belief, I might give that 40%. Whereas there might be another trade where statistics are just as good, but I’m like, I don’t know, I don’t really believe in that one. That gets 20%.

That gives me my baseline, and then I go run all the statistics, and then I say, “Well, okay, I’m shrinking it to this vibes portfolio, because that’s the one I kind of wanted to be at in the first place.” There’s a lot of people who’d be like, “Well, that’s not quantitative. That’s not systematic.” Well, first of all, who gives a shit what they say? You don’t get a medal for being quantitative. You make money or you don’t. Being quantitative is seen as this kind of goal, and it’s not the goal. It’s a tool toward the goal.

Every time you make a portfolio or sizing decision, you are at some point applying that kind of thing. If you’re going to go through all the math, eventually you’re going to have a utility function, and that utility function is going to have a risk aversion parameter. So you’re going to go through all this math, and then at the end it’s going to say, “Oh by the way, what’s your risk aversion parameter?” You have to make that decision somewhere. I’m just doing it at the start. It’s much more of an exogenous thing that’s obvious.

It’s more a reflection of what I think I actually know than a mathematical statement. It’s not a mathematical statement. It’s an epistemological statement.

A lot of people get hung up on “insult found: the discretionary trader.” How is that an insult? Everything you do is discretionary at some level. Literally, you have chosen to do something. You have exercised your discretion to do something. Whether you do that in the math you choose, the problem you attack — you can’t go through life without exercising discretion. That’s actually called knowledge or experience. It’s a useful thing. If you don’t use that, you’re an idiot.

The cost of systematization

There’s certainly this whole idea that if you’re systematized, it’s emotionally easier to stay with things. I don’t think that’s true. I have operated completely systematic stuff when I’ve had jobs, and it loses money. It’s not easier. It just isn’t.

The reason I think you should systematize something is because you have to. Every time you’re removing yourself from interacting with the market, you’re removing yourself from an opportunity to learn. I think you should do things as manually as possible — and I mean it. If you can get away with writing stuff down on a legal pad, then that’s the way to go. If you need to use a spreadsheet, use a spreadsheet. But if you can do it on a spreadsheet, yet you choose to write some massive API call and do the whole thing in some language — you’ve done that because you wanted to, not because you needed to. That’s a mistake.

The places I’ve systematized things are because I wanted to avoid operational risk and key-man risk. If I’ve got a strategy I want to run for an ETF every day, it has to operate if I crash my car and go into a coma. That has to be systematized. But if I’m running something on my PA because I find it interesting and I’m still messing around with it — systematizing that, all I’m doing is removing the opportunity to learn.

It’s one of those things where retail look at institutional and go, “Well, it’s all systematized. I have to do that.” But they don’t think of the reason we systematize things. It’s not to make the strategy better. Honestly, it probably makes it worse, because you’ve locked it in. You’ve said, “This is it, I’m not going to be learning anything more about that now.” But it definitely removes operational risk. It makes it cheaper to run, because you can just let it run on its own. You don’t need a person doing it all the time.

People on the outside are making a judgment of how the things on the inside work, and they’re missing the point.

How do you avoid it? I don’t know. Hopefully if I pointed that out to you now, you’d be like, “Oh yeah, now I get it, this is the important thing.” Think about anything else you’ve ever learned. You’ve learned by doing, by the immersion in it. Trading now, the actual act of trading, is click, click, click, done. You don’t learn anything from that.

But what you do learn about — especially things like adverse selection and the ability to execute — like, “Oh, it looks like there’s plenty of volume. Oh, but every time I put a 100 lot in, all the bids disappear.” That’s the sort of thing you’ll learn so much about just doing it and interacting with it. You’ll never learn that if you automate the trades.

[Kris: To insert something here. I’m reminded by an interview with Agustin Lebron where he talks about getting ideas from just staring at an order book for ours. Seeing how it updates, how bids and offers change after trades, or in the absence of trades but while the rest of the market is doing things. He recommends doing this in medium or low-liquidity names where you can possibly see a story unfold in the order book. I agree with all of this. I mean I don’t think I never explained it quite like that because it’s just the water you swim in. If you trade something relatively illiquid, every movement matters. If some 20 delta put trades on the offer, you have to triangulate whether it’s skew expanding or volatility in general. See Mermaids, Fireflies, and the Bid-Ask Spread.

But there’s a corollary hidden in wisdom of learning from sparse order books: edge is inversely related to liquidity. The more liquid the market, the more whatever’s left looks like a risk premium. SPY is one of the most liquid markets in the world. If you had any edge in trading it, the edge would be so scalable your heirs would be set for an eon. Remind me to rant about this sometime.]

If you’re in a situation where you have to automate the trades, you’re probably playing in a game where you’re not going to win as a retail trader. You shouldn’t be trading, “Oh, I have to run this through the API so I can trade 15 times a day.” That’s one of those quant-envy things where you’re like, “I’m going to do this the way Citadel would do it.” Again, you’re playing basketball one-on-one with LeBron. Good luck. Don’t care how tall you are.

Inefficiency vs risk premia

There’s a split between inefficiencies and risk premia. With an inefficiency, price competition is going to drive it to zero. Nothing’s purely risk and nothing’s purely inefficiency, but market making in options has a lot of inefficiencies. It’s like, “I’m going to buy this option because I can sell this option. Those things should be worth the same. They’re off by a little bit.” That kind of thing. The act of providing liquidity is a risk premium, but the way you do it is looking at all these little inefficiencies.

If you look at the way equity market making has gone in the last 30 years or whatever, ever since the floors have gone and we’ve gone electronic, that has been driven to almost zero. It’s not at the point now where any sort of random person with an off-the-shelf piece of software is going to consistently make money making markets in equity options anymore. It’s gone from profit margins like Tiffany’s might have, and now it’s Walmart. Everything’s driven to zero.

Risk premia are different, though, because they’re there for legitimately a reason. It’s a different utility function. You’re doing something you know is wrong because you don’t want to deal with the risk. You’re buying insurance. You’re not buying insurance because you’re dumb, you’re buying insurance because you want insurance. And I’m selling it to you because either I can take the risk, or I can lay off the risk, or whatever reason — I’m doing it to maximize expected value. You’re doing it to maximize another utility function. We’re both happy.

That kind of thing has got more legs to it. If you look at the variance premium over time it’s been reasonably constant. Obviously it goes up and down. You’d think that after 2008 it would get wider; surprisingly, didn’t much. You’d also think that now, because of all these zero-DTE covered call funds, it would collapse. And it kind of has. But sooner or later people will realize a lot of these funds don’t make money, or some of them will blow up, and it will go back to where it was. All these risk premia are cyclical, but there seems to be a natural limit that doesn’t get driven to zero.

It helps to be able to say: is this an inefficiency that people can’t access, or is it genuinely a thing that everyone knows is there and people are doing it for different reasons? Because that’s the sort of thing you can build a trading life around. Inefficiency — if you find one, good for you, knock it out of the park as hard as you can. It’s not going to be there forever. If it is there forever, Citadel is going to come along and build a desk on it and crush you.

Adaptive markets, but not adaptive traders

Erik: The follow-on, though, even within something like risk premia that we know is there for a reason — an example you’ve given previously is there’s a $20 bill on a busy highway. If I’m standing on the sidewalk and there are five crackheads, and I say, “Hey, I’ll give you 10 bucks to go get that.” And then one of the other crackheads says, “I’ll do it for eight. I’ll do it for five.” And then at some point we get to a floor where the crackheads are like, “I am not going to risk my life for this anymore.” Do you think that with something like risk premia in index options, based on that consistency, we’re generally about at that level? Or do you think it’s likely to shift around more?

Euan: That’s difficult to say. But with your crackhead example — one crackhead might say, “I’ll do it for six bucks,” and he runs out there and he gets killed. And then the next crackhead’s going to come along and say, “Actually, I want seven now, because I saw what happened to this guy.” It’s not the same people all the time.

What we tend to see is a new cohort of traders come in and blow up doing the wrong thing, and then a bunch of other people come in and end up doing the same wrong thing. The market never completely adapts, because it’s never exactly the same group of people who learn. Markets learn, but traders as a whole don’t. For things like this, a lot of it’s propped up by new people coming along making the same mistakes as the old people. Adaptive markets, but not adaptive traders. In a lot of cases people think they’re the same thing. I don’t think they are. There were a ton of floor traders, good floor traders, and then everything went electronic, and they tried to do what they did on the floor on the screen. It just didn’t work. And I don’t know any of those guys who were like, “It’s not working, I’ve got to do this and this and this instead.” That didn’t happen. Instead, a whole bunch of other people — studied computer science at Cornell or whatever — came in. It was a whole different group of people who made money at that point.

options policework

A moontower user sent this [paraphrased] message in our Discord the morning of Jan 9th:

NLR [VanEck Uranium and Nuclear ETF] had a price shock on Jan 2 and has been ‘fast grinding up’ since then. Did I “lose” here because RV climbed up faster than RV and my losses are ‘gamma’ driven?

Now the most important part — what can I learn / what should I do as part of my process?

We are going to do the post-mortem in steps. The first task is to take inventory of the scene. Basic policework. “What happened?”

Once that’s established, we can at least start to disentangle bad luck from decision quality and finally wrap up with risk management/hedging/whether we should close the position or not.

Arriving at the scene of the crime

What do we know? Our friend sold a small amount of 1-month NLR at-the-money straddles in December. To be discreet, I’m going to guess the date to be December 15th and the strike to be the 126 line and IV was ~41%

Below is a simple time series of:

constant maturity 30d IV LAGGED vs 30d realized vol

By lagging IV, we align it with the 30-day realized vol that was experienced in the subsequent month. We can see that the RV (faint green line) our friend experienced far exceeded the IV (dark blue line) of the straddle they sold.

A chart like that is a handy compression, but since it is:

  1. using constant maturity vols (ie interpolated) and
  2. floating (the IV is taken from the .50 delta call each day)

…the chart is not high-resolution to discuss p/l, but can only gesture roughly to its direction. It’s a blurry picture of a license plate as the driver speeds away.

We will get down to the contract level, but first, we want to develop a sense of proportion about notable move sizes.

Realized vol

Jan 2nd was the steepest one-day move: 7.1% or about 112% vol annualized. Nearly 3 standard devs.

 Over 8 days, there was a 13% cumulative rally, or 73% annualized vol.

The calculation: 13% *sqrt(251/8) = 73% annualized vol move

Any funny business under the hood?

  • NLR’s largest component, CCJ, is ~9% of the basket. It rallied a bit over 7% as well which is frankly underperforming the basket since CCJ is a higher vol than the ETF.
  • Its second largest component is DNN ~6% but lots of names in the basket are close to that size. DNN was up 14%…but its normally twice the vol of the ETF already.

In z-score space, the ETF and its 2 largest holdings all moved about the same amount.

All these clowns are riding in the same car. Its a 1-corr move, in a beginning-of-the-year inflow to this sector.

💡For those of you who trade around rebalancing and calendar anomalies, perhaps this is a thread to pull on?

Drilling down to the option contract level

The NLR option volume in the month preceding Jan 9th was not notable. There was a spike in puts traded on Jan 5th, but this was already after the largest single-day move happened

The largest component, CCJ, did not have any noteworthy volume in the prior month either.

What stands out in NLR is how small the open interest is in general. This is not a liquid option name.

Price and P/L

From December 15th to Jan 9th, the Jan 126 straddle expanded from $12.75 —> $15.50 as the stock went to $140. No surprise, the call went to >.90 delta.

So the short straddle position lost $2.75, assuming you did not hedge any of the delta on the way up.

If you’re intent is to trade vol, allowing the delta to ride like that is introducing a lot of noise into your trade expression that was supposed to be about vol.

What if you sold the straddle and hedged the negative gamma daily by bringing your deltas back to neutral? In other words, buying shares after they rallied and selling them as they fell in opposition to the changing straddle delta.

Our service includes an attribution visualizer which allows you to decompose your daily and cumulative p/l due to realized and implied vol changes as the option and stock price move around. It is from the perspective of an option buyer. In this case, we are selling, so just flip the signs. We also need to double the numbers since we are assuming a straddle hedged daily, not a single call or put as the tool assumes.

The total ACTUAL delta-hedged p/l as of January 8, the day before the friend messaged the group, would have been -$.51 per contract or -$1.02 for the straddle. The loss would have been less than letting the straddle ride, since the stock trended up and each rebalance would have forced the hedger to buy on the way up.

If the stock chopped around at 70 vol but still landed on the strike, hedging would have locked in a bunch of negative gamma scalps while the straddle decayed.

Hedging makes your p/l reflect the vol that was realized but whether this is good or bad for you, ex-post, depends on whether the stock chopped or trended.

Ex-ante, you want your hedging to be aligned with the reason for your trade, which in this case is presumably the expectation that IV would have a risk premium above realized, since the trade was selling 1-month atm straddles.

A note on attribution

The chart doesn’t track the sum of unexplained p/l although it is displayed in the summary (not shown). The “unexplained p/l” is the balancer which makes the theoretical attribution tie out with the actual p/l. It is a catch-all for the higher-order greeks, mostly vanna and volga, which reflect the fact that your gamma and vega, respectively, are not constant during a single day’s move.

The bulk of the p/l on that big day is due to realized. It’s fair to say from the summary that realized p/l explains most of the result. This is what we’d expect from an option with only a few weeks until expiry.

No smoking gun

Given the lack of notable action in the option volume in either NLR or its components, the uniform behavior of the moves in the complex, a boring IV chart to close out 2025, and the fact that the move happened on the first business day of the year, that this result was a bunch of methodical but unanticipated sector flow. Approximately 2.8 sigma move in one day, or about 1/200 probability, a bad beat with roughly the same probability of being dealt pocket aces (1/221 because 4/52 * 3/51).

[Stock moves are fat-tailed, so the probability is actually larger than 2.8 sigma would predict, but the fact pattern here still suggests a bad beat. The IV wasn’t suspiciously high in December, there wasn’t any telegraphing flow].

An opportune time to remember one of the reasons gambling and poker experience is helpful…from why poker is used to train traders:

This is one of the great teachings of poker. Short-term results are noise. He explains that in Limit Hold’em, even a high edge hand has only .02 big bets worth of expectancy vs a standard deviation of 2.5 bets.

[Kris: In investing language, a .008 Sharpe for one trial. The SP500 has a daily expectancy of about 3 bps and 100 bps standard deviation for a daily Sharpe of .03. The poker hand has almost 4x the noise of the daily SP500 return.]

Since poker teaches that you will make the right decision and still lose money, it trains you to emotionally decouple decision quality from result quality.

This is a ceaselessly profound concept. Not because it’s so clever, but because of how it resists internalization. It’s easy to understand, it’s hard to apply the understanding to how we receive the world.

As police work goes, there will be no verdict or even charges brought as to whether the decision to sell the straddle was sound. We do get research inspiration. Is sector dispersion especially high on the first of the year? First of the month? Is there more volatility in general on those days? If so, is the median volatility higher or the mean (ie is it being driven by outlier-type moves)? We don’t know if selling the straddle was bad, but we do get new questions. This is what a career in trading looks like. If you don’t like this type of problem, then hooray, I’ve saved you a bunch of time compounded over your life. You’re welcome.

Regardless of the outcome, we still have this business of risk management.

Should our friend have hedged or closed the trade?

We don’t get to snoop forward in time.

The following is true but unknowable in advance:

  • If the stock is trending, you want to hedge aggressively. Buy delta as it rallies, sell it as it falls.
  • If the stock is mean-reverting, you want to sit on your hands.

Your risk approach cannot depend on what you don’t know. And it must depend on what you consider tolerable.

The combination of these constraints will dictate how big your position can be. We’ll call this your limit. From there, you are simply monitoring how big your position is under various scenarios to that limit. If it is greater than the limit, you must reduce it.

I’ll give a simple example, but know this is a vast topic and a chief concern (and unsolved problem…there’s no single answer to this) of any risk-taking outfit.

Let’s say you are willing to tolerate 1% volatility in your total portfolio due to a particular trade on your average day. If you have a $1mm portfolio, that’s $10k. To a first approximation, that means keeping your swings due to delta below $10k. Call NLR a $140 stock with a 48% vol. For a typical day, that corresponds to 3% moves or $4.20.

$10k/4.2 is the daily swings associated with ~2,400 shares or 24 100 delta options. Or 48 50 delta options.

So what is NOT conservative about this risk-based sizing:

  1. “Typical day” is being proxied by 1 standard dev (ie the 3% daily vol). If moves are normally distributed, that means about 1/3 will be greater than that or more than a week out of every month will be composed of bigger moves. And that’s ignoring fat-tailedness
  2. We aren’t accounting for adverse vol changes. If you are short options, trades are negatively skewed so we’ll want to be more conservative still.

What IS conservative about this risk-based sizing:

  1. If you hedge your deltas even once a day, you will not have as much daily variance in your p/l due to delta, which is effectively what we’re describing above.

How does this shake out?

A good starting point!

In the case we’ve been following, if our friend used a rule like this, it woud have prescribed 48 50 delta options or 24 straddles.

The straddle went from $12.75 to $15.50 on a bad beat with no hedging.

The loss = 24 straddles * -$2.75 * 100 = -$6,600

If our friend hedged daily, mirroring the attribution visualizer recipe, the loss would have been:

The loss = 48 contracts * $-.51 * 100 = -$2,448

Notice the constraint:

This makes sure delta is positive for the sake of the calculation AND doesn’t allow you to oversize a position just because you chose a skinny option.

I came to this example from the perspective an option seller. If you are a buyer the most you can lose is your premium if you DON’T delta hedge. You can use your risk tolerance for losing money as your premium spend limit.

If you do delta hedge, you can lose many multiples of your premium. For example, if you buy an OTM put and the stock grinds down slowly to your strike, you will be buying shares all the way down. You will lose not only on your stock trades but also on the premium going to zero. T

I’m going to pause for a second to level with you because I do feel some almost paternal responsibility stemming from the privilege of many smart but also young readers who come to this letter to hear from me because of my gray hair and specifically because I won’t treat options like the next house-flipping get-rich trend.

This topic of risk is so vast that its discovery is an ongoing project throughout your career. You are shaping and being shaped by the rules you create and their feedback, so to think there’s an “answer” is to not appreciate how many facets there are to managing risk across a portfolio of non-linear instruments.

To recap…this was a “3 standard deviation” move and the loss was comfortably below our tolerance. You can season to taste, but this is overall a conservative approach that you can experiment with. This is a point-to-point p/l, so the rule is providing some flex for tough marks along the path. Like I said, a starting point.

🔗If interested, my treatise on hedging If You Make Money Every Day, You’re Not Maximizing

What the risk management decision is NOT about

Whether you should have the trade on in the first place is not the realm of risk management. That’s the alpha signal or whatever you want to call it in your approach. Risk management is concerned with sizing, which is the last layer of defense. (The prescribed size might be tiny, in which case, presumably, you are doing lots of trades.)

I’m saying this because the fact that you already have a trade on is not a reason to keep it on. If you don’t want to put the trade on fresh, you should get out. There’s an opportunity cost to your capital.

If a trade you have on is not bad but just fair, then the decision comes down to whether the variance is acceptable. If there are costs to getting out of a coin flip that you can sweat the risk on, then it’s ok to save the transaction costs. You can refine that a bit to “is the coin flip’s expectancy the same as my cost of capital” yadda yadda, but you get the gist. There’s a cost to reducing variance (ie hedging or closing) and it’s perfectly fine tto avoid it if the risk is tolerable. There are a lot of risks in life you don’t bother hedging.

Finally, rules aside, if you are regularly running risk that makes you lose sleep, impairs your judgement, or threatens to blow you up even 1% of the time, the size is wrong. 1 in a 100 is inevitable if you plan on doing this for awhile.

Positive delta puts in the wild: Avis stock (CAR)

Remember that chart of CAR last week.

(Matt Levine wrote about the fundamentals of the squeeze on 4/15)

So this was Thursday:

TradingView chart
Created with TradingView

 

Also, note that the change in the basis per expiry increased beyond October, meaning the implied carry cost is no longer negative.

When a stock is hard-to-borrow, its options will imply a future price below the spot price, since a market-maker that is getting saddled with long calls, and short puts from the flow must short shares to hedge. The cost to borrow those shares is reflected in the synthetic futures (ie the option combo of long call/short put on the same strike).

The carry rate increased on Thursday, which means puts fell relative to calls on the same strike, as presumably the borrow will loosen as the squeeze subsides. This was another headwind for people who bought downside puts to bet on CAR coming back to earth. Far OTM December puts, like the 70 strike, actually declined in value on the sell-off. A classic example of what I call positive delta puts.

the bias of hedging on implied delta

Tweets

Before we get to today’s meat, here are 2 threads spurred by oil’s advance yesterday.

 

Delta is God

If you’re reading a Thursday Moontower, “you’ve heard the expression vega wounds but gamma kills.” It’s not quite so cut-and-dry. My pushback to that trope is the recent article vega’s finishing moveHowever, I’m sympathetic to “gamma kills” mantra. The running joke I’ve used to say on the desk has a similar energy:

“delta is the only greek”

I wouldn’t take this literally, the joke is bowing to the idea that if you have your hard deltas, ie your shares, pointing in the right direction, you tend to win. The Freudian reading of that statement is I’d rather be good at directional trading than a vol monk.

Today, we give delta its due. Delta is god.

No matter what you think it is, you never quite understand it. The best we can do is understand how it will harm or help us based on the thing we can’t know in advance, but know will affect our p/l — path.

While I’ve been meaning to write about this for awhile, this paraphrased question from a moontower user, bumped this post up the editorial queue:

“I’m backtesting delta-hedged straddles and I’m worried the vol I use to compute my hedge delta is ‘wrong.’ Does the choice of hedge vol bias my P&L, and if so, how?”

Pull up a chair, young Padawan.

I’m going to offer 3 perspectives.

  1. The quant answer.
  2. The quant who speaks “trader” answer
  3. The Moontower treatment

Finally, we’ll see how this idea applies to traders and investors who try to structure an options-like payoff to a trade without using options at all.

So much trader mindshare is fixated on delta-hedging for the same reason we are never happy with the quantity we trade in hindsight. The goal here is to create enough clarity that you can not only make better ex-ante decisions but make your peace with them regardless of the outcome.

Onwards.

The Quant Perspective

We’ll start with the mathematical approach. This is not my wheelhouse, so I’ll save my words for later sections, but if you can’t wait to curl up with notation, then this post is for you (h/t to the Moontower Discord where it surfaced).

I couldn’t help but print the acknowledgements section below. I don’t know stochastic calculus, but I suspect the people involved in this paper might.

Which Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios by Paul Wilmott & Riaz Ahmad

ABSTRACT

In this paper we examine the statistical properties of the profit to be made from hedging vanilla options that are mispriced by the market and/or hedged using a delta based on different volatilities. We derive formulas for the expected profit and the variance of profit for single options and for portfolios of options on the same underlying. We suggest several ways to choose optimal portfolios.

ACKNOWLEDGMENTS

We would like to thank Hyungsok Ahn and Ed Thorp for their input on the practical application of our results and on portfolio optimization and Peter Carr for his encyclopedic knowledge of the literature.

A Quant Who Talks Like A Trader

The next perspective is a bridge. In the incomparable book, Financial Hacking, quant Philip Maymin breaks things down in terms that your common option flow trader will understand.

On hedging to model (forecast) delta vs implied delta

The short-form intuition is this: you bought a call and hedged it. So you are betting on higher volatility. When volatility ends up higher, even if only for random reasons, you benefit, and when it ends up lower, you lose.

That intuition continues to hold even if you hedge at the wrong vol. If, for example, the true vol is 30 but you hedge to 20, you are just introducing noise. The slope between your P&L and the realized vol is still positive, but not as sharply defined.

Philip brings in the practical concerns of, well, having an employer to answer to who doesn’t like loud “noise”.

If you want to minimize your mark-to-market P&L, you may choose to hedge to the market even if you think the market volatility is wrong.

How do you trade-off these two risks, the mark-to-market risk versus the at-maturity risk? Ultimately, you probably will decide based on the maturity of the option you are hedging.

  • If the option will expire in a month or two, you will almost surely be able to weather any intermittent mark-to-market volatility, so you will lean towards hedging to model.
  • If the option will expire in many years, you will likely lean towards hedging to market, at least until the expiry gets closer.

And what do people do in practice? They hedge their bets on how to hedge. One common rule of thumb is to hedge halfway between the model and the market delta. Then you’re never exactly hedged, but you’re never too far away either.

The inability to hedge perfectly continuously impacts your trading by introducing random risk. This risk decreases if you hedge more frequently, but only as fast as the square root. Therefore, if you want to halve your risk, you have to hedge four times as often.

This is a fantastic observation to give a sense of proportion:

Noise from hedging a one-year option on a daily basis instead of continuously is about the same as one volatility point. If you make one volatility point in expected profit and the standard deviation of your profit is one volatility point, then your Sharpe ratio is about one.

And remember…the risk from not hedging continuously can be diversified away.

His final point here echoes what I wrote in a misconception about harvesting volatility.

Which brings us to…

The Moontower Treatment

The original paraphrased question once again:

“I’m backtesting delta-hedged straddles and I’m worried the vol I use to compute my hedge delta is ‘wrong.’ Does the choice of hedge vol bias my P&L, and if so, how?”

My dead-leg-on-the-toilet response:

Here’s the quick answer…the vol that generates your delta introduces bias that you discover after the fact but you can understand how the bias is correlated to your p/l in different scenarios.

For example, if you are long vol and the stock trends, you will wish you hedged on whatever delta was the “lowest” of the reasonable options you could have chosen from…so if the option is ITM you will have wanted to hedge deltas on a high vol, but if it was OTM you will have wish you hedged on a low vol!

I’ve never done this, but you could create a little cheatsheet matrix with:

  • option ITM or OTM
  • market trends or chops
  • preferred vol i wish i would have hedged on = “high” or “low”

By comparing that matrix to your strategy you can see which biases cause you to double down on your implicit exposure vs hedge it (for example, if you are long ITM options and vol expands in a trending market you will hedge on that desirable light delta…but you are already winning on vega so maybe this codependancy is too much “doubling” down which hurts extra if you were short that option)

Of course, I had to make the cheatsheet now that I got a moment to focus on the question. To start, I fed my response to Claude and it whipped something up. I did have to re-work some of its understanding.

[These are Gell-Mann amnesia moments, where it stumbles on things you know well, and wonder about what it tells you in domains you are less equipped to discern.]

Let’s begin with the cheatsheet, memorialized at https://delta-hedging.moontowermeta.com/:

The sheet is self-explanatory, but there are biases we can anticipate. It’s what I referred to as “the doubling-down” in my response to the reader.

Suppose you follow the rule:

“Hedge On Implied Delta”

IF:

[You buy an OTM option because you think IV < forecasted realized]

AND:

[Your vol signal is correct]

THEN:

[Your hedge ratios will be “light”…I buy OTM calls and sell too few shares]

THEREFORE:

If we trendyou will make “extra” p/l beyond the fact that you bought underpriced volatility. This is “doubling-down”.

If we chopyou will make less gamma scalping p/l than you would have with a heavier delta. The forgone p/l will be buffered by the fact that you were right on the vol being cheap.

In this case, hedging on the delta of the implied vol, is doubling down on your vol forecast in the event that we trend, and offsetting some p/l in the event that we chop.

💡Your choice of delta to hedge on begs you to wonder if a high realized vol forecast is more likely to coincide with trend or chop.


Most of the time, options embed a risk premium above the realized vol.

[The bridge between this idea and making money on selling options sways wildly and has a few missing planks. Many have died trying to find the treasure on the other side so take it easy Indiana Jones.]

That said, it’s understandable if you never want to buy an option. But sometimes you want an option like exposure, just like you might want an insurance policy. You want protection against a high-impact event even if you don’t think it will happen.

I discuss this in the Moontower community, where I prefer to hold BTC exposure as options rather than as a hard delta allocation (I actually use a blended approach, but the reasons aren’t germane to this post).

I pick my spots when I buy the options. My most recent call purchases feel validating because I thought the vol was cheap, so despite losing on direction, they were much better buys than the counterfactual of owning hard deltas.

[Welcome to vol trader cope. This is literally what life is like as a vol trader. I lost money but made the right decision. Yay. You only hope that your career lasts long enough to realize the sum of all the right decisions. The alternative of just guessing in a high-variance game and trying to get lucky is good too. If we focus on survivors. And we do. This is America after all.]

But what if you wanted to replicate the call exposure without actually buying the calls?

Replicating a Call When You Think It’s Overpriced

The closest neighbor to the term “portfolio insurance” in a database of vector embedding is “1987” (Did I put those fancy words in the right sequence? Who cares, you get the joke).

Don’t let that taint your mood going into this next section. You know that I know about that history. Calm down, we’ll extract the fruit from replication and point out the poison you can’t eat.

Step-by-step here.

You want BTC call exposure. You look at the options and think they’re overpriced. So you decide to skip the call and instead replicate it dynamically.

How?

You will be delta hedging in reverse. You’re assuming the posture of someone who sold a call and now needs to replicate it. An option market-maker who sells you a call must go out and manufacture it. If they can manufacture it for less than the price they sold it, they make a profit.

In this case, you are taking the role of call buyer, but instead of buying the call, you are going to try to manufacture it yourself, just like the market maker would have if you bought a call from them.

Mechanically, you’ll hold some BTC, intermittently rebalancing your position as spot moves, synthetically tracing the call’s payoff without paying the upfront premium.

How much is some?

You look up the delta of the call you would have bought, and you hold that much BTC.

How does intermittently rebalance work?

As BTC rises, delta increases, you buy more. As BTC falls, delta decreases, you sell some. You’re manufacturing the call’s convex payoff with a series of linear trades.

How often?

How often does a market-maker hedge? This is the question we’ve tackled many times. It’s a trade-off between the “noise” Maymin alludes to as you sample volatility. If you are a market-maker hedging a short option and the market trends, you’ll wish you hedged often (sampling a lower vol than experience from point-to-point).

If it chops, you’ll wish you hedged weekly, sampling a much lower vol than the daily ranges suggest. Both you and the market-maker face the same problem. You are both trying to manufacture an option whereby each time you trade you “sample” a realized volatility. The more you sample, the closer you get to the real vol. The less you sample, the more likely your replication strategy will differ from the real vol and you could get lucky or unlucky to the platonic (and non-existent) continuous vol.

The cost of this replication comes from the adjustments. To replicate a call, you buy more as the market rallies because the option for the strike you’re trying to mimic increases. You sell as the market falls. You are always buying high and selling low. The sum of those round-trips is your premium. You’re just paying it in installments instead of upfront. If you think these installments net of all transaction and slippage costs would exceed the call premium, you should just buy the call.

To feel good about this strategy, you’re rooting for the options to have been overpriced. If realized vol comes in lower than implied, your rebalancing costs less than the call premium would have. You built the same payoff for cheaper.

To determine how much stock you need to buy, you’re computing your delta at some vol, and that choice determines whether your delta is heavy or light. If you hedge at a high vol (say, the implied you think is too rich), you’re holding more BTC than you “should” — heavy delta. If you hedge at a lower vol (your realized estimate), you’re holding less — light delta.

The cheatsheet as an aid to your hedging strategy

The sheet has the posture of someone long an option, who by replicating is manufacturing an equivalent short option. They paid a premium upfront, but hope the sum of their gamma scalp stream exceeds the premium they paid. In other words, their replication posture is the opposite of yours. You are trying to replicate a long option because you think it will cost less than actually buying a call.

So you invert the logic of the sheet!

If BTC chops you want a light delta. Fewer round-trips means less friction eating into the savings you’re generating by not paying the full premium. If you are right about the IV being overpriced but you hedged using the implied delta, then you will suffer a bit because your delta will have been heavy. But this will partially offset the profitable decision to not buy the call outright. If you hedge on your “model” delta (ie the vol based on your realized forecast), then you are doubling down on your prediction that the vol is cheap in the event we chop.

Again, the idea of vol and its coincidence with trending or chopping is lurking beneath but now you are aware of it.

Restriking Your Synthetic Call

Say BTC has run from 70k to 90k. You’ve been replicating a 100k-strike call, but you want to “roll” it up, taking profit and starting fresh with a 130k-strike call.

You can just look up the 130k call at your chosen vol and adjust the delta to match. That will result in monetizing some of your BTC as the 130k call will have a lower delta than the 100k call.

Notice that if you don’t roll your 100k call is closer to ATM with the spot BTC now up to 90k. It has more gamma than your old deep ITM 90k call. More gamma means your rebalancing is more frequent and more costly. You’re “long” a more expensive option. There’s no free lunch. If you substitute your replicated call for a real call, that call’s theta will reflect the higher rebalancing costs you tried to avoid.

So….

What Makes You Wish You’d Just Bought The Call?

This question strikes at the heart of the Black-Scholes assumption of continuity.

Gaps.

The call buyer pays implied vol upfront and owns the path, for better or worse, for the duration of its life. If a stock gaps up 20% over the weekend, the call captures the full move. The gamma which you prepaid for, ensures your delta adjusts automatically.

The synthetic call you tried to manufacture missed buying deltas in the gap. You are not as long as you should be and to make it up you need to buy all your shortfall deltas up 12% as opposed to prices along the way.

Hard optionality is valuable and impossible to replicate. This is why Option Market Maker 101 class teaches you that the only way to hedge an OTM option is with another OTM option. Nobody knows what the SPX down 25% put is actually worth.* You can reason about a relatively tight put spread only because the error is bounded in proportion to the risk you know you are taking beforehand.

(Although we can reason that it commands a premium and likely trades for more than its actuarial value which is not really known. It’s all a bit circular. And you are still left to contend with the fact that the people, as a category, who buy those teenies know a lot more about vol trading than you. There is no non-vol trader buying that option. Also, this paragraph was written in invisible ink to reveal the VIX basis traders on the mailing list.)

Portfolio insurance failed because it was crowded thus blowing up the cost of put replication by feeding on itself. Meanwhile, the owners of the actual puts went on to start the trading firms you know of today.

how to get arbed with perfect information

The “Bridge of Asses”

📺Option Pricing Explained: No Arbitrage + Financial Mathematics from a Quant | 52 min watch

Doug Costa (SIG quant, former math professor, and the teacher I learned Black-Scholes from 25 years ago) builds no-arbitrage derivatives pricing from scratch using a binomial tree. No calculus, pure replication.

The thing I want to point you to is the profound role of the no-arbitrage axiom. It is the basis of derivatives replication and, by my assertion, represents the “bridge of asses” in investing education.

As a reminder, since nobody clicks links, Wikipedia says the pons asinorum or “bridge of asses” is:

used metaphorically for a problem or challenge which acts as a test of critical thinking, referring to the “ass’ bridge’s” ability to separate capable and incapable reasoners

The notion of replication is the pons asinorum of investing education because it is:

the conceptual rails of looking at a web of branching future payoffs, seeing how they could be replicated, and measuring the cost of that replicating portfolio today. It is the formalization of finance’s deepest truth — you cannot eradicate risk, but only change its shape.

You could make an even stronger claim that it lies at the core of decision-making itself, as it formalizes opportunity cost.

And I say this without being able to appreciate its deeper impact. Doug pauses for a moment in the video to marvel: when you add no-arbitrage condition to the standard axioms of mathematics, he says, the entire field of financial engineering “blossoms” out.

His colleague frames the no-arbitrage axiom joyfully:

Either we get a formula [so we win mathematically]. Or it’s violated and we make free money. Either way, we win.

Towards the end of the video, Doug discusses reflexive pushbacks he’s encountered after teaching this.

“One piece of pushback is typically, well, maybe it’s just that with stock prices, you don’t really know the probabilities. So it’s just a matter of knowing the right probabilities— if you could really discover somehow what the true probabilities were, then it would be better to use them [than the risk neutral probabilities].”

Doug’s rebuttal shows how you would still be arbed.

“I’m going to give you an example to debunk that idea. And I call this example the coin flip contract. So I’m going to postulate that there’s a company, a corporation, that finances itself, not by selling stock, but by selling what they call coin flip contracts. And the corporation has gone to great trouble and expense to manufacture a perfect coin, meaning a coin that is exactly 50% to be heads and 50% to be tails every time it’s flipped. So the probabilities are always 1 half and 1 half guaranteed…

You can watch the video, but I paraphrased it here as well. Here’s how it works.

A company issues coin-flip contracts based on a provably fair coin. The contract pays $150 on heads, $75 on tails. These trade in a secondary market at $100. Interest rate is 0%.

So we know everything. The probabilities aren’t hidden or estimated. They’re printed on the coin: p = ½.

Now: what’s the no-arbitrage price of a 110-strike call on this contract?

p̂ = (100 − 75) / (150 − 75) = 

Call value = ⅓ × $40 + ⅔ × $0 = $13.33

Delta = (40 − 0) / (150 − 75) = 8/15 of a contract

Now suppose you say: I know better. The real probabilities are ½ and ½, and I’m not going to ignore them. Expected payoff is ½ × $40 = $20. So you buy the call from me at $20.

Here’s what I do next. I’m short the call. I immediately buy 8/15 of a contract to hedge.

Heads: My 8/15 position gains 8/15 × $50 = $26.67. Plus your $20 premium, I have $46.67. I owe you $40 (I have to buy the contract at $150 and sell it to you at $110). Net: +$6.67.

Tails: My 8/15 position loses 8/15 × $25 = $13.33. But I have your $20 premium. Net: +$6.67.

Every time. Both states. Guaranteed $6.67. I haven’t predicted anything. I don’t care what the coin does.

What did you get? Heads: gain $40 on the option, paid $20, net +$20. Tails: lose your $20 premium, net −$20. You’ve turned a coin flip into a coin flip — a $20 bet where you win or lose based on what the coin does.

If you try to hedge back? Doesn’t matter how you move delta. Win more on heads, lose more on tails. Move it down: vice versa. The best you can do is lock in a guaranteed $6.67 loss.

You had perfect information about the true probability….and you still got arbed buying the calls (you should have bought the contract!).

The market-maker doesn’t need a view on the coin, just the ability to trade the underlying and the derivative simultaneously. And acquiring the knowledge to cross the “bridge of asses.”


A random personal thought:

I suspect is kind of triggering for some people. It offends one’s sensibilities to think

that understanding derivative pricing ends up trumping knowledge about the true odds of things.

It’s like you spend all this time researching and learning and at the end of the day some market-maker knows just enough to not trade at the wrong price with you anyway. I’m overstating that reality, getting picked-off is real and market-makers are rightfully paranoid. But I guess that’s why I’m drawn to replication as a way of thinking. A trader is just looking for some free money when your bid or offer presents a contradiction. And that hunt makes all prices a little smarter, which, is a public good (but also a frustrating result for traders themselves, which is why the job is always uphill. A byproduct of your success is a smaller TAM).

Just to be thorough, this replication thing applies mostly to derivatives. The arb needs to be able to trade the derivative and the underlying and all advantage comes from the relationship between the two. The arb is useless without relative value.

Related learning:

🔗 Understanding Risk-Neutral Probability | Moontower

🖥️Moontower Presentation on Black Scholes “As a Trading Strategy” Slides

[UPDATED]

As I expected, the post how to get arbed with perfect info would trip people up. I didn’t expect confusion because I thought Professor Doug Costa, whose explanation is featured in that post, was itself confusing. But because the concept of replication is hard and feels like it violates the good. It’s triggering. It means you can know the truth and still get arbed. Again, this is why I call it the pons asinorum of finance.

A reader brought it up in our Discord so I’m going to share the discussion here as he felt like our back and forth helped.

Before getting to the conversation, let’s refresh the problem Doug set up:

A company issues contracts based on a provably fair coin. The contract pays $150 on heads, $75 on tails. It trades at $100. Interest rate is 0%.

You calculate the true expected value of the 110 call using the true probability of 50%.

It’s worth $20 because it has a 50% chance of being $40 in-the-money.

You pay $20 for it (but even if you paid a bit less for an unambiguously positive EV trade, this analysis will hold. I just want to stay with Professor’s example)

The dealer sells it to you, hedges with 8/15 of the underlying contract, and locks in $6.67 profit in both states. Pure arb.

You had perfect information about the true probability and you still got arbed. The dealer made money in all scenarios, trading the call at fair value with you.

Doug is showing how the real-world probability doesn’t matter to the derivatives trader if they can also trade the underlying. In this case, the underlying is mispriced, but the dealer doesn’t know that. All the dealer cares about is whether the relationship between the derivative price and the underlying price is mispriced. In this contrived example, the mispricing was more profitable than knowing the true probabilities.

And to add something Doug doesn’t mention…if the investor knew the stock was underpriced and bought that instead, they’d have a positive EV trade (the fair price of the stock is $112.50) but they are still worse off than the dealer who knows the relative value of the 2 securities is wrong and gets to make a profit in all scenarios.

This is a good place to insert the chat.

Reader: I see, so the main point is we can converge a spread by trading two things instead of betting on one.

Kris: In a world with no derivatives, you’re left with having to be good at guessing real-world probabilities, but derivatives are their own source of possible edge that doesn’t inherit from knowledge of the future but from relative mispricings between the derivative and the underlying.

It’s obvious that being able to handicap probabilities would be a source of edge, but it’s quite subtle that once you introduce derivatives and the idea of replication, there becomes a source of profit that doesn’t rely on such an ability.

Reader: Right, so it’s instructive in giving one more spread to look at. If you used a put in your example, then the dealer would lose because they’d be too short. Then a dealer that actually has no information and sells both sides ends up $0. This example is picking the (long) side where it wins.

Kris: Yeah, the underlying in this example is too cheap RELATIVE to the call option.

If the call option was $13.33, then from the vantage point of real-world probability, both the underlying and call are too cheap, but they are priced correctly with respect to each other.

Which makes the point — if a derivative and underlying are correctly priced to each other, then the real-world probability is not important to the dealer. The dealer only cares about the relative values.

You can just compute the Sharpe of buying one vs the other I suppose to see which is better (that’s one lens). The call is more underpriced in % terms, 3.33 when it’s worth 20. But it’s more volatile as it will lose all of its value when it loses.

I’d just stick them both in a Kelly calculator in Claude or something and whichever one it says bet more on is the better one lol.

There are some important implications here. And brain damage — investor brain and derivatives brain collision.

The goal is that derivativesbrainskill.md becomes something one calls as needed, like Neo downloading kung fu. But you don’t wanna get carried away with it and shoot it at everything in life. It’s this weird artificial thing that works in a replication context, but it’s also not artificial in that its violation presents hard cash arbitrage!

That’s the end of the chat, but let me add one more thing to make you feel better if it’s still foggy.

I’ve seen this subtlety trip up seasoned options traders where they take B-S pricing to mean that the forward for a stock is stock grown at the risk-free rate (RFR), but this is ONLY true in a world where you can trade the underlying AND the options. Outside the context of replication, you cannot make that assumption.

Struggling with this idea is entirely forgivable. I mean, the realization that you could use RFR as the discount rate was a revolutionary breakthrough. Bachelier figured out option pricing in the early 1900s, but he and his contemporaries were stumped by what rate to discount the payoffs.

Later academics wondered if you should use something like the required return from CAPM or something, but it was the whole idea that if you trade a derivative vs the underlying against one another, then you can have equivalent payoffs and therefore it’s riskless to go long one and short the other. If it’s riskless, then RFR is the appropriate discount rate.

Warren Buffett sees the necessity of agnostic dealers using the RFR to price options in arbitrage-free ways as an opportunity. He asserts that put options are overpriced because they use too low of a discount rate, but the dealers don’t care so long as they can trade the underlying, they can arb any other rate assumption. Again, so long as “they can trade the underlying.”

This single idea allows derivatives traders who know nothing about the fundamentals of securities to make money in a sea of people who do. It’s quite profound and not a small part behind why I think vol trading is easier than directional trading.

A final follow-up

A reader asked:

How did I get to the 8/15 hedge ratio?

It came from Prof Costa’s setup:

A stock is 50/50 to go to $150 (up) or $75 (down) from $100. What is the no-arbitrage price of a 110-strike call on a one-period binomial?

p̂ = (100 − 75) / (150 − 75) = 1/3

Call value = 1/3 × $40 + 2/3 × $0 = $13.33

Delta = (40 − 0) / (150 − 75) = 8/15 of a contract

Let’s take this apart.

Risk-neutral probability of stock going up

p̂ is the risk-neutral probability of the stock going up.

How do we get that intuitively?

Start with the payoffs:

  • Up_payoff = 1.5x
  • Dn_payoff = .75x

The risk-neutral probability is the one that makes the stock price fairly priced given the possible payoffs. In other words, if you buy the stock, the expected return is 0. It must satisfy this equation:

p̂ (Up_payoff) – (1- ) (Dn_payoff) = 0

Solve for :

p̂ (Up_payoff) – Dn_payoff + (Dn_payoff) = 0

p̂ (Up_payoff + Dn_payoff) = Dn_payoff

p̂ = Dn_payoff / (Up_payoff + Dn_payoff)

Concisely stated:

p̂ = d/(u + d)

p̂ = .75/(1.5+.75)

p̂ = 1/3

When in doubt, you can always set up the expected value equation and solve the algebra. You don’t have to memorize a formula.

But you can also put on your gambler goggles.

If something pays 2-1 odds like a money line of +200 or a prediction market trading at 33, then the implied risk-neutral probability is 1/3.

If you buy this stock, you risk $25 to make $50.

Just remember the intuitive odds to probability converter:

x-to-y odds = y / (x + y) probability

2 to 1 odds = 1/(2+1) = 1/3

This is just symbols for “If I get paid 2-1 when I win, then I must win 1 out of 3 times for this to be fair”

You can practice some more in these posts:

The hedge ratio

Back to the original example…if you’re short one call, you buy 8/15 of a share to be delta-neutral. The numerator is the spread in the call’s payoff across the two states ($40 vs $0). The denominator is the spread in the stock’s payoff ($150 vs $75).

Delta is just option change over stock change. How much the derivative moves for a move in the underlying.

The formula in terms of return:

delta = (C_up – C-down) / S (Up_payoff – Dn_payoff)

delta of the 110 call= (40 – 0) / 100(1.5-.75) = 40/75 = 8/15 = .533

Relative value

To reinforce the main point of Prof Costa’s talk, risk-neutral probabilities are enough to make money if you can find an inconsistency between option prices and the underlying. The real-world probability doesn’t matter in a relative framework.

By understanding the distribution of the stock, we were able to compute a delta for any contract. That distribution implies some probability embedded in the underlying. This is not the same as the real-world probability, which is decreed to be 50/50.

Professor Costa showed that if you buy the stock which was underpriced (although you didn’t know that because you computed it must be fair with a 1/3 probability of going up) as a hedge on the call delta, then if someone paid anymore than the risk-neutral fair value of the call, even if they paid less than what the real-world implied probability price is, the dealer makes free money!

Work through the logic for a bunch of strikes and this is what you get if you sell the calls at the risk-neutral fair value (green) or real-world fair value (purple).

The dealer always wins against the real-world probability!

It is also true that the call buyer who pays some price between risk-neutral and real-world has positive expectancy but they don’t have an arbitrage.

So who loses?

The people selling the stock at $100 when the real-world probability is that it’s 50/50 to be 150 or 75. The stock should be $112.50.

Learn more:

🔗 Kellogg lecture notes that walks through exactly this kind of binomial pricing and hedging

N² – n: why shorting is mathematically cursed

Recall the levered silver flows post where we see the quick math of levered ETFs. For a fund to maintain its mandated exposure, the amount of $$ worth of reference asset they need to trade at the close of the business day is:

x(x - 1) * percent change in the reference asset * prior day AUM

where x = leverage factor

examples of x:
x=2 double long 
x=-1 inverse ETF
x= 3 triple long
x= -2 double inverse

This isn’t just a levered ETF thing. The -1 leverage factor is exactly the same as just a vanilla short position. It’s a sneaky reason why the shorting is mathematically challenged.

The easiest way to think of this as an individual investor is to imagine you have an account value of $100. The account is holding $100 in cash, but it’s the proceeds from shorting a $100 stock (assume you don’t need any excess margin to maintain the short). If the stock falls to $50, your account value is now $150 (your cash + $50 mark-to-market profit on the short). You earned a 50% return on a 50% drop in the stock.

Now what?

If the stock falls another 50%, you make $25.

$25/$150 = 16.7%

If you want to maintain the same exposure so that you make 50% on your account on that second 50% drop, you would have needed to short more shares at $50.

How many more dollars’ worth of stock?

-1 (-1 -1) x -50% x $100 = -$100

You needed to sell an additional $100 worth of stock or 2 more shares at $50. Then on that last leg down, you would have made $25 on 3 shares total or $75.

$75 profit /$150 account value = 50% return

Learn more:

🔗 The difficulty with shorting and inverse positions.

not all averages are created equal

What did we notice?

a * b = Mean² − MAD² (where MAD = mean absolute deviation)

As soon as numbers deviate from the mean, their product is dragged down — even if the mean is unchanged. More deviation, more drag. And what is deviation? Volatility.

Bridging middle school math to investing math

In investing, we compound, or multiply returns. So even if the mean of two returns is identical, the dispersion between them matters. Not just matters. It matters quadratically.

No dispersion: The arithmetic mean of (8, 8) is 8. The geometric mean of (8, 8) is √(8×8) = 8.

With dispersion: The arithmetic mean of (5, 11) is still 8. But the geometric mean of (5, 11) is √(5×11) = ~7.4.

If you earn 10% on an investment and then lose 10%, your mean return is 0, but your actual compounded (geometric) return is 1 − √(1.1 × 0.9) = −0.50%.

Now increase the volatility: earn 40%, lose 40%. Mean return is still 0. Compounded return? 1 − √(1.4 × 0.6) = −8.3%.

The drag on your returns is a function of squared deviation. Put simply:

Compounded Return = Average Return − σ²/2

From Text ➡️ Dashboards

We’ll start with some useful resources for the learners, then move to material for traders ready to do stuff.

CME Trading Simulator

While looking up data on CME’s website I came across this amazing, 100% free learning environment with live ticking data:

https://www.cmegroup.com/education/practice/about-the-trading-simulator

My demo vid:

Implied Forwards and Jensen (not Huang)

As I mentioned a few weeks ago, I’ve been re-publishing educational posts on X Articles which serves as spaced repetition practice for long-time readers or just bringing them to the attention of new readers who would be better served by a steady IV drip (no pun) of archival posts than attempting to raw dog the compendium.

These are 2 I think you’ll like:

From Text ➡️ Dashboards

I bought silver a year ago because of Alexander Campbell’s substack. He does a great job showing his thinking behind ideas with data and charts. This alone is helpful because it reveals “these are the datasets a smart guy pays attention to”.

AI tools are shortening the distance between “Hey, that’s neat, I should add that to my dashboard” and like actually adding it to your dashboard. Even if you stink at the world’s most popular coding tool —- Excel (see Will Claude Eat Excel?)

I used one of Alexander’s recent posts to whip up a silver dashboard. I’ll explain what I did, what I added, and share it with you so you can duplicate it as your own starting template. But the broader lesson is that agents are going to make all content “interactive”, we’re just not used to those patterns. Yet.

It is just another staple in my belief that as the cost of inference approaches zero the value of unique data increases. At one time oil was used for light and warmth. But when the automobile was born it claimed the largest cut of the barrel. If data is oil, more people everday are unlocking the ability to “refine” it by transforming it, building new logic and visualizations.

Let’s get to creating the dashboard.

One giant disclaimer:

Expectations are everything. AI is not going to one-shot this project. I’d estimate it reduced a 6 hour task to 90 minutes. Indulge my parental tone for a sec. It would be a mistake to permit this to let you work less in the spirit of that stupid Genspark AI Super Bowl ad. Instead, you should see this as “I can do 4x as many projects as I could before.” This may sound like hustle-porn (you know it when you see it, right?) but if that’s your attitude I offer 2 observations:

  1. You probably don’t like your work. If you do, then giant increases in productivity allow you to get even closer to the the best parts of your work.
  2. Regardless, this goldilocks period will end, everyone will know how to use the 21st century calculator, and 4x as productive will become the new baseline. Red queen. A very smart guy who used to work with me (he was the one who did a lot of the math and technical stuff that we’d need) works in real estate now. I suspect he’s in the top 1% of nerd in that industry. He recently applied for a job and failed a test that was intended to deomonstrate how resourceful he was in the context of AI tools. Knowing him as well I do, I found this shocking because he’s the kind of person that always does well on formal exams. Granted, he admits he’s not not using AI as much more than a google replacement. That this exam exists and a person like him failed, suggests the goldilocks period may already be drawing to a close. It’s not like real estate companies are living on the bleeding edge either.

On a positive note, I think you learn just as much in the compressed time as if you spent 6 hours. Instead of fumbling around with semicolons and syntax you learn how the internet is stitched together and how technologies talk to each other. Embrace manager mode.

Enough of that, moving on to the meat.

Step 1

Give Claude Alexander’s post Silver Moon.

Tell Claude to generate a dashboard in Google sheets inspired by all the arguments in the article. Examples include:

  • SOFR Rate — funding cost baseline for carry trades
  • Funding Rate — broker-specific borrowing cost (SOFR + spread)
  • ETF Prices — SLV, GLD, UUP, SIL, SILJ for cross-asset context
  • Derived Spot — London silver price via SLV ÷ oz/share
  • Futures Curve — next 5 liquid contracts with live prices
  • Expiry & DTE — days to expiration for roll timing
  • Basis — futures premium/discount to spot ($, %)
  • Annualized Carry — implied yield from contango/backwardation
  • Shanghai Premium — China price vs COMEX (arbitrage signal)
  • COMEX Inventory — registered/eligible silver (physical supply)
  • COT Positioning — commercial vs speculative positioning (sentiment)
  • SLV Shares Outstanding — ETF creation/redemption flows
  • SLV Oz in Trust — physical silver backing
  • Implied Volatility — options market fear/complacency

There are 2 key features that operate the sheet

Control Tab

We include a control tab for sourcing the relevant data. All of Alexander’s sources were public but whether you can automatically connect to them is another matter.

That’s why I like ot have a control tab which triages which sources are MANUAL, API, or SCRAPED.

Google App Scripts

This is the equivalent of VBA behind Google Sheets but it’s in Javascript which Claude will happily write for you whether you want to wire the sheet up to APIs or scrape.

Step 3

Troubleshoot. Claude’s sheet gets you 75% of the way in moments and then you spend 90 minutes on this step.

Most of the scraping failed. Sometime because Claude referenced a stale website. But even when you update the correct URL you quickly find that financial data websites tend to lockdown the ability to scrape.

I worked through each data source, iterating with Claude to find automatic (and free) solutions or writing AppScripts usually falling back to “manual” when necessary.

Finally, as I made changes to the spreadsheet there’s the expected debugging and tracing of formulas that happen whenever you delete stuff from a sheet someone else (in this case a bot) made. Pound ref and N/A always show up for a gangbang.

Step 4

Add spice to taste.

Alexander + Claude leapfrogged a lot of work. But there’s still plenty of room for your own judgement and creativity.

For example, when it comes to COT I use the fantastic tools on the CME website which aggregate both futures and options positioning.

I also added leveraged ETF tickers and logic that estimates how much silver there is to buy/sell based on their daily rebalances and even a first pass at computing market impact (see appendix).

Finally, I included a placeholder picture to compute the implied term structure from the SLV options term structure by backing out hard-to-borrow rates.

from moontower data infra

The google sheet is mostly self-explanatory but even if you get stuck just use Gemini in sheets or the Claude extension in a browser to mentor you along.

Here ya go:

🔗silver_dashboard