As I’ve shared here before, I spun up an investing class for middle and high school kids locally. I am teaching my 12-year-old as it is, so I figured if I formalize it a touch so others could learn as well.

The materials for all the classes live here:

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

There are a few weeks between each session since there’s a fair amount of prep even with AI helping with:

• Claude in PowerPoint was released recently so I gave it a spin. I gave it a stylesheet of colors and fonts as well as an unformatted draft of the lecture, and let it cook. You can see the result below.
• The interactive spreadsheet has a bunch of JavaScript behind it

The class we did this week was a lot of fun. There’s even a video to prove it below (I masked any faces. There were 16 kids in attendance). Most importantly, the kids learned a ton. Parents were texting me with their feedback and it felt good to hear their kids’ gears were turning.

For what it’s worth, I think there was a lot of material in here that parents don’t know either but I’ll leave you to guess what some of that might be.

Investment Beginnings — Class 2: The Math of Investing

Class 1 was about building a business.

Class 2 flips the perspective — you’re the investor now.

Someone is asking you for money. What should you pay for shares? What’s the lowest rate you’d lend at? How do you know if it’s a good deal?

This session covers the foundational math that underpins every investment decision you’ll ever make.

What we covered:

✅ The power of compounding (FV = PV × (1 + r)^n)
✅ The lily pad riddle: why most of the action happens at the end
✅ Early Bird vs Late Starter: why starting 10 years earlier beats investing 3x more money
✅ Warren Buffett: 99% of his wealth came after age 50
✅ Total Return vs CAGR: why doubling your money in 10 years is ~7%/yr, not 10%
✅ The Rule of 72: quick trick to estimate how long to double your money
✅ P/E ratio (multiple) and earnings yield (the reciprocal)
✅ The two levers of stock returns: earnings growth vs multiple expansion/contraction
✅ Zoom case study: great earnings, terrible return — how you can pay too much
✅ The asymmetry of losses: why losing 50% requires a 100% gain to recover

Hands-on:

🕹️ Live bidding exercise: students not only bid on shares of Lamorinda Sneaker Co knowing only that it earns $10/share, but quoted the lowest rates they’d lend at.
🕹️ P/E guessing game: guess the real-world multiples for Tesla, Chipotle, Shake Shack, Lululemon, Nike, and more

Homework:

🔨 Inflation Scavenger Hunt — look up prices from the year you were born vs today🔨 Fee Impact Calculator — compare 0.03% vs 1% fees over 40 years
🔨 P/E Return Decomposition — Pick 5 stocks. For each, look up the price and EPS 5 years ago vs today. 1) How much of the total return came from P/E multiple change vs EPS growth? 2) Then compute the current earnings yield (E/P). Compare it to the trailing 5-year CAGR. 3) Using the Rule of 72: if the 5-yr CAGR continued, how long to double your money? If you earned the earnings yield instead, how long to double?
🔨 Compounding Frequency — calculate FV compounded annually vs semi-annually

Resources:

📊 Slides
📈 Spreadsheet (File → Make a copy to get your own editable version; scripts may trigger a security warning — just advance through it)

Full video:

Money Angle For Masochists

Junior Masochists

Let’s review 2 examples from the class that demonstrate how markets are hard because prices are already forward-looking.

The kids learned how to decompose returns into change in earnings vs change in multiple. Or “what happened” vs “the future” or what I sometimes referred to as “sentiment”.

When I asked the class what stock would have been all the rage during Covid (when many of these kids were only 6 years old 🥹), one boy immediately and correctly responded, “ZOOM!”

I pulled up ZM’s price chart:

I asked…”what do you think happened?”

Kids suggested that less people used Zoom as people went back to offices. I explained that ZM’s earnings actually did skyrocket for the past few years so that’s not the culprit behind the horrible return.

Look at the revenues from this Twitter post:

It’s not just the revenues that are up (although you can see how revenue growth has slowed). EPS has also skyrocketed.

The multiple just got hammered. Great business, but investors just paid too much for it.

Earnings were up >35x, but the multiple is down 99%.

A handy decomposition:

Price return = (1+ percent change in EPS) * (1 + percent change in multiple) – 1

The point of the formula is that your return depends on changes in fundamentals (actual earnings) AND change in sentiment around future growth prospects.

A quick caveat. This is not complete. Imagine a situation where a company is $5/share and EPS of $1 for a P/E of 5. Over the next year, the company’s earnings don’t grow and the stock price doesn’t change. The price return is zero. But the company did earn $1. It’s assets have grown by 20%. You are economically richer by 20% but if they don’t distribute it by other paying a dividend or buying back shares (which would raise EPS) then the formula above did not account for a more holistic total return.

You could estimate:

Total return = (1+ percent change in EPS) * (1 + percent change in multiple) + earnings yield – 1

That would capture the idea that you are economically better off even if it’s not paid out, although management’s allocation decisions are a matter of concern.

As a class, we stumbled into a situation on the opposite side of the spectrum. A boy mentioned he bought Delta Airlines 5 years ago for ~$35. I pulled up the chart and noticed the stock doubled.

First of all, great teaching moment as we covered rule of 72 minutes earlier so I immediately asked the class, what the annual return must be? Proud dad moment as Zak is the first one to say 14.4% which I know he figured by thinking “72 divided by 10, times 2” which is better than I would have done as I would reach for 70/5.

Mental math aside, I asked our young investor, “Why did Delta do well, did the earnings increase or the multiple?” With zero hesitation, he responds that the earnings haven’t grown. So a perfect anti-Zoom example for the class. Delta Airlines coming out of Covid years had sour vibes but even if the earnings didn’t grow, you could make a nice return on the sentiment and therefore multiple improving.

I did go back after the class to see DAL earnings and stock history and I think it makes more sense that the kid bought the stock just 2 years ago, since that is the point in time where the earnings were about the same to now and the stock was about $35.

A crap business that investors sold too cheap.

For our regular Masochists

Since we are talking fundamentals, a mutual on X pointed out that HRB (H&R Block) has recently gotten trashed and that its shareholder yield is ~15%.

Shareholder yield is dividends + net share repurchase + debt reduction as a percent of market value.

News flash, HRB is not a growth business. It doesn’t re-invest much of its earnings versus just distributing the cash. I do find it amusing that the stock could be trashed along with other AI disruption stories when it has already survived the transition from brick & mortar to the internet, the popularity of TurboTax, and the growth of the standard deduction, relieving a wider proportion of the population from filing. With a P/E of 7 and a management that pays out the earnings you make ~15% if its already crap business stays the same.

Shedding 1/3 of its market cap since the start of the year, the implied vol is unsurprisingly jacked. I’m a little nuts and decided this was enough to launch some puts with the “I’ll take the shares if I’m wrong”. I normally don’t like this mentality, but part of the vol selling attitude is that the stock probably doesn’t have a lot of upside which reduces the regret possibility from “I was right on this stock and all I collected was some put premium”. In other words, if the upside is abridged, that’s a statement about the vol of the stock being lower.

Selling puts for yield is pretty aligned with what I’m trading the stock for in the first place — yield. I’m just taking it in the form of options intead of buying the stock because the option market is giving me that, but if the price falls a lot further well, I’ll have to go for that yield in the form of assigned shares.

Never financial advice, I’m just sharing my thinking aloud. As options go I’m currently short covered calls in silver and short cash-secured puts in HRB and long options on TSLA and IBIT. Overall, vols are on the higher end of their range across the market (outside of bond vols), but there’s always relatively cheap and relatively expensive in any market cross-section.

I sent this to our moontower.ai list this week:

If you run a trading or investment book that uses options but don’t have or need the weapons-grade (and weapons-cost) infrastructure that options market-makers have, then you are in our position. We built moontower.ai for us, which means it’s for you.

The various dimensions of options across expiries, strikes, and symbols are impossible to make sense of without the right lens.

Moontower is a bridge.

Everything we build is designed to be “opinionated” — pulling things together the way a vol PM sees them. Not a sea of contract premiums. A coherent picture of what’s typical and, critically, what’s not. What we call “analytics with a point of view”.

Explore Moontower Plans

“Hey, this looks expensive compared to its own history, but cheap relative to prevailing volatility surfaces across the market.”

If you understand that options are about volatility, then that is the type of statement you can make with this lens.

Take It With Your Coffee

We launched the Today’s Markets page in the past few weeks to be the first stop when opening your option view.

Your watchlist loads and the metrics snap to that universe.

• Volume List shows what’s trading.
• Trade Ideas classifies tickers by vol surface signatures into preset ideas.
• Skew Extremes shows 25 delta calls and puts at extreme percentiles
• Filters can exclude earnings and illiquid names to clean the cross-section.

Sector Performance can flag when vol moves against expectations.

Today, the Sector Performance surfaced an unusual dynamic. Crypto implied vols are up on the rally, while SLV vols are down on an up day. Opposite of what you’d expect for both!

The numbers on the bar show the price change in standard deviations;at the number on the end of the bar shows the change in implied strike vol for 1-month options.

Most option users are not dyed-in-the-wool vol traders first. If you are a professional manager refining your option expressions, reach out to hello@moontower.ai or visit us online.

Oil vols and calls skews were up a lot this week as the expectation of the US striking Iran increases. A few pictures:

Polymarket implies only 38% chance that the U.S. does NOT strike Iran by March 31.

Risk reversals, which measure the premium of puts to calls, in USO have shot sharply negative this month.

USO vols are elevated and strongly inverted across the term structure.

Implied vols until late March are ~53%.

You already know to use the free event volatility extractor to compute trading day volatility by removing an expected earnings move from an expiration. Let’s use the calculator in reverse. If we assume a typical trading day volatility of 30%, then if we were certain a strike were to occur, we guess-and-test our way to an 11.3% move size to make the term vol fair at 53%

But this is not earnings. We don’t know if the “event” will occur. We can use the Polymarket probability of 62% that an attack will occur before the end of March. We’ll need to expand the equation we normally use to account for p.

We recall the basic identity:

Term variance = expected event variance + accumulated daily variance.

In math:

where:

DTE = business days til expiry =26

p = probability of strike = 62%*

TermVol = ATM IV from March 27 expiry = 53%

EventVol = annualized vol of strike day = 224%

DailyVol = annualized vol of regular business day = 

*Notice in the case where P =1, the equation would be exactly the same as the one behind the calculator.

Solving for DailyVol:

DailyVol = 40.7%

But, wait, we want to fix the DailyVol to be 30%. We need the event vol that generates a DailyVol of 30% assuming that event only happens 62% of the time, not 100%, as our first calcs assumed.

It turns out to be 14.4% or 285% annualized

Annualizing a move to a vol

• 14.4% x 1.25 x √251
• Why 1.25? Because a straddle or move size is only 80% of the volatility or standard deviation. See The MAD Straddle

In sum, if we treat an Iran strike that satisfies Polymarket’s definition AND we believe the Polymarket odds AND we think it manifests as one large single-day move, then 53% IV suggests that oil will move as normal at ~30% vol but have a single-day shock of ~14%.

This is a highly skewed way of decomposing 53% vol. To assume there’s a bunch of variance concentrated in just a single day. But that 53% vol is also not the market assuming we move ~ 3.25% per day either. It’s some mix of:

• “realized vol is elevated right now because there’s uncertainty”
• “at some point in the near-ish future there’s going to be a lump of variance as oil either relaxes lower (which could easily be 10%) or much higher. The current price of oil is a compromise between 2 states of the world but it’s not the right price in either of them and we don’t know which state it’s going to be”

Thoughts on the Polymarket price

Here’s a more up-to-date snapshot (Substack has a Polymarket integration!)

 

I have zero insight on geopolitics so I’m just going to offer thoughts on prices:

EDIT: The Polymarket prices updated from when this email post sent (a Sunday) and when I wrote it (Friday night)

• The market thinks a strike is coming soon. March 31 is 64% and June 30 is only 68%. Conditional on a strike happening, the market implies 64/68 ~94% chance it happens before the end of March. You can buy June, sell March and only risk $4.
• The dollar volume on these things is small but there are many papers supporting the “marginal trader hypothesis” that it only take a handful of active, well-informed traders to make a market more efficient. This is not suprising. If we played a mock trading game for even zero stakes it wouldn’t take long for you to see how quickly a market converges to a reasonable fair value.
• The volatility risk premium across many liquid markets isn’t abnormal. The market either doesn’t care what oil and Polymarket says or a strike on Iran is not expected to have a material effect on the volatility of equity shares. However, defense names have implied vols in high percentiles (while PLTR vols are tanked btw)

  

Here’s my off-the-cuff impression of the 64% price:

The real odds are probably higher. If this contract were trading for say 10% I’d guess it was overestimating the true probability because of lotto-ticket bias but also because there needs to be a healthy risk premium for seller to enter a highly negative skew trade.

I wouldn’t guess that a bunch of yolo-punting puts a price to 64% for lolz. When someone bids 64%, they are laying odds. Betting nearly $2 to win $1. The price of this contract has doubled in a week…it’s the buyer who likely brings more caution to the order book now.

I could imagine someone buying these as part of a relative value trade against selling oil options but the dollars available means it would need to be retail size and that kind of trade (oil vega vs prediction market?!) doesn’t seem like the kind of thing that would excite the class of trader who expects 20x leverage on crypto perps to get them outta bed in the morning.

If Polymarket depth was big enough to influence stock markets, there’d probably be some interesting scenarios of incinerating a few million bucks, maybe less, to influence the Poly price so you can influence the price of defense stocks where you could make tens of millions. The informational and liquidity linkages between prediction markets and traditional markets will be fascinating (appalling?) to watch as they continue to grow.

 

Last week, in embedding spot-vol correlation in option deltas, I showed how vol paths use anticipated changes in implied vol as the spot moves around to estimate more accurate deltas. It’s a maneuver that respects delta fully as a hedge ratio rather than its narrow textbook sensitivity of “change option price per change in underlying price, all else equal”. We are sure (enough) that all else ain’t gonna be equal, so we can use the knowledge to improve the hedge ratio.

The post explains how a “vol path” takes a slope parameter that dictates how ATM vol changes as spot moves. For example, a slope of -3.0 means a 1% rally drops ATM vol by 3% (ie from 20% to 19.4%, not 3 clicks such as 20% to 17%). It’s like a “vol beta”. We can even see -3.0 slope by looking at a 1-year beta between SPY and VXX based on daily samples.

A stylized view of how this works:

The vol path only affects the ATM vol. If the smile remains the same shape along the path, the vols of all the options also change. That’s not all. The skew, measured as normalized skew or the percent premium/discount of OTM strike vs the ATM strike, is also changing if the shape stays the same but ATM changes.

While vol paths describe the ATM vol, there are skew models that describe how all the options for a given expiry will change. Just like the vol path concept, the goal is to make better predictions of how a portfolio of options will react to stock movements without pretending that vols don’t change. This is an opportune moment to remind you that the presence of a smile in the first place is a patch to the faulty Black-Scholes assumption that vol is constant. If the strike vols don’t change as the spot moves, the ATM vol still does since you have “moved along the smile”.

The entire branch of quant devoted to modeling option surfaces stems from the knowledge that vols change as the underlying moves and that there is value in trying to forecast those changes rather than accept a null prediction of “vols won’t change”.

There’s no controversy about whether there is value in modeling the dynamics of option surfaces. Better models improve:

1. Risk measures. VAR needs assumptions about how the surface reprices when spot moves. Your hedge ratios are direct outputs from portfolio scenario shocks and their assumptions.
2. Market-making. Sound models mean the ability to recognize abnormal kinks within a name or cross-sectional divergences between names. A model gives you a baseline from which to judge “how strange is this surface change”? If the skew rips by X, do I expect that to pop back into line or is this within the realm of normal, given the market’s movements?
3. Option pricing on illiquid names. How do I estimate option values in a name with sparse quotes? A good model fills in the blanks.

My goal with this post, like all of my posts, is not to give it an academic treatment but the non-quant practitioner’s perspective. To offer an intuitive angle to either better organize your understanding, whether this is new to you or if you come from a similar vantage point OR complement the textbook rigor that some readers possess.

As the title suggests, we are going to reduce the topic to 2 basic types of skew modeling approaches — “sticky strike” versus “floating”. The fact that there are 2 is a hint that neither is fully “correct”. Just like skew itself is a kluge, the entire domain of surface modeling is basically a kluge. Beyond hard arbitrage boundaries, the relationships of options to one another is a collection of informed guesses mediating a constantly evolving conversation between models and behavior.

The typical George Boxism “all models are wrong, some are useful” applies. Models are toys by necessity — if they were actual simulations of reality then the reality is simple enough to not need a model. Nothing about the future of a security price satisfies that requirement.

Our procedure here is to assert the model, see what they would predict if they were true, and then watch them break by hypothesizing the trading strategy that would profit from the models NOT breaking (which of course is a blueprint for why they must break).

The nice part is that this is mostly a visual exercise so don’t be discouraged by the post being too long to fit in the email…it’s a lot of pictures.

Floating Skew

A floating skew model says that the percent skew by delta* stays constant. The 25-delta put is always, say, 25% above ATM vol.

Note: Floating skew models are also referred to as “sticky delta” keeping consistent nomenclature with their counterpart “sticky strike”. I always found this similarity in names to be confusing but YMMV.

*Delta is a stand-in for any normalized measure of moneyness. 

Log or percent moneyness itself (ie “a 5% OTM put”) is not normalized for volatility. A 5% OTM option on TSLA is a lot “closer” to ATM then 5% OTM on SPY because TSLA is so much more volatile. 

Delta is a vol-aware unit of distance, but it has the problem of being recursive. We need a volatility to measure distance to measure the vol premium on a strike BUT we delta depends on the very volatility we are looking to parameterize. 

You can use standard deviation based on ATM or .50 delta volatility to measure distance as I do here. I admit this might be cope as I’m just drinking the Heisenberg poison I’ve built immunity to. 

The stylized demos in this post are using the base smile from last week’s post from a SPY snapshot.

Spot = $695
ATM vol = 12.4%
DTE = 31

We also maintain the -3.0 vol path (ie a 1% rally drops ATM vol ~3% and vice versa). This model says the percent premium/discount of a strike’s given delta is preserved.

Looks sensible if we plot vols by delta for green (stock up ~1% to $702) and red (stock down ~1% to $688):

Let’s plot vol by strike:

Hmm.

Let’s look closer. Again, the purple curve is the base curve. The green curve represents the smile if SPY jumps from $695 to $702, or ~1%, and the red curve represents the smile with an ATM strike of $688.

I’ll narrate observations, but it’s best to pre-load your own observations if you’re trying to learn (you’re enabling the technique of ‘hypercorrection’ or ‘surprise learning’).

Observations and notes on breaking

• The down move where vol increases due to vol path, actually leaves us with a lower ATM and downside vols! It’s because the vol path itself is not tangent to the slope of the actual skew of the purple line. In this model, unless the vol path is tangent, vol will underperform on the downside while the OTM calls will outperform. As the stock rallies, vols across the board outperform because the vol path is not as steep as the implied skew.
• If there were no vol path at all (ie vol slope = 0) then these under- and outperformances would be even more egregious. In fact, if that’s how surfaces behaved you would simply sell the slightly OTM puts and buy the OTM calls knowing that whenever the spot moved, the IV spread you had on would profit since the vol would always underperform on the way to your short and vice versa. It’s true that you’d still be exposed to changes in realized vol, but you’d have a giant IV tailwind as compensation.
• If the vol path was as steep as the skew, you’d be much closer to a sticky strike model to be discussed below, along with its own caveats, of course.
• A floating model is incoherent in the extremes. If ATM vol doubles/halves, all strike vols must double/half. Leading the witness a bit, but tails are sticky…which means skew flattens when vols get extremely high, and steepens when it gets its extremely cheap. The floor on a .10d put vol is proportionally higher than the realistic floor of an ATM IV. At the extremes, a single penny can be several vol points as prices get sticky (especially since transaction costs are fixed dollar amounts — think of the fee to sell an option at a “cabinet”.)

By asserting the same percent skew premiums/discounts across the curve, the strike vols themselves are left to vary as our chart shows. This view shows how the strike vols change from the base curve depending on whether the stock went up or down:

 

Sticky Strike

A sticky strike model asserts that vols do not change as the spot moves. The $680 put trades at the same vol whether SPY is $695 or $702.

If we fix the strike vol, what happens to skew?

If strike vols are fixed but spot rallies 1%, your 25-delta put is now a 20-delta put. Same vol. Different delta.

This chart is percent skew by call delta for the base curve. For the shape rotators in the audience, go ahead and guess what happens to the skew at the .75 delta when it becomes a higher delta call after a stock rally.

As a wordcel myself, I’m just going to display the answer.

Zooming in on the actual skew changes:

Put skew flattens (ie gets smaller) on sell-offs while call skew gets trashed. On rallies, put skew firms* and call skew flattens (becomes much less discounted).

*A 2% shift in normalized skew is “small”. If skew is 20% premium and ATM vol is 30%, that’s 6 points of premium. A 20% to 22% move in the degree of premium is 0.6 vol points. Matters to a market-maker but it’s noise to most participants.

Picture of SPY 1m .25d skew for the past year:

Zoomed in, you can see how it flattened during the late Feb to late March sell-off and bottoming ahead of Liberation Day before spiking!

I’m not making stories, but pointing out that it collapsed again on the second leg-down, marking the bottom for the remainder of the year. All hindsight stuff, but overall you can see the range for .25d put skew was about 15% for the year(from about 14 to 29% premium to ATM vol).

In case you need a reminder for why I don’t like trading skew for vol reasons:

a sense of proportion around skew

Reality

Sticky strike predicts flat strike vols.

Floating strike predicts unchanged skew, which describes how strike vols change.

Let’s pause for a second. I’ve done something subtle in how I’ve framed this discussion which might be lost on more novice readers (although I’m not sure just how novice anyone who has gotten this far might be).

Without saying so directly, I am putting a lot of emphasis on what happens to strike vols. For traders as opposed to onlookers who just talk about what vol or VIX is doing, strike vols are the closest thing to what matters — option premium. Strike vols influence option prices directly and prices of contracts determine p/l. “Vol went up today” means nothing if strike vols were unchanged and the stock is simply lower. Telling me that ATM vol is higher doesn’t tell me if a floating model just slid down the curve. “Vol” is an abstraction of strike vol is an abstraction of option premium.

With that out of the way, relating these models to reality starts with observation of strike vols. In fact, this is how such models are generated in the first place. Noticing, then fitting.

You could go crazy with examples, but I will do just a couple to give you enough fodder for your own consideration.

This was a SPY snapshot on 1/20/26 with shares down ~2%

Strike vols are up across the board.

Notice:

• Sticky strike wasn’t true. Strike vols moved.
• Floating skew didn’t hold either. If the strike vols were all up in an approximately even fashion in clicks (ie all vols up 1.5 points give or take .3 for near the money) then skew flattened (think of it this way…higher IV options were up a similar amount to lower IV options).
• You could describe the change as a parallel shift in sticky strike vols. A sticky strike type movement means the skew changes. In this case, the put skew flattened and the immediate call skew became less negative.

A 2% move in SPY is 2 standard deviations. I’m not surprised the surface didn’t adhere neatly to a model. Even vol paths are extremely local (a vol path of slope -3.0 would predict that a 2% sell off would lead to a 6% increase in vol and IV on the original ATM went up double that amount from 13% to 14.5%).

Let’s look at IBIT March expiry on Monday’s selloff. IBIT fell ~6%, about a 2 standard deviation move as well.

Here’s the change in strike vols.

In the belly of the curve, sticky strike was a great description of what happened. Strike vols barely budged, while the signature of the strike vol changes for options that are now OTM calls and puts looks like what a down move with a floating strike model would predict. Call vols up and puts vols down. Sticky strike in the belly, floating skew for OTM.

And while the SPY move looks like it rattled the market as the surface shifted higher, the BTC move had little effect on its surface despite both moves being ~ 2 standard deviations. The SPY move seemed unstable, while the BTC move was stable.

What do you do with this?

If all of this sounds confusing, it’s because it is! This is good news for vol traders. If it weren’t, the market would just be more efficient. In this example, BTC vols underperformed SPY for the same exact type of move. Inverting, that means there’s an opportunity for discernment, as you had 2 assets which had highly correlated underlying behavior but mismatched volatility behavior.

A question to consider given the vol moves…if you buy the now at-the-money BTC vols that haven’t budged or even the OTM puts which actually declined in vol to sell upside SPX or BTC calls, is this an opportunity? This is what vol traders think about for a living. You have desks that see the flow in everything and combine that info with the relative strength and weakness across parts of the surface (it’s the whole idea behind the vol scanner tool).

If you’re a market-maker, you don’t have the luxury of just scanning the markets to cherry-pick. You are deeply embedded in the price formation process since you must post a market. In illiquid names, you must do this with limited flow information. Having a vol surface model to generate fair values to quote around is not optional.

In commodity options, I toggled between sticky strike, floating models with vol paths, and hybrids (basically a floating model with a vol path and a skew correlation that allowed you to rotate or tilt the shape of the curve forward and backward).

Just like a vol slope parameter, these models affect your deltas.

I remember a particularly brutal period where vol was so heavily offered on up moves that when I eventually gave in and ran a much steeper negative vol slope, the change flipped my delta from being flat to short $20mm of oil. And of course, if you are long vol as it’s getting pummeled on the rally, that means your model is now saying you are short on the highs. After all, that’s the problem. The market is rallying, your calls are massively underperforming their delta and you are short futures against them!

But this sensitivity means you can’t be toggling your models all the time because how you model affects your risk. The goal is to model reality, but if you keep changing models like your name is DiCaprio, you’re going to put your risk in a blender.

This is a good place for judgment. You build an understanding of how the surface changes for various types of moves while acknowledging that this depends on the market context and open interest. If investors are well-hedged to the downside like they were in 2022 (the market sell-off and rise in interest rates were extremely well telegraphed), then you might expect put skew to underperform on the way down. You certainly don’t want to run a fixed strike model in that case.

You let the market’s surface changes act as a tell. If the market acts differently on a small sell-off and a big-selloff that’s expected. You don’t really gain information. There’s no null to reject. But if both types of sell-offs have muted reactions, that’s interesting. This is an orderly, expected, and perhaps even stabilizing event.

On the other hand, a stock up-vol up surface move is unexpected. That should inform the model you run. There’s an art to this. How long or how persistent should a behavior be before you can classify which model you should run? There’s no simple answer (again, thankfully!). Open interest and expectations are convolutions that direct whether something is a surprise or not. Surfaces react to surprise. Remember they already know that vol is not constant — it’s the delta in expectations about how volatile the volatility itself is that substantiates new surface behavior. Surfaces anticipate a band of random behavior without reacting because randomness is embedded in volatility.

Sometimes interest rates rise because of growth expectations. But stagflation will do that too. The vol surface will likely care about the difference. Oil might be rallying steadily because China is booming and the global economy looks rosy. It can also rally because there is no peace in the Middle East. The vol surfaces will distinguish between the 2 types of rallies. Your deltas will be vastly different for the same nominal options position depending on the backdrop.

I’ll leave you with this summary that captures what I generally, but loosely, expect when I see the stock market up or down and whether I think the move is stabilizing vs destabilizing:

 

Before we get to the heart of today’s education, this is a video follow up to yesterday’s HOOD: A Case Study in “Renting the Straddle”.

I talk about oil volatility as well and how it shows up in the Trade Ideas tool.

This concept of “spot-vol correlation” gets a lot of airtime from different angles even when it’s not explicit. The mass financial media doesn’t use the exact words but they know enough to call the VIX the “fear gauge”. VIX is a complicated formula that aggregates values representing annualized standard deviations from a strip of inverted Black-Scholes numerical searches with quadratic weights. But all of this gets translated to:

“when stock go down, that number go up”

That travels a lot faster. Even your cat knows that market volatility has an inverse relationship with stock returns.

The more your paycheck depends on option greeks, the more you will need to zoom in on this concept. Mostly because the relationship between vol and prices changes your actual risk. The Black-Scholes world assumes vol is constant, but we know better. The sensitivity of options to various market inputs (greeks are measures of risk) is naive without adjusting for behavior that is predictable enough for your cat make a better guess than random about what will happen to vol when stocks move.

How can use this cat knowledge to estimate better deltas so when our model says we are long $50mm worth of SPY, we aren’t suprised when it seems to act like we are only long $40mm worth?

There isn’t a single way to do this but I’m going to show you how I did it as a calculus-challenged orangutan.

Before we get to numbers and pictures, I want to mention one last thing.

There’s a riddle in the world’s best trading book Financial Hacking. An imaginary bank trader calls a meeting with management and says he’s found “greatest trade in the world”. He sits them down for a presentation and says he can buy calls for 20 vol and sell puts at 40 vol, delta hedge until expiry, and make a 20 point “arb”.

What’s the problem?

There are several, perhaps many, option traders reading this right now who have thought about the holy grail of long gamma, collecting theta. Look you can go do this right now.

1. Sell a strangle on 1-month oil futures and buy a ratio’d amount of 12-month CL straddles.
2. Buy a ratio time spread in a name that has a major event coming up
3. Trade SPY risk reversals

All of these trades will give you the “desired greeks”. But these are illusions. In order:

1. The lower vol on the deferred future makes the gamma of those options look higher than it is, but you need to weight the gamma by the lower beta those futures have to spot oil prices
2. The decay you think you collect on the near-dated short is unadjusted for the “shadow theta” or glide path of IV increasing as the upcoming event is a greater proportion of the variance remaining as each second elapses
3. Spot-vol correlation means that theta number is not just the cost of gamma but vanna. The owner of the put is getting more than gamma.

Ok, time for less words and more F9.

I grabbed a SPY IV curve from earlier this week. 31 DTE.

The spot price was $695.27 at the snapshot time but we are just going to keep things simple by ignoring any cost of carry and saying that spot is $695. I just wanted a sensible IV curve for demonstration purposes.

The ATM IV on the $695 strike is 12.40%

Fetching the strike vols and using a vanilla Black-Scholes calculator with 0% cost of carry and 31 DTE, we get this self-explanatory table:

The Naive POV

Those deltas answer the question:

“How much will the cValue change if the stock goes up $1?”

But those deltas don’t know what our cat knows? Vol will fall if the market goes up. It’s not a certainty, but I’m happy to lay you even odds on the proposition if you think it’s random.

If vol falls, the option is going to underperform roughly by the change in vol on the strike * option vega.

Greeks are useful insofar as they describe our actual risk. If my cat-instincts know that the call will underperform if the market goes up, then I probably don’t want to sell quite as many shares against it to be neutral on a naive delta.

Likewise, if I sell those calls and the market falls, the increase in vol will mean I won’t make as much money on my call short as the naive delta predicted.

Notice that whether I buy or sell the call, I am better off having hedged it with less delta than the naive model predicts.

We are just trying to incorporate what the cat already knows to dial in better hedge quantities. We are folding expected vega p/l into delta because the empirical relationship between spot and vol changes is strong enough to bet on it.

We need some parameter, some concept of beta, that describes the strength and sign of the relationship between spot change and vol change. In SPY, the sign is negative because of the inverse relationship. In silver, the sign is positive. It is a “spot up, vol up market”.

An important note. We are speaking in generalities — any market has a general spot-vol signature, but it can flip for periods of time and the strength of the relationship also bounces around. These empirical relationships reflect flows. The supply and demand of options as the spot moves around. God doesn’t assign them. Academics will talk in terms of capital structure and how when a company falls, it’s more levered, and therefore mechanically more volatile, equity is a call option on the highest and best use of the company’s assets, yadda yadda. There’s truth to this, but its not the most useful lens for thinking about option surfaces which are tangible projections of an order book of shares across price and time.

A detour with a purpose

I started in commodity options just before the listing of electronic options markets. When I first stepped into the trading ring, many market-makers were still using paper sheets. We had spreadsheets on a tablet computer, but heard of a fledgling software called Whentech. Its founder, Dave Wender, was an options trader who saw the opportunity. I demo’d the product, and despite it being a glorified spreadsheet, it centralized a lot of busy work. It had an extensive library of option models and it was integrated with the exchange’s security master so its “sheets” were customized to the asset you wanted to trade.

I started using it right away. Since it was a small company, I was able to have lots of access to Dave with whom I’ve remained friends. I even helped with some of their calculations (weighted gamma was my most important contribution). I was a customer up until I left full-time trading. [Dave sold the company to the ICE in the early 2010s. It’s been called ICE Option Analytics or IOA for over a decade.]

The product evolved closely with the markets themselves. Its nomenclature even became the lingua franca of the floor. Everyone would refer to the daily implied move as a “breakeven” or the amount you needed the futures to move to breakeven on your gamma (most market-makers were long gamma). Breakeven was a field in the option model. Ari Pine’s twitter name is a callback to those days. Commodity traders didn’t even speak in terms of vols. They spoke of breakevens expanding and contracting.

What does this history have to do with a spot-vol correlation parameter?

This period of time, mid-aughts, was special in the oil markets. It was the decade of China’s hypergrowth. The commodity super-cycle. Exxon becoming the largest company in the world. (Today, energy’s share of the SPY is a tiny fraction of what it was 20 years ago.)

Oil options were booming along with open interest in “paper barrels” as Goldman carried on about commodities as an asset class. But what comes with financialization and passive investing?

Option selling. Especially calls.

Absent any political turmoil, resting call offers piled on the order books, vol coming in on every uptick as the futures climbed higher throughout the decade.

A little option theory goes a long way. Holding time and vol constant, what determines the price of an ATM straddle?

The underlying price itself: S

straddle = .8 * S *σ√T

If the market rallies 1%, you expect the straddle price at the new ATM strike to be 1% higher than the ATM straddle when the futures were lower. Since the “breakeven” is just the straddle / 16, you expect the breakeven to also expand by 1%.

But that’s not what was happening.

The breakevens would stay roughly the same as the market moved up and down.

If the breakevens stay the same, that means if the futures go up 1%, then the vol must be falling by 1% (ie 30 vol falling to 29.7 vol)

It dawned us. Our deltas are wrong.

If we are long vol, we need to be net long delta to actually be flat.

When your risk manager says why are you long delta and you explain “I need to lean long” to actually be flat, you can imagine the next question:

“Ok then, how many futures do you need to be extra long for this fudge factor?”

We need to bake this directly into the model because it’s getting hard to keep track of. Every asset and even every expiry within each asset seems to have different sensitivities between vol and spot. The risk report can’t be covered in asterisks detailing thumb-in-the-air trader leans.

Whentech listened.

Vol paths

Whentech introduced a new skew model that allowed traders to specify a slope parameter that dictated the path of ATM IV. Their approach was simple and numerical. It was some version of this:

ATM Vol path = ATM IV × (100% + vol slope × moneyness)

Let’s say I set my vol slope parameter to -1.0

SPY ATM vol is 12.4%

If SPY goes up 1%, what’s the new ATM IV?

New ATM Vol = 12.4% x (1+ -1*1%)
New ATM Vol = 12.4% x (99%)
New ATM Vol = 12.28%

A -1.0 vol slope corresponds to a “constant breakeven” regime. If the stock is up 1%, vol falls 1%.

This is a table of vol paths for different vol slope parameters:

Keep in mind that the vol path is only for ATM vol. You can think of the ATM region of a smile sliding up and down a ramp of slope -1.0, -3.0, and so forth.

💡Notice that all of these ATM vol paths suggest a lower vol ATM vol at say the $675 strike than the actual smile implies. That is really a separate discussion, since skew is not really a “predictor” of vol anymore than a back-month future is a predictor of spot price. It is just a value that clears the market so it has risk-premiums embedded. It’s just another example of “real-world probabilities do not equal risk-neutral probabilities”. Even if that’s not satisfying, you could think of the skew as needing to average any number of price paths approaching a strike. If we drop $40 overnight, ATM IV is going to be higher than what the current $40 OTM put vol. If it takes 2 weeks, maybe not.

SPY skew is quite steep compared to most assets. A vol path that is tangent to the skew curve (-9.0 parameter) would be a very aggressive spot-vol correlation, especially considering that -1.0 is constant breakeven. Anything more negative means, as you rally, the value of an ATM straddle shrinks. That’s a strong clue that this slope idea is highly localized. If SPY doubles, the new ATM straddle isn’t going to be worth less than the current one, nevermind 0.

Zooming in on the strikes that are $5 around the ATM $695 strike:

How vol paths affect your delta

Once we’ve chosen a vol slope, we can compute the vol path, which in turn alters our model deltas. We can do this numerically, instead of deriving new formulas for greeks.

We are going to make a simplification, which is to assume that for a small spot move, changes in vol affect all the strikes by the same proportion. You are invited to think of what that would mean for implied skew. I plan to tackle that in a later article, but we’re building up in steps.

Let’s zoom in on the 695 call in the case when SPY goes up $1.

In the naive model, the 695 call goes up by its delta or $.507

But based on the different vol slopes, we know IV is going to fall from 12.4% to anything from 12.38% (-1.0 slope) to 12.24% (-9.0 slope). When we reprice the option with the lower vol, we see our profit is less than $.507. The difference, which is mechanically due to negative vega p/l, is being used to convey an “effective delta”.

If the market behaves as if the vol slope is -5.0, then instead of hedging the ATM call on a .507 delta, you should have used .44 delta.

[This is the topic I’m talking about at minute 37 in the context of estimating dealer hedging flows]

I show the vega p/l just to make the decomposition tie out between the recomputing of the option vs what it’s worth if IV was unchanged.

Vol beta

We’ll close by tying this dynamic back to hedge ratios in “delta one” vol products like VIX futures and ETPs.

VIX depends on a strip of options, not just ATM. But let’s stick with our simplification that IV changes proportionally across strikes such that if ATM vol decreases 10%, VIX falls 10% (not 10 percentage points but 10%…like 20 vol going to 18).

This is our IV projections according to different vol slopes for SPY shares up 1%:

The vol slope parameter can be thought of as a vol beta. As in, what’s the beta of VXX shares to SPY?

[ I wrote about this last year during Liberation Day because on the sell-off, I bought both ES futures and VX futures but I needed to estimate the right ratio to buy them in.]

Running the regression for the past year in moontower.ai shows a VXX/SPY beta of -3.25:

The rolling one-month beta is more volatile and would correspond to vol slopes between -1.5 to -5

Related video:

📺VXX Beta explained via Moontower Hedge Ratio Tool

I bought June/Feb13 put calendar in SLV a few weeks ago when the vol spread inversion went nuclear.

That was a disaster.

SLV dumped 30% 2 days later.

The Feb puts I’m short are of course 100 delta, so the effective position is long a June OTM call synthetically.

💡If a stock is $80 and you own the 100 put for $25 and 100 deltas worth of the stock, then you are synthetically long the 100 call for $5. If you don’t believe me, look at your p/l payoff for the portfolio of long puts and stock at expiry for stock prices of $90, $103, and $120 vs what it would be if you just owned the 100 call.

We understand the position and the risk. But we don’t talk about taxes much here so I’ll use this example to introduce the complexity of the real-world.

Let’s say I roll my June puts.

Consider the tax implications.

I will realize a gain on the appreciated puts.

The puts I’m short that are now the risk equivalent of being long shares because they are so far ITM. I have a mark-to-market loss on these puts, but it’s not realized. This is a problem. The entire trade has been a loser, but if I roll my June put,s I crystallize a short-term tax gain. Ideally, I need to crystallize the short-term loss on the puts I’m short by buying them back.

If I don’t buy them back and get assigned, I don’t realize the loss. Instead, I acquire shares with a basis of the strike price minus the premium I collected when I sold them. If I sold the 100 put at $5, my cost basis is $95. The shares are $70, but my loss is still unrealized until I sell the shares.

The problem might not be immediately obvious, so let me break it down.

• If I roll my June puts instead of closing the entire position out, I have a trade that has been a loser, but the tax accounting shows a short-term gain + an unrealized loss.
• To crystallize the loss, I must buy my put back or sell the shares once I’m assigned. But, both of these trades sell lots of SLV delta. If my intention is to maintain a synthetic long call position (long stock + long ITM puts) I’m stuck with an accounting gain.

⛔Because of the wash sale rule I cannot sell my SLV shares then immediately buy them back.

• You can envision a scenario where SLV rallies up again, my synthetic call position recovers the economic loss but I have a taxable gain on the rally. My p/l on all the activity is a wash BUT I have loads of short-term taxable income!

Not picking up your matched short-term loss is leaving a dead soldier behind.

(Ok, that was dramatic. I’m sorry enough to say so, but not enough to delete it. I want to imprint it.)

There are a few choices whereby you can roll the puts, achieve the desired risk exposure but I’m not an accountant and this is not advice. There’s no wink here. Talk to an accountant.

Goal: crystallize short-term loss without getting rid of your long silver delta

Possible solutions

1. Once you are assigned, sell your SLV shares and replace the long with a highly correlated silver proxy such as other ETFs or silver futures. From an IRS interpretation of the wash sale rule, the futures are probably safer since COMEX is NY silver and SLV is London deliverable. But again, not an accountant.
2. Replace your length with assets highly correlated to silver, like miner stocks. The basis risk is obvious.
3. Close your puts and buy the stock at the same time, effectively buying a worthless synthetic call.

Let’s talk about #3 a bit more.

If the stock is $70 and the 100 put is only worth intrinsic (ie there’s no time value left in the 100 call), then that package is worth $100. The stock price plus the $30 put. Now you wouldn’t expect a market-maker to fill you at fair value.

I figured a market-maker might fill me for a penny of edge. When I was looking at the quote montage, the 99 strike call was offered at a penny so by arbitrage the 100 call should be offered at $.01

I tried to pay $100.01 for the package.

No dice. Nobody wanted the free money. I didn’t raise my bid, figuring I would try again on expiration day since perhaps a seller didn’t want to bother with the inventory. If they traded it on expiration day, the whole position would offset at settlement, and they would collect their easy penny.

Well, what happened?

My short put got exercised early! I got stuck with the shares and now have to sell the shares to crystallize the loss.

The interesting thing to point out is that paying up a penny to lock in a short-term accounting loss is a type of trade that’s win-win. The market maker sells a worthless synthetic option, I get my tax situation aligned.

This is a screenshare constructing a synthetic call in IB’s strategy builder, then adding it to the quote panel so you can see the bid/ask for the structure.

In the spirit of spaced repetition, I published The Gamma of Levered ETFs as an article on X. Seemed relevant given silver’s 30% selloff on Friday.

Here’s the short version of the math of levered ETFs. To maintain the 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

Applying this to silver:

AGQ, the ProShares Ultra Silver ETF, is 2x long. It had ~$4.5B in assets at the close on Thursday.

For the underlying swap to maintain the mandated exposure, at the close of Friday (assuming no redemptions) the swap provider must trade silver. How much of it?

2(2-1) * -30% * $4.5B

or -60% of $4.5B.

-$2.7B worth of silver in forced flows. Negative = sell.

There’s an UltraShort 2x ETF, ZSL, that had about $300mm of AUM going into Friday.

Rebalance trade:

-2(-2-1) * -30% * $300mm = –$540mm

Assuming no redemptions, these levered ETFs needed to sell ~$3.25B worth of silver into the close.

In a typical environment, silver volumes are mostly split between London’s spot market (LBMA) and COMEX futures (NY deliverable) with Shanghai (SHFE), India (MCX) and SLV (London deliverable, US traded ETF) combining for less than 10% of total volumes.

At the NY close, SLV and COMEX represent all the liquidity that’s open.

COMEX futures traded nearly $150B of volume Friday and SLV traded ~$50B which is on the order of 10x the dollar volumes silver used to trade a year ago at lower prices. Still, those forced sales, if they are happening in the few hours of trading may represent something like 5-10% of the liquidity.

I’m guessing readers who are actually on metals desks have a better guess.

Silver futures margins, after being raised again this week, are about 15% of the contract value (although your broker may ask for more. IB asks for twice that, which was prescient!)

If Shanghai futures, which were closed, have a similar requirement, that means the exchange doesn’t have enough collateral to cover the 30% move if Shanghai futures match the COMEX move.

I don’t know how that exchange works (many exchanges have an insurance pool where some of the losses are socialized across clearing members), but one thing that would be interesting is if Shanghai exchange officials have the authority, balance sheet, and ability to have sold COMEX futures as a hedge. I doubt that, it’s just a speculative musing, but if such a thing did happen, their Sunday evening unwind trade would be to buy back COMEX futures as they liquidated Shanghai holders. Again, this is just a ridiculous musing, but I look forward to seeing how it all shakes out.

In any case, I think a useful takeaway from all this could be to add expected levered rebalancing flows to your dashboards (of course, this is a recursive problem because the price at any point in time reflects some people’s knowledge of these flows. Pre-positioning always opens the door to backfiring if enough arbs think the same way).

Today is about option execution. It’s a blanket response to a host of misunderstandings I find in talking to investing practitioners who don’t come from the market-making side.

Before that we’ll cover a couple things I found interesting.

Option volume explosion

This table is from the OCC:

From 2005-2007 option volume doubled.

It took 13 years, until 2020, to double again.

And just 5 years to double still again.

If you’ve been paying attention to the options world, you know that the last 5 years have seen both a massive increase in retail participation which coincided with the very successful product launch known as 0dte.

A friend with a senior role at a MM has described the last few years as “money raining from the sky”.

---

Trader Challenges

I loved this post by Rob Carver:

Wordle (TM) and the one simple hack you need to pass funded trader challenges

In fact, I’m going to be meta and write about this post because of its pedagogical value. Rob saw a situation in the wild, turned it into a real-life word problem, and solved it. We’re going to step through what was unsaid because that process would be helpful to learners who want to get better at recognizing the nature of a problem and constructing a solution.

Excerpt from the intro:

There has been some controversy on X/Twitter about ‘pay to play’ prop shops (see this thread and this one) and in particular Raen Trading. It’s fair to say the industry has a bad name, and perhaps this is unfairly tarnishing what may pass for good actors in this space. It’s also perhaps fair to say that many of those criticising these firms, including myself, aren’t as familiar with that part of the trading industry and our ignorance could be problematic.

But putting all that aside, a question I thought I would try and answer is this – How hard is it to actually pass one of these challenges?

The rules of the Raen challenge are this:

• You must make 20%
• You can’t lose more than 2% in a single day. There is no maximum trailing drawdown. So if you lose 1.99% every day forever, you’re still in the game.
• You must trade for at least 30 trading days before passing the challenge
• It costs $300 a month to do the challenge. This isn’t exactly the same Raen which charges a little more, but as a rounder number it makes it easier to directly see how many months we expect to take by backing out from the cost per month. I assume this is paid at the start of the month.

His post is totally free. You should read it. But I want to branch from this point where he laid out the rules of the challenge.

The first thing you need to recognize

To even begin answering the question of how it is to pass the challenge you need to state it in terms of “how hard is it to pass given some [expected daily return] and [volatility]?”

Since the variables of concern are return and volatility, or reward vs risk, the concept of a Sharpe ratio immediately comes to mind.

If you have an expected daily return of 20% with 0% volatility, then your chance of success is obviously 100%. That’s an infinite (undefined?) SR.

If you are even remotely near the investing world you probably have some sense that the SP500 has something like a .5 SR and an SR of something like 2 would be very high. If SP500 is 15% vol, then you are talking about an investment with SP500 vol but gets you 30% per year. You do that for an extended period of time and everyone knows your name.

We can make a reasonable matrix of values for our 2 values of interest.

Since giving a go at trading involves work and buying SPY does not, then having a SR greater than .50 to warrant the effort sounds table stakes. Any number from 1.0 to 2.0 feels like an appropriate starting point, even if it’s arbitrary. Rob starts with 1.5. Fine.

 

Restating the problem:

If you have a Sharpe Ratio of 1.5 and trade with 15% annual volatility, what’s your chance of passing a funded trader challenge that requires:

• Hitting 20% profit
• Without any single day losing more than 2%
• While paying $300/month

Rob recognizes that this is best solved with a simulation, but that’s actually an enlightened reflex. Let’s not take it for granted.

Could you solve this with math in a closed-form way?

Let’s try to solve this with formulas. What we can assume:

Daily returns ~ Normal(μ, σ) where:

• μ = (SR × Vol) / 252 = (1.5 × 0.15) / 252 = 0.0893% per day
• σ = Vol / √252 = 0.15 / √252 = 0.945% per day

Question 1: How long to reach 20%?

Simple approach: 20% / 0.0893% = 224 days

Hmm. This seems wrong for several reasons:

1. You don’t go straight up
2. The bust-out rule: any day < -2% resets you

We need to bring volatility and path dependency into consideration. We ask another question.

Question 2: Probability of hitting -2% on any given day?

P(return ≤ -2%) = Φ((−0.02 − 0.000893) / 0.00945) = Φ(−2.21) ≈ 1.4%

This seems useful. Now what?

Question 3: What’s the probability of reaching 20% before hitting -2%?

The lyrics to Steppenwolf’s Pusher come to mind:

tombstones in my eyes

I have no idea how you solve this analytically.

Fortunately, this question comes to mind 5000 years since Indian mathematicians invented Arabic numerals and 3 years since Anthropic released the tireless teacher known as Claude.

It (he? she? they?) says:

This is a “gambler’s ruin” problem with two absorbing barriers: +20% (win) and one day at -2% (lose), but after bust-out, you reset to 0% and try again!

He goes on to explain that solving this problem analytically requires:

• Solving partial differential equations (diffusion processes)
• Handling the reset mechanism (not a standard boundary condition)
• Tracking cumulative costs over multiple attempts
• Computing time-dependent probabilities

But Claude, how would I know this stuff?

Oh young Padawan, if you wanted to solve this without simulation, you’d need to learn:

1. Stochastic Calculus

• Brownian motion
• Geometric Brownian motion (for compounding returns)
• Itô’s lemma
• First passage time problems

2. Partial Differential Equations

• Kolmogorov forward/backward equations
• Boundary value problems
• Absorbing barriers and reflecting boundaries

3. Renewal Theory

• For the “reset and try again” mechanism
• Markov renewal processes
• Expected costs with renewal

Estimated Learning Time: 1-2 years of graduate-level probability theory

Oh.

It’s gonna take more effort than watching a Veritasium at 2x speed? Bruh, I don’t have cave time on my hands here.

I assume Rob reacted to this trading challenge word problem like a linebacker diagnosing pass vs run in a split second. This is a problem for simulation.

Anyway, I just thought it would be helpful to explicate the unsaid.

And one las thing. Since Rob was generous enough to post his code, I made Claude work overtime for no pay (inner monologue: does this mean I would’ve owned slaves 400 years ago?). Claude’s yield:

https://trading-sim.moontowermeta.com/

---

Option execution

While stock execution is a vast topic depending on how finicky you want to be about HFT, microstructure, game theory, counterfactuals, lit vs dark, block trading, etc etc, most institutional algos coalesce around some concept of VWAP or TWAP where the goal is to minimize market impact by making sure you are not an outsize percentage of the volume.

Option execution has no equivalently popular Schelling point benchmark such as a VWAP. This is partly because option premiums are themselves moving targets depending on the interplay of time, implied volatility, and moneyness. This is not a big issue, you could harmonize measurements in any number of ways including by the greeks. You’ll get some sense of how we can do that below. But the fact that you can pay more for an option but be purchasing a lower IV is hint enough that VWAP-ing according to premium is incoherent.

The biggest issue is that the intent of an option order has a different time horizon from a typical stock order. If you are an investor, you probably don’t care that it takes 3 days to accumulate your position as long as there are no catalysts in the execution window. But options, when used as they should be, with a basis in volatility assessment (in other words, discernment of the magnitude of a move over a certain period of time), naturally require more time-sensitive execution.

Options are also less liquid, especially outside the top 50 names or so. Given the sheer number of expiries, strikes and symbols, it is far more likely you are selling the 62 DTE, .24 delta call against a market-maker rather than someone else who woke up that morning thinking they’d like to invest that particular option. This means, you want to minimize your encounters with the order book so even if you could VWAP your order, if the counterparty is a MM you are far more likely to be leaking info that you have an order big enough to make piecing it out worthwhile. This information can and will be used against you.

The best you can do with respect to option execution is not be stupid. Your order is easy to spot. It’s the one whose limit doesn’t tick with the stock (we’ll address counters to this below!). You are always going to lose to your execution in expectation. You can tattoo that on your face. We’ll shed some light on what’s happening on those screens.

Edge functions

At any snapshot in time, market-makers have a fair value for what an option is worth. They stream a bid/ask whose width depends on an “edge function”. The general specification for this function is “how many cents of edge to I need to compensate me for hedging the delta and the vol risk”?

Delta risk

If I believe that buying 100,000 shares of stock XYZ will push it up 8 cents from the current stock offer, then I need to pad my option offer in the .25 delta call by $.02 just to breakeven on expected slippage on selling 1000 options. Note that the offer I’m streaming is not using “last sale” or “mid-market” as the S in the option model. It’s using “stock ask”. The call bid I’m streaming, is using the “stock bid” since I will be selling shares to hedge if I buy calls. You can step through this exercise yourself for put bids and put offers.

Vol risk

Let’s say this same option has a vega of $.10 meaning if IV increases by 1 point, the option premium, all else equal, goes up by $.10.

If this is an asset whose IV moves about 1 point per day, I might be fine accepting something like 1/2 a vol point of edge or $.05 cents to open a position. If the IV moves 10 points a day I’ll demand $.50 per contract to compensate me for the vega risk.

So at the very least, the bid/ask spread reflects the market maker’s willingness to warehouse the vol risk and there cost to hedge at the point of sale. Of course, a multitude of dealers with differing fair values and leans creates a tighter market than if there is only a handful.

If the market is more volatile, then slippage assumptions increase not just becasue the underlying shares are wider, but because there will be less liquidity at each price level. The consumer should expect wider spreads. Volatility itself will also more volatile. Again, wider spreads.

Edge functions are a tax on all option-related strategies when markets get crazy. Including strategies that are biased long vol but generally takers. They still need to trade. Edge functions are like COGs. Market-makers’ costs, including their funding spreads, are passed on to everyone else.

Order types

Like stocks, there are limit and market orders in options. But there is a class of contingent orders that makes sense in light of option premiums being moving targets.

Delta-adjusted orders

A delta-adjusted order may take the form of:

“If the stock is $100.50 bid or higher, I’m willing to offer the 99 put as low as $.50”

You can even peg the offer to the stock bid so that it keeps cancel/replacing according to its delta (so if it is .25 delta, every time the stock drops 4 cents your offer increases a penny) down to some ultimate low premium you’d accept.

You can even “auto-hedge” this type of order. So if you get filled on the puts, it triggers an order to sell or short shares on the bid subject to some tolerance. If the stock moves lower quickly after you sell the puts the auto-hedger will have parameters that you set about how far to “chase”. This is important because you should expect to get filled on the puts when the stock drops quickly and a market-maker snipes your stale offer before your broker has a chance to pull.

In other words, your fills are constant reminders of adverse selection. I’ve explained this in the broader discussion of limit orders:

From Reflections on Getting Filled:

My Bayesian analysis of being filled on a limit order vs market order

Imagine a 1 penny-wide bid/ask.

If you bid for a stock with a limit order your minimum loss is 1/2 the bid-ask spread. Frequently you have just lost half a cent as you only get filled when fair value ticks down by a penny (assuming the market maker needs 1/2 cent edge to trade). But if you are bidding, and super bearish news hits the tape (or god forbid your posting limit orders just before the FOMC or DOE announce economic or oil inventory), your buy might be bad by a dollar before you can read the headline.

If you lift an offer with an aggressive limit (don’t use market order which a computer translates at “fill me at any price” which is something no human has ever meant), then your maximum and most likely loss scenario is 1/2 the bid-ask spread.

Do you see the logical asymmetry conditional on being filled?

Passive bid: best case scenario is losing 1/2 cent

Aggressive bid: worst case scenario is losing 1/2 cent

This is why exchanges offer rebates for posting bids/offers — the payment incentivizes liquidity which nobody would ever offer otherwise because of adverse selection concerns. When you are not a market-maker you have the luxury of “laying in the weeds” until you spot the “wrong” price and then strike.

Vol-adjusted orders

This is an order that pegs your option bid or ask to a model implied vol. So if you are 25% vol offer in that 99 put your offer will adjust upwards if the stock goes down, or lower as the stock rallies. It sounds like a delta-adjusted order but it’s slightly different in that the delta adjusted order has no concept of theta embedded in it.

For example, the vol-adjusted offer will automatically accept a lower price at the end of the day vs the start of the day since a 25% vol option is worth less after time passes.

The delta-adjusted order relies on some underlying model’s delta for price adjustments but it’s not affected by the passage of time (well, technically there is some modest charm effect as time passing infuences delta). A delta-adjusted offer will tend to drift away from being marketable as time passes and theta kicks in, as you are effectively offering a higher IV. Likewise, a delta adjusted bid will become more marketable as time passes which effectively means you are bidding a higher volatility. The vol-adjusted order solves this, but lets the premium float (although these order types can sometimes allow additional constraints).

In general, these types of orders try to limit the adverse selection you are guaranteed when you place an ordinary limit order in options, where you only get filled when the stock moves against you. In fact these are the order types market makers themselves use for streaming logic.

More advanced considerations: “Mark to cross section”

This is one I’ve never seen anyone talk about but it’s obvious once I point it out.

Suppose you are working a large bid using a vol order. You want to pay 25% IV for the 40 strike in XYZ. It’s chipping away little by little and then suddenly you are filled on the entire thing at an average of 24.9% vol.

How do you feel?

Like you always feel when you get filled fast. Like the position is what your one-night stand partner looks like in the morning light. You can’t chew your own arm off fast enough to get out of it.

The first thing you’ll check is whether you got picked off because the stock gapped. But you find the stock hasn’t done anything weird. And then you notice a different chart on your dashboard…

The VIX futures have collapsed in the last 30 minutes.

The reason you got filled on your vega is that vol is much lower across the market. And this doesn’t show up in conventional execution metrics like mark-to-arrival. In fact, you got filled better than your vol limit of 25%.

Your order was recognized and when it was clear that vol was much lower, that any number of vols across the market were a good relative buy compared to what you were bidding, the market maker said “Yours”.

In other words, you wish you could mark your fills to the cross-section of other vols in the marketplace (or at the very least maybe VIX or SPY vol).

This is a whole other layer of attribution. If you sold a basket of vols but SP500 vol was down far more than your basket, do you feel good about your vol-picking skills?

It’s the same issue in active investment management. It’s a waste of time to earn beta. You need to outperform on a risk-adjusted basis to justify effort and identify skill.

Discussion

There isn’t so much a solution to execution as there is an understanding of the levers to think about what approaches work best for you.

At the end of the day, there is a price to entice someone else to warehouse a risk they didn’t wake up asking for. You are trying to execute at the lowest price that satisfies that threshold. Every cent you pay above that threshold is additional surplus to the market-maker above the minimum they require.

You can put out feeler orders on small size to probe how far into the bid-ask they are willing to execute. You are sussing out their edge function. It’s not foolproof. They understand that it’s not worth it to give up that information for small size. They may allow a high stale bid to sit because if they hit it they make a negligible amount of expectancy but what if they leave it there?

Let’s make this concrete with a simple, highly stylized example. An option is worth $1.00, and there’s an opaque distribution around where the next bid might come in, centered at $1.00 ± $0.04. If a one-lot bid comes in for $1.01, the market maker earns one cent in expectancy by hitting it.

But now there’s some chance that the next bid is higher than $1.01. And while that probability is below 50%, the expected bid conditional on waiting may actually be higher than it would have been had the $1.01 bid never appeared. And it might be for a larger size. Depending on what that distribution looks like, the market maker may have more “pot equity” in letting the market remain framed as-is rather than pasting a bid for a single contract.

The more liquid a name is, the less room there is for such games. If it’s highly liquid, there will be enough mm or customer flow to just whack that one lot stray.

The question is always, will I get a better average price in vol terms* if I hit or lift, vs going slowly and giving away info so that the screens mold to me. If trading against your order increases a market-maker’s inventory, they will pad their edge functions even more. If it decreases their inventory, then you probably want more market participants to see it so that it hastens their built-in desire to trade against you. If you want a bidding war for your flow, then you want to make sure everyone sees it. Be more inclined to “advertise” it by showing it as opposed to using electronic eyes (delta orders that are hidden and snipe when a marketable order appears against it).

[*Again, practice proper benchmarking. If I get a good price on my order to buy calls, it could be because the stock fell over the window when I was accumulating them, but on a delta-adjusted basis I could have been getting worse fills on the way down. In other words, I paid more in IV terms.]

Improving execution, to find the minimum edge that someone will accept to trade with you, is a hard problem. At a minimum, I’m trying to help you conceptualize the problem better with respect to adverse selection and the basic hygiene of benchmarking aspects of your fill due to vol changes vs delta. Execution is a classic domain where people do “resulting” where they let their outcomes adjudicate whether they made good decisions. The problem is hard enough without inviting that particular mistake to dinner.