Benn posted a great question:

First, there are many responses in the thread that bring up terrific points regarding how taxes and transaction costs would treat these strategies very differently. Those were the caveats I thought of too.

But there is a glaring issue that’s very hard to spot. It did not pop out to me nor many others.

Try to find it yourself.

Alright, I’ll hand it off to Nick, one of the few who saw it right away:

What’s tricky about this comes back to what Benn says…it’s not an intentional chart crime. We probably get fooled by this all the time. As Nick and Benn explain in the thread, the trick is to use log scaling on the y-axis because we are dealing with a compounding (ie exponential) process.

Let’s do that.

Step 1: Extract the returns at annual intervals using plotdigitizer.

I pasted the chart in the app, labeled the axes, and simply click on a date in early January each year. The app returns the x and y coordinates for export. (It’s tedious because I had to do it for each line separately but I think you can buy the software and it will auto-trace it.)

Step 2: Chart in Excel

I simply charted the tables starting from the start of 1998 when the lines were about to start diverging (I re-denominated all the data back to the start of 1998). Here you can see both the 1998-2023 chart on the original Y-axis and on a log Y (base 10) axis:

The log scaling reveals that the early lead of the call overwriting-strategy does not widen over time as the original chart suggests. In other words, all the gains were in the beginning.

This is not surprising to option traders. Vol selling has been wildly in vogue since the current millennium became a teenager. The asset management world noticed that it performed well and then created a ton of product based on those results.

Using the same data, here’s a rolling 5-year CAGR which shows the story. The yellow section is the outperformance/underperformance of call-selling vs buy-and-hold.

 

The broader lesson: your eyes will be more trustworthy, if you plot compounded returns with log scaling.

 

💡Learn more

• Chart Crimes (2 min read)
• Getting Comfortable With Log Charts (3 min read)

A few days ago I got the idea to do a screencast where I use an option chain and greeks explain a bunch of vol trading concepts.

None of my front-ends really look like what I had in mind so I spent Wednesday building a minimal viable version to allow viewers to look over-my-shoulder as I explain some stuff.

On Friday, I just turned the camera and started blabbing. No prep. I had an open afternoon so no time constraint. I just let it rip. On a Twitter livestream.

I hear it was helpful. I decided to call it Years worth of option education in under 90 minutes. That was the most click-baity title I could give it and still live with myself.

I re-watched it to chronicle what you actually can learn. Turns out it’s a lot of stuff that’s pretty hard to come across if you haven’t spent time on a prop desk.

Give it a gander. Love to know what else can help.

Modeling a vol curve

• Computing a forward
• Specifying a vol curve with standard deviation gridpoints
• Computing the gridpoints
• Inputting skew parameters at the points to fit the market
• Using Excel’s linest function to get the coefficients of an n-order polynomial
• Using the curve to estimate IV for any strike

Option valuation

• Implementing Black Scholes for European-exercise style options
• Includes greeks and N(d1) and N(d2)
• Numerical methods for estimating gamma and theta

Interpreting skew

• How large skew values lead to counterintuitive probabilities as the implied distribution balances probability with magnitude
• Using vertical spreads to see the implied distribution
• Changing skew parameters to watch the spread prices change and the distribution shift
• How skew “corrects” the Black Scholes distribution to match empirical distributions
• Comparing implied distributions to “flat sheet” distributions

Understanding vol changes day over day

• The difference between fixed strike and “floating” strike vol changes
• How fixed strike vols change arise from the interaction of spot moves and skew parameters change
• Why fixed strike vol changes drive your p/l

Dissection

• How market makers actually use classic option structures and synthetic relationships
• Option traders “chunk” their positions to understand them just as seasoned chess players don’t see random configurations of pieces but see “mini-themes” that they understand deeply. For option traders these themes are structures like butterflies and condors
• How market makers “take structures out of the position” to minimize hedging costs

Decomposing vol p/l from greeks

• Learn how to use your gamma and theta to estimate the realized vol portion of your p/l
• Learn how to use your vega to estimate the implied vol portion of your p/l
• See how delta p/l comes form options and share positions
• Understand how the tug-of-war between gamma and theta relates to the stock’s move on the day

Uncategorized

• Pulling market data into Excel
• why the late 90s tech bubble was not irrational and how option markets understood that
• bubble distributions from the lens of the option market
• Put-call parity
• An intuitive way to estimate gamma p/l from middle school physics math: delta = velocity, gamma = acceleration, price change = time passage, and distance = p/l
• This shows why p/l is a function of the stock move squared

Earnings are a highly concentrated source of volatility for public companies because besides reporting results they give guidance on the future, discuss what they are seeing across business lines, as well as risks and opportunities for growth. Earnings reports are a rich source of information and in the Claude Shannon sense of the word, information is volatility.

As expected, option prices that include the earnings date command a premium implied volatility as the market expects the stocks to move on the burst of new information. The observation of a premium earnings IV leads investors and traders to important questions.

• How much is the premium? In other words, how do I disentangle the amount of volatility that is “normal” vs the amount coming from the market’s expectation of how much the stock will move?
• If I am a volatility trader focused on the relative value of options between names or I am a dispersion trader who cares about the relative vol levels between and index and its components how do I compare the volatility between a name with earnings (or a event specific to the name) to other names?

Our task is beckons. We must extract earnings from the vol surface.

That probably sounds like a tedious, quanty operation. But it’s not. It’s actually a pretty simple procedure once you understand the building blocks. In fact, the procedure is an implicit review of 2 main topics. Because this topic encompasses* the prior topics it acts as a test of your knowledge as well as a step forward.

Prerequisite Building Blocks

I won’t review the building blocks here but I’ll point you directly to the relevant calculators which document the procedures.

1) Implied forward volatility

Given 2 expirations we can effectively subtract the volatility of the near dated expiry from the later dated expiry to imply a forward volatility or the amount of volatility implied in between the 2 expirations.

2) Event Volatility Extractor

When the market anticipates events like a stock’s earnings date, it often factors increased volatility into the affected option expirations.

Traders analyze this implied volatility by separating it into the volatility for the event day itself and the typical daily volatility.

To do this, a trader estimates an expected move size for the event.

The unintuitive impact of events

It’s worth emphasizing how important events to understanding an option surface. It’s one of those things that intuition is a poor guide to. The arithmetic is worthwhile.

Consider this situation.

A straddle has 40 business days until expiry. The name typically moves 1.5% per day. We’ll just use trader math to estimate a fair annualized volatility of 24% (1.5% x 16 because 16 is approximately √251).

However we get 2 new pieces of info.

1. The IV is actually 36%
2. Earnings occur in 35 business days.

We can estimate an earnings vol by acknowledging that term vol includes 39 “regular” days and 1 “event” day.

We presume that a regular day has 24% annualized vol. So what “event vol” makes the term vol worth 36%?

We are basically solving for what event vol reconciles these facts given that we know the average vol (the term vol) and the “regular” vol.

[Keep in mind variances are additive but not volatility. Variance is simply vol squared.]

Term variance = regular variance + event variance

.36² * 40 days = .24² * 39 days + X² * 1 day

Solve for X.

x = event vol = 171%

The event is a 171% vol event for a single day but this is in units of annualized volatility.

Convert back to daily volatility by going in reverse — divide by 16. (I’m resisting a reference to the Spaceballs vacuum scene).

171%/16 = 10.7%

Remember that’s now a daily vol (aka standard deviation). We should convert it to a straddle as a percent of the underlying because that corresponds to the what people actually talk about — “expected move size” on earnings.

Just multiply by .8 since a straddle is the same as the mean absolute deviation.

.8 * 10.7% = 8.6%

[To review, see 😈The MAD Straddle]

Let’s take inventory.

• The stock moves 1.5% per day which would correspond to a 24% vol name.
• However, the vol is 36% implying that on earnings it’s expected to move 8.6% on that single day.

The variance coming from all regular days is 39 * .24² ~ 2.25 (unitless, unintuitive number)

Event variance is 1 * 1.71² ~ 2.94

Despite earnings being 1/40 or 2.5% of the weight in day terms, it’s 2.94/(40 x .36²) ~ 57% of the total variance until expiry. That day has more option premium associated with it then all the other days combined. The bulk of the straddle decay occurs on that day.

This also means the theta of the preceding days is lower than you think. In practice, what happens is the vol creeps up every day offsetting some of the model theta. You can think of a glide path where as you get closer to earnings the average vol per day increases as “low vol days” peel off and the earnings day drives bulk of the straddle. This same mental image can help you understand why an event very far in the future doesn’t show up so strongly in the terms structure — its impact is diluted by the sheer quantity of regular days before it.

[These concepts underpin the trading strategy known as Renting the Straddle.]

* See educator and MathAcademy architect Justin Skycak’s explanation of encompassing vs prerequisite graphs as well as Principles of Learning Fast

Now you are convinced that this is some part important, some part interesting and you already have a taste of the most complicated math it requires (6th grade). We just need to pull it together.

Extracting earnings from a term structure of implied volatility (as opposed to a single expiry) requires using our building blocks in conjunction. The same technique can be extended to multiple earnings as well as any kind of event.

This is a good time to remind you that much of the trading is about making apples-to-apples comparisons. Normalizing data so that the comparisons are relevant is so much of the work to be done. It’s more grindy than sexy. But it also shifts the focus from what novices think investing is about to the work that actually needs to be done — measurement not prediction or “seeing the present clearly”.

As we step through an earnings extraction, I will point to real-life examples of what I mean by measurement not prediction.

A few selling points on this post:

1. The building blocks do the heavy lifting so this won’t take long.
2. The yield is insane — this is one of those topics that opens lots of mental doors.
3. I provide a link to a spreadsheet so you can play with the ideas yourself or extend them as desired.

An “ugly” term structure

I fetched .50d IV’s for NVDA at end-of-day 1/17/25.

 

This is an ugly term structure.

There are 2 primary reasons.

1. Market widths and leans in option bid-ask will shop up as artifacts in your surface fits.
2. There are events in the option surface. Most notably there is an event embedded in the 2/28/25 expiry. We know that because the IV jumps 10 points from the prior week.

Option market makers are like blue-collar household help. Their job is to “iron out the kinks”. Buy the cheap IV and sell the expensive IV when they see a wrinkled term-structure.

But if you did this by looking at the graph, you’d be selling the following expiries:

• 2/28/2025*
• 6/20/2025*
• 9/19/2025*
• 1/31/2025

You’d be buying:

• 2/21/2025
• 1/24/2025

Here’s the term structure again but we simply change the x-axis to DTE instead of expiry and annotate our naive buy and sell axes.

The forward vol matrix is a granular way to get the same idea.

I’m obviously using leading words like “naive” to indicate something important is missing which is creating all these kinks.

We are going to address the most glaring kink, the 2/28 expiry which jumps 10 vols from the prior expiry and showing up as a 1 week forward vol of 80%. The other kinks are naturally handled from the transitive logic of how he handle the big kink. Always handle your big kinks first ;-P

Accounting for events

Your reflex when you see a bump in the term structure is “what known event is happening between the expiry dates?”

In this case, NVDA report earnings on Wednesday, 2/26/2025. The 2/28 expiry “captures” earnings most acutely. The earnings vol is embedded in every expiry from 2/28 and beyond it’s impact is attenuated as DTE grows. That earnings even becomes a smaller percentage of the total variance in the term. If you are looking at a 5 year option, the earnings vol will be invisible whereas the 2/28/2025 expiry has the bulk of its variance coming from that single day.

Here’s the plan.

1) We use the event volatility extraction formula which takes in DTE, IV and a guess for the event straddle (ie move size) to translate all the vols with earnings into “ex-earnings” vols. I’ll use the terms “ex-earnings vols” and “base vols” interchangeably.

💡A note on nomenclature

I’ve heard “ex-earnings vols” called:

• clean vol
• base vol
• regular day vol
• non-event vol
• non-earnings vol

The point is that you are looking at a surface where known events have been removed. This allows higher fidelity comparisons between names. The event that is extracted is an implied or consensus move size. If you buy a vol with an event inside it you might be betting on the base vol being too low OR you believe the consensus move size is “too cheap”. Isolating what your betting on comes down to trade structuring. Maybe it’s a calendar spread, maybe it’s a “rent the straddle” glide path trade.

Depending on your trading lineage, even this nomenclature can be confusing. In my background I also referred to vols using a 365-day tenor as “dirty vols” and vols implied from a custom tenor as “clean”. For example, if you treat holidays and weekends as a 25% variance day your tenor will be about 280 days. I don’t want this post to encompass too much, but if you’re a glutton see Understanding Variance Time.

2) We can now chart the term structure of the base vols and feed them back into the forward vol matrix!

The goal of the guess is to see a smoother chart and matrix. Smoother. Real science-y stuff here.

Based on simple guess-and-test (which is easy to do with the spreadsheet I’ll provide) I came up with an earnings straddle of 6.5%.

The matrix looks much better (the smaller kinks are still present since we only dealt with one, albeit large event, but I’ll address that later. It’s easier to focus on one major thing at a time.)

Notice how the 2/28 expiry, stripped of earnings, now follows a gentle up-sloping term structure instead of stickin’ up like a sword from a stone.

[Bonus observation: power law functions handle vol term structures well. Remember a power function can be converted to a line using a log-log transformation where your variable Y is vol and X is DTE so you can fit a linear regression. You can start to imagine a wider infra where you have a well-defined event calendar, extract implied events sizes everywhere, and fit base vol term structures to identify kinks, ie buy and sell signals. As it dawns on the reader what relative value vol trading looks like. Throw in layers of execution topics and you can see the basic truth — there isn’t any magic sauce it’s just fastening a thousand submarine doors before the thing can go anywhere. And every day the state of the art of little door details inches up.]

Let’s see the chart if we try 5% and 8% earnings move respectively. The first chart keeps the dirty earnings curve and ex-6.5% earnings curves for reference.

The second chart zooms in.

And the matrices:

If you assume an 8% earnings move, the 2/28 expiry looks cheap. If you assume 5%, it looks rich. The first step is to find what makes the curve look well-behaved and then based on your view on base vol, earnings vol, or both you can isolate how you should trade it.

Beyond a single event

Even with the adjustment, I’d readily admit the term structure is pretty kinky. But the ugliness is useful because it’s an opportunity to step-through what actually happens in practice. Let’s talk about how to iron the kinks that remain and see what’s left over.

Let’s step through some notable points on this chart. I’ll be clear in my explanation but I’ll do them a bit out of order.

Point #1: This doesn’t stand out as being too cheap relative to the rest of the curve because there’s no reason to assume that upward sloping-term structures are “wrong”. But there’s a technical reason it might appear so low…these vols use a 365 day model. This snapshot is taken on Friday. Vols “appear” to go down on Fridays and shoot up on Mondays but its a sawtooth artifact of a model which treats every day equally. This is explained thoroughly in Understanding Variance Time.

Point #2, a thru c: The 1/31/2025 expiry is a busy macro week — inflation data, jobless claims AND an FOMC meeting. If we extract event straddles from this expiration the base vol will fall to line up much better with the power function. The expiries behind it (b and c) also contain that busy week but its effect will be diluted while a contains the brunt of it. So b will be higher than a but less than c which will fall the least. See how it’s creating upward steps!

We’ll come back to #3 and #5.

Points #5 and #6: June and September expiries. Notice June has a bigger bump than September. Have a guess?

Earnings! Although the earnings dates are in May and August they fall AFTER those expiries and are captured by the following month!

September has a smaller bump than June is because it’s further out in time. The impact of a single day move is proportionally smaller for a longer-dated option than a shorter-dated one.

Back to #3: This is the 2/21 expiry. This one is interesting. If we impose a lump of variance for FOMC week while it definitively has a larger impact on #2 a thru c, it will still have some impact in the 34 DTE 2/21 expiry which means as low as it looks, it’s even cheaper than it looks. If the snapshot is accurate, 2/21 looks like a candidate to buy vol. If you thought earnings vol looked “expensive”, you could sell 2/28, buy 2/21, then cover your 2/28 short as 2/21 expired. When I say “you” this is pros who’d kick their grandma down a flight of stairs for a tick of edge. You can throw trades like this onto the pile of tiny edges. I doubt the juice is worth the squeeze for retail. However, if you are looking to buy or sell options outright, then understanding this can help you on the margin. It’s a “I’d go for it on 4th and 3 but not 4th and 5” type of knowledge.

[An alternative thought…does the week before earnings structurally deserve a lower vol because the chance of the company saying anything material is close to zero? It’s good hygiene to wonder what you’re missing whenever something looks cheap or expensive. If you cleaned up all surfaces for events would you find the week preceding earnings to look cheap across the board?]

And finally #4: The 3/21 expiry looks expensive even after adjusting for earnings. Again, you’d want to doublecheck the vol returned by your snapshot.

If you thought earnings were expensive, a more oblique way to express it would be to buy #3 (2/21 expiry) and sell #4 (3/21 expiry) which corresponds to a 51.5% forward vol NET of adjusting for earnings. Most of that “value” does seem to be driven by 2/21’s cheapness rather than 3/21’s expensiveness (something you can notice by observing how the 2/28-2/21 forward is more stretched than the 3/21-2/28 forward despite us thinking when we anoint 2/28 as fair. This can inform your weighting of a calendar — you sell 3/21 maybe you buy twice as much 2/21 if you think that’s the best leg. This is where having additional info about the flows that are pushing options around and the general “art” of trading is apparent.)

Pretty pictures

If you clean every event, iron out all the kinks you might just find a well-behaved curve. “Listen” to the market carefully It’s a call and response:

You: “Look a kinky opportunity”

Market: “Nah, there something coming up. This is what people expect.”

You: “Ah, thanks for the heads up, I’ll incorporate it”

Market: “Aren’t you gonna cast your vote?”

The response is up to you, but see the present clearly. Remember, measurement not prediction.

I’ll leave you with a spreadsheet so you can play with an event size and see how it propagates through the term structure. Smooth curves = smooth forward vol matrix.

💾Moontower Event Vol Matrix

Spreadsheet screenshots:

 

Final thoughts

Trading vol around events is a major topic.

At scale, quants will have more “proper” methods for doing this but I can tell you that a significant portion of my career earnings have come from understanding this stuff. (It was 20 years ago, about 2005, that I was starting to build this infra. All in Excel by the way.)

The techniques improve. I’m not a quant as I’ve said many times. I don’t know the state-of-the-art but with some simple math and yea a lot endurance, observing, noticing you can go quite far.

Is this gonna turn you into SIG or Jane? Hell no, but these are the ant trails that take you to the questions. To a frame of mind that measures for and seeks contradiction. Notice how little broad opinions matter. Instead, you are trying to turn market prices into mini-hypothesis. Trades are tests against hypothesis.

But it starts with measurement.

Here’s a few questions that option traders are asking every day.

How does the surface/consensus synthesize knowledge about:

• prior earnings moves?
• seasonal earnings moves?
• time-series of implied earnings moves?
• how earnings vol is cross-sectionally priced broadly and by sector?
• how implied correlation is priced during earnings season?
• how VRP’s look ahead of earnings? after earnings? if we clean implieds to get to base vols, do we clean realized vols after known events have passed to have base realized vols?

These are all active areas of inquiry. They are not solved problems. They will be eventually but then the conditions of their solving will have meant another set of opportunities will emerge.

If you are an aspiring pro, insert yourself somewhere and just start chopping wood.

A reliable way to have Nassim Taleb come through your window to call you an iiiiidiot is to see a stock crash and say something like “That was a 10-standard deviation move!”

On #voltwit, the corner of fintwit with an option fetish, a surefire way to get QVR’s Benn Eifert to QT you is to quote VIX changes in percent terms.

The topic resurfaces every time VIX makes a large move, voltwit predictably reacts so I figured Grok (X’s LLM chatbot) must know about this.

I asked Grok some questions and sure enough it seems like it’s been paying attention.

“Is it wise to quote VIX changes in percent terms?”

• Volatility of Volatility: The VIX itself is inherently volatile, so small percentage changes might not always reflect significant shifts in market sentiment or conditions.
• Misinterpretation: Large percentage changes from a low base might exaggerate perceived market fear or complacency. Conversely, small percentage changes at high levels might understate the market’s condition.
• Contextual Information: Percentages without context (like the baseline level of the VIX) can be misleading. A 5% increase in VIX might mean different things at different times.

“Do vol traders dunk on tourists who quote VIX percent changes?”

In the trading community on platforms like X, there’s often a playful or critical banter between experienced volatility (vol) traders and those perceived as “tourists” or less experienced traders.

• Misinterpretation Risks: Vol traders often emphasize that quoting VIX in percentage terms without additional context can lead to misinterpretations. For example, a 10% increase from a very low VIX level might not be as significant as the same percentage increase from an already high level.

   

Benn’s primary gripe with VIX percents is the behavior of VIX is level-dependent. Its distribution is not congruent at high and low levels of vol.

Notice how the Y-axis is VIX vol points not percents.

In chatting with Benn about this article he pointed out a basic mechanic that makes vol level-dependent:

Volatility is inherently about squared returns, so you can have a very low base level of realized vol but all it takes is one big-ish sized return and because we’re squaring it (along with all the other little returns in the window) it’s going to have a massively outsized impact on window realized volatility. That makes vol very jumpy from low levels.

Another vol manager, Kris Sidial of Ambrus, explains it simply. Note my response below it.

There are multiple contexts in which it is quite useful to measure percent changes in volatility. There are tradeoffs, as you’d expect with any measure. But I’ve always been forceful about the need to slice things from different angles. It’s a healthy way to identify mixed signals, but it’s also affirming when sufficiently different angles agree.

A good example of this “multiple angles” idea is the 2 part series:

• Volatility term structure from multiple angles (I)
• Volatility term structure from multiple angles (II)

Let’s get into a few reasons to measure vol in percent changes.

Cross-sectional comparison

As a relative value options trader, I would typically have an “axe list”. These are vols in various names in various parts of the surface I thought were relatively cheap or expensive.

[The idea of an “axe list” is covered in the Moontower Mission Plan]

Armed with my opinions, I would then buy the options I thought were cheap on days when their strike vols were underperforming, and sell the expensive ones on days the strike vols were outperforming.

Because I’m looking at vols cross-sectionally it makes sense to look at the percent changes in the vol. A one-point move in SPY is much larger than a one-point move in TSLA.

[See Understanding The Vol Scanner for a full explanation.]

Notes and caveats

1) Measuring percent changes in vols work well “locally”.

For example, it was common in modeling spot-vol correlation in oil to assume that as oil futures went up 1%, that vol declined 1%.

[This dynamic corresponds to a “constant ATM straddle regime”. It is easily visible from the straddle approximation formula.]

But nobody believes that doubling the oil price will suddenly lead to a halving in vol. The model only works “locally.”

2) Percent vol changes can be further refined by normalizing for “vol of vol”

If SPY vol changes from 20% to 21%, a 5% change in vol level might still be more significant than TSLA vol changing from 60% to 63%, also a 5% change, because TSLA vol of vol might be higher. After all, it might be common for TSLA vol to move 5% per day.

The analogy to regular investing would be the difference between a dollar-neutral position and a beta-weighted position. If you are long $100 of TSLA and short $100 of SPY, your portfolio will act like it’s long even though in dollars it’s flat. You are long beta because TSLA is more volatile.

[I’m ignoring the correlation aspect of beta because it’s not central to the argument.]

3) An extra note on “vol of vol”

If you measure vol of vol based on changes in ATM vol you are getting a confounding reading. Like if you measured your pulse with your thumb.

Why?

ATM vols are “floating” strike vols. If SPY drops 1% and ATM vol increases by 1 point, that might just be movement along the vol curve. The vol on the 99% strike might have simply been 1 point higher than the prior day’s 100% strike. On a fixed strike basis, the vol didn’t change. In this case, the appearance of a vol change merely reflected a change in the underlying.

For vol trading purposes, you usually care about fixed strike changes (ie curve shifts not movement along the curve) because that’s what drives the vega p/l of the attribution.

Risk and P/L measurement

The second reason to care about percent changes in vol only applies to vol traders. Vol traders defined as traders who run a delta-neutral book and make their edge from isolating cheap and expensive vol.

That said, the discussion should be highly educational for anyone trying to learn options or as a useful self-test for traders who might be interviewing and expected to talk about managing a book.

Let’s back up to consider vol risk. Specifically, vega, the sensitivity of your option p/l due to changes in implied vol.

We start with a scenario. Assume the ATM and at-the-forward (ATF) strikes are the same.

You buy 100 December ATM straddles in stock A and short 100 December ATM straddles in Stock B.

Stock A and Stock B trade for the same price.

Stock B has 2x the implied vol of Stock A.

Are you vega-neutral?

Are you theta-neutral?

[You can look at the greeks from an option calculator to help but if you are an experienced option trader you shouldn’t need to.]

Ok, let’s get to the answers.

You are vega-neutral. Recall the straddle approximation:

Since vega is just change in option (in this case straddle) price per 1 point change in vol, then:

vega = .8 * S * √t

Look at the formula — ATF vega has no dependence on vol level!

Since S and t are equal then your long and short vega perfectly offset.

[Note: OTM option vega DOES depend on vol level. They have volga or “vol gamma” which is what fuels vol convexity.]

Ok, you’re vega-neutral.

Are you theta-neutral?

Again we don’t need an option model. If Stock B is 2x the vol as Stock A its straddle is 2x the price. If both stocks don’t move until expiry, all options go to zero. Necessarily, Stock B experienced 2x the decay.

If you are short the straddle in Stock B, your portfolio collects theta. It is NOT theta-neutral.

Vol traders will often think in terms of vega. “I bought $50k vega in ABC today”.

At the same time, they often try to run a roughly theta-neutral book.

[See Weighting An Option Pair Trade for a discussion about vega and theta weighting and how the weighting should be matched to the expression of your bet — proportional vs spread].

In the riddle above, being vega-neutral did not mean theta-neutral. But we can actually transform vega so that a vega-neutral position is correlated to a theta-neutral position!

Another way to measure vega: “vega per 1%”

Let’s say the vega per straddle was $.50

If you buy 100 straddles your vega is 100 x $.50 x (100 multiplier) =

$5,000

If vol increases by 1 point you make $5,000 from the change in implied vol.

Assume that the implied vol of the straddle is 25%

Multiply the vega by the vol:

$5,000 x 25% = $1,250

Watch what happens if we raise the vol by 1% or .25 points instead of 1 point:

Vega p/l: $5,000 x .25 points = $1,250

Remember when we raised vol by a full point from 25% to 26% (or a 4% change in the vol) you made $5,000 or $1,250 x 4)

By multiplying the vega by the vol itself we have created a new measure:

Vega per 1% measures the vega p/l per 1% change in the vol.

Let’s return to the original riddle.

We now assign implied vols to the straddle. You are long 100 straddles of Stock A at 25% vol and short 100 straddles of Stock B at 50% vol.

While this is vega neutral, it is NOT vega per 1%-neutral

Stock A “vega per 1%”: +$5,000 x 25% is +$1,250

Stock B “vega per 1%”: -$5,000 x 50% is -$2,500

You are net short $1,250 vega per 1%

This perspective is useful for a few reasons:

1) Linear estimate of p/l with respect to percent changes in vol

If vol is up 3% your p/l is simply 3 x vega per 1%. If you are using a view like “vol scanner” to see all the percent changes in vol cross-sectionally the changes will map easily to your vega per 1% risk

2) Vega per 1% proxies a theta-weighted position which is how vol traders often think about their risk and the idea that they are betting on relative proportional vol changes.

If you are short vega per 1% you are collecting theta

Multiple angles

Looking at vega in both the conventional way (p/l sensitivity per 1 point change in vol) and vega per 1% reveals features of a position.

If you are long vega per 1% but short vega, what does that mean?

Any combination of the following:

• You are short time spreads,
• You are long high vol options and short lower vol options. Owning skew or vol convexity are both examples of this.
• Cross-sectionally you are long high vol names and short low vol.

[Note in all these case it’s possible to be paying theta and short gamma locally. But if you shocked the position in a scenario analysis you likely make a ton of money. The relationships between Greeks are all clues as to what is lurking in a complex portfolio.]

In the riddle scenario, to be flat vega per 1%, you must ratio the trade and be short 1/2 as many high vol straddles. Note you will be net long vega. You will win if all the vols parallel shift higher (ie they all go up 10 points), but if they maintain their .5 relationship the p/l will be flat, consistent with the meaning of flat vega per 1%.

Understanding your greeks means understanding what you’re rooting for. You’d be surprised to know that sometimes option traders don’t even know what they’re rooting for.

When you get down to it, any large percentage change in vol is going to require multiple angles to understand. Your p/l isn’t going to line up because vega itself will change as the underlying changes and vol changes interact. Measuring percent changes on small numbers is usually a bad idea and requires transformations to find divine anything worth mentioning.

Does it make sense to talk about a 75% change in VIX from a base vol of 8%? Of course not. One of the reasons you know that is because it can’t fall 75% from 8%. That’s a clue right there that “standard deviation”, a concept we learned about from symmetrical pictures in HS math texts, is not in charge.

In sum, percent changes in vol can be useful measures but you have to know how to wield them and where they break down.

Unless you want to be the main character on voltwit for a day (and have to fix your broken window). But if you’re ok with that at least go the extra mile and do some technical analysis on VIX.

Final caveat

When you use vega per 1% you implicitly assume that both assets have the same vol of vol. In other words, if a 15% vol name’s IV bounces around 1 vol point per day, then a 30 vol name bounces around 2 vol points per day. This may or may not be true but it’s a better guess than raw vega weighting, which would show being long 100 straddles in A (25% IV) and short 100 straddles in B (50% IV) as flat.

Given where markets are these days, there are a lot of investors, often former or current employees and execs of Mag7 names that are sitting in large, concentrated position at a low cost basis.

In English, they’re as rich as celebrities but standing standing right next to you giving out Pocky on snack duty for 3rd grade soccer.

They are reluctant to sell because the tax hit is immediate. One possible solution to “have their cake and eat it too” is to stay long but collar the stock. This is typically presented as buying a put option financed by a covered call.

Here’s an example based on closing TSLA option prices on 1/28/2025 for the Jan 15th 2027 expiry (ie 717 DTE).

The stock closed around $396.65.

We can just round numbers, call it $400.

You can buy the 25% out-of-the-money put, the 300 strike, for about $56 and sell the 25% out-of-the-money call, the 500 strike, for about $108.

To be perfectly clear — you can buy the put for protection, sell the upside call and COLLECT about $52 or about 13% premium.

Think about the risk/reward for a moment.

If the stock drops $100 in 2 years you are stopped out at $300 but you collected $52 so your net loss is only $48 or about 13%.

If the stock climbs to $500, you will get assigned on the short call so you’ll make $100 on the long stock position but still get to keep the $52 premium for a total gain of $152 or about 38%.

In other words, you can stay long the stock but you get paid 3x what you lose on a $100 up move vs $100 down move.

It sounds like free money.

The prices come from option theory’s arbitrage-free (ie risk neutral) pricing.

This is a checklist of forces that seem to create the illusion.

✔️The forward price is actually $430

We know that because if you look at the option chain, despite the $430 call being ~ $33 out-of-the-money, it’s the same price as the 430 put which is in-the-money.

The reason for this is because if it didn’t you could put on a reversal or conversion trade to arbitrage the funding rate on the stock.

Think of it this way, if the 430 call cost $20 more than the $430 put you could sell the call, buy the put and collect $20. At expiry, since you are short the $430 synthetic stock you are guaranteed to sell TSLA at $430 (either you exercise the 430 put if it’s ITM or get assigned on the 430 call if it’s ITM). So you can buy the stock today for say $397 which would be a (mostly) riskless position since you are long the stock and short the synthetic. The cashflows would be:

• Collect $20 on the synthetic (remember you sold the call for $20 more than the put)
• Ensure a profit of $33 by expiry (you bought TSLA for $397 and will sell it at expiry at $430)
• Forgo ~$32 interest on $397 for 2 years (assume 4% rfr)

Net arbitrage profit: +$21 in excess of funding costs!

If the 430 call traded $20 UNDER the put you would do the arbitrage in reverse. You’d buy the call, sell the put and be guaranteed to buy the stock for $430 at expiry. To hedge you would short it today at $397 and collect $29 on the cash in your account.

So at expiry you are buying the stock for $430 that you shorted at $397. Cash flows:

• -$33 on buying TSLA synthetically and shorting it today
• +$32 in interest on cash proceeds from the short
• +$20 in option premium (remember, you sold the put $20 higher than the call you bought)

Net arbitrage profit: +$19 in excess of funding costs!

If the RFR is 4% (which it approximately is) then the 430 call and put must traded around the same price for there to be no arbitrage.

Therefore $430 is the 2 year at-the-forward strike.

✔️Despite both option strikes being $100 or 25% away from the spot price, the call is much “closer”

Part of this has to do with the forward being $430. Referencing the 430 strike the 500 strike is only 16% OTM while the 300 strike is now 30% OTM.

The option that is “closer” has a higher delta and worth more due to moneyness.

But the other reason comes from the fact that Black-Scholes assumes a lognormal distribution of returns (which is a positive skew distribution).

Why? If a stock is bounded by zero but has infinite upside the OTM call will be worth more than the equidistant OTM put. The distribution is balanced around a median stock expectation that is dragged lower by volatility (if you make 25% then lose 25% you are net down over 7%).

In TSLA’s case the 300 put has a -.20 delta while the 500 call has a .60 delta!

(TSLA also has an inverted skew — the call IV is touch higher than the put IV but that has a minor effect on the cost of the collar in the context of this discussion.)

Here’s a summary table including the collar price if the IV was the same for the 300 put and 500 call:

💡What this post “encompassed”

If you understand this post you have implicitly reviewed:

• Lessons from the .50 Delta Option (moontower.ai)
• Using Log Returns And Volatility To Normalize Strike Distances (moontower.ai)
• Teaching option basics with live data (moontower YouTube)
• Synthetics: Alternate Realities (Market Jiujitsu)

---

I called this post “arbitrage is a hall of mirrors” because no-arbitrage pricing theory created this situation where the risk/reward of the collar looks incredibly attractive.

Part of that is theory explicitly incorporates the opportunity cost (the risk-free rate) while our intuition tends to gloss over it. Opportunity cost is an easy topic to understand when someone explains it to them, but it’s trickier to apply in live decision-making scenarios. Look no further than rich people who clip coupons or drive 10 miles out of their way for Costco gas.

The output of arbitrage-pricing can be dissonant to our eyeball tests. It was one of my favorite topics to write about because it does feel so warped.

🟰Understanding Risk-Neutral Probability

This is my guide to the subject. It’s full of nested problems, Socratic method, and even financial theory as philosophy. I’ll re-print one of the nested sections:

👽Real World vs Risk-Neutral Worlds

No-arbitrage probabilities allow us to price options by replication

The insight embedded in Black-Scholes is that, under a certain set of assumptions, the fair price of an option must be the cost of replicating its payoff under many scenarios. Any other price offers the opportunity for a risk-free profit. Have you ever wondered why the Black Sholes “drift” term for a stock is the risk-free rate and not an equity risk premium (like you’d expect from another type of pricing model — CAPM) or the stock’s WACC? A position in a derivative and an opposing position in its replication is a riskless portfolio. Therefore that portfolio only needs to be discounted by the risk-free rate. Option pricing derived from a no-arbitrage replication strategy means we should use the risk-free rate to model a stock’s return.

‼️What seasoned option traders get wrong: Outside of the option pricing context, the risk-free rate is the wrong assumption for drift!

From Philip Maymin’s Financial Hacking:

One of the most common mistakes that even highly experienced practitioners make is to act as if the assumptions of Black-Scholes (lognormal, continuous distribution of returns, no transactions costs, etc.) mean that we can always arbitrarily assume the underlying grows at the riskfree rate r instead of a subjective guess as to its real drift μ. But this is not quite accurate. The insight from the Black-Scholes PDE is that the price of a hedged derivative does not depend on the drift of the underlying. The price of an unhedged derivative, for example, a naked long call, most certainly does depend on the drift of the underlying. Let’s say you are naked long an at-the-money one-year call on Apple, and you will never hedge. And suppose Apple has very low volatility. Then the only way you will profit is if Apple’s drift is positive; suppose Apple has very low volatility. Then the only way you will profit is if Apple’s drift is positive…if it drifts down, your option expires worthless. But if you hedge the option with Apple shares, then you no longer care what the drift is. You only make money on a long option if volatility is higher than the initial price of the option predicted. The drift term of the underlying only disappears when your net delta is zero. In other words, an unhedged option cannot be priced with no-arbitrage methods

💡Takeaway: Arbitrage Pricing Theory

Sometimes called the Law of One Price, the idea contends that the fair price of a derivative must be equal to the cost of replicating its cash flows. If the derivative and cost to replicate are different then there is free money by shorting one and buying the other. This approach is how arbitrageurs and market-makers price a wide range of financial derivatives in every asset class including:

• Futures/Forwards

• Options

• ETFs and Indexes These derivatives are the legos from which more exotic derivatives are constructed.

A Source of Opportunity

Let’s recap the logic:

1. Arbitrage ensures that the price of a derivative trades in line with the cost to replicate it.

2. A master portfolio comprising:
   1. a position in a derivative
   2. an offsetting position in its replicating portfolio
   3. This master portfolio is riskless.

3. A riskless portfolio will be discounted to present value by a risk-free rate otherwise there is free money to be made.

4. The prevailing prices of derivatives imply probabilities.

5. Those probabilities are risk-neutral arbitrage-free probabilities.

But those probabilities don’t need to reflect real-world probabilities. They are simply an artifact of a riskless arbitrage if it exists.

This can lead to a difference in opinion where the arbitrageur and the speculator are happy to trade with each other.

• The arbitrageur likely has a short time horizon, bounded by the nature of the riskless arbitrage.

• The speculator, while not engaging in an arbitrage, believes they are being overpaid to warehouse risk.

Examples

1) Warren Buffet selling puts

The Oracle of Omaha engages in oracular activity — not arbitrage. Warren is well-known for his insurance businesses which earn a return by underwriting various actuarial risks. Warren is less famous for his derivatives trades. [The fact that he rails against derivatives as WMDs might be the most ironic hypocrisy in all of high finance but as I always say — we are multitudes.] Like his insurance business, the put-selling strategy hinges on an assessment of actuarial probabilities. In other words, he believes that real-world probabilities suggest a vastly different value for the puts than risk-neutral probabilities. The major source of the discrepancy comes from the drift term in Black Scholes. Warren is pricing his trade with an equity risk premium in excess of the risk-free rate that a replicator who delta hedges would use.

The option traders who trade against him can be right by hedging the option effectively replicating an offsetting option position at a better price than the one they trade with Berkshire. Warren is happy because he thinks the price of the option is “absurd”. In Warren Buffett is Wrong About Options, we see this excerpt from a Berkshire letter during the GFC:

Jon Seed writes:

Warren’s assumptions aren’t crazy. In fact, they seem to be pretty accurate. As Robert McDonald derives in the 22nd Chapter of his 3rd addition of Derivatives Markets, a 100 year put for $1bn assuming 20% volatility, a long-term risk free rate of 4.4% and a dividend rate of 1.5% implies a Black-Scholes put price of $2,412,997, close to Buffett’s $2.5 million. But Warren isn’t discussing risk-neutral probabilities, those assumed in Black-Scholes and imputed by volatility assumptions. He’s evaluating the model’s probabilities as if they were real, actual probabilities. If we, (really Robert McDonald), evaluates Black-Scholes using real probabilities by also incorporating our best guess of real equity discount rates, we see that the model is consistent with Warren’s common sense approach. Assuming stock prices are lognormally distributed and that the equity index risk premium is 4%, we would substitute 8.4% for the 4.4% risk-free rate, obtaining a probability of less than 1% that the market ends below where it started in 100 years. Buffett also assumes that the expected loss on the index, conditional on the index under-performing bonds, will be 50%. This again is a statement about the real world, not a risk-neutral world, distribution. With an 8.4% expected return on the market, the implied expectation of $1 billion of the index conditional on the market ending below where it started is $596,452,294, or 59.6% of the current index value. Again, this is close to Buffett’s assumption of 50%.

2) FX Carry

FX futures are derivatives. Their pricing is a straightforward output of the covered interest parity formula. I think I learned this concept on the first day of trader class back in the day.

The key to the formula is recognizing that the value of the future is just the arbitrage-free price arising from the difference in deposit rates between the 2 currencies in the pair.

If a foreign currency offers a higher interest rate than a domestic currency, you expect its future to trade at a discount. We won’t bother with the math since the intuition is sufficient:

If you borrowed the domestic currency today to buy the foreign currency so you could earn the yield spread for say 1 year, you’d have a risky trade — you’d be exposed to the foreign currency, and its associated interest income, devaluing when you try to convert it back to the domestic currency.

therefore, to make the trade riskless, you need to lock in the forward rate today by selling the future. You know what that means — you expect that forward rate to trade at the no-arbitrage price

The higher-yielding currency must therefore have a lower forward FX rate.

The carry trade is basically a speculator saying:

“I know the future FX rate should trade at a discount to the spot rate but I’ve noticed that the future rate rolls up to converge to the spot rate, rather than that lower rate being a predictor of the spot FX rate in one year.

So I’m going to buy that FX future and hold it for a profit.”

The carry trade is not an arbitrage or riskless profit. It’s a risky profit. But the opportunity arises because the futures contract would present an arbitrage at any price other than the risk-neutral price.

If you don’t know what MicroStrategy (MSTR) then congrats, you have won life. Close this tab and go back to sliding down rainbows and swimming with otters.

For those who remain you likely know that Saylor has been financing his BTC purchases from sale of convertible bonds.

I have nothing to add to that conversation but I have a trade idea. It’s gonna take some background to build up to it.

First, there are 2 required reads. They aren’t long and they’re excellent. The best combination. I will highlight some key points from them.

Convert of Doom: Microstrategy and the dark arts of ‘volatility arbitrage’ (6 min read)
by Alexander Campbell

This post explains how Saylor is effectively arbitraging the MSTR’s volatility by issuing convert that pay zero interest. This works because a convert is just a bond with an embedded call option. By delta hedging the implied vol in the embedded option, dealers or investors can earn a return if the realized vol exceeds the implied vol. The expected return presumably compares favorably or at least similarly to if MSTR just issued interest-bearing debt but Saylor, is effectively transmuting volatility into interest payments.

[In general, when a convert is first issued it’s common for both the stock and vol to decline as dealers hedge both the delta and often the implied vol by selling long-dated options to offset some of the vega they’ve bought at a discount.]

Campbell is both educational and insightful showing how:

1) the Merton model can be used to understand why MSTR is so much more volatile than BTC — the MSTR’s premium to NAV is positively correlated to BTC!

(In Battle Scars As A Call Option, I explained how one of my most painful trades occurred when I was long UNG vol when it went premium to NAV. In that case, the sizeable premium was inversely correlated with the price of gas. The exact opposite scenario of MSTR’s juiced vol today!)

2) this is a regulatory arbitrage.

Quoting Campbell:

Result: Retail buys MSTR shares at 150% premium while sophisticated investors arbitrage vol differentials and MSTR books the diff between all these trades as profitable transactions.

Here’s the irony: We require hedge funds to register with the SEC, spend $50-500k annually on compliance, and limit themselves to accredited investors with millions in the bank. Yet retail investors can freely buy MSTR shares through Robinhood.

And therein lies all the difference. There’s nothing wrong with what MSTR is doing, but it’s a good example of the law of unintended consequences.

Regulators block retail from ‘risky’ hedge funds while inadvertently pushing them into something potentially more dangerous.

By restricting crypto access for years, regulators left retail investors few options. Bitcoin futures required $300k contracts with 50-100% margin. ETFs were obscure or nonexistent. So people bought MSTR instead – a far more complex and potentially risky vehicle.

In trying to protect retail investors, the SEC has inadvertently funneled them into a potentially much riskier product.

Which brings us to the next required reading:

Moonshot or Shooting Star? A Volatile Mix of MicroStrategy, 2x Leveraged ETFs and Bitcoin (7 min read)
by Elm Wealth

Oh how I love the existence of levered ETFs on concentrated ideas. This post echoes a very real possibility of XIV’s “volmageddon”.

Something we’ve discussed ad nauseum in this letter is volatility drag and how geometric returns diverge much lower from arithmetic returns as we increase volatility. The divergence is proportional to variance or volatility squared.

The article links to a neat calculator which offers hands-on lesson in volatility drag.

💡Learn more💡

• A Simple Demonstration of Return Vs Volatility
• The Volatility Drain
• Well What Did You “Expect”?
• Geometric vs Arithmetic Mean In The Wild
• Path: How Compounding Alters Return Distributions

And linking these to options which is where we are heading:

• Lessons from the .50 Delta option

Exponents are good, wholesome fun. And this post was certainly that, inspiring the the trade idea we’re building towards.

The long quote below (emphasis mine) cuts to the heart of the matter.

Now let’s use some data to look at the probability of going bust just from a single really bad day. The price of a 2x leveraged ETF should go to zero if the price of the stock underlying the ETF goes down by 50% or more in a single day. The probability of such an event is a function of the variability of the MSTR stock price. If we assume the volatility of MSTR will be about 90% (or 5.6% per day), then we could think of a 50% decline in the stock price in one day as being a roughly 9x daily volatility move. A natural question is how often do stocks with very elevated variability, like MSTR, experience days when they decline by 9x their daily variability in returns?

We looked at about 1500 US stocks over the past 50 years, chosen so that at some point they were within the top 1000 stocks by market-cap. We found that the annual probability of such stocks experiencing a one-day price decline of 9x daily volatility was about 6%.

[Kris: The fatness of the tails should swipe you like a dragon. In Mediocristan, 9 standard dev moves don’t happen.]

This isn’t quite the final answer though, as we need the probability of a stock dropping by that much some time during the day, rather than just close-to-close. The usual estimate for the probability of touching a level over some time interval is to simply double the probability of being below that level at the end.

[The explanation of this is the same logic we’ve discussed whereby we estimate the probability of a one-touch by doubling the delta. Here’s Elm Wealth explaining:

To see why this is true in a simple random walk without drift, note that for every path that finishes below the level at the end of the period, there is another path where it hit the level and then followed a path that was a mirror of the path that finished below the level. So, for every path that finished below the relevant level (here a 50% drop), there’s another path that touched the level but then reflected and wound up above the level at the end.]

So, assuming MSTR volatility of 90% per annum, the probability of a down 50% intra-day move occurring at least once over the next year is about 12%.

If we use the MSTR volatility implied by the options market of 160%, then down 50% is only 5x daily volatility. The same data as above yields a close-to-close annual probability of about 30%, which we estimate as about a 60% probability of an intra-day drop that would send the ETF to 0.

There are a number of alternative perspectives one could take in trying to estimate this probability: for instance, trying to estimate the probability of a large one-day drop in Bitcoin and how that might impact the MSTR premium to BTC. For example, a 25% one day drop in BTC and a 33% collapse of the MSTR premium would imply a 50% drop in the MSTR share price.

[Kris: This hints at the MSTR premium vs BTC correlation Campbell wrote about]

A more complex analysis might try to estimate whether it is possible for these leveraged ETFs to become large enough that their daily rebalancing trades could themselves drive the price down 50% in one day. For example, imagine that MSTR rapidly triples in price due to some combination of BTC rally and an increase in MSTR’s premium to the BTC it owns, and the assets in the MSTR leveraged ETFs go from $5 billion to $30 billion. The market capitalization of MSTR could be about $270 billion and the leveraged ETFs would be owning $60 billion, or 22%, of MSTR stock outstanding.

Now imagine for some reason, MSTR stock drops 15% during the day – which, given MSTR volatility, would not be unusual. The leveraged ETFs would need to sell $9 billion of MSTR stock at the closing price. Recently, MSTR daily average trading volume at the close of the day has been about $2 billion, so this would be quite an impactful amount of MSTR to sell at the end of the day. For every 1% the price declines further than the 15%, the ETFs will need to sell another $500 million of MSTR, and if that pushes the price down by another 1%…well, you can see this doesn’t have a happy ending for owners of the leveraged ETF or MSTR.

[Kris: see The Gamma of Levered ETFs]

Bottom line, we think there’s a pretty decent probability – somewhere in the range of 15% to 50% – that these 2x leveraged MSTR ETFs are effectively wiped out in any given year if they are not voluntarily deleveraged or otherwise de-risked sooner.

Towards a trade idea

The 2x ETF is MSTU and the 2x inverse ETF is MSTZ. Unless these are delevered, if MSTR [falls/rises] by 50% in one day [MSTU/MSTZ] goes to zero.

I’m going to walk you through my stream of consciousness as I reached the end of the article.

1) I’ll accept Elm Wealth’s logic , my first question is…um, are there options listed on the levered ETFs?!

Checkmark✔️

MSTZ is thin but MSTU has over 350k contracts of OI.

2) We are not in Kansas anymore. The distribution is extremely discontinuous.

On a hellacious down day in BTC coinciding with premium compression (that positive correlation that Saylor has been monetizing being his undoing is the kind of poetry markets like to write) and the telegraphed, reflexive ETF rebalance flows can take MSTU straight to zero XIV style.

And this can happen on any day.

3) So the next thought in the chain was to consider buying 0DTE puts. Like every morning before brushing my teeth.

This is a non-starter for 2 reasons.

i. 0DTEs are not listed on MSTU

ii. ODTEs don’t capture the overnight vol so you don’t own “all” the risk. This is especially important in BTC since it’s a 24 hr market.

4) We’ll come back to the question of expiry. I’m just adhering to the sequence of my thinking for better or worse (feel free to debug my mental compiler).

So what strike do I want? The bet only hinges on a Boolean outcome — did MSTR fall 50% or not?

If the thrust of the trade is so starkly binary then the put I want is lowest strike on the board that you can pay a penny for. I only care about maximum odds. The strike is the payoff, the premium is the outlay. So if I buy the $5.00 put for a penny I get 499-1 odds.

[Since we are thinking in a risk-budgeted binary way rather than in continuous option terms, a parallel framing would be the $5/$0 put spread]

It’s worth noting that this is a bit weird compared to typical investment scenarios. You really only care about the distance of the strike from 0 which determines the payoff and the premium. The price of the stock doesn’t matter because your payoff depends on a certain percent move happening. No matter what nominal price MSTU trades for, if MSTR goes down 50% MSTU gets wiped out.

Let’s start by thinking aloud about constructing a bet and work from concrete to abstract before we bring it back to concrete again.

Buying a 2-week put

Suppose you spend $1,000 2x per month to buy puts that cost $.01.

(Because of the 100x multiplier this translates to 1,000 option contracts)

If you are buying the $2.50 strike, you will get paid $250,000 if MSTU hits zero.

Over the course of the year, following this strategy will cost $24,000 ($1000 twice a month).

Say it hits on the last trial, your net profit is $226,000 (payoff – cumulative outlay). Call it 9-1 odds.

If MSTU has a 10% chance of hitting zero within the year, this is a fair bet. If the probability is higher, you have positive edge, lower you have negative edge.

This a good place to pause for birthday problem math. It allows us to convert into a useful unit of probability per day.

MSTU trades 251 days a year. If we think it’s 10% to hit 0 one of the days we can compute the probability of it NOT hitting zero on any given day like this:

(1 – p²⁵¹) = 10%

p = 99.958%

Converting to odds:

.99958 / ( 1- .99958) ~ 2382

The odds against MSTU hitting zero on a random day is 2382-1.

If there were 0DTEs if you could buy say any strike from $24 or higher for a penny you would have edge to your model probability of “There’s a 10% chance that MSTU hits zero this year”.

Using the daily probability to compute the chance of MSTU going to zero in 10 business days (roughly what a 2-week option encompasses).

1 – .99958¹⁰ = 99.581% or 238-1 odds.

If you can buy a 2-week $3 strike put for a penny you’d have edge to this probability.

We can extend the reasoning above to construct useful tables based on a range of assumptions about the “probability of MSTU going to zero with a year”.

 

Let’s step through an example.

If you believe MSTU has a 20% chance of going to zero in a year, then you need to 56-1 odds on a 1-month put for it to be fairly priced (and assuming the ETF getting zero’d is the only way to win).

[To compute the payoff ratio: strike / (strike – premium)]

If you could buy the 1-month $.57 strike (yes, a very low strike!) for a penny you would get the 56-1 odds.

I started with this whole “what option can I pay a penny for” reasoning because my intuition told me that for a trade like this you will want a strategy that trades an a very near-dated option for a teeny price because that’s probably where you are going to find the best odds in this framework.

But I should not get to anchored to either this penny idea or the notion that the near dated is absolutely the right way to play this.

At this point, it’s time to look at some data to see if:

1) the prices are ever attractive

2) can we narrow down an expiry range

Market prices

The first thing I did was pull up an option chain for the regular monthly expiry — Jan2025. Good news. While the far OTM puts markets are sometimes wide, several strike are 0 bid, offered at $.05 but critically last sale is $.01. There’s someone who sells these things for a penny.

[This is not the case for the puts in the less liquid MSTZ double short ETF. The markets are also much wider. What does this tell you?]

I fetched fitted end-of-day put prices for MSTU options from 9/24/24 to 1/6/25, filtering for all puts below 1.5 delta. FAR out-of-the-money puts (the puts that correspond to the 98.5% call).

• I computed the payoff ratios for all puts less than 1.5 delta by comparing strike and premiums as explained earlier.
• The color coding corresponds to expiry buckets in calendar days (ie 1-10, 11-20, etc)
• I added a penny to all the premiums. So an option fitted to be $0 is marked at $.01

Right away 2 things stand out:

1. The chart has trash scaling because of point #2
2. As expected, you are going to get much better payout ratios on near-dated options. If there’s 2 DTE and the stock is $100, buying the $50 put for a penny offers 4999-1 odds.

So the intuition about the near-dated being better bang for the buck seems correct but the scaling is obscuring the picture and there’s another problem (we’re going to get to it but if you feel up to it, try to guess what it is. One hint is it’s not about transaction costs. That’s important and I’ll say a word about it later as well but that’s not the angle here.)

Let’s fix the scaling. Log base 2 compresses this range nicely and is easy to interpret (every tick mark doubles the payoff).

Ahh, much better. Now we see a smooth descent of payout ratios. To be clear, a y-value of 10 is 2¹⁰ or a payoff ratio of 1024-1. An 11 is 2048-1, and so forth.

Here’s the payoff table reproduced in log base 2 terms:

You can see how the puts suggest the market thinks the probability of MSTU disappearing is somewhere in the 10-15% range (but probably less since you can win on the puts without the ETF zero’ing).

The risk/reward on these near-dated puts is much higher than the deferred puts which is expected since we require much higher odds. Remember if we thought that there’s a 10% chance of MSTU zero’ing in a year, a $10 strike put can trade for $1 (ie 9-1 odds) and be fairly priced. But these short-dated options need to offer much better odds to compensate for a much smaller probability window of MSTU going under.

We need to compare the payoff ratios with the probabilities we inferred from the annual probability (the birthday math) for the stock zero’ing in 1 week, 1 month, etc.

But before that we can address the mystery problem.

By comparing the strike & premiums we can identify if an option is cheap or expensive compared to our model probability but we can’t assess the validity at the strategy level. In other words, we can’t answer whether it’s better to spend $24k on long-dated puts or $2k a month on 1-month puts.

To handicap that we need to adjust our payoff ratios by how often we need to trade so that we can now compare all the strategies on the same measuring stick — “if my annual probability of MSTU zero’ing is X, what’s the best approach”.

So we divide the payoffs by the number of times you must trade per year.

[Used some simple rules, ie for 1 dte, we divide by 251, for 30 dte by 12]

We don’t need to use log scaling for the strategy level chart.

The way to read this is if you think the annual probability is greater than say 20% (see the horizontal dashed lines) than all opportunities above the line are positive expectancy. There’s a lot more opportunity in the nearer-dated confirming the original intuition but every now and again it looks like a 2-3 month put gets fairly cheap.

The median payout ratio normalized to annual odds is 7-1 or 12.5% implied probability of MSTU offing itself.

 

In summary:

1. MSTR is highly volatile
2. If it moves 5-10x it’s daily standard deviation in one day to the downside, MSTU can go to zero.
3. Those size moves historically (via Elm’s article) happen about as often someone rolls a 10 with 2 dice if we say MSTR is 100 vol. If we use it’s implied vol which is more forward looking, we’re it’s more like rolling a 7.
4. The market seems to price the puts in-between those possibilities but we see that the price moves around quite a bit so you can scoop some when they get offered cheap.
5. The more aum MSTU gathers the larger the end of day rebalance trade. Something to keep in mind.

Keeping a close eye on this, perhaps building a monitor around this idea is a nice way to grab a convex outcome. Especially one that I suspect has reflexive properties that are conservatively ignored in this independent events “birthday math”.

Endnote on execution

I assumed $.01 slippage on these options. If you pay $.02 for an option that we computed the payoff based on a penny, you’re getting half the odds. So when talking about really long odds and teeny probabilities and option prices, costs matter. Regardless, you have all the knowledge you need to compare max payoff to your own execution prices to bridge this fully to reality.