how reading moontower makes people feel

In a recent survey, I asked:

Don’t think too hard about this one…in a word or phrase, how does reading Moontower make you feel?

I am heartened by how hany responses are some variation of “curious”. That makes me feel like this is a good place. So I’ll take the encouragement.

I am amused by how many responses were basically “smarter” or “dumber”. Framing is everything, right?

And the one that made me laugh out loud: “Overpaid”

In the emotional zero-sum game, I guess I’m pretty relieved they didn’t say “Underpaid”

You deserve the readers you get. I appreciate you all.

Here’s the full list of responses:

warm
Informed about derivatives
Smarter every time
Listening to a very intelligent and interesting friend
Cozy
Thoughtful
Stimulated
Curious
Stimulated
Interested
smarter
interested
Learned
Good about humanity and less intellectually alone
Good
Growing my knowledge / mental frameworks
Smarter
like you’re smart and i need to catch up
I subscribe to a ton of newsletters from all sorts of topics and enjoy opening my inbox and surfing the serendipity
Groovey
Interested
Challenges conventional wisdom
Well rounded, great articles that cover a range of topics
smart
Like I have a lot to learn about options trading
Smarter
Intelligent
Intimidated; need to get my act together; I’m failing as a parent
Mentally stimulated
Educational
curious
Helps me take a step back, contemplate our human existence with contentment and joy
Dumb. But interested to test things out to understand
smarter
Groovy
educated mostly, positive / optimistic for culture and economic topics
Educational
Always learning
Behind!
Informed or Frustrated
Inquisitive
More knowledgeable
Educated
Little overwhelmed but interested and appreciative of the learning community
Grounded
overpaid
Informed
Smarter
Enlightened
Like I know you from a previous life
Empowered
engaged / reminder of how sharp derivative markets are
Awesome
Dumb
Glad I learned something
engaged
smarter
Included – I have no friends or peers in my niche
Curious
Smart
Good
Willing to learn
Like a dinosaur
Cool, like having a beer or coffee with a chill and knowledgeable person
Engaged
lazy
Smarter
Engaged
Energized
Today I learnt
Informed
Good shit
Makes me think better
Like I have a mountain to climb
That I too can be a stoner dad one day
positive
Alright alright alright
challenged
Enlightening, a little wistful
I know I am going to learn something when I open it
Informed
Smart
Interested
Undereducated, sort of envious of anyone this smart
Less dumb
More curious to deep dive
Excited but also depressed by how easily this writing comes to you
Feels like learning from a school friend who understands the subject better
Eager to learn more
Understood / Inspired
Informed
go in circles (in a good way)
More informed, new perspective
Enlightened
smarter
Curious
Indifferent
Like speaking to a good friend with shared interests
Dumb
Curious
Like I’m discovering interesting ideas
Fucking stoked to have the privilege
immature
Refreshing
More confident
Learning at my own pace
Wiser
Inspired – advice the younger generation should hear
Gut check to optimize something
Informed, sharper trader
Hope for folks who moved past their primary vocation
Informed
Stimulated
Insightful
Sparks my thinking
Enlightened
curious
Similar journey
Like I should sell my software company and trade options
Inspired
behind =)))
Wavy
Smarter
smarter
Sometimes it’s too long
Not as smart as I wish I were
Open minded
curious
exhausted
Time for some brain pain
Too much to learn
Joyful
curious
Hopeful
Invigorated
smarter
Self aware
obsolete
Enlightened
Informed
Tickles the curiosity
Smarter
Needs better organization
Smarter
Inadequate, need to know more about options
Better
Math is not my strong point
Curious
Inspired to think harder
Challenged intellectually
Intellectually invigorated
More confident investor
Curious
Less intelligent
Smarter
Like talking to a nerd friend
Down to earth
Challenged with a smile
Usually makes me think
Happiness
Underachieving
Dumb
Thoughtful
Stimulated
More educated
Food for thought
inspired
Determined
Inspired – “I wanna be like this guy someday”
Insightful and introspective
Overwhelmed in a good way
Need to mentally prepare to read
Interested
Introspective
Interested
smarter
Mentally draining (in a good way)
Cognitive
Performance engineering disguised as prose
Informed
Engaged
Introduced to a very intelligent friend
Happy but frustrated by complexity
Mentally taxed – part of the appeal
Smarter, with more to go
Elucidated
It makes me think
Stupid
better
Learned
smart
Inspired
Smarter
Empowered
wiser
Motivating
Curious
Inspired
Happy
connected
Smart
Not as smart as this guy
Makes me pause and think
Seeing through someone else’s lens
Inept
Great but complex
Productive and intellectually satiated
Opened my eyes to new parts of options
Deep dive in trading
Alive
Grounded in reality
Like talking to the cool uncle I want to be like
Interested
Step-by-step understanding of options
Curious
thoughtful / reflective
learning how to adult
Productive
Like I have a lot to learn and you’re the goat
Part of a club
Curious
motivated
Motivated
educated
Interested
Sane
Curious
Analytical
Validated as a burnt out market maker
resonance
older brother
Informed
Inspired to do more
smart
Chance to learn something
Respected
Calm; in control
Enriching
Interested
Real – unique voice
Comfort food for the brain
learning
Wish I traded options more
Satisfied
Like a student
Informed and educated
Hopeful for new ways of thinking
Interested
validated
curious
Smart
informed
Smart and bold
20 years late
Interesting
connected, familiar, curious
Smart
Continuing lifelong learning
Smarter
Curious
Conscious
informed
Better understanding of options while knowing little
Better informed
Trading with a soul
smarter
Stupid
Curious
Thought provoked
Know nothing about markets
Upbeat, positive, enriching
Inquisitive
Quirky
Insight without too much math
Educated
Interested
Inquisitive
Improving my skill set
Curious
Enlightened curiosity

 

 

the middle class is no place to stay

If you are on X or just do a lot of investment-reading online, you are quite aware of the “$140k of income is the new poverty line discourse”.

Mike Green kicked it off with probably the most viral Substack post of the year:

My life is a lie: How a Broken Benchmark Quietly Broke America (Yes, I Give A Fig)

Of course, there are critical counters:

Criticisms notwithstanding, the article struck a nerve, so naturally finds thousands of supporters. I link to Adam Butler a lot, so here’s an example of his:

Marking the Household to Market: Why Mike Green’s $140k Benchmark May Be Conservative

And the article has been picked up and commented on by major media outlets as well. Nothing attracts a crowd like a crowd.

I thought the article was good, even if I reject the blunt claim that $140k of income is the poverty line. The article is trying to grapple with the American malaise from an economic and cost-of-living point of view. It’s hard to disentangle economic opportunity from any discussion of culture. Chris Arnade has a uniquely interesting voice on these issues, given that he’s been literally doing the footwork for over a decade on the ground (see Why are Americans Unhappy?). Green himself has pointed to Peter McCormack’s Nick Fuentes vs The World – No Country for Old Rules which centers more on the theme of a broken promise between generations. I personally find Kyla’s writing in 2025 to be exceptional with respect to diagnosing the Gen Z condition. See her recent post Everyone is gambling and nobody is happy.

I got nothing rigorous or data-driven to add. This is no obstacle to shooting from the hip. Let’s make it personal.

Immediately after reading Green’s article, I texted my mother.

The median household income in the US in 1990 was $29k or about $70k in 2025 dollar (source DQYDJ). It was about 30% higher in New Jersey. Our household income was $40k. Solidly middle class.

What did that mean?

  • We had 1 car, a 1980s Dodge Lancer that we bought used. Brand new, they were $12k.
  • Childcare? That’s called unpaid grandparents. When they couldn’t be there, we did have a babysitter who lived across the street who would come help my younger sister get ready for school when I was 12 (and make sure I actually woke up). She came by for about 30 minutes at a $5/hr rate. We were classic latchkey kids. Dinner time was the first occasion to see my mother each day.
  • My mother deeply valued education, but we were not in a good school district. So my sister and I went to 12 years of Catholic school. My high school was a La Salle high school, Christian Brothers Academy in Lincroft, NJ. “CBA” in the text. CBA allowed me to attend one of my years for free because my mom pleaded that she didn’t have the money. Tuition was a massive burden BUT we were able to do it and still eat.
  • We had lots of gifts under the Christmas tree every year. How? Credit card debt. But we were given nothing outside of Christmas and our birthdays, so this was prioritized. Not easy, but a choice nonetheless.
  • We went on one vacation a year. In the early 80s, it was always Wildwood, NJ or Niagara Falls but by the late 80’s we went to Florida. I even got a boogie board and we were allowed to get any junk food we wanted from the supermarket on vacation. We still didn’t really eat out, we’d stay in time-share hotels where mom could cook or we’d eat sandwiches. We went to Disneyworld, SeaWorld, MGM Studios. I saw Batman (the Michael Keaton one) in a theater in Daytona Beach. It was our favorite 2 weeks of the year.

Money issues were a cloud over my whole childhood growing up. Still, I knew I was safe and I knew I wasn’t exactly poor. If we didn’t eat our food our elders would tell us to think of the “poor” so we knew there were levels to this. Plus, I saw the Sally Struthers commercials with the African kids who don’t even swat the flies from their eyelids. It took me 3 years of begging to finally get the bike I wanted, but I eventually got a bike.

In general, I knew that it would be selfish to ask for things because we were constantly reminded that money was scarce. In Catholic school, I felt like I had less than the other kids on average, although I’m sure there were some kids in the same situation as me. As I got older, my awareness of this grew. The bar to think someone else was rich was low. Did they have a pool, even above-ground? A second car? Did they get Skidz or Cavariccis when they were popular or long after they went out of style?

It’s extremely clear what my mother’s financial algorithm was. Budget ruthlessly while prioritizing what she thought were a few must-haves:

  1. education
  2. vacation at the beach (this is universal across my family and I wonder if there’s an Egyptian undercurrent to it)
  3. And specifically in my immediate family, Christmas gifts. I had a sense not only that my mother loved us, but I think she wanted us to have a concentrated moment of joy, even if it was once a year. She was hard on us, at least compared to how we saw “the American kids” get raise,d but Christmas always felt a bit extra. Like she was saying, “I know it’s hard around here most of the time, but life is supposed to have joy. You’re not brats and I see you. This is our little deal. Bear with me the rest of the year and I’ll make it up to you.”

The point is, being middle-class is hard. You cover basic needs and triage just a few wants. There’s very little slack. If mom loses her job, does the credit card debt bury us? Look, my parents split. Mom gave dad what little money she could afford out of sympathy. He had nothing, his single-livery-car business going bankrupt a few years earlier than he went bankrupt personally after a short time on his own. It’s all so precarious.

But it is obviously precarious. So much so that you burn with desire to escape it. By the time I was in high school, I knew I wasn’t going to live like this when I got older. I’ll do whatever pays because I know this sucks as a state-of-being, even if it’s not destitution. This is my hot take — there was nothing ever comfortable about being middle-class. It’s not supposed to be the basin of an attractor curve. It’s a transition to a better life or moment during a freefall. It sucks to be middle-class according to these articles. Well, guess what? It sucked 40 years ago, too. And if you are in it, the only thing you should be doing is trying to escape it.

The literati, finance-footed, and cultural observers go on about the plight of the middle-class — causality, diagnosis, comparing what life is like for those in the middle-class. That’s fine in some academic, sense-making context. Maybe it’ll even affect policy. But this is all you need to know — being middle-class sucks. It will always suck. And all the discussion about it is under some guise that if we [insert policy] it won’t suck OR to make you feel that the plight is exaggerated. The first is lie and the second is patronizing.

Poor people don’t need to be told it doesn’t suck as much as they think. Poor people don’t think there’s some intervention that will make being poor suddenly acceptable. They just want to be unpoor. It would be adaptive to adopt that mindset if you are one rung away from being poor, too.

Am I being harsh? I think I’m just being realistic, but I know it sounds harsh. My perspective is meant to be individually pragmatic because there’s never going to be rest for the middle-class as a cohort. As pointless as they feel, I have my sympathies. When I was growing up, I believed that through education, I was going to escape. And with reasonable odds. Like being top 5% in my class was a sure ticket to a better life. Not easy, but an amazing payoff reliably predicated on effort and persistence.

Today, there is a profound sense that you can do “all the right things” and that only earns a ticket to a capricious, opaque lottery. College application stress in 2025 is societally pathological. Meanwhile, on the backside, new grads face cloudy prospects and high living costs.

Being middle-class sucks. To be stuck in it without a legible path out is but a dormant revolt. “Do everything to be in the top 5% and escape” vs “do everything to be in the top 5% to be allowed to enter a lottery with a 5% hit rate” is a giant deterioration of the American bargain in just a single generation.

To wrap up, remember, it may sound self-contradicting because the middle-class is defined by centrality and encompassing the masses, but it’s not a place to stay. The resolution of the statement is dead simple — being average stinks. It always has and it always will. You might find solace in the fact that being average here is better than being average elsewhere. But you’ll probably stay average if that’s of any comfort. The world is indifferent to an American’s complaints. The human condition reminds us that it is a luxury to be heard. Accept that and act accordingly.

Moontower #296

Friends,

If you are on X or just do a lot of investment-reading online, you are quite aware of the “$140k of income is the new poverty line discourse”.

Mike Green kicked it off with probably the most viral Substack post of the year:

My life is a lie: How a Broken Benchmark Quietly Broke America (Yes, I Give A Fig)

Of course, there are critical counters:

Criticisms notwithstanding, the article struck a nerve, so naturally finds thousands of supporters. I link to Adam Butler a lot, so here’s an example of his:

Marking the Household to Market: Why Mike Green’s $140k Benchmark May Be Conservative

And the article has been picked up and commented on by major media outlets as well. Nothing attracts a crowd like a crowd.

I thought the article was good, even if I reject the blunt claim that $140k of income is the poverty line. The article is trying to grapple with the American malaise from an economic and cost-of-living point of view. It’s hard to disentangle economic opportunity from any discussion of culture. Chris Arnade has a uniquely interesting voice on these issues, given that he’s been literally doing the footwork for over a decade on the ground (see Why are Americans Unhappy?). Green himself has pointed to Peter McCormack’s Nick Fuentes vs The World – No Country for Old Rules which centers more on the theme of a broken promise between generations. I personally find Kyla’s writing in 2025 to be exceptional with respect to diagnosing the Gen Z condition. See her recent post Everyone is gambling and nobody is happy.

I got nothing rigorous or data-driven to add. This is no obstacle to shooting from the hip. Let’s make it personal.

Immediately after reading Green’s article, I texted my mother.

The median household income in the US in 1990 was $29k or about $70k in 2025 dollar (source DQYDJ). It was about 30% higher in New Jersey. Our household income was $40k. Solidly middle class.

What did that mean?

  • We had 1 car, a 1980s Dodge Lancer that we bought used. Brand new, they were $12k.
  • Childcare? That’s called unpaid grandparents. When they couldn’t be there, we did have a babysitter who lived across the street who would come help my younger sister get ready for school when I was 12 (and make sure I actually woke up). She came by for about 30 minutes at a $5/hr rate. We were classic latchkey kids. Dinner time was the first occasion to see my mother each day.
  • My mother deeply valued education, but we were not in a good school district. So my sister and I went to 12 years of Catholic school. My high school was a La Salle high school, Christian Brothers Academy in Lincroft, NJ. “CBA” in the text. CBA allowed me to attend one of my years for free because my mom pleaded that she didn’t have the money. Tuition was a massive burden BUT we were able to do it and still eat.
  • We had lots of gifts under the Christmas tree every year. How? Credit card debt. But we were given nothing outside of Christmas and our birthdays, so this was prioritized. Not easy, but a choice nonetheless.
  • We went on one vacation a year. In the early 80s, it was always Wildwood, NJ or Niagara Falls but by the late 80’s we went to Florida. I even got a boogie board and we were allowed to get any junk food we wanted from the supermarket on vacation. We still didn’t really eat out, we’d stay in time-share hotels where mom could cook or we’d eat sandwiches. We went to Disneyworld, SeaWorld, MGM Studios. I saw Batman (the Michael Keaton one) in a theater in Daytona Beach. It was our favorite 2 weeks of the year.

Money issues were a cloud over my whole childhood growing up. Still, I knew I was safe and I knew I wasn’t exactly poor. If we didn’t eat our food our elders would tell us to think of the “poor” so we knew there were levels to this. Plus, I saw the Sally Struthers commercials with the African kids who don’t even swat the flies from their eyelids. It took me 3 years of begging to finally get the bike I wanted, but I eventually got a bike.

In general, I knew that it would be selfish to ask for things because we were constantly reminded that money was scarce. In Catholic school, I felt like I had less than the other kids on average, although I’m sure there were some kids in the same situation as me. As I got older, my awareness of this grew. The bar to think someone else was rich was low. Did they have a pool, even above-ground? A second car? Did they get Skidz or Cavariccis when they were popular or long after they went out of style?

It’s extremely clear what my mother’s financial algorithm was. Budget ruthlessly while prioritizing what she thought were a few must-haves:

  1. education
  2. vacation at the beach (this is universal across my family and I wonder if there’s an Egyptian undercurrent to it)
  3. And specifically in my immediate family, Christmas gifts. I had a sense not only that my mother loved us, but I think she wanted us to have a concentrated moment of joy, even if it was once a year. She was hard on us, at least compared to how we saw “the American kids” get raise,d but Christmas always felt a bit extra. Like she was saying, “I know it’s hard around here most of the time, but life is supposed to have joy. You’re not brats and I see you. This is our little deal. Bear with me the rest of the year and I’ll make it up to you.”

The point is, being middle-class is hard. You cover basic needs and triage just a few wants. There’s very little slack. If mom loses her job, does the credit card debt bury us? Look, my parents split. Mom gave dad what little money she could afford out of sympathy. He had nothing, his single-livery-car business going bankrupt a few years earlier than he went bankrupt personally after a short time on his own. It’s all so precarious.

But it is obviously precarious. So much so that you burn with desire to escape it. By the time I was in high school, I knew I wasn’t going to live like this when I got older. I’ll do whatever pays because I know this sucks as a state-of-being, even if it’s not destitution. This is my hot take — there was nothing ever comfortable about being middle-class. It’s not supposed to be the basin of an attractor curve. It’s a transition to a better life or moment during a freefall. It sucks to be middle-class according to these articles. Well, guess what? It sucked 40 years ago, too. And if you are in it, the only thing you should be doing is trying to escape it.

The literati, finance-footed, and cultural observers go on about the plight of the middle-class — causality, diagnosis, comparing what life is like for those in the middle-class. That’s fine in some academic, sense-making context. Maybe it’ll even affect policy. But this is all you need to know — being middle-class sucks. It will always suck. And all the discussion about it is under some guise that if we [insert policy] it won’t suck OR to make you feel that the plight is exaggerated. The first is lie and the second is patronizing.

Poor people don’t need to be told it doesn’t suck as much as they think. Poor people don’t think there’s some intervention that will make being poor suddenly acceptable. They just want to be unpoor. It would be adaptive to adopt that mindset if you are one rung away from being poor, too.

Am I being harsh? I think I’m just being realistic, but I know it sounds harsh. My perspective is meant to be individually pragmatic because there’s never going to be rest for the middle-class as a cohort. As pointless as they feel, I have my sympathies. When I was growing up, I believed that through education, I was going to escape. And with reasonable odds. Like being top 5% in my class was a sure ticket to a better life. Not easy, but an amazing payoff reliably predicated on effort and persistence.

Today, there is a profound sense that you can do “all the right things” and that only earns a ticket to a capricious, opaque lottery. College application stress in 2025 is societally pathological. Meanwhile, on the backside, new grads face cloudy prospects and high living costs.

Being middle-class sucks. To be stuck in it without a legible path out is but a dormant revolt. “Do everything to be in the top 5% and escape” vs “do everything to be in the top 5% to be allowed to enter a lottery with a 5% hit rate” is a giant deterioration of the American bargain in just a single generation.

To wrap up, remember, it may sound self-contradicting because the middle-class is defined by centrality and encompassing the masses, but it’s not a place to stay. The resolution of the statement is dead simple — being average stinks. It always has and it always will. You might find solace in the fact that being average here is better than being average elsewhere. But you’ll probably stay average if that’s of any comfort. The world is indifferent to an American’s complaints. The human condition reminds us that it is a luxury to be heard. Accept that and act accordingly.

Money Angle

I keep adding calculators and tools for your use and education here.

This week, you’ll find a new American Options Early Exercise Calculator. It works for both calls (for dividends) and puts (for interest). Education is embedded in the documentation.

Ethan Kho host of the terrific Odds On Open podcast published our chat. The YouTube comments give high praise, but one of them accuses me of being drunk. I remember being on only a few hours of sleep that day, but bruh, drunk? C’mon. My fault for reading the comments.

Money Angle For Masochists

I also whipped up this calculator in response to a question I am asked often — how do you weight the legs of an option trade? Again, the education is embedded in the documentation/links.

Option Pair Trade Calculator

Benn Eifert’s Options Threads

@quant_spence compiled a document of all of Benn’s threads and interview snippets with respect to options. If you don’t know who Benn is, you are in the wrong section of this newsletter.

Benn’s Option Thread of Threads

Coastline

I’m boosting Thursday’s article once more because even if you aren’t interested in options but just returns and volatility, it will get your gears turning, which is always good for some inspiration on how to think about risk, sizing, and measurement broadly. Plus who doesn’t like a good ol “how do fractals relate to markets” section.

The Coastline Paradox in Financial Markets

Webinar

Quant Insider is hosting a live webinar this morning:

Systematic Mean Reversion & Cointegration: From Statistical Tests to Trade Execution

Moontower readers get 50% off with this coupon.

 

 

Stay groovy

☮️

Moontower Weekly Recap

Posts:

Moontower Binomial Tree Explainer

Last week, in American options are not vanilla, we covered not only the concept of early exercise for American options, but rules for “optimal” early exercise.

If you want this broken down in video form, I direct you to Sheldon Natenberg’s explainer in CBOE’s educational series:

📽️ Early Exercise of American Options (CBOE, video lesson)

Today we’ll not only get into the common model used to price American-style options (you can use them for European-style as well, while Black-Scholes only works for European), but you can get hands-on to see how they work.

Just to tie a bow on last week’s post and not give you a false impression that early-exercise rules are dry calculations, here’s a shower thought I had laid out in a progression:

  1. American reversal/conversion values are lower than European R/C because of early exercise. Basically, the expected value of T is smaller than T itself. “DTE” in the world of American-style options is not deterministic.
  2. The spread between the Euro vs American R/C is a function of interest rate volatility. But I’ve never seen the spread directly modeled because the R in option models is taken as constant and “managed” at the level of portfolio Greeks [and general judgement].
  3. I heard a few years ago a big MM took a hickey on early exercise mispricing during the 2022 rate hikes. That feels like a clue. I’m guessing they overvalued calls/undervalued puts because their R/Cs turned out to be too high. Amercian R/C values turned out to be much lower than the Euro values as they were assigned. In other words, the spread, which represents the value of the option to exercise early, was greater than expected.
  4. I don’t know the details of what happened at that MM but I’m just guessing. If anyone wants to enlighten me you know how to reach me. I’m purely curious.
  5. If this didn’t make sense, but you want it to, you like to be nerdsniped, which I appreciate. But this is definitely not something to be practically concerned about.

Before we go on to the tree models, how’s this for an oblique, albeit grim, option play via Darkfire Capital LLC:

The survivor option:

Ok, here’s your free money of the day tweet – on your deathbed, instruct the trustee of your trust to buy as many brokered CD’s with the lowest coupon/longest maturities possible.

Once the death certificate is issued, forward it to broker and have them exercise the survivor’s option – bang, that CD priced at 88 is now par.

Laugh heartily from your coffin.

 

Tree models

Natenburg tells us that tree models are easier to grasp than Black–Scholes and can price both European and American options. He explains that the Cox–Ross–Rubinstein (CRR) binomial model remains one of the most popular implementations of trees to this day.

They work by pricing options just before expiry then working backward to today. At each node you ask: exercise now or wait?

Another SIG bootcamp exercise was to build these in Excel from scratch.

I used an LLM to help me code up both a tutorial and simulator so you can learn this stuff without signing a non-compete 🙂

Step 1: Build the Stock Price Tree (forward in time)

💡CRR Parameters — where they come from

I’ve written a step-by-step explainer of the risk-neutral probability formula if you want to build up from intuition to math:

📐The General Formula to Back Out The Risk-Neutral Probability (moontower)

Step 2: Backward induction (the magic)

For European options, you skip the max with intrinsic (no early exercise), using only the Hold value.

A word on convergence

The binomial tree is a discrete way to approximate continuous price movements. As you increase the number of steps:

  • Each time slice gets smaller.
  • The tree gains more branches, resembling a smooth diffusion.
  • The option price converges toward the “true” theoretical value.

Claude’s “rule of thumb” shows diminishing returns since you’re doing roughly 10× more computation for that extra 0.9% improvement:

  • 100 steps ≈ 99% accuracy
  • 1,000 steps ≈ 99.9% accuracy

Get your hands dirty

🌲Moontower Cox-Rubinstein Binomial Tree Lab: A self-explanatory demo

The green nodes represent early-exercise candidate conditions. A nice way to explore the tool is to see where the clustering occurs based on the inputs to build your understanding of what makes an option more or less likely to be exercised early.

 

🖥️Black-Scholes and Cox-Rubinstein side-by-side calculator

 

American options are not vanilla

I’ve always found it amusing that the most commonly traded options, American-style equity options contracts, are considered “vanilla”. Because they can be exercised early, their valuation is an instance of a famously difficult problem — explore/exploit. The only reason I might refer to them as “vanilla” is not for them being simple, but simply common.

The last time you bought a TV you encountered this problem — do I keep researching or pull the trigger? From shopping or channel-surfing to giant commitments like who to settle down with or businesses to pursue, the problem sits at the heart of decision-making across all domains though it tends to be more formally considered in fields like computer science, finance, game theory, and operations engineering.

💡See my notes from Brian Christian talking about Algorithms to Live By to see how explore/exploit is seen in everything from “bandit” problems to child development and learning strategies.

The Black-Scholes formula is elegant. It’s a closed-form equation that you can implement in a common financial calculator. As trainees, we programmed it into our bootcamp standard-issue 12C:

But Black-Scholes doesn’t work for American-style options.

💡If you need accessible, non-formal refreshers on Black-Scholes, see:

The equation is a factory — it takes in raw material and squeezes out a hot dog on the other side. But it has no visibility on the path between input and output. But that path is key to the core question:

What is the optimal stopping time of an American-style option?

When should we exercise the right to “stop” the option early?

While American options let you exercise anytime before expiration, you usually shouldn’t. The value of optionality (your right to wait) is typically greater than the small benefit of early exercise.

This discussion is not only useful but fun since we invoke microeconomics in real-life.

It starts with 2 key questions.

WHY would you exercise early?

  • Puts: to collect interest sooner on the short stock position.
  • Calls: to capture a dividend

WHEN is it worth it to exercise early?

We check 2 tests in sequence:

a) Is the benefit worth more than the optionality you give up?
Compare the gain from early exercise to the value of the out-of-the-money option at the same strike.

Examples:

  • Put: If a stock is $100 and you own the 120-put, you can exercise the right to sell/short the stock at $120. Is the interest you earn on the $120 until expiry worth more than the 120-call, which you are effectively selling at 0?
  • Call: Let’s say this $100 stock pays a $1 dividend and you own the 80-call. Is the 80-put, which you are giving up, worth less than the dividend?

The cost of the exercise (the time value of the option) vs its benefit (interest or dividend) is just the first step. But now you need to zero in on the when.

This is where we have to go to “marginal” cost/benefit. In other words…

b) Is one day’s cost of waiting (theta) greater than one day’s benefit (interest or dividend)?

When daily theta decay exceeds daily interest/dividend gain

Let’s get more concrete.

Stepping through the tests

We start with assumptions.

✔️Stock price = $100
✔️DTE = 60
✔️RFR (the rate you earn on cash in your account) = 5%
✔️Implied volatility = 20%
✔️ No dividends

We will analyze the 105 strike put.

I used a Black-Scholes European style calculator to compute the option values. You’re supposed to use an American calculator, but since I’m trying to explain the exercise rules, that would mask some exposition. Pointing out the European calculator’s mistakes will be better for learning.

Ok, we start with this table of our option values for each day until expiry.

Column explanations:

  • Put theo: Black-Scholes value for the 105 put given our assumptions
  • Call theo: Black-Scholes value for the 105 call given our assumptions
  • Total interest: interest you’d collect until expiry if you exercised the right to sell shares at $105. Computed as 105e^(.05 * DTE/365) – 105
  • theta from the option model
  • 1 day’s interest computed as 105e^(.05 * 1/365) – 105
  • Test 1: value of the call – total interest. This is a blunt total comparison of the put I own’s time value (represented by the call) vs the interest I’m forgoing by not exercising
  • Test 2: 1 day interest that I forgo vs 1 day optionality represented by theta. This represents the marginal comparison of the interest vs optionality for 1 day.

At 12 DTE, the European model is telling us that the 105 put is worth LESS than its intrinsic value of $5. That’s a clue!

That’s the point at which the time value of the put (ie represented by the call on the same strike…remember put/call parity means the call is “in” the put) is LESS than the interest you’d earn if you exercised early.

💡The European put can and will trade under intrinsic. The American-style option should not trade less than $5 because if it did, you would simply buy the put, buy the stock, exercise immediately and have a risk-less profit of the amount it traded under intrinsic. So if for some reason you could buy the American-style 105 put for $4.92, you’d buy the stock for $100, then exercise the put which allows you to sell the stock at $105. Between the stock and put you’ve laid out $104.92 but your proceeds from the sale are $105. You pocket $.08 with no risk.

At 12 DTE, the 105 put has passed Test 1 (interest > option value). Visually, you can see this on the upper chart. But look at the lower chart…

The lower chart is the visual of Test 2 (1-day interest exceeding 1-day optionality).

5 DTE is the point of optimal early exercise.

Let’s do this again a bit faster with the 108 strike.

Since this put is deeper ITM, the corresponding 108 call is headed to zero earlier and the interest one earns on $108 is a bit more than the interest on $105 so without working through any math we expect to exercise the 108 put sooner than 5 DTE.

For the 108 strike, Test 1 (total interest exceeds the call value) is satisfied with 55 days until expiry, but the optimal exercise point isn’t until 20 DTE.

Interest “rent” is constant; theta is NOT

The daily interest benefit is linear. It’s basically total interest divided by DTE. But you may have noticed the theta values are curved. It’s intuitive that theta for an ATM option increases in an accelerating way until expiry. After all, with 1 DTE, the ATM option is entirely extrinsic value and you know in 24 hours the time value goes to zero.

For the 108 call, it has no value several days before expiry so there’s nothing to decay. The option went through the steepest part of its depreciation days earlier. Test 2 depends on when 1d interest and 1 theta “cross”.

For reference, this is a visual of theta vs DTE for strikes of various moneyness. Remember, the stock is fixed at $100. The closer the option is to ATM, the later it experiences its steepest decay. With a week to go, the far OTM 109 call has no decay. It’s already worthless.

Optimal exercise of American calls

You exercise calls early to capture a dividend. You must be the shareholder of record on the “record date” to be entitled to the dividend. When the stock goes “ex-dividend” it means any holders of the stock are NOT entitled to the dividend.

Owning a call option does not give you rights to the dividend since you are not a shareholder. That’s why the cost-of-carry component of option pricing discounts calls by the amount of the dividend.

When a stock goes “ex”, meaning the dividend has been paid out, the shares fall by the amount of the dividend which makes sense — the balance sheet has shed X dollars per share of cash.

The owner of the call will experience the drop in share price without any dividend receipts to make up for it.

💡If a $100 stock pays a $1 dividend and the shares open at $100 your brokerage or data vendor will say the stock is up $1 on the day. Unchanged would mean the stock should open at $99. If you own a dividend-paying stock it’s not “extra” return. The company just chiseled off a piece of its value and gave it to you in cash. It’s economically a wash. If they didn’t pay you the cash the company would retain it, the enterprise value would be unchanged and your return is the same. Your cash flow is different but of course you could have sold 1% of your shares to the same effect. In fact, that’s more tax-efficient. Of course, these are all first-order mechanical considerations. The properties of companies that pay or don’t pay dividends is a separate point of debate. If you do not believe stocks fall by the amount of the dividend, meet me at the corner of Trinity and Rector. I’ll be in a trenchcoat with a suitcase of Euro-style call options to sell you on a lovely selection of fat dividend yield aristocrats.

The optimal exercise of ITM American calls is easier. Test 1 is simple. Is the dividend I’m receiving greater than the value of the OTM put I’m giving up? [Again the put value tells me the time value or optionality of the ITM call I actually own]

What about the optimal timing of the exercise?

The marginal thinking represented by Test 2 is straightforward. The benefit of exercising the call early is discrete — it’s a dividend on a specified date. If the dividend is worth more to me than the time value of the call, I shouldn’t give up the time value until the last moment I have to capture the benefit. I should exercise on the day before the stock goes ex-dividend so I’m the shareholder of record.

Real-world considerations

  • Early exercise decisions are directly dependent on interest rates (for puts), dividend amounts (for calls) and volatility (which influences the optionality you are surrendering when you exercise the option). Just think of the benefit you receive vs what you are giving up and what influences those quantities.
  • Stock settles T+1. If you exercise a put on Thursday, your short share proceeds from the stock hit your account Friday, meaning you collect interest over the weekend. When I started in trading, stock settlement was T+3, so Tuesdays were “put day”. That’s the day you’d exercise to capture interest over the weekend. From 2017 to 2024, “put day” was Wednesday as the standard settlement was T+2. Microstructure nerds might be aware of a famous pick-off trade in the early aughts where a SIG alum bought shares from a NYSE specialist requesting T+1 settlement knowing that the company was going to pay a giant special dividend the next day. This ended up being very expensive to the seller. And eventually, to the buyer as this maneuver landed them in court. The option world is littered with dividend shenanigans. The range of ethical codes is wide and can certainly extend to “a moral obligation to relieve dumber people of their money” or “legal fees are part of my expected value calculation”. Having spent time in the trading world, I’m not surprised to when I notice these familiar moralities in tech, but a distinction in trading is pro vs pro violence was ok, ripping off customers was killing the golden goose.
  • Sometimes companies announce a large dividend suddenly that the exchange will treat as special. Strike prices will be revised lower to account for the special dividend keeping the economic impact on options unchanged. That said, incremental changes in dividend policy are risks to option holders. Increased dividends lowers calls/raises puts all else equal.

We have an option calculator that allows you to compare the “early exercise premium” of American to European options:

https://www.moontower.ai/tools-and-games/option-pricing-calculator

Holiday coziness = backgammon

I taught both my boys backgammon this past week. This is the younger guy playing with mom.

I used to play a lot with my colleagues during the year I was a broker on the NYSE floor. Every time I came back to the “booth” (that’s the area with the phones on the sides of the exchange floor where clerks take orders from customers), I’d make a move on the Jellyfish software. When I was at Parallax, I worked with 2 of the best backgammon players in the world (one of them was the best in the world iirc) and Parallax was actually founded by Roger Low, who was a world-class player himself. Roger had retired by the time I joined so I never got to know him.

Backgammon is especially neat if you like options because the doubling cube begs you to price volatility.

💡Fun fact

When I had my phone interview with SIG (this is the round that happens in between campus interview and the final on-site) I was asked a basic question regarding the cube. Since accepting the cube doubles the stakes of the game and rejecting it forfeits the current value of the game, what is the minimum probability of winning you need to accept it?

When looking at the board you need to evaluate how risky the position is. Like how likely is the tide to turn? When you offer the cube you are selling your opponent an option and its value depends on their chance to come back. But if you offer it to them when they have no chance they will reject it and be bailed out because you have now sacrificed your own possibility of getting a gammon (double points) or backgammon (triple points) if you beat them by a large margin.

It’s a great game for kids because they get to practice dice math. I was playing last night and knew I’d win as long as my next roll wasn’t a [1,1], [1,2], [2,1], [3,3]. So the kids would first need to figure out that’s what they are rooting for and then I ask them the probability. 4 out of 36 or 1/9. Plus the little guy figured out that the 7 is the most common roll and how to count the number of ways to get each number (he was delighted by the pyramid pattern when he realized it).

We always play more games during the cozy holiday season and I’m irrationally pleased that this season it’s backgammon. I recommend it.

[Also, if you are just learning, play a bot on your phone. You learn the game very quickly getting trounced by bots. Ultimately, I’m just a casual player, I never studied it, but would just play for $5 a game with my NYSE squad. Of course, with the doubling cube and possibility of gammons that can multiply pretty quickly. The real hitters play for four and five figures per game.]

“markets vs democracies” in the wild

On Nov 26th, Imran asked his followers would what more likely to double in the next year — gold or BTC?

I looked at the result of the poll just a few hours after it was posted and it was BTC 52% to gold 48%.

By the time the poll closed with 750 votes, BTC had garnered 2/3 of the votes.

I don’t know if me a jerk had anything to do with this but when I saw that the vote was almost a coin flip I chimed in.

Focus on the last part.

The poll should be nowhere near 50/50 because you would be able to lock in a great trade by selling gold in this proposition, buying gold call spreads financed by even more expensive BTC call spreads.

This is a classic difference between markets and democracy. It’s a perfect example of the Dinosaur Markets post in real life. The markets in the options reflect the volatility and the cost of replicating these bets. Money-weighted votes are interested in the truth where opinions are cheap as sand.

It’s very difficult to have opinions that are above replacement value about liquid assets. If you’re truly good at this, then being Scrooge McDuck rich based on consistently betting on these fantastic opinions is the only proof of such a skill. Few people are rich because of a crystal ball.

A good way to make a living in finance is to find the people that voted for gold in this poll and offer to trade with them. You need to do this in the dark because if you tried to do it on a public exchange, you’d be undercut by traders competing to sell the gold proposition to these opinionated people and it would drive the price down to a non-arbitrageable price.

Public markets protect overconfident people with arbitrageable opinions from their own ignorance and stupidity. Private or non-transparent markets are nice ways to shove a vacuum hose into their bank account.

There are many places where there’s alpha in projecting your opinion. This is the stuff you spend your time on in life. Where you have self-knowledge, private info, competitive advantage, skills, taste and so on.

But when it comes to markets, remember what we learned here just a few weeks ago:

the arbitrage reflex is more profitable than the opinion reflex

Moontower #295

Friends,

First of all, here’s the slides I shared in this vid:

I hope that video was helpful. I watched it again because I think the topic is complicated, not technically, but because it’s counterintuitive. Trades like that were bread-and-butter through my pro career (and very much why I “spotted” it despite it not sticking out in the tools other than the Trade Ideas filter which is highly validating of our algorithm! I’ll allow myself to be happy about my own work for 1 second. Ok moment’s passed, let’s continue.)

Before we head on to Money Angle stuff I’ll share 2 things I enjoyed.

Fermat’s Last Theorem Documentary (link)

This 45-minute video follows the story of Andrew Wiles’ quest to prove Fermat’s Last Theorem. I didn’t know the history of the theorem or even what it stated. It claims that for any whole number exponent other than 2 there are no values of A,B,C that would satisfy Aˣ + Bˣ = Cˣ. You can’t prove this computationally because there’s an infinite number of possibilities. The proof requires mathematical logic.

Fermat allegedly had a proof but hadn’t written it down. Turns out Wiles’ proof would have been impossible for Fermat to have known (this will make sense when you watch it). Anyway, I loved it. Yinh fell asleep. YMMV.

Holiday coziness — games!

I taught both my boys backgammon this past week. This is the younger guy playing with mom.

I used to play a lot with my colleagues during the year I was a broker on the NYSE floor. Every time I came back to the “booth” (that’s the area with the phones on the sides of the exchange floor where clerks take orders from customers), I’d make a move on the Jellyfish software. When I was at Parallax, I worked with 2 of the best backgammon players in the world (one of them was the best in the world iirc) and Parallax was actually founded by Roger Low, who was a world-class player himself. Roger had retired by the time I joined so I never got to know him.

Backgammon is especially neat if you like options because the doubling cube begs you to price volatility.

💡Fun fact

When I had my phone interview with SIG (this is the round that happens in between campus interview and the final on-site) I was asked a basic question regarding the cube. Since accepting the cube doubles the stakes of the game and rejecting it forfeits the current value of the game, what is the minimum probability of winning you need to accept it?

When looking at the board you need to evaluate how risky the position is. Like how likely is the tide to turn? When you offer the cube you are selling your opponent an option and its value depends on their chance to come back. But if you offer it to them when they have no chance they will reject it and be bailed out because you have now sacrificed your own possibility of getting a gammon (double points) or backgammon (triple points) if you beat them by a large margin.

It’s a great game for kids because they get to practice dice math. I was playing last night and knew I’d win as long as my next roll wasn’t a [1,1], [1,2], [2,1], [3,3]. So the kids would first need to figure out that’s what they are rooting for and then I ask them the probability. 4 out of 36 or 1/9. Plus the little guy figured out that the 7 is the most common roll and how to count the number of ways to get each number (he was delighted by the pyramid pattern when he realized it).

We always play more games during the cozy holiday season and I’m irrationally pleased that this season it’s backgammon. I recommend it.

[Also, if you are just learning, play a bot on your phone. You learn the game very quickly getting trounced by bots. Ultimately, I’m just a casual player, I never studied it, but would just play for $5 a game with my NYSE squad. Of course, with the doubling cube and possibility of gammons that can multiply pretty quickly. The real hitters play for four and five figures per game.]

Money Angle

On Nov 26th, Imran asked his followers would what more likely to double in the next year — gold or BTC?

I looked at the result of the poll just a few hours after it was posted and it was BTC 52% to gold 48%.

By the time the poll closed with 750 votes, BTC had garnered 2/3 of the votes.

I don’t know if me a jerk had anything to do with this but when I saw that the vote was almost a coin flip I chimed in.

Focus on the last part.

The poll should be nowhere near 50/50 because you would be able to lock in a great trade by selling gold in this proposition, buying gold call spreads financed by even more expensive BTC call spreads.

This is a classic difference between markets and democracy. It’s a perfect example of the Dinosaur Markets post in real life. The markets in the options reflect the volatility and the cost of replicating these bets. Money-weighted votes are interested in the truth where opinions are cheap as sand.

It’s very difficult to have opinions that are above replacement value about liquid assets. If you’re truly good at this, then being Scrooge McDuck rich based on consistently betting on these fantastic opinions is the only proof of such a skill. Few people are rich because of a crystal ball.

A good way to make a living in finance is to find the people that voted for gold in this poll and offer to trade with them. You need to do this in the dark because if you tried to do it on a public exchange, you’d be undercut by traders competing to sell the gold proposition to these opinionated people and it would drive the price down to a non-arbitrageable price.

Public markets protect overconfident people with arbitrageable opinions from their own ignorance and stupidity. Private or non-transparent markets are nice ways to shove a vacuum hose into their bank account.

There are many places where there’s alpha in projecting your opinion. This is the stuff you spend your time on in life. Where you have self-knowledge, private info, competitive advantage, skills, taste and so on.

But when it comes to markets, remember what we learned here just a few weeks ago:

the arbitrage reflex is more profitable than the opinion reflex

 

Money Angle For Masochists

Let’s continue on the same theme because these threads are going to become far more frequent for anyone who cares about markets. “Thinking in Bets” has a long runway, the way this country is headed, before it jumps the shark, so you might as well get used to it. From “Everything computer?” to “everything casino?”

Prediction Market Arbitrage: Using Option Chains to Find Mispriced Bets

Horse tweeted:

The moment you see a bet on a platform like Kalshi, Polymarket, or the soon-to-be Robinhood+SIG exchange, your mind should jump to the options chain.

The tweet says the Kalshi market is pricing a 9% chance of BTC hitting $250k

The options market can offer a quick sanity check. BTC is about a 55% vol. We are just being very approximate so not worrying about the term structure. I just want to show you my automatic mental response to the tweet.

Without hesitating, I pulled up the calculator on my phone and entered:

ln(250k/89k) / (.55 * sqrt(13/12))

Why?

We want to compute how many standard deviations away the $250k strike is to get a z-score which we can then convert to probability. Standard deviation depends on volatility and time. The more time or volatility you have the “closer” some percent return is. A strike that’s 100% away is extremely “far” if the asset needs to get there by tomorrow. If you have 10 years to get there, it’s not super far at all, you only need to go up 7% per year. Likewise, if an asset only varies by 5% a year, 100% is “far”, but if it moves 50% per year, 100% feels much “closer” or possible.

The formula above is simply dividing the percent return to get to the strike by the annualized volatility scaled by root(time) to find the distance.*

*Standard deviation or volatility as a quantity is proportional to the square root of time. Or you can say variance, the square of standard deviation, is proportional to time. The easiest way to remember this is to recall that when you compute the standard deviation of anything, you have an intermediate step of summing the squared deviations to get the variance, then divide by N. But to get back to the standard deviation, you take the square root of the ratio. The ratio in the intermediate step was variance/N. The final answer, the standard deviation, was the ratio of sqrt(variance) / sqrt( N). In our computations, N is replaced by time.

At the time of the tweet, BTC was 89k and there was 13 months until 2027. I assume 55% volatility.

Solving:

ln(250k/89k) / (.55 * sqrt(13/12)) = 1.80 standard deviations

We then use a standard normal table or normdist in Excel to see that 1.80 standard deviations encompasses about 96.4% of the cumulative distribution. Therefore, the probability of BTC going HIGHER than 1.8o standard deviations must be 3.6%

This is fully explained in Using Log Returns And Volatility To Normalize Strike Distances

The computation of this distance, besides being dependent on an estimate of volatility which we can borrow from the options market, assumes the asset is lognormally distributed. If you believe, as the options market certainly will if you look not at the at-the-money vol, but the far out-of-the-money call vols, that there is more positive skew than a lognormal distribution then our 3.6% estimate is too low.

But that logic is moving us in the right direction. We want to take the intel embedded in the options market when considering the price in the prediction market. We expect the liquid options market with much more volume and money behind it, to be the best guess as to the “fair price” of a proposition. If there’s an edge, it will be in the mispriced prediction market.

A prediction market bet can take a binary flavor. For example, “Probability that BTC settles above X by some date.”

It can take a “one touch” flavor. “BTC to touch but not necessarily settle above X by some date.”

Of course, “touch” is more likely than “settle” because “touch” encompasses all the times BTC settles above X, but also includes all the cases where it breaches X and falls back below X by expiry.

We can get information about the price of both binary and one-touch scenarios from the option market.

1. The Binary Bet: Price the Terminal Outcome with Vertical Spreads

Pricing: To find the true market-implied probability of the event, use the price of the spread:

Vertical spread price/Distance between the strikes ~ probability of asset expiring above he midpoint of the spread

Potential arbitrage if…the probability implied by the options chain is lower than the price offered on the prediction platform, you can buy the vertical spread and take the under in the prediction market or vice versa.

Further Reading: A Deeper Understanding of Vertical Spreads

2. The Path Bet: Account for Skew and Volatility with the One-Touch Rule

Pricing: You can estimate the path probability using the trader’s rule of thumb: take the delta of the vanilla option at that strike and multiply it by 2. This naturally takes into account the option implied skew because the delta is derived from the implied volatility at the strike.

The mechanics of an arbitrage here are complicated as it requires dynamic hedging. If that sounds interesting, perhaps you are born to be an exotic options trader. I have never tried replicating a one-touch option so while I could certainly “financially hack” a model, the main point I want to convey is that the pricing of the one-touch can be inferred from the vanilla options market.

Further Reading: one-touch

 

Stay groovy

☮️

Moontower Weekly Recap

Posts:

Prediction Market Arbitrage: Using Option Chains to Find Mispriced Bets

Horse tweeted:

The moment you see a bet on a platform like Kalshi, Polymarket, or the soon-to-be Robinhood+SIG exchange, your mind should jump to the options chain.

The tweet says the Kalshi market is pricing a 9% chance of BTC hitting $250k

The options market can offer a quick sanity check. BTC is about a 55% vol. We are just being very approximate so not worrying about the term structure. I just want to show you my automatic mental response to the tweet.

Without hesitating I pulled up the calculator on my phone and entered:

ln(250k/89k) / (.55 * sqrt(13/12))

Why?

We want to compute how many standard deviations away the $250k strike is to get a z-score which we can then convert to probability. Standard deviation depends on volatility and time. The more time or volatility you have the “closer” some percent return is. A strike that’s 100% away is extremely “far” if the asset needs to get there by tomorrow. If you have 10 years to get there, it’s not super far at all, you only need to go up 7% per year. Likewise, if an asset only varies by 5% a year, 100% is “far”, but if it moves 50% per year, 100% feels much “closer” or possible.

The formula above is simply dividing the percent return to get to the strike by the annualized volatility scaled by root(time) to find the distance.*

*Standard deviation or volatility as a quantity is proportional to the square root of time. Or you can say variance, the square of standard deviation, is proportional to time. The easiest way to remember this is to recall that when you compute the standard deviation of anything, you have an intermediate step of summing the squared deviations to get the variance, then divide by N. But to get back to the standard deviation, you take the square root of the ratio. The ratio in the intermediate step was variance/N. The final answer, the standard deviation, was the ratio of sqrt(variance) / sqrt( N). In our computations, N is replaced by time.

 

At the time of the tweet, BTC was 89k and there was 13 months until 2027. I assume 55% volatility.

Solving:

ln(250k/89k) / (.55 * sqrt(13/12)) = 1.80 standard deviations

We then use a standard normal table or normdist in Excel to see that 1.80 standard deviations encompasses about 96.4% of the cumulative distribution. Therefore the probability of BTC going HIGHER than 1.8o standard deviations must be 3.6%

🔗This is fully explained in Using Log Returns And Volatility To Normalize Strike Distances

The computation of this distance, besides being dependent on an estimate of volatility which we can borrow from the options market, assumes the asset is lognormally distributed. If you believe, as the options market certainly will if you look not at the at-the-money vol, but the far out-of-the-money call vols, that there is more positive skew than a lognormal distribution then our 3.6% estimate is too low.

But that logic is moving us in the right direction. We want to take the intel embedded in the options market when considering the price in the prediction market. We expect the liquid options market with much more volume and money behind it, to be the best guess as to the “fair price” of a proposition. If there’s an edge, it will be in the mispriced prediction market.

A prediction market bet can take a binary flavor. For example, “Probability that BTC settles above X by some date”

It can take a “one touch” flavor. “BTC to touch but not necessarily settle above X by some date”

Of course, “touch” is more likey than “settle” because “touch” encompasses all the times BTC settles above X, but also includes all the cases where it breaches X and falls back below X by expiry.

We can get information about the price of both binary and one-touch scenarios from the option market.

1. The Binary Bet: Price the Terminal Outcome with Vertical Spreads

Pricing: To find the true market-implied probability of the event, use the price of the spread:

Vertical spread price/Distance between the strikes ~ probability of asset expiring above he midpoint of the spread

Potential arbitrage if…the probability implied by the options chain is lower than the price offered on the prediction platform, you can buy the vertical spread and take the under in the prediction market or vice versa.

Further Reading: A Deeper Understanding of Vertical Spreads

2. The Path Bet: Account for Skew and Volatility with the One-Touch Rule

Pricing: You can estimate the path probability using the trader’s rule of thumb: take the delta of the vanilla option at that strike and multiply it by 2. This naturally takes into the account the option implied skew because the delta is derived from the implied volatility at the strike.

The mechanics of an arbitrage here are complicated as it requires dynamic hedging. If that sounds interesting, perhaps you are born to be an exotic options trader. I have never tried replicating a one-touch option so while I could certainly “financially hack” a model, the main point I want to convey is that the pricing of the one-touch can be inferred from the vanilla options market.

Further Reading: one-touch

“dancing animals”

Option trader and professional shtpoaster P4 tweeted this quote I’ve been sitting on a for a month:

Kurt Vonnegut tells his wife he’s going out to buy an envelope:

”Oh,” she says, “well, you’re not a poor man. You know, why don’t you go online and buy a hundred envelopes and put them in the closet?”

And so I pretend not to hear her. And go out to get an envelope because I’m going to have a hell of a good time in the process of buying one envelope.

I meet a lot of people. And see some great-looking babies. And a fire engine goes by. And I give them the thumbs up. And I’ll ask a woman what kind of dog that is. And, and I don’t know. The moral of the story is — we’re here on Earth to fart around.

And of course, the computers will do us out of that. And what the computer people don’t realize, or they don’t care, is we’re dancing animals. You know, we love to move around. And it’s like we’re not supposed to dance at all anymore.”

My wife will sympathize with Kurt’s wife. I will leave the house, forget the stamps, and come back telling her about the friend I just made who works in the produce section where we just discussed apple varietals for 30 minutes. Tangos are my favorite.

I’ve always thought this was a bit of a bug. It is quite inconvenient since I easily lose track of time. But “dancing animals” is the redemption I didn’t know I needed. It’s a redemption we all need.

It is grace.

It’s a little reminder that we are messier than our identities or affiliations or goals. Your strengths contain your weaknesses. Your beliefs are self-contradicting. You thought you were so smart 10 years ago and now cringe at what you didn’t know.

And still we will try to figure it all out. And we should.

But if we ever forget that we are animals and if we forget to dance, all that we figure out will not be for us but for our sedation.