Codenames Telepathy

In a week, I will have known Yinh for 17 years. We don’t complete each other’s sentences. We still have plenty of stories to tell each other, although CoVid lockdown is burning that fuse a bit faster. (I always think of the Chris Rock joke about a wife telling her spouse to “get kidnapped and come back with new stories”). We have thus far deferred marital mind-meld.

But you would not know this by watching us play the 21st-century version of the Newlywed game…Codenames.

The Joy Of Codenames

If you play any party games you know Codenames so I won’t re-hash it. If you don’t play party games then you should know that blacking out right after dinner at family functions is anti-social. You should play Codenames instead.

Being in sync with a Codenames teammate is successfully web-crawling their brain. You hop from the axon of one idea to the dendrites of another as you stretch to find how they linked words on the Codenames’ grid to the clue that launched the brain scan in the first place.

We tend to work best when she gives the clues because in our relationship I tend to be the one filled with more random nonsense. This is a bug in times when being distracted is a penalty, since I can bike-shed with the best of them. But in Codenames, being a central repository for mutual references that unlocks with a single word is a decided advantage.

Here’s an example from Friday night. Yinh gave the clue “Empire, 2”. This clue was supposed to unlock:

  • “strike”

    Easy enough. Empire was a reference to “Empire Strikes Back”.

  • “chair”

    Ok, so why did she think “empire” would lead me to “chair”?

    I had a theory as to why she connected these words, so to test it I asked her what her logic was. She stumbled. She had forgotten why these words went together, which made me think my theory was even more correct.

    I’ll give you a hint. It wasn’t an “empire” -> “throne” -> “chair” pathway.

    Here’s the actual pathway:

    “empire” –> what empire comes to mind? –> Roman or Ottoman –> Ottoman = “chair”

    Here’s the best part. Her logic was both not original thinking or explicit. It was a subconscious reference to Eddie Izzard’s Dressed to Kill stand-up special that we’ve seen together. When I reminded her of Izzard’s joke that linked the Ottoman Empire to furniture she immediately realized what she had done.

I always get a kick out of her screwing up a reference but still understanding what she means. Just this week, she asked me what the latest on the Skrillex vs NY Times drama was going to which I responded, “You mean Slatestarcodex?”. She laughed at how she butchered it and I was happy to know she listens to me when I talk about the random crap I find interesting.

Check out Codenames. Read your partner’s mind to crush your in-laws at your next family game night.

Lessons From The Layup – Corner 3 Spread

During an interview with Ted Seides, investor Andrew Tsai recounts an internship at the well-known trading firm Susquehanna in the mid-90s (disclosure: I worked there for 8 years after college). In particular, he remembers a company outing to a dog track that summer:

I’m sitting next to one of the partners and I’m looking at the sheet of all the races, and he’s like “How are you gonna bet?” I respond, “Well, I’ve never really done this before but this dog looks like he’s got a good track record and he’s been running strong lately.”

The guy looked at me like I was a complete idiot.

He’s like, “What are you talking about, ‘How is this dog doing?'”

Andrew is perplexed. Well, isn’t that kind of what we’re talking about.

The partner starts to explain, Look at the relative value of this dog and that dog.

The lightbulb went on for Andrew.

“We started talking about spread trading and trying to capture that basis and I’m like ‘These are my guys’. It was really this culture of dissection that I loved.”

Relative Value Goggles

One of my favorite Twitter follows is the anonymous account @econompic. He’s in my top 5 and you should follow him too (only about 15% of my followers follow him which is basically as stupid as a butterfly trading for a credit). Go for the finance stuff and stay for takes on breakfast cereal, Weezer, and the NBA. Oh and the polls. You see, Jake’s polls act like the Susquehanna partner while Andrew is the rest of #fintwit. They are cleverly designed to surface mispricings in how people think about risk or relative value.

His relative value instincts are well-tuned. It’s like he has goggles that allow him to filter the world through prices. It’s a lens that’s critical for trading. One of his recent tweets is a great example of this. I’ll withhold the full tweet for now since it has spoilers. Let’s start with this screenshot:

So which shot do you take?

(take note of your answer and reasoning before continuing)

Spread Perception

The first thing that should leap off the screen is the gap between the free throw and the top-of-the-key 3. Using NBA dimensions, that’s a 15′ shot vs a 23’9″ shot. And you are rewarded 5X for it from the benevolent genie offering this bet. The reflex you need to hone is that:

Prices imply probabilities


Because of expected value. Expected value is the probability of payoff times its magnitude. Would you pay Best Buy $50/yr to insure a $1,000 TV? If there’s more than a 5% chance that it fails you might. If there was a $500 deductible then the benefit is cut by half and you need to think there’s at least a 10% chance the TV fails. And if you think you get more TV per buck every year thanks to innovation then purchasing insurance implies an even greater defect rate.

So when you weigh the cost of different choices (insure vs not insure, fix vs replace, cheaper product vs more durable product) you are implicitly weighing probabilities. Making that explicit can expose mispricings.

Let’s go back to basketball.
Dissecting the basketball shot.

Just to get a hang for the reasoning let’s start with a simplifying assumption. You are 100% to make the layup.

  • Free Throws
    How confident do you need to be from the free-throw line to forgo the certain $50,000 you’d make from a layup? At least 50% confident. If you can shoot a free throw with a better percentage than a coin flip the free throw “has more equity”. If you are a 60% free throw shooter than that option is worth $60,000.
  • Top-of-the-key 3
    $500k to make this shot. You only need to be 10% confident to justify forgoing the layup for a chance at some big money.

    Ok, here is where the probabilities should really get your senses tingling. The free throw implies a 50% probability and the top-of-the-key 3 implies 10%. Are you 5x more likely to make a free throw than this 3-pointer?

    Unless you are 7 and literally can’t heave a ball from the 3-point line, it’s hard to imagine your chance of making these shots to be so far apart. In fact, if the 7-year-old can’t reach the rim at all from long range, I have my doubts they can shoot consistently shoot 50% from the stripe in the first place. But I’m willing to concede that possibility. For an adult, that spread is too wide. You either can’t hit free throws with a .500 percentage or your chance of making a top-of-the-key 3 is greater than 10%.

    To take an outside view, consider NBA players. Guys who shoot about 40% in games, can shoot between 65-75% in practice. HS coaches can tell you that a 30% 3-pt shooter can make about half their shots in practice. Since free throw percentages are bounded by 100% you are talking about no more than a spread of 2x between free throw and 3-pt percentage. Your margin of error on the spread could be 100% and you’d still only have a spread of 4x. These shots are priced at 5x!

    An exactly 50% free throw shooter be a 12.5% 3-point-shooter using the most conservative estimates and this top-of-the key 3 is still “too cheap”. And remember, there is a conditional probability aspect to this since we are dealing in relative pricing. If you are certain you need a miracle to hit an uncontested 3-pointer there is almost no chance you are truly a 50% free throw shooter.

  • The rest of the table

To amateurs the corner-3, without a view of the backboard or the chance for a lucky bank shot, is daunting. But are you really half as likely to hit a corner-3 vs the key-3? As we get into the low probability shots it’s reasonable for a person who really knows their habits to potentially parse these odds but it takes quite a bit of experience to know that you are really 100% better at top-of-the-key 3s then corner 3s. Without that conviction, I’d take the better implied odds in the corner-3.

The entire payoff schedule suggests that you should either take a layup or a corner 3 as you are being offered very cheap relative pricing on those options. You can check out the rest of the tweet for the comments and replies. (Link)

What If You’re Broke?

If you read the thread there’s mention about how being broke can push you towards the layup even if the expected value of another choice is higher. This is a great opportunity to bring ideas like “risk aversion” or “diminishing marginal utility of wealth” into practical consideration.

The expected value framework above is an optimal case. It assumes every dollar has equivalent value to the player. The fancy term for this is “risk neutral”. If you have $5,000 and making another $5,000 has a “happiness value” that is equal and opposite to the “sadness value” that you experience if you lose $5,000 then you are risk-neutral. Since you are not a robot and need to eat, you are not risk-neutral. You would not bet all your money on a 50/50 coin flip. And you probably wouldn’t do it if you had a 60% of winning the flip. You are “risk-averse”.

A related concept is the diminishing value of additional wealth. This is pretty obvious. Jeff Bezos’ first million probably felt good. Today, it would be an imperceptible amount on his Mint dashboard.

Without knowing the lingo we all understand the intuition. If you are a broke college kid you might always opt for the layup. A sure $50k might mean getting out from under that 15% credit card APR, while $100k is ‘nice to have’, not ‘need to have’. That first $50k can be life-changing by getting you off the wrong path.

Likewise, the rich gal with a vacation house in Malibu is not so constrained. She can rely on the optimal pure expected value prescription. Just as a trading firm with a huge bankroll is willing to bet large sums on small edges. They will optimize for EV when the bet sizes are small relative to capital.

Our intuition moves us in the right direction. It tells us that the college student will be more conservative in choosing which shot to take. By mixing in a simple concept like “utility of wealth”, we can actually re-price all the probabilities implied by the shot payoffs.

Adjusting Probabilities For Risk Aversion

Linear vs Concave Utility

  • Risk-neutral utility curves are linear.
    If you are the risk-neutral robot every dollar you make is worth exactly the same to you. Your second million is as sweet as the first. That’s a linear utility function. Those are the curves embedded in any expected value proposition which simply spits out “pick the highest one”. I presumed such a framework in the prior table that said: “Min Probability To Accept Shot”.
  • Risk-averse utility curves are concave
    If you are risk-averse, every additional dollar is not worth quite as much as the one before it. And every extra dollar you lose hurts just a tad more than the one before it. Losing your rent money hurts more than losing your Ferrari money. So instead of a linear function, we need a function that:

    1. Is always increasing to reflect that more money is always better than less money (‘Mo Problems and other first-world complaints notwithstanding).

    2. Slope starts out faster than the linear model then flattens as we make more money.

    Luckily, there is a simple function that does exactly that. The log or natural log function. People who study “risk-aversion” and diminishing marginal utility of wealth don’t think about it linearly. They don’t presume $5,000,000 is twice as “useful” as $2,500,000. They might say it’s only 1.75 as “useful” ( ln 5 / ln 2.5 = 1.75).


Re-computing Minimum Probabilities As A Function Of Starting Wealth

  • 25 year old with $10,000 to his name.
    The guaranteed layup increases his wealth by 6x and log wealth by 2.8x.
    The free throw increases his wealth by 11x but his log wealth by only 3.4x!

    Look how much it raises the minimum probabilities for him to accept various shots if he has a log wealth utility preference. He needs to shoot 3s as well as a good [contested] NBA shooter to gamble on the big money instead of the layup!

  • Give that guy a $10,000,000 bank account, and he’ll choose according to Spock-like expected value prescriptions.
  • Finally, check out the implied minimum shot probabilities for various levels of wealth. The larger your bankroll the more you can rely on probabilities imputed simply by expected value. If you are fabulously rich, you aren’t paying up for life insurance, home insurance, and so forth. You’ll deal with those bills as they come. For most of us, calamities mean financial ruin.
    How we decide depends not just on the expected value but on our own situations. The more secure we are (on the flatter section of the log wealth curve) the more we can afford to act optimally.

    (There is quite a bit of fuel for liberal policymakers here. They will realize that this is another example of Matthew effect or accumulated advantage. Richer people can avoid negative EV trades like insurance. Another thought. The inflection point on the so-called Laffer curve is probably much further to the right if we re-scale the axis in terms of log wealth suggesting we may tolerate much steeper graduated tax brackets. I’m not making a political opinion so don’t @ me. I’m just observing things that I’m sure have been discussed elsewhere.)


Prices impute probabilities. By taking the extra effort to make this explicit we can de-fog our relative value goggles. This improves our decision making in trading and life.

Since we are not “risk-neutral” robots the correct decisions are often theoretical. Translating the prescription to your own situation is an extra step that we typically leave to our intuition. This is quite reasonable. At the end of the day, we aren’t going to define our own wealth functions in Excel (log wealth is just one example of a non-linear function that seems to accommodate our intuition but the actual slopes and smoothness can vary quite a bit from person to person).

I recommend following Jake. His polls will help you tune your intuition.

Thinking Like A Trader

Robin Hogarth would call investing a “wicked” learning environment. Information is hidden in such a way that we struggle to identify causality. I contrast investing a bit with trading and games which are fertile ground for reasoning, probability, and logic. With large enough samples, you get tighter feedback loops, a critical input to learning. I don’t know much about digital marketing or Google AdSense but my guess is market-making has more in common with SEO than it has with buy-and-hold investing.

Because of my interest in topics like learning, reasoning, and money matters, I sometimes have younger readers ask me how to “think like a trader”. I’ve pointed to some places in the past.

  • A few months ago I shared the books I’ve crowdsourced from traders: The Investing Pros Library (Link)
  • I’ve narrowed a list to target beginners in particular. (Link)
  • A collection of resources to teach kids business. (Link)

While these are all useful, the single best way to adopt a trader mindset is to do what Annie Duke recommends:

Think in bets.

Betting is really about decision-making.

Every decision you make is actually a bet that the chosen action is better than all the alternative options. Annie’s recent book Thinking In Bets is about making better decisions when you don’t have complete information. Making better decisions is so obviously important living a happy life it feels silly to even state that. Naval Ravikant says that good judgment is the most important attribute of anyone with a high leverage position like a CEO. Or the director of NIAID. If the impact of a decision is levered 10x, then a 1% better decision is an order of magnitude more valuable than the inferior one.

We spend years learning formal grammar or cherry-picked history yet most of our decision-making skills come from trial-and-error. In a complicated world where causality is opaque and noise is abundant, it’s simply too easy to learn the wrong lessons. Look around. The post-hoc fallacy is everywhere. “Since event Y followed event X, event Y must have been caused by event X.” You took an herb and your cold went away. Must have been the herb. You played video games when you were a kid, and now you have a good job. I guess Halo was good job training. There are many outcomes that people say are attributable to X when it could be even more likely that the outcome was in spite of X. Your reason for something could be the exact opposite and you don’t even realize it.

Michael Mauboussin, during a 2019 talk, said:

The best probabilistic decision-makers have more in common with each other than with the average decision-maker in their profession…Warren Buffett has more in common with Annie Duke than he has with the average investor. (Link)

Betting As ‘Decision’ Practice

A strong trading education will include an explicit education in decision-making. Markets become the proving ground for that education.

Marc Andreesen in a recent (and rare) interview:

One of the things you find about professional gamblers – they may play poker at night but what they do during the day is they hang out together and they make side bets for large amounts of money. And it’s literally a side bet of sitting in a diner and betting on whether there are going to be more red cars than blue cars passing by. What they’re doing is ‘steeling’ their own psychology to be able to pull the trigger on bets like that with a purely mathematical lens and with no emotion whatsoever. They’re trying to steel themselves to be able to be completely clinical. And so as contrast, what they then hope for that night when they sit across the table from someone is to hope they’re dealing with somebody who’s super emotional. Because the clinical person is going to just slaughter the emotional person. (Link)

I’d respectfully edit this line: “They’re trying to steel themselves to be able to be completely clinical.”

Actually, they just think they have an edge. The byproduct of that betting process is the “steeling”. This loops until they are quite clinical about the risk. This was the nature of trader training. There were many hours of poker hands and mock-trading deconstructed. Bets are hypothesis tests. The constant feedback calibrates how you map future hypotheses to future situations. Betting is a practice that tightens that loop.

In contrast, big decisions in life are hard because we don’t get much practice at them. You don’t get a lot of reps when it comes to picking a spouse (Elizabeth Taylor notwithstanding).

Ways to Learn More About Decisions

I recommend Annie’s interview with Ted Seides on his pod. I thought it stood out in the sea of “behavioral” content. I jotted some notes to jog your memory on key points. (Link)

If you agree that decision-making is a discipline that should be explicitly taught you’ll be pleased to hear there’s an organization called the Alliance for Decision Education. Its board and advisors include many notable backers including Mauboussin, Annie Duke, Prof. Daniel Kahneman, author Brian Portnoy, and several former Susquehanna execs including a founding partner. Learn more about their mission to spread decision best practices. (Link)


  • I didn’t read Duke’s book despite really enjoying the pod.

    The content was the air I breathed in training. The principles would be a refresher for sure but hard to prioritize when I struggle to make it through my existing book queue.

  • There are diminishing returns to studying decision-making explicitly vs just practicing.

    Kahneman himself doesn’t think the awareness of bias inoculates you from it. I don’t fully agree. Naming the biases provides you with the red challenge flags to throw in the moment. I’ve had enough discussion about trades to see the value in surfacing the subliminal. To the extent that we still make mistakes and always will, Kahneman’s point is well-taken.

  • There are many schools of thought.

    Naturalistic decision making is championed by Gary Klein. The ergodicity crowd led by Ole Peters thinks many of the so-called behavioral biases like ‘risk aversion’ are an artifact of the assumptions baked into studying how people decide in contrived lab settings. There’s a lot of brain damage to be had if you dig. This doesn’t even get into more common sense questions: the limits of transferability. Are poker players actually good decision-makers away from the table?

  • A final point (and partial confession).

    There is nothing more insufferable than a trading trainee. They are so eager to tell you what decision bias you are falling for. It’s like people who just learned about nutrition or exercise commenting on your form or what you are eating for lunch. You want to chase a Pop Tart with a glass of Fruit Loops milk in front of their bulging eyes just to spite them.

    We don’t need people leaning into crappy decisions because you come off like a pedantic hack.

I mentioned that trader training included playing games and mock-trading. If you want a glimpse of what mock trading options was like, check out my recent post:

Mock Trading Options With Market Makers (Link)

If you’d like to try your hand at mock-trading futures or options with your friends or family here’s a game we used to play in either group interview settings or with brand new trainees.

You Can Mock Trade With A Deck Of Cards (Link)

The Why And How OF Taking “Discoverable Notes”

I have been an active note-taker for years and a fan of how meta Tiago Forte gets about the process of taking notes. Tiago’s Building A Second Brain course is very popular in productivity circles. While I have never take it, I recently came across his essay Progressive Summarization: A Practical Technique for Designing Discoverable Notes. (Link)

It’s an outstanding framework for understanding the whys and hows of taking notes. I, by accident, have arrived at a very similar system so it was interesting to see someone explain it thoroughly as only Tiago can.

This is a summary of what resonated with me.

The Why Of Taking Notes

The Right Info At The Wrong Time

What you read is good and useful and very important, you’re just reading it at the wrong time.

The challenge is knowing which knowledge is worth acquiring. And then building a system to forward bits of it through time, to the future situation or problem or challenge where it is most applicable, and most needed.


Bridging The Acquisition And Use Of Knowledge

It’s too mentally expensive, if not impossible, to internalize all or most of the information we consume. A good system is intended to bridge the time between when you discovered the information to when you use it.

At that future point, when you’re applying that knowledge directly to a real-world challenge…By the time you’re done solving a real problem with it, book knowledge has become experiential knowledge [which you carry forever].


The How Of Taking Notes

Defining The “Second Brain”

An external, integrated digital repository for the things you learn and the resources from which they come. It is a storage and retrieval system, packaging bits of knowledge into discrete packets that can be forwarded to various points in time to be reviewed, utilized, or deleted.

Designing The “Second Brain”

Goal: You are trying to triage information in an organized way. You read something you know is interesting and you want to be able to reference later.

Challenge: You need to file it quickly, make it discoverable, and emphasize why it’s important so “future you” can make sense of the notes efficiently.

Tiago says:

A note-first approach to knowledge management means we have to think about design. You are, in a very real sense, designing a product for a demanding customer — Future You. Future You doesn’t necessarily trust that everything Past You put into your notes is valuable. Future You is impatient and skeptical, demanding proof upfront that the time they spend reviewing notes will be worthwhile.

Balancing Tradeoffs

  • Discoverable: Digestible notes. So needs to be compressed
  • Understandable: Context including sources, examples, details

Getting the balance between compression and context right is not a trivial matter. When the time comes for Future You to decide whether or not to review this note, seconds count.

When you fail, you successfully sent a packet of information forward through time, but not in a state where it could survive the journey… You have to summarize the note without knowing what it will be used for.

The Progressive Summarization System

  • Layer 0 is the original, full-length source text.
  • Layer 1 is the content that I initially bring into my note-taking program. I just capture anything that feels insightful, interesting, or useful.
  • Layer 2 is the first round of true summarization, in which I bold only the best parts of the passages I’ve imported. Keywords, phrases, sentences
  • Layer 3, I switch to highlighting, so I can make out the smaller number of highlighted passages among all the bolded ones. This time, I’m looking for the “best of the best”
  • Layer 4, I’m still summarizing, but going beyond highlighting the words of others, to recording my own…restating the key points in my own words
  • Layer 5 (as needed): Remix. for a tiny minority of sources, the ones that are so powerful and exciting I want them to become part of how I think and work immediately, I remix them. After pulling them apart and dissecting them from every angle in layers 1–4, I add my own personality and creativity and turn them into something else.

My Own Accidental Version of Progressive Summarization

  • Layer 0 is usually just the link without the text which is risky since the link can break. (With Slatestarcodex site being taken down I’m experiencing this firsthand)
  • Layer 1 same as Tiago
  • Layers 2 and 3 are combined. Mix of bold and italics.
  • Layer 4 is paraphrasing often drawing connections to other ideas. While time-consuming because it requires thinking I am rewarded by an easier retrieval stage. More selective about what notes I do this with.
  • Layer 5 usually means pasting the note in other notebooks when the content has multiple contexts


Notes from Capital Allocators: Annie Duke


About Annie: Professional poker player and author of Thinking in Bets

All decisions are a bet

When you choose x you forgo y. The decision is a bet that x is a better outcome than not x.

Beliefs are formed then confirmed

Dan Gilbert’s known for happiness research but his 90s research which is lesser-known was focused on ‘belief formation’. We are hardwired to not vet beliefs since beliefs are typically perceptual. Hallucinations and mirages are rare. However, abstract beliefs that emerge from our social interactions, language, symbolism are incorporated via this same mental machinery which was really designed to assimilate perceptual beliefs.

Gilbert showed that by default we accept the belief is true BEFORE we vet it.

Research shows:

    • We often fail to later vet the belief
    • If we do vet it, we are biased
      • Kahneman’s idea of “motivated reasoning”: our beliefs drive how we vet the belief
      • Confirmation bias
      • Blindspot bias
      • Smart people are often more extreme in their biases because they rationalize with a greater repertoire
        (This was tested by first evaluating subjects’ statistical prowess then comparing how they handicap a neutral versus emotionally-charged outcome)

How do we improve?

  • Frame decisions as bets. 

    1. Assigns probabilities to outcomes
    2. Define what we explicitly are evaluating
    3. Invites others into the truth-seeking process which is also good for social reasons
      • When acting very certain we can suppress or intimidate other’s views
      • Avoid the pitfall of confusing certainty with accuracy
      • Makes the communicator more believable
      • Avoids biasing others before they start the vetting process

  • Define winning as being more accurate inoculating ourselves against self-serving bias.

    • Use Mertonian norms which comprising the ethos of science (acronym: CUDO)
      • Communism: Standardize how data is presented so members of the community cannot present the group biased picture
      • Universalism: Ideas have objective truth regardless of the messenger; “Don’t shoot the message”
      • Disinterested: Do not infect the group with your beliefs
      • Objective Skepticism: seek counterfactuals and dissent
        • In groups, use ‘red’ and ‘blue’ teams
          Red team’s function is to rebut or dissent. This instantiates a role in which being a team player is actually to challenge.

  • Be aware of our tendency to “temporally discount”.

    • A dollar today appears worth way more than in a year
    • Tonight’s wine is tomorrow’s hangover (Seinfeld’s “Day Jerry” vs “Night Jerry”)

If we associate better-calibrated beliefs with better outcomes and a happier life we should strive to be honest with ourselves while reasoning even if it sacrifices our ego/fun in the moment.

Risk Management

  • Why we fail to apply the principles of Kelly betting:

    1. Garbage In/Out: Poorly calibrating the edge and/or variance
    2. Our ability to apply rational System 1 rules are compromised when we are in an emotionally charged state (“limbic system firing”), which is when the rules matter most. “Stacking Irrationality”.

  • The merit of risk limits irrespective of risk/reward or expected value:

While irrational, they are less damaging than allowing yourself to continue betting when your “emotionally unfit”. A bias which gives us the chance to play again tomorrow when we are not emotionally unhinged is adaptive for the long-run. (Me: Reminiscent of ergodicity discussions about maximizing compounded expectation)

Tells and body language

  • Good to follow: Joe Navarro. A FBI operative specializing on body language

  • How poker players read opponents:

    1. If never faced them before, start with base rates
    2. As you learn how they bet, Bayesian update base rates
    3. Merge tells with updated probabilities

  • Signs of being relaxed vs discomfort:
    • embodied by distance from table
    • self-soothing behavior

The Antidote To Abstraction

All kids want their mama. And for good reason. Mamas are the best. It’s no contest. Of course, that doesn’t take anything away from fathers’ impacts. And if we sum the total hours fathers spend with their children I’ll bet it’s less time than they spend with mom. So in the spirit of a father’s impact-per-hour, I’ll be brief.

Read Charles Eisenstein’s essay The Age Of We Need Each Other (Link) (Link with my highlights)

It’s one of the best things I’ve read in a long time. I suspect different people will get different things out of it. I’ll share my own reading of it:

Its message is the antidote to all the abstraction that makes us feel helpless. If all the top-down strife is leading to daily bottoms-up anxiety this essay has a remedy.

It reminds me of the advice to break giant problems or endeavors into manageable bites. Except the giant problem is not “making a living” or “getting healthy”. It’s finding the purpose of your life. Sometimes we just don’t know what we are “supposed” to do right now. I think the answer lies right here in this essay. It affirms a timeless truth. A truth so simple it’s easy to forget:

You are, can be, or will be somebody’s world. It might just be one person. And that is not just enough. It’s everything.

Most of the people I consider heroes are people I personally know. They are unsung. And this becomes more true as I get older. I think that truth is a clue to what you are “supposed” to do.

Anyway, read the essay yourself. See what you get.

In a similar vein, I’ll share the Michael Crichton quote I keep on my main Notion dashboard.

If you want to be happy, forget yourself. Forget all of it — how you look, how you feel, how your career is going. Just drop the whole subject of you.

The quote continues.

We all know this is true because…

If you want to know how the whole essay goes check out Happiness. (Link)

You Can Mock Trade With A Deck Of Cards

Here’s a mock trading game I learned as a trainee to simulate futures and options market making. This game was commonly used as a day 1 exercise in trading class or when interviewing cohorts of college grads during recruiting “combines”.

The Futures Game

What you need:

  1. A deck of cards
  2. Nerdy friends (the more the better)
  3. A paper and pen per person to use as a tradelog


You want to deal out enough cards to players (these are the market makers) so that there is about 25 remaining in the deck. There’s some leeway here.


  • You have 6 players. So deal them each 4 cards leaving 28 cards undealt.
  • Market makers may look at their hands but don’t share info.
  • The undealt cards are known as the “public pile”. They should be evenly divided into 4 or 5 sub-piles ideally (again there’s leeway depending on how many cards there are).
  • The sub-piles are going to represent “trading days”.
  • The cards themselves are news flow which will move the futures prices.

Description of futures prices:

  • The futures are the 4 suits. There’s a club’s market, a spades market, etc.
  • The final settlement price of the futures will be the sum of the ranks of cards in the public pile. (Ace =1 thru King = 13). So the maximum any future can be worth is 911

    It’s best to define the tradeable universe to keep the liquidity centralized.

    So you could have a diamond market, a spades market, and a “reds” market (which would be an index settling to the sum of diamonds and hearts).

    How To Play

    The first trading day

    • Reveal the cards in the first public sub-pile.
    • Market makers make bids and offers for the various markets. Tight 2 sided markets should be encouraged/required. For example:John: “I’m 65 bid for Hearts and offered at 68”

      Jen: “I’ll pay 67 for 5 Hearts contracts” (perhaps Jen is holding no Hearts in her hand)

      John: “Sold you 5 at 67” (John is holding 16 points of Hearts in his hand)

    • Record all your trades on your own pad or paper:1. Which contract you bought/sold
      2. Quantity of contracts
      3. Price of contracts
      4. Counterparty

    So for example, if I paid 51 for 4 “clubs contracts” from Mary I would record that information on my paper. Mary would record her sale of the 4 contracts at 51 on her card with me as the counterparty.

    • The trading is open outcry. There are no turns.

    Settling the trading day

    1. When the trading peters out for that “day” everyone should check their trades against their counterparties to make sure there are no so-called breaks or “outtrades”.
    2. On a central eraseboard or paper the “closing price” of each market can be recorded. So if the King of clubs and 3 of clubs were revealed from the sub-pile, then clubs “settled at 15”. Clubs might have traded 53 last in the expectation that more clubs will be revealed on subsequent days.
    3. Repeat this process for all remaining tradings days

    The last settlement

    • Compute “P/L” for all trades.

    If I bought 4 clubs contracts for $51 and clubs final settlement was $63 then I made a profit of $12 x 4 or $48. Mary’s loss would match that amount for that trade.

    The total P/L of all traders should sum to zero at the end of the game.

    Options Variant

    • Either the same group or a different group of people could choose to trade calls and puts on the final settlement price of the futures.

    So if I paid 3 for Clubs 55 calls and the final settlement was $63 then I profit the difference between the $63 and the strike ($55) minus the premium I outlayed:

    $63-$55 – $3 = $5

    • You could even get fancy and trade “vol”. You could sell say 10 clubs calls and buy 5 clubs futures to hedge the delta.
    • This game is played the same way the futures game is played or in conjunction. Repeat the process for all trading days then compute P/Ls at the end. Again if there are no errors the game should be zero-sum.

Mock Trading Options With Market Makers

I got into options trading straight out of college. In 2000, the option exchanges were bustling. The Amex in NYC (where I was based), the PHLX, the P-Coast, and of course the CBOE. As a trainee, your day consisted of assisting the option market makers and specialists. Building spreadsheets, running risk reports (hitting a macro, then killing some trees), and the worst part of the job — the pre-open routine of reconciling positions and breaks. Hopefully, you’d finish before your trader sauntered into the office hungover.

During the actual trading day, your duties were pretty limited. I remember going to Cafe World at the corner of Trinity and Rector with a diagram of where my trader wanted each dish from the buffet arranged on his plate. Although you aren’t paid much you are still a liability for your first 6-12 months.

Mock Trading

Your main purpose in the cocoon phase is to learn. After the market close, you’d attend “mock” which was short for mock-trading. Mock would be led by senior traders. “Senior” basically meant a market-maker that was now “on a badge” the credential you needed to trade on the floor. You were getting taught by people that ranged from 1 to 5 years older than you which should tell you a) how start-uppy the culture and b) how much every day’s hundreds of trades added up to valuable experience quickly.

At my firm, mock was basically hunger games. You’d stand around for an hour shoulder to shoulder with 15 guys (yes it was mostly guys) in front of a dry erase board as 3 or 4 senior traders posed as brokers barking out orders and moving the stock and option prices around setting up opportunities for the trainees to spot arbitrages.

You’d have to hedge your trades (nothing like selling one of your teacher’s some puts as another teacher announced the stock bid was now 25 cents lower), lean markets based on what prevailing bids or offers were “resting” on the exchange book, read body language, remember all the verbally announced orders that might have been announced but were not in play until the stock moved. Memory, pattern recognition, and extremely fast mental math. In fact, everyone in the room would play a timed put/call parity computer game during the day to prepare (I actually trained during the tail end of the fractions era).

So for fun, I thought I’d share an example of what mock trading would be like.

Spot The Edge

Requirements and assumptions:

  • Stay delta-neutral. If you want to buy or sell the stock you must cross the spread.
  • Options markets are all 500 up, meaning the bids and offers have 500 contracts on them.
  • Cost of carry = 0%
  • 90 days until expiration
  • You will need to know Put/Call Parity

    Call = (Stock Price – Strike Price) + Put + Cost of Carry

    Since there’s no cost of carry let’s restate this more simply:

    Call = Intrinsic + Put

Ok, here’s the option’s board:

A broker walks into the pit and announces:

I have 200 XYZ 55 straddles offered at $4.15!

I’ll get you started with a hint. Be the first person to yell: “Buy em!”

Now go figure out why.

Here are the exercises you can do with the information above.

  1. Compute the implied volatility.
  2. Find the arbitrages or best series of trades in conjunction with the broker orders that are being shouted into the pit.
  3. Report your remaining position and at what average price it was established.

    Extra credit: Compute your P/L. You may reference an option model after the mock trading session ends.

It’s all spoilers ahead so if you actually want to do this, don’t scroll further until you are done.


  • Compute the implied volatility

The approximation for the ATM straddle is given by the expression1 :

Straddle = .8Sσ√T

where S = stock price
σ = implied volatility
T = time to expiry (in years)

Let’s use mid-market of the 55 put and put/call parity to get the call price.

C = Intrinsic + P

C = 0 + $2.10 = $2.10

Since the straddle is just C + P we get $4.20 for the straddle. Plugging into the approximation:

$4.20 = .8 x $55 x σ x √.25

Solving for σ we get an implied volatility of 19%

  • What series of trades do we do?

    1. Buy 200 55 straddles for $4.15
    2. Sell 200 55 calls at $2.15
    3. Sell 400 60 calls at $1.05
    4. Buy 400 65 calls for $.05
    5. Sell 200 65 puts at $10.10
    6. Sell 3,000 shares of stock for $54.95

    Whoa. That’s a lot of trading. Because of put/call parity, traders can collapse their thinking and position by strike. A call is a put and a put is a call. You can always convert one into the other by taking the opposing delta in the underlying.

    Let’s summarize these trades by strike.

    65 Strike

    Buy 400 65 calls for $.05
    Sell 200 65 puts at $10.10


    1. Buy 200 65 calls for $.05 and sell 200 65 puts at $10.10. Buying a call and selling a put on the same strike is known as a combo. It is the same thing as synthetically buying the stock. Why? Think about it, no matter what happens you will be buying the stock for $65 at expiration. You’ll either exercise the call or be assigned on the put. But you collected $10.05 today to make that commitment so you effectively bought the stock for $65 – $10.05 today or $54.95. Sweet.

    So this can be summarized simply as buying 20,000 shares for $54.95

    2. You also bought 200 extra 65 calls for .05

    60 Strike

    Sell 400 60 calls at $1.05

    55 Strike

    Buy 200 55 straddles for $4.15
    Sell 200 55 calls at $2.15

    Since you bought 200 straddles, you bought 200 calls and 200 puts. The calls cancel out and you are left long 200 puts at a net price of $2.00 (spent $4.15 200x in straddle premia and collected $2.15 200x in call premia).

    Now remember we synthetically bought 20,000 shares for $54.95 via the 65 strike combos.

    Back to put/call parity.

    C = (S-K) + P
    C = ($54.95 – $55) + $2.00
    C = $1.95

    So the combo plus these 200 55 puts means you legged buying 200 55 calls for $1.95

  • What is our residual position and at what average price?

    Let’s do what option traders do and show the net position by strike. That’s how we see what we actually have on. It allows us to make sense of the complexity at a glance.

  • 1. First, we can see the 200/-400/200 pattern on equidistant strikes (ie they are each $5 apart). That is a butterfly. A relatively low-risk distributional trade that has very little vega, gamma, and theta with several months until expiration.

    What price did we leg it for?


    1. We bought 55 strike call synthetically for $1.95
    2. We sold 2x as many 60 calls at $1.05
    3. We bought the 65 calls for $.05

    Adding up, $1.95 + (2 x -$1.05) + $.05 = -$.10

    Negative 10 cents?

    Correct. You just legged buying a structure that can never be worth less than zero for a credit. Arbitrage.

    What is the delta of our total position?

    Option traders want to stay delta-neutral. So estimating the deltas (or having Black Scholes spit them out) we compute the delta contribution of each strike and find we must sell or short 3,000 shares to be delta neutral.

  • Extra Credit: What’s the P/L?

    Butterfly P/L

    Using a flat 19% implied vol I get a Black Scholes value of $.93 for the butterfly. We actually got paid $.10 to own it. So our theoretical profit or edge is $1.03 times 200 contracts.

    $1.03 x 200 contracts x 100 multiplier = $20,600 profit

    Combo or Synthetic Stock P/L

    We bought 20,000 shares of stock synthetically for $54.95 via the 200 65 strike combos. If the stock is marked at mid or $55.025 then we made $.075 on 20,000 shares or $1,500.

    Stock P/L

    We did need to sell 3,000 shares at $55.00 (the bid) to hedge 3,000 shares or deltas. If the stock is marked at mid or $55.025 then we lost $.025 on 3,000 shares or $75

    Total profit: $20,600 + $1,500 – $75 or $22,025!

Wrapping Up

Back in those olden days, we’d play this game after market hours but you can imagine multiple brokers shouting orders at the same time and more months than just a single expiry. We studied many different types of arbitrage relationships so we could spot mispricings from many angles.

You’d take what you learned from these games and apply it during the trading day. You’d watch how market makers and brokers in the pits reacted to different orders as you start to piece the matrix together. At my firm, the people who performed best were sent to a Philly suburb for 3 months. This was known as “class” and it was held 4 times a year. “Class” was theory and option nerd stuff until lunch then mock for the rest of the afternoon. Mock had a simulation environment with electronic overhead screens just like the exchanges and everyone held a tablet PC with stock trading software and a proper option model. This is where you started going beyond mock and getting into more game theory and real-life trading scenarios.

The faster you got into a “class” cohort the faster you got your own “badge”, P/L, and risk budget (not to mention enough comp to rent a 400 sq ft studio without a roommate).

Times have changed. The game isn’t about mental math and yelling loud and having the best memory. But this was how my intuition was built up and the lessons still permeate how I think about trading today.

Finding Vol Convexity

In this post, we will learn what it means for a position to be convex with respect to volatility.

In preparation for this post, you may want a refresher.

  • Vega is the sensitivity of a P/L to changes in volatility. This is the exposure volatility traders are taking active views on. It requires tremendous attention since changes in vol directly affect P/L via vega but also impacts or distorts the “moneyness” of all options in a portfolio. In that way, large vega exposures are signs that the risks under the hood of a portfolio are especially dynamic.

    Refresher Post: Why Option Traders Focus On Vega (Link)

  • Convexity is the idea that there are non-linear P/L sensitivities within a portfolio. The curvature of the P/L derives from the fact that the exposures change as the market moves. Option deltas are not constant. That means deltas derived from options, as opposed to deltas derived from so-called “delta one” instruments like common stock or futures, are subject to change as the market moves. Vol convexity is the same phenomenon. Instead of applying to a delta, it applies to vega.

    Refresher Post: Where Does Convexity Come From? (Link)

In Moontower style we will do this without anything more than middle school math. This 80/20 approach provides the intuition without the brain damage that only a relative handful of people need to know.

Mapping Directional Trading To Volatility Trading

Directional Traders

Most investors are looking to profit from the direction of stocks.  Stated another way, most investors are taking active delta exposures. The size of their delta determines the slope of their P/L with respect to the market’s movement.

Directional Convexity

Some of these investors use options to make directional bets. This gives their positions convexity with respect to the changes in stock (also known as gamma). The convexity derives from the fact that their delta or P/L slope changes as the stock moves.

Volatility Traders

Now consider another, much small, class of investor. The option traders who try to keep delta-neutral portfolios. They are not seeking active delta exposure. They have no alpha in that game. Instead, they are taking active vega exposures. The size of their vega determines the slope of their P/L with respect to changes in implied volatility.

Volatility Convexity

Like the directional traders who use options, vol traders maintain convex exposures with respect to changes in the stock. Again, that’s gamma. But vol traders are much more focused on vol convexity. The reason vol traders focus on this more than directional traders is that vol traders typically run large portfolios of options across names, strikes, and tenors. These portfolios can include exotic and vanilla options. The presence of vol convexity means vol changes propagate through the entire portfolio in uneven ways. Risk managers model how vega exposures morph with vol changes.

For directional traders with just a few line items of options on their books, vol convexity is going to be much further down on the list of concerns. Somewhere in between “What’s for lunch?” and getting flamed by intern on Glassdoor.

Maximum Vega

Vol traders often think in terms of straddles. In fact, in many markets, brokers publish “straddle runs” every few hours. This is just a list of straddle prices and their implied vol per expiration.

At-The-Money Vega

A handy formula every novice trader learns is the at-the-money straddle approximation1:

Straddle = .8Sσ√T

where S = stock price
                        σ = implied volatility
                                     T = time to expiry (in years)

So if there is 1 year until expiration, the 1 year ATM straddle on a 16% vol, $50 stock is $6.40 (.8 x 50 x .16).

So if implied volatility goes up 1 point to 17% how much does the straddle change?

.8 x 50 x.17 = $6.80

So the straddle increased by $.40 for a 1 point increase in vol. Recall that vega is the sensitivity of the option price with respect to vol. Voila, the straddle vega is $.40

More generally this can be seen from re-arranging the approximation formula.

Vega = Straddle/σ = .8S√T

Ok, so we have quickly found the ATM straddle price and ATM straddle vega. Look again at the expression for the straddle vega.


There are 2 big insights here. The first can be seen from the expression. The second cannot.

  1. The vega of the ATM straddle does not depend on the level of implied vol.

    The vega only cares about the stock price and time to expiration. So whether you are talking about a $50 crazy biotech stock or a $50 bond ETF the 1-year vega is exactly the same even if the straddle prices will vary according to the implied vols.

  2. The vega of the ATM option is the maximum vega of any option in that expiry.

    This statement implies that the vega of an option varies by strike. All of the other strikes have a lower vega. They are less sensitive to vol than this one. That makes sense. This option has the greatest extrinsic value.

    (I have a confession. The maximum vega actually occurs at the 50% delta option strike, not the at-the-money or at-the-forward. I used ATM because it is more intuitive. The hand-waving should not trouble you. Going forward I will use the .50 delta option for the charts. If you need a refresher see my post Lessons From The .50 Delta OptionDon’t worry, the intuition is not going to change if you fail to appreciate the difference)

Vega Across Strikes

While we were able to compute the vega for the ATM straddle to be $.40 from the straddle approximation, how about the rest of the strikes?

For those, we need to rely on Black Scholes. You can find the formula for vega anywhere online. Let’s feed in a $50 stock, 0 carry, 16% vol, a 1-year tenor, and a strike into a vega formula. We will do this for a range of strikes.

Here’s the curve we get:

This chart assumes a single option per strike which is why the vega of the .50 delta strike is $.20 (not $.40 like the straddle vega).

The big takeaways:

  1. The vega of a non-.50d option does depend on the level of vol.
  2. There is a maximum vega any option can have and it occurs at the .50d option

The Source of Convexity

If option traders’ profits are a function of vol changes, then their vega positions represent the slope of that exposure. If the vega of the position can change as vol moves around then their position sizes are changing as vol moves around. The changes in exposure or vega due to vol changes create a curved P/L.

Let’s see how changes in volatility affect vegas.

When Vol Increases All Strikes Become Closer To .50 Delta

Here’s the vega by strike chart the same stock. The blue line assumes 16% vol across all strikes. The red line is 32% vol across all strikes.

In fact, imagine overnight, the stock’s vol doubled from 16% to 32%.

The maximum vega at any strike is still fixed at $.20, it just occurs at the new .50 delta strike. The .50 delta strike moved up $2 or about 4% but look how the vega of the options at nearly every strike increased. This is intuitive. If you double the vol then a strike that used to be 1 standard deviation away is now 1/2 a standard deviation away. All the OTM deltas are creeping closer to .50 while of course, the .50 delta option remains .50 delta.

Watch How Your Position Changes

You can start to see the reason why a position can be convex with respect to changes in vol. Imagine you were long the .50 delta option and short the way OTM 90 strike call.

  • At 16% vol the call you are long has $.20 of vega and the call you are short has 0 vega. You are unequivocally long volatility. Even if you are long 1 .50 delta call and short 10 90 strike calls you are long vol (1 x $.20 + (-10) x $0). Your portfolio’s net vega is long $.20 of vega

  • At 32% vol, the call you are long has slightly less than $.20 of vega since the .50 delta option has shifted to the right. Let’s still use $.20 to make the point. The calls you are short now possess $.05 of vega. Your new position vega computed as (1 x $.20 + (-10) x $.05) or -$.30 of vega. You are now short vol!

Your vega which represents your slope of P/L with respect to vol has changed simply by the vol changing. The higher the vol goes, the short vol you become.


  • The strikes near the meat of the distribution can only gain so much vega. Remember, maximum strike vega is only a function of spot price and time to expiry.
  • Further OTM options become “closer” to 50d. This pushes their vega up relative to the ATM option.

This chart shows vega profiles across strikes over a wider range of vols. At extreme vols lots of strikes look like .50d options!

Trade Examples

Long ATM option, short OTM option. (Long vega, short “vol of vol”)

Starting conditions:

Stock price =  $50
Implied vol = 16%


Long leg

        • 1 .50 delta call @$50.64 strike (approximately ATM)
        • Vega =$.20
        • Premium = $2.90

Short leg

        • Short 1 .14 delta call @$60 strike (approximately 20% OTM)
        • Vega = $.11
        • Premium = $.55

Summed as a vertical call spread

        • Premium = $2.90 – $.55 = $2.35
        • Vega = $.20 – $.11 = $.09
        • Note the position is long volatility

Now let’s change implied vol up and down.

It’s a busy picture. Let’s walk through the scenarios:

We start at 16% vol and increase vol

        • Call spread value increases (solid green line) because the position is long vega. Your P/L is rising since this is the position you are long.
        • However, the ATM call vega (blue dash line) stays relatively fixed while OTM call vega increases (red dashed line) causing the call spread vega (green dashed line) to decline from its initial value.
        • As vol increases your vol length is decreasing. Whoa, this looks like negative gamma with respect to vol!

We start at 16% vol and decrease vol

        • Call spread value decreases (solid green line) because the position is long vega. Your P/L is falling since this is the position you are long.
        • However, the ATM call vega (blue dash line) stays relatively fixed while OTM call vega decreases (red dashed line) causing the call spread vega (green dashed line) to increase from its initial value.
        • As vol declines your vol length is increasing. Whoa again, this looks like negative gamma with respect to vol!

Ratio Trade. Short ATM option, long extra OTM options. (Vega neutral, long “vol of vol”)

Same starting conditions:

Stock price =  $50
Implied vol = 16%

We are targeting a vega-neutral portfolio

New Portfolio

Long leg

        • Some amount of .14 delta calls @$60 strike (approximately 20% OTM)
        • Vega per option =$.11

Since the short leg has $.20 of vega and our long leg has $.11 of vega we need to buy 1.75 of the 60 strike OTM calls ($.20 / $.11) to have a net flat vega position.

        • Total vega for the long leg: $.20
        • Premium per option = $.55
        • Total premium = $.9625 ($.55 x 1.75 contracts)

Short leg

        • Short 1 .50 delta call @$50.64 strike (approximately ATM)
        • Vega = $.20
        • Premium = $2.90

Summed as a ratioed vertical call spread

        • Premium = $2.90 – $.9625 = $1.9375
        • Vega = $.20 – $.20 = $0
        • The position is flat volatility

You know what’s coming. Let’s change the implied volatility and look at the structure price. Remember you shorted the ATM option at $2.90 and bought 1.75 OTM calls for a total premium of $.9625.  

In other words, you shorted this structure for an upfront premium of $1.9375. Watch what happens to its value when you raise or lower the implied vol.

To understand why the structure behaves like this, look at the scenarios.

We start at 16% vol and increase vol

        • While both your longs and shorts increase in value, your longs pick up extra vega, while your shorts are already as sensitive as they will ever be to vega. So every uptick in vol causes the structures net vega to become long vol.
        • By being short this structure you are getting longer vol as vol increase. If your vol exposure gets longer as vol increases your exposure is convex with respect to vol.

We start at 16% vol and decrease vol

        • The opposite scenario occurs. As vol declines your short option leg has a fixed vega while your long vol leg that is OTM “goes away” as it’s vol declines. If vol is very low that call is extremely far OTM in standard deviation space. Imagine the extreme downward vol shock scenario — the stock is taken over at its current $50 for cash. All the options go to zero, the structure goes to zero, and you simply collect the premium you sold the structure at.
        • By being short this structure you are getting shorter vol as vol declines. If your vol exposure gets more short as vol falls your exposure is convex with respect to vol.

One last chart to drive it home. The green line is your P/L as vol changes. Notice that your max P/L in the vol declining scenario is $1.9375, the entire value of the structure. It is unbounded on the upside. It looks like the more familiar picture of being long a straddle! The fact that the P/L chart is curved and not linear is convexity and as we know, results from the size of exposure changing with respect to vol.

The blue line shows exactly how that vega exposure changes with respect to vol. You started vega-neutral. As vol increased you got longer. As vol fell you got shorter.

Caveats And FYIs

This was intended to be an introduction. But here’s a non-exhaustive list of “gotchas”:

  • Option surfaces usually have a skew. OTM options often trade at a premium volatility to ATM options which reduces the spread of vegas between the options. Less room for relative narrowing.
  • We haven’t talked about the cost of those premium vols. Short gamma, paying theta anyone?
  • In these examples, we shocked the vols up and down uniformly across the strikes. I’ll leave it to you to consider what adding a fixed amount of variance per strike would do to a vol surface.
  • I completely ignored the fact that as you change the vols you are changing the location of the .50 delta option, or for that matter the delta of every option. In other words, I showed fixed strike vol behavior assuming a uniform shock. Adjusting for that is recursive and frankly unneeded for the intuition.
  • Volga is the term for the sensitivity of an option’s vega with respect to vol. Volga itself changes as vol changes. That .50 delta option has starts with little sensitivity to vol. But if we crank vol higher that option moves further from .50 delta as the new .50 delta strike has moved somewhere to the right. So it follows that the old .50 delta is picking up volga, or sensitivity to vol. Again, no need to go full Christopher Nolan I’m just leaving breadcrumbs for the committed.

The main intuition I want you to get is that OTM options are sensitive to the vol of vol because their vegas can bounce around between 0 and the maximum vega. ATM options are already at their maximum vega. So structures that own extra options relative to be being short the ATM are convex in vol.


Vol convexity is important because changes in vol influence much of the greeks. Understanding the concept can be used for defense and offense. Vol directly impacts option prices according to their vega. But it also changes their vega.

Who should care?

  • Anyone who wonders how a nickel option can go to $20.

When you combine the convexity of options with respect to vol (volga) with the convexity of options due to changes in the stock price (gamma) you get nitroglycerine.

  • That segment of the market who takes active views on volatility, not direction.

Remember even when dealing with non-linear instruments, a snapshot of a portfolio at a single point in time might show it to be vega-neutral. But a photo of a car can make it look parked. Only the video can show how fast the car can move.

Hertz Roulette

In case you didn’t hear, Hertz got approval to sell $1B of common stock to the public which is bidding up its shares. Hertz however is in bankruptcy with its debt trading for less than $.50 on the dollar. Issuing stock in such a situation is unprecedented. These are unprecedented times I guess. The fact that stock traders are bidding for shares which are subordinate to debt that isn’t even going to be made whole feels awfully kangarooey.

  • Matt Levine’s write-up is funny and educational. (Link)
  • Alex Danco describes Hertz as the first example of new type of bubble — a view that the future doesn’t matter. This type of bubble is a new branch “on the financial tree of life”. He also expects this type of bubble to only happen once. (Link)

Danco’s description of the bubble dynamic in Hertz is a sign of just how meta-weird the world can feel sometimes. The dynamics of markets have become their selling point. Not their role. Automobiles were a transformative breakthrough as a transportation function. But anything that covers distance that quickly is even more fun to race and crash. To thrill-seekers, the convenience of faster transport isn’t even a secondary attribute. It doesn’t even figure into their framework of “what is a car for?”. If you find enough of those people, you have found a source of demand for the same product for different reasons. Hertz as roulette not equity. No veil of “investment” pretense.

So the question that remains — has anyone ever bought chips to play in a bankrupt casino? That’s what those new shares will be. Just don’t be the last one to cash out at the cage.