In this issue:
- math in the car with kids
- trader quick math
- from straddle to gamma
Friends,
My older kid is getting braces in a few weeks. Based on the expected time he has to wear them, it’ll cost about $350/month. That’s a car lease. I’m not complaining (God: “he’s complaining”), I just suffer from chronic numeraire substitution. I’ll come back to the braces thing in a bit, but let’s chat some other stuff for a bit.
My sons are in 4th and 7th grade. A nuisance I will one day miss is shuttling them and their friends all around. We talk about lots of stuff, but stuff is often made of numbers, so I end up teaching them how to reason numerically about real-life stuff in an organic way in the context of things they find interesting. Yay. Except they groan because they know it’s coming. But I believe in osmosis and their future selves will be thankful. Or at least have endearing stories at my funeral about their old man being a crank who also happened to love them. And since they might have kids of their own one day, appreciate, just as I do now when I think about my parents, that we’re all just making shit up as we go.
Where were we before my inner monologue took over, ah yes, car convo. I got the boys in the car with another friend headed to practice. The 7th graders said they were learning scientific notation. Shouldn’t have told me that. Immediate quiz. Represent 1/50th in scientific notation.
I was impressed. I listened to his friend reason aloud for about 20 seconds before getting 2 x 10⁻²
Zak got the answer faster than I did. The Math Academy lessons are paying off.
Why is scientific notation useful?
To torture us.
Besides that.
“I don’t know, [proceed to fumble around for explanations before landing on something that tracks]. Because we need to measure stuff in micrograms? Is there even such a thing as micrograms?”
Very good. That makes sense. From the stars to bacteria and atoms scientists deal with things that are really big or really small. It’s right there in the name: SCIENTIFIC notation. We talked about how insane the idea of a light year is for a bit before arriving at the gym but not before I told them next time they watch YouTube, instead of watching Jesser we’re gonna learn about the Fermi Paradox which they theorized naturally but didn’t realize it was a famous contradiction.
On the way home from practice, the kids started talking about IQ. I forget what the comment was, but it indicated that they did not understand that an IQ of 100 is normalized to be the average. Sweet. We get to learn about bell curves right now.
I explain that 15 points is 1 standard deviation which encompasses 68% of the population. So to be greater than 1 standard deviation means being in the top 16%, since the 32 remaining percent have to be split between the lower and upper parts of the population, leaving 16% ABOVE 1 standard deviation.
2 standard deviation outperformance means top 2%.
I note that my scientific notation quiz asked for 1/50. Your father is psychic.
[Between that and the fact that I predicted that Axl Rose, who’s friends with AC/DC and lives in LA is probably at the Rose Bowl concert we were at last May, only to have him walk out from backstage about 60 seconds after I said that, they might think I’m a witch.]
Then we do 3 standard deviations. That encompasses 99.73%. For just the upper, it’s about 1.3 per 1000; let’s call it 1 in 750.
Given the size of your middle school, there are probably 2 kids that smart.
Except for that your school isn’t a random draw from the population.
We’re a long way from where I grew up. That night I explained to them that the test they took in 3rd grade, where they got 2 standard deviations above the mean, wasn’t even close to getting accepted to the local GAT program. Sorry boys, you’re not Asian enough and that’s on me.
Wanting to change the topic from IQ, I brought up height. After all, we just left hoops. I invented some numbers. The average adult American male is 5’9 with a standard deviation of 3”.
We stepped through the progression.
A 6’0 man is taller than 5 out of 6 men. (1 st dev)
6’3 and you’re 1 in 50. (2 st dev)
6’6 and you’re 1 in 750. In the running for the tallest boy in H.S. (3 st dev)
(Although selection effects need another nod here).
7-footers are 5 sigma. Using just the right-tail probability that’s 1 in 3.5 million.
This was a chance to apply their newfound knowledge of scientific notation.
How many 7-footers do you expect in the world if there are 3.5 billion adult men?
A million is 6 zeros. 10⁶. A billion is 9 zeros.
9 zeros divided by 6 zeros leaves us with 3 zeros.
We expect 1,000 7-footers.
Google says it’s estimated that there is “2,800” 7-footers in the world which the CDC statistically extrapolated using a standard deviation of 2.9 to 3. Small differences add up when you start adding sigmas such that our final estimate is off by a factor of 3. But hey, the right order of magnitude.
While we were countin’ sigmas the 9-year-old wants to know how Wemby exists. Wemby is officially listed as 7’4. There’s online debate as to whether his height is underreported and if it’s really 7’5. We’ll use that since it’s 6 standard deviations.
Siri, what’s the probability of an event beyond 6 standard deviations? 1 in 500mm. One-tailed, 1 in a billion. Wemby.
Statistically speaking you wouldn’t expect to have enough 7’4 mutants to assemble a starting 5 lineup but in reality you there’s enough of them to at least field a football team. Waves hand in the shape of epsilon.
Anyway, in service of handy takeaways, it’s useful to remember that a 3 standard deviation extreme on 1 side of a bell curve occurs about 1 in 750. For quick math, call it 1 in 1,000 or 10³. So if you’re talking about the American population of 3 x 10⁸, the number of 3 sigma people on a particular trait would fit in an MLB stadium.
Or about the same number of people who subscribe to moontower. See, you’re all 3 sigma! ❤️
Speaking of…
Moontower is 7 years old.
The first issue was March 17, 2019. This is Moontower #307, Munchies is up to #146, there’s been 96 paywalled posts, plus possibly the single largest archive of options blog posts on the internet. (ChatGPT mentions Larry McMillan and Kirk DuPlessis as being similarly if not more prolific.) Fyi, nearly everything I’ve ever published is indexed here and religiously updated so when robots erase me it is in totality. Thanks for following. I never expected to be writing this long. I didn’t expect anything.
Addendum on braces:
I wore braces from freshman to senior year of high school. My son will get his off about a month into freshman year. How’s that for generational progress.
The braces thing conjures something of a subway platform riddle for demographics where I can’t tell if the world is moving or me. My little guy got “spacers” in 3rd grade and will wear a retainer for 2 years. I’m like, is getting braces twice a new thing, or something I just never would have seen in my strata growing up?
I’ve noticed 2 other versions of this demographic subway platform riddle.
The older kid is now past the halfway point of middle school and I still never hear of fistfights. Growing up, at least every other week, the beacon went up, “FIGHT!!!”. Social class or changing times?
Finally, skiing. This one isn’t a riddle but it’s so jarring. I was 20 years old the first time I stepped on a mountain. Here’s school in the winter feels optional. Everyone has a cabin in Tahoe, all the dads are metereologists, and an expert on MTN stock.
Cold, heights and ski lifts, driving on dangerous roads?
I think I’ll just binge Nelly & Ashanti: We Belong Together thank you very much.
[We did knock this out in 2 nights. Plenty of time to cancel the 1-week Peacock subscription it required. I friggin’ love Nelly and so much more after watching the show. He comes off as an amazing father, raising both of his own as well as his sister’s kids when she passed at a young age. The only thing that bugs me is how good he looks at age 50. Save something for the rest of us bruh.]
Money Angle
Dean Curnutt graciously invited me to be on his outstanding podcast. His prompts led the conversation towards useful stuff. The description:
We begin with developments in commodity markets, particularly crude oil, and silver, where geopolitical tension and speculative flows have led to sharp changes in volatility surfaces. Kris explains how option skew in underlyings like oil can reprice rapidly during shock events, leading to inverted termstructure and a well bid call skew. These dynamics create unusual behavior in vertical spreads and probabilities implied by option prices.
Kris describes how the relationship between spot moves and volatility changes across market environments, emphasizing that traders must continually recalibrate their models. What appears to be a stable relationship—such as the familiar beta between the S&P 500 and the VIX—can shift quickly depending on positioning and market structure.
A major focus of our conversation is on the mental math traders use to interpret option prices without relying on models. Kris walks through several shortcuts that allow traders to move quickly between volatility, straddle prices, and probability estimates. These approximations help traders identify when prices look unusual and whether options markets imply probabilities that diverge from other markets.
Finally, we discuss the work Kris is doing on financial education. Inspired by teaching his own children about investing and compounding, he has begun running small classes for students and sharing the materials publicly. The goal is simple: introduce younger investors to concepts like time value of money and long-term compounding earlier in life.
If you are interested in a step-by-step breakdown of how I found an estimate of an out-of-the-money put like I did in that interview this post is for you:
🔗building an option chain in your head
Money Angle for Masochists
A topic I could have rattled on for much longer in that interview with Dean is trader mental math devices. By now y’all know option traders have the ATM straddle approximation burned into their retina:
straddle ≈ .8 Sσ √T
A useful approximation I did not explain in the interview is the similar-looking ATM gamma formula for a Black-Scholes straddle:
Γ ≈ .8 / (Sσ√T)
The three things that shrink gamma are in the denominator:
Higher S (price): The same $1 move is a smaller percentage move on a more expensive underlying.
Higher σ (vol): The option is already “priced for action.” The curvature of the price function gets spread over a wider range of expected outcomes. More vol → flatter curvature near the money → less gamma.
Higher T (time): Same logic as vol. More time spreads the curvature out. The more time to expiry the less a given move influences the delta of the option. The delta of 10-year option is not going to change much based on how the underlying changes day-to-day.
A couple of educational points:
- Take note of the scaling. Double the vol, gamma roughly halves. You need to quadruple DTE to get the same effect.
- As always, a good habit when trying to understand greek levers, is to take examples to extremes. If you raise DTE or vol to infinity, all options go to their maximum value. For calls, that’s the spot price itself. For puts, it’s their strike price. That means calls go to 100% delta since they move dollar-for-dollar with the spot. Puts go to 0 delta. It doesn’t matter where the spot price goes, the option is already at its max value. It doesn’t change. If a call is 100% delta and a put is 0% delta, the option has no gamma. Its delta doesn’t change with respect to the spot.
Going back to those formulas for a moment:
straddle ≈ .8 Sσ √T
Γ ≈ .8 / (Sσ√T)
The denominator of gamma = straddle/.8
Substituting:
Γ ≈ .8 /(straddle/.8)
Γ ≈ .8 /(straddle/.8)
Γ ≈ .64 /straddle
So when you want to do mental math you take “2/3 of the inverse of the straddle.”
This might sound obtuse, but taking inverse or “1 over” some number should be one of the fastest mental math operations anyone dealing with investing does. After all, when you see any ratio like P/E or P/FCF you are immediately flipping that to a yield where it can be compared with things like interest rates or cap rates.
If a straddle is $5, the gamma is 2/3 of $.20 or ~.13
And we know that doubling the straddle halves the gamma so you can just memorize that a $10 straddle has ~6.6 cents of gamma and linearly estimate gamma for any straddle price relative to that (ie $20 straddle is about 3.3 cents of gamma and $15 straddle is in the middle of 3.3 and 6.6).
And of course there’s time scaling. To find an option that has double the gamma you need to cut the DTE by 1/4.
Keep flipping this stuff over in your head, it’s satisfying, and it thickens the myelin around whatever brain cells you sacrifice to options damage.
If the 9-year-old can do it, so can you.
(I’m kidding. I just found this moment of deep thought cute. Between both kids’ basketball lives, the gym has become my office. Max does his Math Academy after his practice while waiting for the bro to finish. He recently discovered my Kindle is a scratchpad which has made my “no math without scrap paper” rule less of a nuisance. I adopted that rule from Math Academy’s recommendation, my affinity for mental math notwithstanding.)