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.

The single biggest adjustment to get my head around when I crossed the chasm from equity options trading to commodity futures options was the idea that every option expiry was actually its own underlying.

In equities, a 3-month option on TSLA and a 1-month option on TSLA refer to the same underlying. The 3-month vol encompasses the 1-month vol. A 3-month option with the same strike as a 1-month option cannot trade cheaper than the 1-month option. Said otherwise, the calendar cannot trade below zero (well, with American-style options anyway).

This is not true in commodity options. A 3-month 75 call on WTI can technically trade below a 1-month 75 call on WTI even if they are the same IV simply because the 1-month future could be $15 higher than the 3-month future and therefore have $15 more intrinsic value. That example feels like cheating though.

Consider a more interesting case. I’m writing on the evening of 3/10/26:

The Nov16’ 2027 expiry 66 call, which is close to ATM, is about $6.25 at ~17.5% IV

The Nov17’ 2026 70 call, also close to ATM, is about $9.25 at ~ 42% vol

The shorter-dated call, which has less than half the DTE of the longer-dated call, is 50% more expensive! The futures price is 70/66 or 6% higher so it’s not the futures price driving the bulk of the difference.

It’s the extreme vol differential. If this was an equity, the implied forward volatility would be negative! Another way of saying this would be arbitrage.

Your equity option intuition is of no help here.

[A personal note here…this is also my favorite stuff. Equity options with their corporate actions and dividend headaches. Meh. Give me futures spreads and options on commodities all day. I loved building infra for this and trading these things. Those markets are very smart at pricing options but it also teaches you a lot about vol and risk.]

Measuring the forward vol in commodity options is a tricky problem. It was a pretty hefty component of how I’d trade commodity vol. I’m not giving away how I’d do it although I’ve hinted in prior futures-related posts at things that could get one started. This post will even fall under that category, but I’ll leave it at that.

Still, without getting into forward vols, there is a lot to understand about the risk of an option time spread in commodities. WTI, here and now, is putting on a clinic for I’m sure countless clueless option punters. And when it eventually dies down, many time spreaders are going to find themselves unpleasantly surprised as the surface finds a way to reveal that the obvious trade was but a trap.

Here’s a snapshot of 1M and 12M constant maturity IVs from CME QuikStrike. On March 9th, the ATM vol spread was 80 points wide. Prefer ratios? Fine, M1 was 3.5x the IV of M12

I’m going to look at realized vol data for the past year, data that is more conservative than this insane snapshot, to show how crazy you would be to think that this time spread is any way tradeable in a relative value sense.

What to expect today:

• How gamma works differently when your two legs settle into different futures contracts.
• h²: a single number that tells you how much gamma work your back-month leg is actually doing in front-month terms.
• I walk through what I’ll call the raw gamma paradox: M12 actually has more gamma per contract than M1. Except it’s a mirage.
• Why the fix of just buy more M12 vol detonates your vega and what this means for trading time spreads.

Data study setup

The analysis in this post is based on WTI M1 and M12 futures from
March 2025 to March 2026. The details and code can be found in the appendix.

The key features is we construct our own continuous contract for M1 and M12 and we estimate the gamma corresponding to constant maturity 1-month and 1-year ATM calls

Address the temptation head-on

You’re looking at crude oil options. We’ll take the vols down a notch, but if you receive my points with this more benign treatment, then it will make the current oil landscape hit that much harder.

Say M1 implied vol is sitting north of 60%. M12 is under 20%. You come from equity vol land, every instinct screams buy the back, sell the front. Look, this section is behind the paywall so there shouldn’t be any kids around:

Well, minister, don’t sully the cloak for a dream. The only prophesy your filling is the inevitable penance when M1 vol rips higher and M12 just sits there. Two things are working against you simultaneously. One of them shows up in your vega P&L.

The other one hides in a measure I refer to as h².

You need this measure to weight your option model’s gamma. To derive it, we’ll combine several concepts I’ve written extensively about.

Gamma revisited

A quick review is in order.

Gamma is curvature. Your P&L on a delta-hedged option over a single move is:

P/L = ½ · Γ · (ΔS)²

The ATM gamma formula for a Black-Scholes option:

Γ ≈ .4 / (S · σ · √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.

Back to our setup, you’d expect the long-dated M12 option to have less gamma than the short-dated M1 option since there is more time in the denominator. But in WTI right now, M12’s 1-year ATM gamma is actually higher than M1’s 30-day ATM gamma. Per contract, the back month has more curvature.

It will come back to that denominator in 2 ways:

1. The 12-month price is lower
2. Remember the scaling, DTE effect on gamma is less than vol’s effect

But we can account for all of this by updating hedge ratios.

We are going to review then expand on what exactly a hedge ratio is.

Hedge Ratio Squared: Mapping M12 Gamma Into M1 Move Space

To compare gamma across two different underlyings, you need a translation layer. You need to know: when M1 moves $1, how much does M12 move? In practical terms, if you’re long 1 M1 contract and want to be gamma-neutral with M12, how many M12 contracts do you need on the other side?

We start by recalling that beta (𝛽) is a vol ratio times correlation. A correlation of .70 means:

“If A moves 1 standard deviation, B moves .7 of its own standard deviation”

The vol ratio effectively normalizes the standard deviations of each asset. If the vol ratio is 1, then if A moves 1% then B moves .70%.

Review: From CAPM to Hedging

This allows us to express M12 exposures entirely in terms of M1 price moves.

This chart pulls all of this together.

• We see that the # of M12 contracts (1/h) you need to hedge M1 is exploding as the beta collapses.
• Beta is collapsing mostly due to the vol ratio plummeting as opposed to the dip in price ratio and correlation.

h is the hedge ratio for delta.

Before we derive hedge ratio for gamma, we need a quick review of gamma p/l.

Gamma P/L

M1-Equivalent Gamma

The M1-equivalent gamma of the M12 option is therefore:

Notice how:

• Delta scales with h
• Gamma scales with h²

Based on our data, and letting realized vols also stand-in for implied vols, we get this table:

h² has collapsed to its all-time low in this dataset. The 1-year mean is 38.6%. We’re at 2.15%:

We have a very practical question we need to answer with all this arithmetic:

What does this mean for the risk of a time spread?

The Raw Gamma Paradox

The adjustment of “hedge ratio squared” is so powerful it can flip a sign.

Look at the raw gamma numbers:

M1 30-day ATM gamma: 0.0241 per $1 move

M12 1-year ATM gamma: 0.0315 per $1 move.

M12 has 1.3x more gamma per contract than M1. And this is comparing a 1-year M12 option to a 30-day M1 option.

The longer-dated option has more curvature.

How?

Remember the formula: Γ ≈ .4 / (S · σ · √T)

M12 has a lower price ($67 vs $85) and much lower vol (18.7% vs 66.9%). Both of those boost gamma. The price and vol effects are swamping the time-to-expiry effect. M1’s 30-day option should have screaming gamma from the short DTE, but the vol is so high it crushes the curvature. Meanwhile, M12 is a lower-priced, lower-vol contract where the gamma can concentrate even at the 1-year tenor.

You might look at that and think: great, I’m long the gamma-rich leg…until, of course, we impose the h² adjustment.

The hedge ratio (h) is only .0215.

M12 1y gamma in M1-equivalent terms = 0.0315 × 0.0215 = 0.000677.

Instead of the back month having 30% more raw gamma per contract (ie .0315 vs .0241) it has 97% less (.00067 vs .0315).

You need .0315/.00067 or about 46x more M12 contracts than M1 contracts to be “gamma-neutral”. In other words, you need “the square of the hedge ratio” quantity of contracts to be gamma neutral.

💡In the context of turning the hedge ratio into contract, quantity we use the inverse (ie recpriocal) of the hedge ratio. The hedge ratio (h) is telling us that M12 is only offsetting ~2% of the risk of M1 so we need 1/2% or ~50 contracts to hedge

Typically, h is about 1.4, requiring only a 2:1 option hedge ratio (1.4² = 2)

What does this do to your vega?

The vega of a 12-month ATM option is √12 or ~3.5 greater than the vega of a 1-month ATM option. If you are long a 1-year option time spread you are long vega. But if we assume that vol changes themselves are proportional to √T then you could argue that your scaled or normalized vega is flat.

If you want to be gamma-neutral, you’d typically need about 2x as many 12-month options because of the typical h². You can’t solve for being gamma-neutral without being long vega. But now the conceit becomes especially ridiculous when h² collapses to .0215. You’d need to be long an outrageous amount of vega to be gamma-neutral.

The position being completely uncomfortable tells you something. These options have nothing to do with each other. The two risks are knotted together by h², and when h² is at 0.0215, they’re not touching. You might as well be spreading options on 2 different assets.

It’s the same problem with pair trading vols. In a normal circumstance, 2 assets might have a reasonably strong correlation. But once one leg has an idiosyncratic episode, it turns into the equivalent of M1 in our analogy. You can mitigate some of this by not pair trading vols on individual equities, as inter-equity correlations will be more volatile than inter-sector or inter-index.

[For folks on exotic vol desks, you will remember some pretty insane dispersions in international index vols circa 2018 coming out of the worldwide vol depression of 2017].

Spread Gamma

Mechanically, your unadjusted option model might show your long time spread is long gamma. But as oil rallies and your front month delta gets short relative to your back months, you are, in the parlance of commodity trading, “short spreads”. You are short M1 and long M12 due to gammas as M1 goes up much faster than M12. So your headline greeks might say you are “long gamma” but a commodity trader would immediately recognize that this position is short “spread gamma”. It’s not exactly the same as being short calendar spread options (topic for another day) but it’s similar so long as the spreads have a positive beta to the M1 future. In other words, if M1 always moves more in dollar terms than the months behind it, whether it’s to the upside or downside.

Real-life risk

One of the great features in the ICE Option Analytics software (formerly Whentech) was the multiplier column in the futures configuration. It allowed you to enter a hedge ratio for each term. So, for example, if you thought that M12’s hedge ratio was .50 then your software would say that long 100 M1 and short 200 M2 was a flat delta in the summary risk. You would, of course, still pay attention to the spreads you had underneath.

On any given day, the futures spreads might underperform or outperform the hedge ratio parameter, introducing noise into the p/l you expected for a given futures move. But critically, the software also adjusted your gammas in each term by the square of the hedge ratio.

[You can thank me for this. When the product was still in beta days (no pun) around 2005, I was the one who spotted that gammas were only being adjusted for the hedge ratio, not its square. You notice these things when your p/l doesn’t seem to line up with your expectations based on your greeks.]

Manually updating the h’s in your model is a hands-on way to feel just how volatile they can be. I would keep a separate spreadsheet with realized vols and correlations and revise the hedge ratios once a week or so.

[For seasonal commodities, h is not just a noisy function of DTE, but depends critically on what month you are in. A “3-month” option in WTI is always kinda the same thing, but a 3-month option on corn in Sep is very different from a 3-month option on corn in May. That spreadsheet had more hair on it for the seasonal names.]

Wrapping up

Today you learned how to properly weight your model gammas. If you plan to trade option portfolios in a professional setting you will impale yourself without understanding how gammas stack.

These ideas will help you group gammas in related names to summarize risk more intelligently, but it will also alert you to when the risks that you think are related simply aren’t.

Appendix

METHODOLOGY
===========

Universe:        WTI crude oil futures, M1 (front month) and M12 (12th month)
                 Contracts roll monthly (CLK5, CLN5, CLJ6, etc.)

Period:          2025-03-28 to 2026-03-09 (237 trading days with complete data)

Returns:         Daily log returns on M1 and M12 settlement prices
 
Realized vol:    20-day trailing annualized based on daily close-to-close
                 Computed separately for M1 and M12

Beta:            20-day rolling return correlation * vol ratio 12m/1m

Hedge ratio(h):  M12 contracts needed to delta-hedge 1 M1 contract
                 1/(beta * 12m price / 1m price)

h²:              gamma multiplier
                 Γ_M12 in M1-equivalent terms = Γ_M12 × h²

Gamma:           Black-Scholes gamma for ATM call option on [M1,M12]
                 S = price, σ = trailing 20d RV, T = [30/365, 365/365], r = 0

Caution:         Implied vol set equal to trailing 20d realized vol
                 (i.e. options are priced at current realized, not market implied)

Code:            https://github.com/Kris-SF/data-pipelines/blob/main/wti-futures/wti_m1_m12_returns.ipynb

Data Source:     IB API

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

The materials for all the classes live here:

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

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

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

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

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

Investment Beginnings — Class 2: The Math of Investing

Class 1 was about building a business.

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

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

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

What we covered:

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

Hands-on:

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

Homework:

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

Resources:

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

Full video:

Money Angle For Masochists

Junior Masochists

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

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

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

I pulled up ZM’s price chart:

I asked…”what do you think happened?”

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

Look at the revenues from this Twitter post:

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

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

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

A handy decomposition:

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

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

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

You could estimate:

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

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

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

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

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

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

A crap business that investors sold too cheap.

For our regular Masochists

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

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

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

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

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

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