Certain topics re-infect popular social media discourse like an engineered cold virus. Like when the NY Times is having a slow week and drops the “Couple Can’t Make Ends Meet in NYC on $500k a Year”

This is not a fresh take. Sorry to be crass, but NYC is a beacon for dreamers around the world who want to partake in a life tournament. It is entirely indifferent to your needs.

Plus, $500k is not what it was. Also not news.

[The number of millionaires in the US has doubled twice in 20 years. For the investment-brains Rule of 72’ing that in their heads, it’s 7% growth in millionaires per year.

Also when you break out their expenses, it always includes “private school”, “max 401k contribution”, and a parade of “needs” that insult normal people’s definition of “ends meeting”. The headline should say “I make $500k and don’t feel rich”. And for that our nice couple that got As in school should be banished with their expectations to either 1992 or literally any place that is NOT the final table at the Main Event. Sorry, but aliens only need apply. ]

The fintwit version of “discourse that doesn’t die even though everything we know about the situation is evergreen” is whether one should use a stop-loss.

I’ve watched this conversation come and go so many times. I can no longer restrain the impulse to do that thing that nuisance colleagues do in meetings… raise my hand to hear my own voice.

[You know as soon as that hand goes up you’re just “omg please kill me what set of life decisions brought me to this moment in the universe where I have to hear this pick-me baby perform this routine in a conference room in front of a weary audience of which 2 are muted on a zoom call with their camera off in their car on the way to Starbucks for the will to finish Wednesday” ]

With that pep rally…

let’s talk stop-losses

First, the obvious:

A stop-loss is a risk management tool, not an alpha tool.

The purpose and effect are to change the shape of your P&L so you can endure.

The cost of a stop-loss, aside from slippage and transaction fees, is sometimes cutting eventual winners.

Euan Sinclair gives this a proper treatment with math.*

The tribal intuition is closely related to the idea behind a one-touch option.

Remember this from Sunday?

You can also analogize to options trading.

Gamma hedging is not an edge. It’s a hedge. It reduces position size. It’s a cost.

[See a misconception about harvesting volatility]

Stop-losses, like gamma hedging, are a kind of passive flow — they’re independent of signal and discernment.

They’re forced in the sense that the time of trade chooses you, not the other way around.

If you chose the time because you thought you had an edge, you’d evaluate it based on alpha criteria. But since it’s risk management, you benchmark it against other ways to manage risk — like starting with smaller size.

If you place a resting bid or offer and it gets filled, you’re probably losing to that trade. It’s a stale order.

But — if you’ve decided that the sum of losses from stops is less than the counterfactual of trading smaller from the start, then it’s accretive.

My gut is that most people have no idea whether that’s true for their strategy.
Which is just a sub-instance of a bigger issue: they don’t fully understand their edge.

Not a damning criticism — the admonition is a matter of degree. Having an edge is hard enough. Appreciating all the contours of that edge is even harder.

If it were easy, professionals wouldn’t blow up. But they do.

Even market-makers, whose businesses hinge on understanding risk and tradeoffs, sometimes blow out. They weren’t clueless, but every serious professional still has open questions about their strategies that bump up against the types of tradeoffs we examine when constructing risk rules.

The best firms are probably closest to the efficient frontier of those tradeoffs.
Clueless tourists are far below it — and often don’t even see the problem.

For the narrow audience of vol traders, I think the traditional stop-loss framework makes little sense.

I explained why in this 5-minute clip from The Trade Busters.

To wrap up…

I’m not a directional trader, so maybe this doesn’t mean much, but I’ve never placed a hard stop order in my life.

If I want something with a stop-like profile, I use options. Otherwise, I size the risk appropriately at the start (imperfect, but I prefer this to overbetting and then using a negative expectancy maneuver more frequently as a risk management tool. If you have been on an options desk you know that traders are obsessed with “how do I hedge less?”).

If the risk grows, which happens because vol is not constant — then I check:

Is the risk bigger than what’s allowed?

If so, reduce the position.

Follow the risk protocol.

It’s not about price levels.

It’s not about P&L memory.

It’s a binary: Is the risk too big or not?

I go deeper on that in this short clip.

I’ll leave it there. I’m done with this topic. Conceptually it’s not hard. It’s a trade-off and the details of that trade-off matter with their relevance varying with the strategy.

[Trend-following is a good example of a strategy that strongly lends itself to stops. It’s built into the premise. It’s managers understand exactly what the trade-offs are in quasi-replicating an option that samples vol over longer periods because they know more frequent sampling understates vol in the presence of auto-correlation.]

If someone is religious about the utility of stops in some general sense without parsing the properties of the underlying risk, I’m suspicious that they are parroting some guru. Or trying to make their NYC rent by selling investment tips.

Eh, who are we kidding… these YouTubers are always telling you to smash the subscribe button from a subdivision in South Florida.

*Euan discusses stop-losses in chapter 9 of Positional Option Trading. This is just a blurb but the chapter is more technical:

• stops are complicated:
  • Many trades that would have been winners will have been stopped out, so it is not as simple as assuming that you are just cutting off the left tail of the distribution. You would need to know how many trades would cluster at the stop threshold [This is a question of path].
  • Simple simulations show that the expected value of a strategy will fall if you use a stop, although you shed the large losers.
  • Trailing stops cost even more than a fixed stop because they are always in play, as opposed to a fixed stop which gets further away.
  • Stops don’t just stop losses. They drastically change the shape of the return distribution and can lower the average return. Adding stops won’t transform a losing strategy into a winning strategy. The only reason that we would add a stop is that we prefer the shape of the stopped distribution.
    • Stops make more sense if we are trading momentum and less sense if we are trading mean reversion. [Kris: note that much of option trading is mean reversion on some meta-level]
    • A position should be exited when we are wrong. Sometimes this will coincide with losing money. In this case, a stop is harmless. But sometimes losing money corresponds to situations for which we have more edge. Here, a stop is actively damaging and contrary to the idea behind the strategy. [Kris: Fully agree and why I believe in risk rules that are independent of P/L for option trading and more thoughtful of ex-ante risk shocks]

2 weeks ago I shared this tweet:

By saying I’d buy that proposition I’m saying “I’d buy that vol”

Andy, responded with a joke about what’s the “one touch” option worth, which I asked him to delete because I really would have liked a fool to sell me that proposition and Andy’s comment gives away the sauce.

[X’s killer app would be to escrow bets, but that’s another convo altogether]

I’d be a size buyer of a 50% probability that we get back to the highs before the end of the year.

It’s not bullishness. It’s vol trading. If you could buy that probability for 50% you could arb it vs the value of a one-touch call that pays off “if the stock ever touches or exceeds the strike price before expiry’.

That call will trade for a higher implied probability than 50%.

I’ve alluded to the rule of thumb before, but a one-touch probability is approximately twice the delta. At the time Andy tweeted, the highs would have corresponded to the 10% out-of-the-money call strike.

If we double the delta of that call, we estimate the one touch probability. Given SPX vol, I knew immediately that a 10% OTM call expiring on Dec 31 has a higher delta than 25% so the one-touch probability would certainly be at least 50% bid.

So as a matter of education there’s multiple lessons in this simple exchange.

Estimating the one-touch probability

Using our rule of thumb, we just need to estimate the delta.

@quantian’s got that trader reflex — do the napkin math:

What did he do in steps:

1. Used VIX as a proxy for implied vol which is annualized
2. Scaled it to 9 months using √time or √.75… 25% * √.75 ~21.7%
3. 10% / 21.7% ~ .46 sigma
4. 1- normdist(.46) ~ .35 probability [assumes probability ~ delta]
5. Double the probability —> 70% chance of one-touch

😈Possible enhancements to the estimate if you are a masochist:

• With VIX at 25 I’d expect the term structure to be inverted so 9 month vol is lower. This would lead you to be more conservative on your delta. Sigma is directly proportional to our vol estimate so if we use 75% of the vol our sigma increases by 1/3 (since we dropped lowered the vol by 1/4). Normdist is not a linear function so still use the calculator and that pushes the delta down to .27 or 54% probability.
• The delta depends on the the moneyness of the forward price not the spot price and since interest is greater than dividends for SPY the delta is a touch higher.
• We’ll assume the positive skew embedded in Black-Scholes lognormality assumption offsets negative call skew. In sum I’d sell quantian’s delta of .35 and buy a delta of .27 so let’s call the fair delta .31 and the probability of the one-touch as 62%. I love getting even money odds when I should be laying 5-3.

Someone else knows their options btw:

My response to quantian showed another way to conservatively estimate that the probability was higher than 50%

Intuition for why one-touch probability is 2x the probability of expiring ITM

I’ve discussed the shortcut before in crossing over zero. But I sketched an intuitive example using a trader’s favorite binkie — the binomial tree.

The image is self-contained explanation and follows from a simple assumption of a stock 50/50 to go up or down $1.

Starting from $100 and traveling 5 periods what’s the probability of the stock expiring $99 vs the probability of it “touching” less than $99?

Learn more:

Estimating the probability of a stock expiring above or below a strike from the delta works for relatively low vol or short dated option. The below post explains the real meaning of delta and why/how that estimate can break down:

Lessons from the .50 delta option

 

Trader Math

I’ve included the one-touch probability in the wiki-style collection:

Math Shortcuts Traders Know By Heart

The “weekend effect” in options refers to the tendency of implied vol to increase on Mondays.

Why does it happen?

First, perplexity.ai provides not only a decent start. In fact point #5 is quite impressive and clue-y.

The trading implication, point #6, is somewhere in between incomplete if you know what you’re talking about to dangerous if you are a novice reader. I’ll address later in the post.

Perplexity does admirably but it doesn’t get to the core.

My condensed summary:

The weekend effect is a mathematical artifact that presents itself as “implied vol increases on Monday”.

Why?

Empirically, from Friday’s close until Monday’s open, options typically do not decay as much as the model’s theta would predict.

Therefore, to fit the the Monday a.m. option price, the implied vol must necessarily be higher than it was on Friday.

This post will explain why this is a mathematical artifact as opposed to a real change in the IV.

If the upward change in IV is an artifact, then does that suggest that an unchanged IV means vol is actually down?

Yep. Sure does.

Right now this tweet is a puzzler:

I’m going to explain it in detail.

And for your part, you’ll walk away understanding weekend theta and volatility time in a new way.

This post is going to get you over the line by zooming in on weekends in a basic way.

However, you could reconstruct all the ideas if you deeply understand this benchmark post:

⌛Understanding Variance Time

In that post, we start high and you can use it pinpoint how to think about weekends.

In this treatment, we zoom in on a weekend and you could use that to scaffold your own construction of the high-level post.

 

Day counts

Conventional option models such as Black Scholes accept a days to expiry parameter. This input is represented as a fraction of a year.

When you encounter an option calculator in the wild, this fact is abstracted away. A calculator usually asks for a trade date and expiry date. If the trade date is March 1, 2025 and expiry is June 1, 2025 there is 92 DTE.

The model, if it uses a 365-day tenor, will convert 92 DTE to 92/365 = .252 years. This is known as a calendar day model.

You can have models that specify other tenors. A common variation is the business day model. It will have a 251-day tenor because it starts with 365 days but subtracts weekends and holidays (don’t forget Juneteenth).

From March 1 to June 1 there are 63 business days. DTE for that model is 63/251 = .251

There is slighty more DTE in the calendar day model than the business day model. Therefore, for a given option price, the calendar day model will have a slightly lower implied vol than the business day model. After all, both models are looking at the same option price but the first model thinks there’s more time to expiry.

On January 1, there is 1 year to expiry (365/365 or 251/251) for both models. But as soon as the clock starts ticking, the DTE between the 2 models diverges. The divergence depends on the ratio of business to non-business days until expiry.

How much time has elapsed?

Armed with this knowledge, let’s measure how much time has elapsed from Friday’s close to Monday’s close.

1. Calendar day model

3 days have elapsed.

Friday pm to Saturday pm, Saturday pm to Sunday pm, Sunday pm to Monday pm

Using the calendar day tenor, 3/365 or .82% of a year has elapsed.

2. Business day model

1 day has elapsed.

From Friday pm until Monday pm is just 1 business day.

Using the business day tenor, 1/251 or .40% or half as much time as the calendar day model!

Therefore from Friday’s close to Monday’s close, relative to a business day model, a calendar day IV will need to be ratcheted higher.

Relativity

What’s more “correct”, a calendar day model or business day model?

Wrong question.

There is no “correct”.

Consider what we know.

• Most conventional models use a 365-day tenor. The VIX calculation, the representation IV most broadly tracked, uses a 365-day tenor.
• Market observers are so used to seeing IV increase on Monday that they give it the name “weekend effect”

I’ll let you think about it. Based on the comparison with a 251-day model can you explain the effect?

 

Ok.

Here’s the logic…relative to a 251-day model, a conventional calendar day model must see its IV increase on Mondays. If this is a recurring trend, we should deduce that the market’s consensus is not decaying the options as much as the 365-day model predicts. The market’s pricing behavior strongly suggests that it believes time passes more slowly over a weekend.

I used the 251-day model to be extreme. The 251-day model assumes time doesn’t pass at all over a weekend. Time did not tick until Monday transpired. From Friday to Monday’s close it thinks 1/2 as much time to expiry elapsed.

It’s not that the market uses a different model. The market is not a monolithic entity with one model setting option prices. Consensus does. And the sum of everyone’s opinion suggests that time passes more slowly over a weekend than calendar models predict.

You can also appreciate this by inversion.

If you use a business-day model the weekend effect is inverted…IV looks like it falls on Mondays!

It makes sense.

The 251-day model thinks no theta happens on the weekend days, so the fact that option prices are lower on Monday am than Friday pm makes their model think that IV must have fallen.

It seems like we have a 2-sided market.

A Saturday is neither a full trading day or a zero. It’s worth something between 0 and 1 day.

Split the difference

What if I say that weekends and holidays are half days? Said otherwise, variance time passes 1/2 as fast.

If we continue to denominate our basic unit, a full trading day, as 1.0 and weekend days or holidays as .5 we get the following tenor:

251 x 1.0 + 114 * .5 = 308

We have a 308-day model that decays days at different rates.

When a Tuesday rolls off the calendar 1/308 or .3% of a year elpased.

When a Sunday peels off, .5/308 or .15% of year elapsed.

Turns out if you use a calendar like this, weekend effects are dampened. Which means your model is decaying time closer to the market’s consensus. The IVs in your model need less adjustment to match the market.

Overnight

We need to talk about one more topic before I can explain the tweet above.

It starts with a question.

In a calendar day model, how much time has elapsed from say Thursday’s close until Friday’s open? In “wall” or clock time it’s 4pm est to 9:30am est or 17.5 hours.

But if time passes slower on a weekend, it’s reasonable to assume time passes slower overnight if we start getting to the hour level. After all, I don’t think 17.5/24 or 73% of an option’s daily decay occurs just as the market opens.

If we define 1 day as close-to-close then we need to defined close-to-open as some proportion of 1 and 73% sounds way too high.

The US is the last market to open after Asia and Europe so it’s reasonable to assume that the overnight includes more than zero volatility (in the spirit of “if a tree falls in a forest”, even if the US was the first to open the overnight would be worth more than 0. The world happens when the market is closed. At the very least BTC volatility could be used as a ruler to measure with).

You could do a big study where you measure the ratio of close-to-open variance to close-to-close variance but for the purpose of this post I’ll use .25. It’s in the right ballpark.

Back to our question: how much time has elapsed from say Thursday’s close until Friday’s open?

.25

How much time elapses from Friday’s close to Monday’s open?

Friday PM to Sunday PM = 2

Sunday PM to Monday AM = .25

2.25 days elapse from Friday PM to Monday AM in our calendar day model.

How about in a 308-day model?

Friday PM to Saturday PM = .5

Saturday PM to Sunday PM = .5

Sunday PM to Monday AM = .25

1.25 days elapse from Friday PM to Monday AM in our 308-day model.

Explaining the tweet

Translating:

1. On the floor I used a 365-day calendar model.
2. On Monday, the market seemed to only decay the options by 2/3 of what my model theta expected.
3. If I divide 1/3 the model’s theta for the weekend by the option vega that translates cents to vol points. Raising the vol in my model by that many vol points usually fit the market.
4. I would think of that as my benchmark for “vol is unchanged”. Differences in IV from that benchmark would be interpreted as “change in vol”

Measuring vol in this way means “vol does not structurally go up on Mondays” which makes sense. It probably doesn’t. It only looks like it does if your measuring stick assumes time passes as quickly on non-market days as business days.

Again, from Friday PM to Monday AM:

🕛Calendar day model: 2.25/365 or .6% of a year elapses

🕛Blended 308-day model: 1.25/308 or .4% of year elapses

The blended model says time passes at 2/3 the rate of the calendar day model.

Hence adding back 1/3 of the theta in vol points seems to line up well with assumptions of overnight .25 and weekend days being .5 each (inclusive of the overnight, just as 1 business day is inclusive of .25 overnight).

[Weekend is .5 weight but made up of weekend day .25 and overnight .25. You could argue weekend overnights are worth less because foreign markets are also closed, but since .5 seems to align with how much I’d have to adjust vols it’s a fine assumption even if I’m attributing the overnight and weekend day incorrectly. Hand-waving the deomposition would be a problem if I had to price an option on a Saturday.]

Bringing it altogether in an illustration

The percent of theta added back is in the ballpark of 1/3 depending on DTE but if you look at the error in vol points, it’s quite small (10% of 10 cents theta is a penny…for an option with 8 cents of vega this is .12 of a vol point).

Wrapping up

I’ll leave you with an assortment of considerations:

• If you are comparing IVs on assets with the same schedule the biases “cancel out”. Who cares if you are looking at relative vols and everything is biased high or low proportionally? This is the case for many option users. In other words, for almost everyone you can ignore this knowledge.
• (But moontower is written for an audience over a wide range so many of you will care and then there’s all the readers who just like weird technical stuff. And vol-time relativity is as worthy a nerd-snipe as anything else yea?)
• If you trade options on different expiry calendars (for example a USO option vs a oil futures option) then these differences will distort the IV spreads you’re observing.
• The distortions are amplified as event vols are incorporated into differing expiry calendars or am vs pm expirations.
• This matters far more for nearer dated options (ie < 1 month).
• Notice how I measured days from PM to PM. This is an artifact of working backwards from the expiration time which is typically after the close. But the more granular you get, the more the meaning of “when does this option actually expire?” matter. Can you contrary exercise? Does your broker have an earlier or later cutoff than other brokers. If AAPL pins on expiration but the index futures dive on geopolitical news after the close, then you would abandon any long calls on the pinned strike, exercise puts…or if you are short options on the strike expect to be stuffed with APPL shares by your counterparty. In other words, there is optionality after 4pm est.
• Spicier take: Trump having an interest in the assets open all weekend has implications for how fast time passes over the weekend. If weekends have more volatility, you’ll notice it in more decay transpiring from Friday to Monday. Unlike that last perplexity.ai bulletpoint’s assumption, this is not free money. The weekend gaps are larger (ie Friday pm to Monday am variance as a percent of Friday pm to Monday pm variance increases).

At the close of Friday, I was looking at far OTM gold put spreads.

With GLD at $275, the April 251/249 put spread is similar to a binary bet that GLD expires below $250 at expiry or about a 10% sell-off.*

You get 39-1 odds.

This is using market-based pricing, net of skew.

At-the-money vol is 16%.

Turns out 39-1 is about the right odds if we use 16% as a standard deviation. When I say “about right” I simply mean the gold left tail is basically conforming to an out-of-the-box Black Scholes distribution. I don’t mean B-S is “right” just that the put spread odds conform to what the B-S would say.

I wouldn’t expect that for SPX puts, but gold downside doesn’t have that much skew or tail fatness.

See the table mapping ATM vol relationship to the likelihood of ~10% selloff in a Black-Scholes world:

But a lesson that is far more important jumps out:

Implied volatility—the second moment of the return distribution—has almost no direct connection to the probability of catastrophic price moves!

Look how likely a 1-month 10% selloff is as a function of implied vol. If vol is 15% instead of 12% such a sell-off is 4x as likely and if it’s 18% instead of 12% it’s 10x as likely.

Just think of VIX…it can easily be 12 or 18, right? But what you should really take away from that is that volatility tells us very little about the tails.

Option Pricing on the Wing

A few things option traders learn early about pricing tails:

1. At-the-money volatility tells you nothing about the true probability of large moves.

2. Tail risks are not just “scaled-up” versions of small market moves.

3. Historical data is almost useless for extreme events because of sample size is too small to build robust probability estimates.

Instead, traders price tails based on…other tails.

Not backtests.

I invite you to keep your eyes open for sales pitches that pretend they know the likelihood of extreme moves and how they claim to know it. And if the sample sizes weren’t small enough, marketers might try to condition the probabilities — “when the market does X the probability of an extreme event does Y” which reduces the sample sizes even more. Godspeed with analysis that looks like that.

Inferring tail behavior from the meat of a distribution is so sensitive to small changes in volatility — so sensitive that it’s a clue that they’re not strongly related. I’m not making a math argument (I’m sure many readers could say what I’m saying formally). It’s a common sense observation that falls out of the volatility itself being volatile and sensitive to sampling.

The true odds of extreme moves are unknowable but there’s no reason to think they move increase an order of magnitude when VIX goes from 12 to 18, or for that matter, get cut by an order of magnitude when VIX falls from 18 to 12.

*The concept explained in a deeper understanding of vertical spreads

This post is an excerpt from the finance part of why home prices could fall with mortgage rates. 

I extracted it because it is a tidy explainer of something you feel but might not be able to articulate.

The reason you feel that selling your home to buy another these days feels like you have somehow incinerated money, is because you are.

It’s just bond math — you are buying a loan that is trading at a massive discount back for par when you pay off your mortgage. We can compute just how much money you are incinerating.

Quite unfortunate because along with low-supply and bottlenecks this is financial force that also conspires to remove supply and liquidity.

Excerpt below…

---

Houses are worth very different amounts to existing homeowners vs buyers. And since many buyers are homeowners (I’m talking about people changing primary residence not second homes), the split valuation even exists within the same brain. You’re Hyde when you list and Jekyll when you bid.

Let’s walk through it.

Homeowners who secured low-interest-rate mortgages years ago effectively “shorted bonds”. As interest rates rose, the value of these loans plummeted, embedding equity into these “short bond” positions. This means that the mortgage itself has become an asset that is highly valuable to the current owner. It’s like “it’s equity in a mark-to-theo short”. But that equity is trapped. It’s specifically tied to that home. The new buyer doesn’t get it because they must finance at the higher current rates.

This mechanically alters fair value of the asset depending on who owns or doesn’t own it. The homeowner might not be able to articulate it but they have two assets: the physical home and the valuable low-interest-rate mortgage. If they were to sell the home, they would have to buy back the mortgage at its face value, rather than its current impaired value thus losing the accrued profit from the “short bond” position.

Let’s make it concrete with a numerical example.

You bought a $500k house 5 years ago with a $100k down payment. You borrowed $400k at 3%.

What do you owe today?

Step 1: Calculate the Monthly Payment on the Original Mortgage

The formula for the monthly payment of a fixed-rate mortgage:

Where:

• M = monthly payment
• P = principal loan amount = $400,000
• r = monthly interest rate = 3%/12 = 0.0025
• n= number of payments = 360

Monthly payment = $1,686.42

The mortgage still has 25 years (or 300 payments) until maturity.

The remaining balance after 5 years is $355,625 (from this calculator)

In our fake world that’s pretty similar to the real one, a lot has changed in 5 years. Interest rates have doubled to 6%.

What is the value of the outstanding mortgage?

Step 2: Calculate the Present Value of the Remaining Payments

We need to find the present value of these remaining payments, discounted at the current 6% market rate.

The formula for the present value of an annuity is:

• M = $1,686.42 (monthly payment)
• r = 6%/12 = 0.005 (new monthly interest rate)
• n = 300 (remaining payments)

PV = $261,744

The present value of the remaining ~$356,000 mortgage, when discounted at the current 6% market rate, is approximately $261,744.

This significant difference highlights the additional equity embedded in the homeowner’s “short bond” position due to the lower interest rate. The homeowner has an intuitive sense that they are losing when they sell the home because they will have to pay the bank $356k to close the loan when it’s only worth $262k. Eww.

The additional $94k of equity that the homeowner has at prevailing interest rates represents almost 20% of the value of the $500k home!

If mortgage rates fall, the conventional wisdom that marginal demand to buy should increase is a fair assumption. However, rates falling cuts directly into this shadow equity that owners feel compared to a high-rate environment. I suspect this will actually “loosen” a bunch of trapped supply as the bid/ask spread narrows as the homeowners embedded equity in their “bond short” shrinks.

[I’m using the word “shadow” but it’s quite real vs the alternative of buying the same house for $500k at the higher interest rate. It’s “shadow” because the only way to monetize it is to let time elapse until the mortgage eventually goes away. Your lower cost of living relative to someone who doesn’t have a low interest-rate mortgage on the same property is the only way to realize the equity.

Active solutions to this illiquidity trap is allow homeowners to somehow port their mortgage to a new property or allow them to buy back their mortgage at the current value instead of the remaining principal amount.

I already hinted at a passive solution. Let the clock run. As time progresses, homeowners continue to pay down their mortgages. With each payment, the principal balance of the mortgage decreases, and the equity in the home increases. Over time, the impact of the low-interest-rate mortgage diminishes as the remaining balance shrinks. This gradual reduction in the outstanding mortgage balance reduces the value of the “short bond” position, making it less of a factor in the homeowner’s decision to sell. Eventually, as the mortgage balance becomes smaller relative to the home’s value, the embedded equity becomes less significant, narrowing the bid-ask spread.]

If mortgage rates fall in concert with the economy and employment weakening (pretty standard backdrop to falling interest rates), then supply may loosen in combination with general demand shortfall. It feels like a downside risk…but by now I’m also resigned to believing home prices won’t fall. We don’t have enough of them. Lending standards are conservative. There’s nothing frothy about the supply/demand balance. At the same time, it’s illiquid and unaffordable. My selfish position is I’d like to see prices ease but I’d happily settle for a wider selection of homes, even if they are overpriced.

Last week, we launched the Portfolio Visualizer in moontower.ai

It’s a tool for using vertical spreads to make directional bets.

Input: You enter a target price and expiration

Output: It shows you a matrix of every out-of-the-money strike combination up to the target in terms of what odds it pays.

A quick refresher on the utility of vertical spreads

I’ve written a lot about them but as a refresher, vertical spreads are clean ways to bet on a terminal price by a certain date.

• You know your max downside so you can put them on without worrying about being shaken out of them by marks. I call this “risk budgeting” a trade
• Spreads, especially tight ones, have largely offsetting greeks. When you use an outright option to bet on direction you are expressing a view on a basket of parameters in addition to direction — most notably volatility and its doppelganger time. You can be right on direction and wrong on the other parameters. Using options demands a view on volatility (ie the “volatility lens” I’m always droning about). Vertical spreads discard this requirement because the vol and time exposures are heavily neutralized.

The fussy reader will raise their hand, as they should.

“What about skew?

Doesn’t the implied skew impact the vol differential between the 2 strikes of a vertical spread?

How do I know if the odds are a good deal?”

These are awesome questions. Let’s get to work.

The goal: develop a sense of proportion about how much “high” or “low” skew impacts the odds

Payoff Visualizer

Let’s start with some odds.

We’ll look at the payoff odds for call spreads on SLV and USO etfs on trade date 2/25/2025.

We are looking at approximately .50d – .25d call spreads for expiry 4/17/2025.

In other words, buying roughly an ATM call vs selling a .25 delta call expiring in nearly 2 months.

SLV

Lower strike: $29.50

Lower strike implied vol: 24.9%

Higher strike: $32

Higher strike implied vol: 27%

Measured skew = .27 / .249 – 1 = +8.4% premium

The call spread is marked at $.57 vs a maximum value of $2.50 offering 3.4-1 odds if SLV expires above $32.

💡Note that if the 32 strike’s implied vol increased, all else equal, the spread would decline in value, the skew premium would be higher, and the buyer of the spread would get even better odds. It seems counter intuitive but by increasing the skew and pumping up the 32 strike call, the market is saying 2 things at once:

• the magnitude of the upside of the distribution is fatter (the call is expanding in value)
• the probability or hit rate of SLV going higher is lower — you are getting better odds to be long this binary outcome

This seemingly offsetting sentiment makes sense. If the upside magnitude is higher AND the hit rate is higher then the stock price must also be higher. But if we hold the stock price constant and just move the implied skew, then we are adding clay to the middle & downside of the distribution AND the further upside BUT removing it from the nearer upside.

In sum, silver has positive call skew and the .50d-.25d call spread with 2 months to expiry offers 3.4-1 odds.

Now let’s look at oil.

USO

Lower strike: $76

Lower strike implied vol: 27.9%

Higher strike: $81

Higher strike implied vol: 26.9%

Measured skew = .26.9 / .279 – 1 = -3.6% discount

The call spread is marked at $1.59 vs a maximum value of $5.00 offering 2.1-1 odds if USO expires above $81.

🔬What you should notice

USO vols are similar to SLV but the call spread .50d-.25 delta call spread is much more expensive (and pays less than 2/3 the odds of the SLV call spread). Skew is driving the disparity. In USO’s case, you are selling the topside call at discounted IV to ATM, but in SLV you selling a premium IV.

Weighing in on whether this is a structural mispricing of the probabilities is not where this post is going. That’s more of a research question. Knock yourself out, if you want to dig up the empirical distribution of returns. I can point to reasons why the skews look like this.

1) Spot-vol correlation

SLV vols tend to increase as silver rallies. Part of the precious metals as risk-off, fiat hedge rubric. Oil is a risk-on asset usually with demand for energy correlated with global demand growth. Like stocks, it has a negative spot/vol correlation. However in times of middle east geopolitical stress the spot/vol correlation can flip to positive.

2) Hedging

Producer hedging in oil markets tends usually involves buying puts or put spreads and selling calls. While consumers (ie airlines and refineries) are natural buyers of oil, their hedging requirements are typically swamped by producers.

[The vol inclined reader will notice that these reasons explain the skew but don’t rule out the call spreads being mispriced. The implied skew reflects the supply/demand for vol at various moneyness, but that’s not the same as saying the implied skew predicts the distribution. If you want to bet simply on directionality in a given time frame, taking advantage of the skew is a perfect use case for the risk-budgeted vertical spread. If you are not dynamically hedging, you are not concerned about the conditional behavior of vol as spot moves around.]

The key lesson of the discussion so far:

The level of skew, the percent premium/discount, is a major driver of the vertical spread price and therefore the payoff odds and implied probability.

An inferred lesson, that is not really surprising, is that different assets have different skews. Nobody is ever going to price a SPY surface with silver skews. Both the distributions and spot/vol correlations have different properties.

To develop a sense of proportion around how the range of measured skew affects payoff odds requires looking at assets idiosyncratic skew behavior. In moontower.ai, we display both time series for skew parameters as well as percentiles

Between knowing if the skew is “high” or “low” compared to history and seeing the payoffs in the visualizer we come to a practical questions that tie it altogether:

If .25d puts are in the 10th percentile vs the 90th percentile, how much is that going to change the odds offered on my put spread hedge?

If skew is at “average” levels what kind of odds should I expect on a 2-month .50-.25d call or put spread?

The remainder of the post will dive into these questions so you can walk away not only with a sense of proportion but be able to form your own rules of thumb so you can quickly handicap values like how much extra your paying for a put spread when skew is “low” or how much more your getting for selling a call spread when the OTM calls are cheap?

Here’s a few thoughts for everyone before we get to the paywall:

1) Tradeoffs

For a directional bet, I don’t really see which spread to buy as a question of “what’s optimal?” You’d need an very fine-grained view of the distribution to identify that. Instead, I see a menu of tradeoffs between hit rate and payoff which the matrix displays naturally. In fact, just looking at the screenshots above of the matrix is very educational.

The matrix is the view I’d construct ad-hoc when I want to take a shot. I didn’t map the whole thing but I’d I basically run the same payoff calculation in my head by eyeballing a bunch of strikes, perhaps according to which option markets were tightest or have meaningful OI for liquidity purposes. It makes life easier to just have it organized in a matrix this way.

2) The hips and the fist

In any fighting sport you learn that power comes from the hips. The fist is just conduit that channels the power. When thinking about a directional bet, the work really is upstream of the options. The options are just the fist. It’s the easiest part. Most of the alpha power comes from the directional analysis. Or sticking your finger in the air.

Developing a sense of proportion about skew

We need to do 2 things to answer the practical question of how much does “low” or “high” skew in name change the value of the vertical spreads in real-life:

1. We need to see how much skew varies in a name
2. We need to run extreme skew parameters thru an option pricer to see how the spread’s price range (and therefore payoff odds) varies with the skew parameter’s range

Skew Variation

I did a small study of 3 names with different skew properties. I looked at 3 years of data to find the .50d IV as well as the .25d put and call skew parameters for options with about 2 months until expiry.

🗒️Notes on method

a) I allowed a tolerance of 50-70 days until expiry (that means there wasn’t necessarily a qualifying expiry for each trade date but there’s enough data points to get the gist).

b) Skew parameter = .25d IV / .50d IV – 1 … you can interpret that as a percent premium or discount to the .50d IV.

If .50d IV is 30% and the .25d put is 33%, then skew = +10%

💡Sometimes you’ll hear the term “clicks” or “vol points”. In this example, a 10% premium corresponds to 3 vol points or clicks.

Let’s just jump to the data which I think is mostly self explanatory now that you know the definitions. For each name I show the scatterplot of skew vs .50d vol level for:

a) .25d put

b) .25d call

c) .25d put – .25d call risk reversal

Remember unless it says “clicks” the skew parameter is in percent of .50d IV

USO

QQQ

SLV

Substack’s not the greatest for delivering these charts but you can always right click on the image and “open in new tab”.

The charts are nice because you can see that skew can vary with vol level. That’s discussed in the “key observation” stickies. Those stickies also show how skew is not some abstract idea — its specific behavior matches the specific properties of the underlying asset. Sometimes oil panics up or down. The tech index doesn’t panic up (even if single stocks in the index might).

We can get lost in reflecting when our goal was to see how much the skew varies in a name. This table will make it clear:

We can ignore the risk reversal. I included it because it took no extra effort. Instead, let’s focus on .25 skew. Again, just eyeballing, we can see that the interquartile ranges for put and call skews are typically around 5% wide (silver is a bit wider). Meaning that from the median to the 75th or 25th percentile you are only talking about changing the OTM option’s IV by about 2 or 3% of the .50d vol.

So if median put skew in QQQ is 16.2% premium to .50d IV and skew blows out to the 75th percentile than it goes to 19% premium.

If QQQ ATM vol is 20% your talking about a put IV going from 23.2% to 23.8%

How if QQQ IV is in the 25th percentile?

If ATM vol is 20% then the put is 13.3% premium or 22.7%

If skew went from the 25th percentile to the 75th, the put vol increases from 22.7% vol or 23.8% or just over 1 click.

Is one click a lot?

That’s the question we’re after.

Put spread sensitivity

We can test this using a Black-Scholes calculator.

We can compute a 60 day .50d put at 20% IV vs a .25d put at 22.7% IV.

This represents “cheap” put skew in the 25th percentile.

We will do this on a hypothetical $100 stock.

The .50d put will be actually be in-the-money slightly as opposed to at-the-money.

[See Lessons from the .50 Delta option for an explanation]

We end up with the 100.50 strike vs the 94.50 strike. This $6 wide put spread represent the .25d wide put spread.

When the skew is “cheap”, the IV differential between the strikes is 2.7 vol points (ie 22.7% – 20%).

The put spread is worth $2.03

[The 100.50 put is worth $3.50 and the 94.5 put is worth $1.48]

If we raise the skew to the 75th percentile, the 94.5 put goes to 23.8% IV and a price of $1.62

The put spread drops in value to $1.89

Again, notice when the put skew expands, the put spread drops in value! The left tail is getting fatter but the intermediate down move is losing distributional mass.

This is the put spread as a function of the vol differential between the 2 strikes. As the differential widens (the smaller put increasing relative to the .50d put) the spread gets cheaper.

What happens in odds space?

If the put spread is $2.03 and can be worth as much as $6.00 a buyer is getting 1.96-1 odds.

When the put spread gets cheaper, the buyer gets better odds of 2.17-1

In probability space, you go from 33.8% to 31.5% probability of expiring lower by expiry.*

The put spread changed in value by about $.15 on a $2 put spread as skew went from the 25th to 75th percentile all else equal. Given the variation of skew in QQQ it’s like every 3 cents in the put spread is 10 percentile points.

I won’t step thru it as slowly as I did with QQQ but here’s oil put spreads varying from 5% to 10% to 15% skew for the 100-91.5 put spread (.25d wide put spread):

Context is everything

When I was trading oil, a giant move in skew would mean, on a delta-neutral basis, a $5 wide spread would move 3 cents. If you made a nickel wide market on a put spread you’d be accused of making a market the broker could “drive a truck through”. In percentile space, you can see why.

But then again, you could argue that the exchange fees and broker commission represented a few percentile points.

If you are an options market maker, translating what low or high means to actual prices is important. It gives your market width context with respect to the surface parameters that you track. It lets you estimate how much edge you need to pad your market compared to how wrong you can be about how the surface might reprice.

On the other hand, if you are a purely directional trader the the difference in skew being low or high might be immaterial to your decision to take the odds unless your are able to discern probabilities down to just a few percent.

Now when you see a skew time series for a fixed maturity you know you can put the parameters in an option model and see just how much it changes the spread value.

You can derive your own sense of proportion customized to the context you trade in.

I’ll wrap by emphasizing that I’m writing from the perspective of mapping skew parameters to actual spread pricing. Trading skew on a delta-neutral basis because you think realized or implied vol will out or underperform as spot price travels across various strikes is a different animal. That is less about distributional outcomes and more about dynamic vol behavior. You care about what IV does when your vegas expand and contract, and what realized does as spot moves through your dollar gamma profile. It’s highly path-dependent. The opposite of a terminal risk-budgeted bet. In fact, the 2 different approaches can prescribe opposite trades meaning the risk-budgeted trader and the dynamic hedger can happily trade with each other. A classic example is the 1×2 ratio spread. The directional trader might use jacked put skew to buy a 1×2 put spread creating a highly attractive payoff profile in most scenario, while a dynamic hedger might be happy to own the 2 options because they expect vol to scream higher as the stock goes down.

For fun: What the most you’d be willing to pay for a $15/$10 1×2 put spread? The least? If you paid zero for it, can you lose?

Update 8/2025:

I was asked if I had written anything on the impact of events on skew percentile measures in the Discord. Sharing widely:

I haven’t, but most events are just one-day pricing problems. The effect of skew from 1-day pricing is heavily diluted if you are looking at 1-month skew and beyond. If you are looking at percent skew in like 1 week options, well all kinds of measurement issues anyway:

1. IV on 1-week options alone is a hairy topic since assumptions about how much vol time remains become impactful. Which is why if I’m trading near-dated, I really just think in terms of straddles and the price of vertical spreads. For the latter if a stock is trading 102.50 with 3 days to expiry you simply ballpark that the 100/105 call spread should be about $2.50 which is about the same saying the 105 call and the 105 put are equal (HW: prove this with put-parity). All the weird lognormal Black-Scholes math really melts away in the short term and you are in the realm of common sense handicapper.

2. The vega of options gets small in near-dated options so…maintaining a sense of proportion if important. If the ATM vol is 20% and the skew is 10% (ie 22 vol) vs 15% (23 vol) and the vega is 0.005 you haven’t even changed the value of the vertical spread by a cent even though the skew is 50% larger. I have written about this “sense of proportion” stuff. People get caught up in metrics that when you translate back into price space matter on the order of “it’s like paying an extra commission charge to IB on the trade”. In other words, if the difference changed your decision to trade, then the rest of your infra better be medical grade accuracy bc you’re trading for slivers.

On Jan 28th, one of my favorite topics came up in the Moontower Discord: natural gas options!

I was asked why the VRP was so low.

Recall VRP is the comparison of implied vol (forward looking) to realized vol (backwards) looking. In this datapoint,

One month IV = 49%

One month RV = 86%

The strange datapoint is a perfect icebreaker for discussing:

• seasonality (which happens to be most strongly expressed in natural gas)
• a common pitfall (and potential correction) for VRPs

   

Background

Before discussing options, we need to understand the shape of seasonality and the fundamentals that drive it. I’ve said many times that I’m not a fundamental trader. That means I don’t position based on any views about fundamentals. But a basic understanding of fundamentals is necessary to make sense of why an asset’s vol surface looks the way it does.

Let’s begin…

 

A Brief Primer on Natural Gas Dynamics

Natural gas prices follow a seasonal cycle, with volatility peaking in winter due to heating demand and spiking again in summer due to electricity consumption.

Seasonality volatility

Winter is the most volatile season.

• Heating demand: Cold weather drives demand
• Supply constraints: Limited storage or pipeline capacity can trigger price shocks.
• Weather uncertainty: Forecast swings can cause sharp market reactions.

Summer is the next most volatile season.

• Electricity demand: Power plants burn more gas for cooling
• Hurricane season effect on supply: Storms in the Gulf can disrupt production

The Storage Cycle

• Injection Season (April–October): Gas is stored for winter, with October marking peak supply. If storage maxes out, prices can collapse.

  [In 2009, this was a major risk with the Oct $2.00 put price surging to an insane vols on extremely heavy volume. I remember feeling terrible for leaving my business partner to deal with that expiration on his own because I had to be in Mexico for my wedding week. He was able to join the festivities after making sure we weren’t going to need a cave to take delivery. I distinctly remember computing that the stretched IVs still never reached the extreme levels of realized vol that accompanied that expiry. On a hedged basis, every option except the ones where Oct gas expired were a buy. The market found the path of maximum pain.]

  Injection season is often traded as a package of futures of options known as the “J-V strip” based on their futures month codes. In trader language, the “ape-oct strip”.

• Withdrawal Season (November–March): Stored gas is used to meet demand, with March marking peak depletion—low storage levels can drive price spikes.

  This season is also traded as a package — the “X-H strip” or “Nov-March”

…which brings us to the “widowmaker”.

 

March-April Spread: A Market Tightness Gauge

The March-April futures spread more affectionally known as the “widowmaker” or simply “H/J”:

• High March premium: Indicates low supply and potential scarcity.
• Weak or negative spread: Suggests ample gas and lower risk.

I’ve written at length about this spread and the options on it.

🔗What The Widowmaker Can Teach Us About Trade Prospecting And Fool’s Gold

You can learn a lot from its vol surface that can be applied to any asset with a “bubble” distribution. Moontower’s ever-so scientific definition: a price that most likely collapses but can reach any arbitrarily high price before it tanks.

🔗What Equity Option Traders Can Learn From Commodity Options

 

Supporting evidence in pictures

 

Exhibit A: Natural Gas Inventory 5-Year Seasonality Chart

Exhibit B: A historical snapshot of the gas futures term structure

Exhibit C: Realized vol by month

Exhibit D: Despite the strong realized vol seasonality the range of volatilities both across and within years is itself quite volatile.

Exhibit E: The March/April futures spread

The “widowmaker” expires this month. In commodity land, futures spread are usually quoted as near month – back month. So in a backwardated market the spread would be positive (ie March > April).

In equity markets the convention is reversed. The price is quoted as back month – front month. The charts below are from IB which uses equity market convention.

You can see the “winter premium” come out of the spread as April is now trading close to parity with March but the spread was negative (ie April < March for the past several months).

This is typical behavior. The spread usually goes to parity as the fear of a cold winter subsides. But it’s dangerous to short early in the season because the spread can go extremely negative (if you’re looking at the price using the IB quoting convention…which burns my eyes but whatever).

Here you can see the spread for 2026. April is trading at about a 32 cent discount to /March (~10%). If next winter is mild, you’d expect the gap to close.

In the below charts we respect tradition by using the quoting convention of March – April…you can see the destination of the spread is usually ~ 0:

This was the spread in 2007 when John Arnold became a legend stuffing Amaranth’s Brian Hunter’s attempt to squeeze H/J:

 

Option surface dynamics

Let’s see how these fundamentals influence the vol surface.

Skew Feature

🔀Inverted skew

Call IVs typically trade at premiums, often steep premiums to puts. Because gas is prone to squeezes it often maintains a “spot up, vol up” dynamic. On the downside, gas can find incremental demand via “coal-switching” whereby gas prices become competitive with coal as a input to electricity generation. This source of demand dampens vol as futures fall reducing the probability of a complete collapse in an oversupplied market.

This is a chart of Feb gas options (note that Feb options expire in January…in commodities the contracts are named after their delivery month not their expiry month). The curves represent roughly 2 weeks and 3 months to expiry. You can see the skew inversion.

💡Learning moment: Note how skew looks steepens when DTE is smaller. Much of this is an artifact of the X-axis being in strike space not delta delta space. Why does that matter? Because how “far” a strike is depends on time. If gas is $3.00, then the $3.20 strike is much further (ie lower delta) with 2 weeks to go than 3 months to go. Low delta options usually command premium IVs.

 

Term Structure Feature

📅Seasonality in the forward vols

We know that realized vols are higher in the Winter. Look at the vol term structure I pulled from the futures options via CME.

2 things to note:

1. LNF6 and LNG6 slope upwards indicating a higher volatility than Q42025 vols. This makes sense those 2026 capture December and January coldness while LNZ5 December options only capture through Thanksgiving. This tracks, nothing weird.
2. Those winter vols implied vols are LOWER than Spring 2025 vols (ie LNK5 or May). What the heck?!

You have 2 forces colliding from opposite directions.

a. We absolutely expect vol to be higher next winter than this spring

b. Deferred futures contracts don’t move as much as prompt ones.

In other words, the beta of those winter contracts to whats happening now is low. Today’s supply/demand balance for gas has only modest impact on future prices. This is actually a universal effect in commodity futures. This is easy to deomonstrate if we consider a long duration between contract months.

The price of oil today has little impact on what the 5-year future does. Which makes sense. Near term drivers of oil could be weather, refinery outages, shipping logistics, and the current economy. Longer term, oil prices depend on drilling projects, regulations, and the state of economy which is anyone’s guess from our current seat. A price spike today, can lead to more investment in oil, which would increase supply in the future. Nobody thinks that a near term squeeze should have an equal response in the deferred month.

The quantitive observation that deferred months have lower realized than near months is called the Samuelson effect. Stated otherwise, contracts become more volatile as expiry approaches

💡This is a strong effect as a contract travels from being a 12 month contract to a 1 month contract but effect is smaller as a contract goes from being 10-years to 9-year or from 20 days to 1 day. The schedule of how variance decreases as DTE increases looks like a curve. Hold this thought.

The stronger the Samuelson effect, the more downward sloping you’d expect the term structure. The fact that the deferred months are only slightly lower vol and the curve is flat actually implies an ascending term structure if you adjusted for Samuelson effect.

The schedule of how Samuelson unfolds has a tremendous impact on what you believe the term structure actually says. If there was zero Samuelson effect, then the term structure you see is the actual term structure which you are then free to extract forward vols from. That’s the case of equities where all the options are struck on the same underlying as opposed to each expiry referencing a different deferred future.

The stronger the Samuelson effect, the more true term structure and forward vols ascend. If your 1 year future trades at the same implied vol as your 1 month future despite the fact that the 1 year future is moving less, means that the market is implying more variance in the coming months. If there was no Samuelson effect than you’d assume a flat forward vol not an ascending one.

Armed with this knowledge, I present the UNG vols.

The bottom panel shows UNG vols ascending thru 2025, not descending like the futures options were.

UNG options are struck on a single underlying just like regular equity options, but that underlying ETF maintains exposure to the front month natural gas futures.

In the futures options, the January expiry references a deferred future and expires in December. It is an option that references a contract that will not be moving very much for the next few months, but will be whipping around like crazy near the end of its life.

The January UNG option is referencing a prompt future that moves around a lot and will converge to the futures option in the month of December when the ETF is holding the same contract January futures option references.

The key to translating the futures options vol into a UNG equivalent vol is a schedule for the Samuelson effect. This creates a model that allows market makers to relative value trade the futures options vols vs the ETF vols. (This was central to my strategy which required normalizing commodity vols into something that can be coherently compared to other vol surfaces).

Some food for thought:

✔️Instead of designating a Samuelson schedule, you can invert the problem. What Samuelson schedule needs to be true to make futures options and UNG options be relatively in-line?

✔️How does that schedule compared to how vol unfolded in prior years?

✔️In what ways is the fundamental context of this year different from prior years?

I’ll close this section with one more picture.

That is the forward vol matrix from UNG courtesy of moontower.ai on 2/4/2025

Recall this chart:

Winter vols average in the low 50s. The forward implied vols are pricing upper 50s. That is a normal premium and if you track the forward vols every day you’ll notice that winter vols 6-12 months out will price somewhere between mid 50s to mid 60s in the case that the upcoming winter supplies look tight.

💡Winter volatility is right-skewed so if the realized avergaes low 50s the median can be assumed to be lower but sometimes the vol is much higher than the 50s. You can see the “polar vortex” in Feb 2014. You can also see just how low the vol can be in the winter:

That chart shows how deferred forward vols are like fair point spreads but have giant error bars compared to the range of realized outcomes.

 

Additional thoughts

✔️Circling back to the question that launched the post: why was the VRP (ratio of IV to realized vol) so low on January 28th?

Like looking at VRPs after earnings it’s an instance of a VRP failure mode — you are comparing a forward-looking numerator to a backwards-looking denominator. As we change seasons, nobody expects the recent bout of volatility to repeat in the next month.

✔️An example of a trade I did in the mid 2010s

I don’t remeber the exact year but in the early spring, summer vols were getting smashed as producers were selling calls as part of large hedge programs, They were bombing summer call strips. For example, if they sell 5,000 J-V $5 calls they are selling 5,000 calls in each of April, May, June, July, Sep, Oct.

30k call options in total.

I don’t remember how many total calls were sold or how long the program lasted but at some point the vols looked quite tasty.

I eyed July 4 calls which a broker was offering for 10 ticks. 10 ticks = 1 penny. So to breakeven gas had to go to $4.01

A natural gas tick is worth $10 so each option cost $100. I bought 10k for $1mm of premium. Gas was trading in the low $3 range at the time. I decided the best way to manage the trade was to risk budget it instead of delta hedge it. The options were a very cheap vol but I didn’t want to invite the path risk of selling deltas on a grinding rally where a summer risk premium started to emerge if it was looking like a hot season. Instead, I would play for a spikier move. I remember pulling up some historical charts and figuring based on an admittedly low sample size that the chance was somewhere around 20% but if it happened I’d conservatively make 9-1 on the calls. I also asked some sell-siders about whether there was anything in the fundamental context that differentiated this year from prior years.

[There’s a concept I call “analog” years where some years look like others. “In 2007 corn plantings were at this stage by this time of year and everyone thought that summer was going to be hot and the vol surface liked like this”, etc.

I didn’t check on these things so much to ideate a trade as to just check if I was missing something baked into the common knowledge the underlying market when I’m reacting to a trade.]

All told, it was reasonable to risk $1mm in premium, all-or-nothing.

So did the calls hit?

No. But, I was right about the vols being low. It was still early spring and the futures weren’t going anywhere, but the calls stayed penny bid for the next month. A month elapses, spot goes nowhere which means the IV on the strike clearly increased.

From there I had choices, I could re-asses if I thought the vol was still cheap. If not I could roll the calls up, or sell some closer-to-ATM vols and still be long vol-of-vol. If I thought the vol was still seasonally cheap, I could roll them down and have more gamma as the futures started to roll up the Samuelson curve (ie move around more). The point is there was a new set of decisions but they have nothing to do with the original trade which was “buy the cheap vol, decide how to manage it”.

In the end I was right but didn’t make a sum of money that stands out in my memory. This is not especially unusual. Trading be like that.

Extensions to think about

✔️Earnings seasonality

Market-maker’s starting point for thinking about earnings straddles will be:

1. How much has the stock moved on prior earnings dates
2. How was the earnings straddle priced going into those dates

I expect they also consider earnings seasonality. If a retailer makes most of its money in Q4 then the process for pricing earnings won’t be uniform every quarter. The sample size of relevant earnings history will be even smaller as each year maybe there’s only 1 day that is a true analog for appreciating how many days of volatility should be baked into the earnings straddle.

I’d totally expect that the market handles this well. I’m just relating the idea of seasonal volatility to assets outside commodities.

✔️Ags and softs

It’s not surprising that I took the nat gas framework and pointed it at cotton, cocoa, sugar, coffee, soybeans, corn, and wheat. Each of these markets has its own idiosyncrasies. Examples:

• peculiar option to future expiry mapping
• seasonality drivers
• concepts like “old crop” vs “new crop” which means Samuelson curves are discontinuous
• import/export features => currency correlation considerations
• unique natural flows

As a vol trader you are doing some mix of modeling and qualitative adjustments to estimate the implied forward vols in these markets.

💡It’s critical to understand how the distribution of variables that roll up to those numbers effect how the forward vol estimates are distributed…the weaker you think the Samuelson effect is the more you will be inclined to buy time spreads. Are you buying time spreads while your modeling of Samuelson is on the low end of its range? Then realize that your position is more vulnerable to near term stress than your headline greeks would suggest.

If you use percentiles to measure skew, vol or any other metric are you conditioning them on season?

Would it make sense to condition on the degree of backwardation or contango in the market?

Ok, I’m going to leave it there.

If you are interested in commodity vol stuff either directly or just to expand your own option-thinking toolbox check out:

• What Equity Option Traders Can Learn From Commodity Options (13 min read)
• Crossing The Commodity Chasm (13 min read)
• Financial Hacking: ETF vs Negative Oil Futures (a step-by-step explorable)

Benn posted a great question:

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

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

Try to find it yourself.

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

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

Let’s do that.

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

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

Step 2: Chart in Excel

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

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

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

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

 

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

 

💡Learn more

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

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

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

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

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

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

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

Modeling a vol curve

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

Option valuation

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

Interpreting skew

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

Understanding vol changes day over day

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

Dissection

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

Decomposing vol p/l from greeks

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

Uncategorized

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

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

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

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

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

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

Prerequisite Building Blocks

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

1) Implied forward volatility

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

2) Event Volatility Extractor

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

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

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

The unintuitive impact of events

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

Consider this situation.

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

However we get 2 new pieces of info.

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

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

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

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

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

Term variance = regular variance + event variance

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

Solve for X.

x = event vol = 171%

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

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

171%/16 = 10.7%

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

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

.8 * 10.7% = 8.6%

[To review, see 😈The MAD Straddle]

Let’s take inventory.

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

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

Event variance is 1 * 1.71² ~ 2.94

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

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

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

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

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

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

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

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

A few selling points on this post:

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

An “ugly” term structure

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

 

This is an ugly term structure.

There are 2 primary reasons.

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

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

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

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

You’d be buying:

• 2/21/2025
• 1/24/2025

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

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

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

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

Accounting for events

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

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

Here’s the plan.

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

💡A note on nomenclature

I’ve heard “ex-earnings vols” called:

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

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

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

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

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

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

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

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

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

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

The second chart zooms in.

And the matrices:

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

Beyond a single event

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

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

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

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

We’ll come back to #3 and #5.

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

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

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

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

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

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

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

Pretty pictures

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

You: “Look a kinky opportunity”

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

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

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

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

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

💾Moontower Event Vol Matrix

Spreadsheet screenshots:

 

Final thoughts

Trading vol around events is a major topic.

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

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

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

But it starts with measurement.

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

How does the surface/consensus synthesize knowledge about:

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

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

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