A moontower user sent this [paraphrased] message in our Discord the morning of Jan 9th:

NLR [VanEck Uranium and Nuclear ETF] had a price shock on Jan 2 and has been ‘fast grinding up’ since then. Did I “lose” here because RV climbed up faster than RV and my losses are ‘gamma’ driven?

Now the most important part — what can I learn / what should I do as part of my process?

We are going to do the post-mortem in steps. The first task is to take inventory of the scene. Basic policework. “What happened?”

Once that’s established, we can at least start to disentangle bad luck from decision quality and finally wrap up with risk management/hedging/whether we should close the position or not.

Arriving at the scene of the crime

What do we know? Our friend sold a small amount of 1-month NLR at-the-money straddles in December. To be discreet, I’m going to guess the date to be December 15th and the strike to be the 126 line and IV was ~41%

Below is a simple time series of:

constant maturity 30d IV LAGGED vs 30d realized vol

By lagging IV, we align it with the 30-day realized vol that was experienced in the subsequent month. We can see that the RV (faint green line) our friend experienced far exceeded the IV (dark blue line) of the straddle they sold.

A chart like that is a handy compression, but since it is:

1. using constant maturity vols (ie interpolated) and
2. floating (the IV is taken from the .50 delta call each day)

…the chart is not high-resolution to discuss p/l, but can only gesture roughly to its direction. It’s a blurry picture of a license plate as the driver speeds away.

We will get down to the contract level, but first, we want to develop a sense of proportion about notable move sizes.

Realized vol

✅Jan 2nd was the steepest one-day move: 7.1% or about 112% vol annualized. Nearly 3 standard devs.

✅ Over 8 days, there was a 13% cumulative rally, or 73% annualized vol.

The calculation: 13% *sqrt(251/8) = 73% annualized vol move

Any funny business under the hood?

• NLR’s largest component, CCJ, is ~9% of the basket. It rallied a bit over 7% as well which is frankly underperforming the basket since CCJ is a higher vol than the ETF.
• Its second largest component is DNN ~6% but lots of names in the basket are close to that size. DNN was up 14%…but its normally twice the vol of the ETF already.

In z-score space, the ETF and its 2 largest holdings all moved about the same amount.

All these clowns are riding in the same car. Its a 1-corr move, in a beginning-of-the-year inflow to this sector.

💡For those of you who trade around rebalancing and calendar anomalies, perhaps this is a thread to pull on?

Drilling down to the option contract level

The NLR option volume in the month preceding Jan 9th was not notable. There was a spike in puts traded on Jan 5th, but this was already after the largest single-day move happened

The largest component, CCJ, did not have any noteworthy volume in the prior month either.

What stands out in NLR is how small the open interest is in general. This is not a liquid option name.

Price and P/L

From December 15th to Jan 9th, the Jan 126 straddle expanded from $12.75 —> $15.50 as the stock went to $140. No surprise, the call went to >.90 delta.

So the short straddle position lost $2.75, assuming you did not hedge any of the delta on the way up.

If you’re intent is to trade vol, allowing the delta to ride like that is introducing a lot of noise into your trade expression that was supposed to be about vol.

What if you sold the straddle and hedged the negative gamma daily by bringing your deltas back to neutral? In other words, buying shares after they rallied and selling them as they fell in opposition to the changing straddle delta.

Our service includes an attribution visualizer which allows you to decompose your daily and cumulative p/l due to realized and implied vol changes as the option and stock price move around. It is from the perspective of an option buyer. In this case, we are selling, so just flip the signs. We also need to double the numbers since we are assuming a straddle hedged daily, not a single call or put as the tool assumes.

The total ACTUAL delta-hedged p/l as of January 8, the day before the friend messaged the group, would have been -$.51 per contract or -$1.02 for the straddle. The loss would have been less than letting the straddle ride, since the stock trended up and each rebalance would have forced the hedger to buy on the way up.

If the stock chopped around at 70 vol but still landed on the strike, hedging would have locked in a bunch of negative gamma scalps while the straddle decayed.

Hedging makes your p/l reflect the vol that was realized but whether this is good or bad for you, ex-post, depends on whether the stock chopped or trended.

Ex-ante, you want your hedging to be aligned with the reason for your trade, which in this case is presumably the expectation that IV would have a risk premium above realized, since the trade was selling 1-month atm straddles.

A note on attribution

The chart doesn’t track the sum of unexplained p/l although it is displayed in the summary (not shown). The “unexplained p/l” is the balancer which makes the theoretical attribution tie out with the actual p/l. It is a catch-all for the higher-order greeks, mostly vanna and volga, which reflect the fact that your gamma and vega, respectively, are not constant during a single day’s move.

The bulk of the p/l on that big day is due to realized. It’s fair to say from the summary that realized p/l explains most of the result. This is what we’d expect from an option with only a few weeks until expiry.

No smoking gun

Given the lack of notable action in the option volume in either NLR or its components, the uniform behavior of the moves in the complex, a boring IV chart to close out 2025, and the fact that the move happened on the first business day of the year, that this result was a bunch of methodical but unanticipated sector flow. Approximately 2.8 sigma move in one day, or about 1/200 probability, a bad beat with roughly the same probability of being dealt pocket aces (1/221 because 4/52 * 3/51).

[Stock moves are fat-tailed, so the probability is actually larger than 2.8 sigma would predict, but the fact pattern here still suggests a bad beat. The IV wasn’t suspiciously high in December, there wasn’t any telegraphing flow].

An opportune time to remember one of the reasons gambling and poker experience is helpful…from why poker is used to train traders:

This is one of the great teachings of poker. Short-term results are noise. He explains that in Limit Hold’em, even a high edge hand has only .02 big bets worth of expectancy vs a standard deviation of 2.5 bets.

[Kris: In investing language, a .008 Sharpe for one trial. The SP500 has a daily expectancy of about 3 bps and 100 bps standard deviation for a daily Sharpe of .03. The poker hand has almost 4x the noise of the daily SP500 return.]

Since poker teaches that you will make the right decision and still lose money, it trains you to emotionally decouple decision quality from result quality.

This is a ceaselessly profound concept. Not because it’s so clever, but because of how it resists internalization. It’s easy to understand, it’s hard to apply the understanding to how we receive the world.

As police work goes, there will be no verdict or even charges brought as to whether the decision to sell the straddle was sound. We do get research inspiration. Is sector dispersion especially high on the first of the year? First of the month? Is there more volatility in general on those days? If so, is the median volatility higher or the mean (ie is it being driven by outlier-type moves)? We don’t know if selling the straddle was bad, but we do get new questions. This is what a career in trading looks like. If you don’t like this type of problem, then hooray, I’ve saved you a bunch of time compounded over your life. You’re welcome.

Regardless of the outcome, we still have this business of risk management.

Should our friend have hedged or closed the trade?

The following is true but unknowable in advance:

• If the stock is trending, you want to hedge aggressively. Buy delta as it rallies, sell it as it falls.
• If the stock is mean-reverting, you want to sit on your hands.

Your risk approach cannot depend on what you don’t know. And it must depend on what you consider tolerable.

The combination of these constraints will dictate how big your position can be. We’ll call this your limit. From there, you are simply monitoring how big your position is under various scenarios to that limit. If it is greater than the limit, you must reduce it.

I’ll give a simple example, but know this is a vast topic and a chief concern (and unsolved problem…there’s no single answer to this) of any risk-taking outfit.

Let’s say you are willing to tolerate 1% volatility in your total portfolio due to a particular trade on your average day. If you have a $1mm portfolio, that’s $10k. To a first approximation, that means keeping your swings due to delta below $10k. Call NLR a $140 stock with a 48% vol. For a typical day, that corresponds to 3% moves or $4.20.

$10k/4.2 is the daily swings associated with ~2,400 shares or 24 100 delta options. Or 48 50 delta options.

So what is NOT conservative about this risk-based sizing:

1. “Typical day” is being proxied by 1 standard dev (ie the 3% daily vol). If moves are normally distributed, that means about 1/3 will be greater than that or more than a week out of every month will be composed of bigger moves. And that’s ignoring fat-tailedness
2. We aren’t accounting for adverse vol changes. If you are short options, trades are negatively skewed so we’ll want to be more conservative still.

What IS conservative about this risk-based sizing:

1. If you hedge your deltas even once a day, you will not have as much daily variance in your p/l due to delta, which is effectively what we’re describing above.

How does this shake out?

A good starting point!

In the case we’ve been following, if our friend used a rule like this, it woud have prescribed 48 50 delta options or 24 straddles.

The straddle went from $12.75 to $15.50 on a bad beat with no hedging.

The loss = 24 straddles * -$2.75 * 100 = -$6,600

If our friend hedged daily, mirroring the attribution visualizer recipe, the loss would have been:

The loss = 48 contracts * $-.51 * 100 = -$2,448

Notice the constraint:

This makes sure delta is positive for the sake of the calculation AND doesn’t allow you to oversize a position just because you chose a skinny option.

I came to this example from the perspective an option seller. If you are a buyer the most you can lose is your premium if you DON’T delta hedge. You can use your risk tolerance for losing money as your premium spend limit.

If you do delta hedge, you can lose many multiples of your premium. For example, if you buy an OTM put and the stock grinds down slowly to your strike, you will be buying shares all the way down. You will lose not only on your stock trades but also on the premium going to zero. T

I’m going to pause for a second to level with you because I do feel some almost paternal responsibility stemming from the privilege of many smart but also young readers who come to this letter to hear from me because of my gray hair and specifically because I won’t treat options like the next house-flipping get-rich trend.

This topic of risk is so vast that its discovery is an ongoing project throughout your career. You are shaping and being shaped by the rules you create and their feedback, so to think there’s an “answer” is to not appreciate how many facets there are to managing risk across a portfolio of non-linear instruments.

To recap…this was a “3 standard deviation” move and the loss was comfortably below our tolerance. You can season to taste, but this is overall a conservative approach that you can experiment with. This is a point-to-point p/l, so the rule is providing some flex for tough marks along the path. Like I said, a starting point.

🔗If interested, my treatise on hedging If You Make Money Every Day, You’re Not Maximizing

What the risk management decision is NOT about

I’m saying this because the fact that you already have a trade on is not a reason to keep it on. If you don’t want to put the trade on fresh, you should get out. There’s an opportunity cost to your capital.

If a trade you have on is not bad but just fair, then the decision comes down to whether the variance is acceptable. If there are costs to getting out of a coin flip that you can sweat the risk on, then it’s ok to save the transaction costs. You can refine that a bit to “is the coin flip’s expectancy the same as my cost of capital” yadda yadda, but you get the gist. There’s a cost to reducing variance (ie hedging or closing) and it’s perfectly fine tto avoid it if the risk is tolerable. There are a lot of risks in life you don’t bother hedging.

Finally, rules aside, if you are regularly running risk that makes you lose sleep, impairs your judgement, or threatens to blow you up even 1% of the time, the size is wrong. 1 in a 100 is inevitable if you plan on doing this for awhile.

Tweets

Before we get to today’s meat, here are 2 threads spurred by oil’s advance yesterday.

Delta is God

If you’re reading a Thursday Moontower, “you’ve heard the expression vega wounds but gamma kills.” It’s not quite so cut-and-dry. My pushback to that trope is the recent article vega’s finishing move. However, I’m sympathetic to “gamma kills” mantra. The running joke I’ve used to say on the desk has a similar energy:

“delta is the only greek”

I wouldn’t take this literally, the joke is bowing to the idea that if you have your hard deltas, ie your shares, pointing in the right direction, you tend to win. The Freudian reading of that statement is I’d rather be good at directional trading than a vol monk.

Today, we give delta its due. Delta is god.

No matter what you think it is, you never quite understand it. The best we can do is understand how it will harm or help us based on the thing we can’t know in advance, but know will affect our p/l — path.

While I’ve been meaning to write about this for awhile, this paraphrased question from a moontower user, bumped this post up the editorial queue:

“I’m backtesting delta-hedged straddles and I’m worried the vol I use to compute my hedge delta is ‘wrong.’ Does the choice of hedge vol bias my P&L, and if so, how?”

Pull up a chair, young Padawan.

I’m going to offer 3 perspectives.

1. The quant answer.
2. The quant who speaks “trader” answer
3. The Moontower treatment

Finally, we’ll see how this idea applies to traders and investors who try to structure an options-like payoff to a trade without using options at all.

So much trader mindshare is fixated on delta-hedging for the same reason we are never happy with the quantity we trade in hindsight. The goal here is to create enough clarity that you can not only make better ex-ante decisions but make your peace with them regardless of the outcome.

Onwards.

The Quant Perspective

We’ll start with the mathematical approach. This is not my wheelhouse, so I’ll save my words for later sections, but if you can’t wait to curl up with notation, then this post is for you (h/t to the Moontower Discord where it surfaced).

I couldn’t help but print the acknowledgements section below. I don’t know stochastic calculus, but I suspect the people involved in this paper might.

Which Free Lunch Would You Like Today, Sir?: Delta Hedging, Volatility Arbitrage and Optimal Portfolios by Paul Wilmott & Riaz Ahmad

ABSTRACT

In this paper we examine the statistical properties of the profit to be made from hedging vanilla options that are mispriced by the market and/or hedged using a delta based on different volatilities. We derive formulas for the expected profit and the variance of profit for single options and for portfolios of options on the same underlying. We suggest several ways to choose optimal portfolios.

ACKNOWLEDGMENTS

We would like to thank Hyungsok Ahn and Ed Thorp for their input on the practical application of our results and on portfolio optimization and Peter Carr for his encyclopedic knowledge of the literature.

A Quant Who Talks Like A Trader

The next perspective is a bridge. In the incomparable book, Financial Hacking, quant Philip Maymin breaks things down in terms that your common option flow trader will understand.

On hedging to model (forecast) delta vs implied delta

The short-form intuition is this: you bought a call and hedged it. So you are betting on higher volatility. When volatility ends up higher, even if only for random reasons, you benefit, and when it ends up lower, you lose.

That intuition continues to hold even if you hedge at the wrong vol. If, for example, the true vol is 30 but you hedge to 20, you are just introducing noise. The slope between your P&L and the realized vol is still positive, but not as sharply defined.

Philip brings in the practical concerns of, well, having an employer to answer to who doesn’t like loud “noise”.

If you want to minimize your mark-to-market P&L, you may choose to hedge to the market even if you think the market volatility is wrong.

How do you trade-off these two risks, the mark-to-market risk versus the at-maturity risk? Ultimately, you probably will decide based on the maturity of the option you are hedging.

• If the option will expire in a month or two, you will almost surely be able to weather any intermittent mark-to-market volatility, so you will lean towards hedging to model.
• If the option will expire in many years, you will likely lean towards hedging to market, at least until the expiry gets closer.

And what do people do in practice? They hedge their bets on how to hedge. One common rule of thumb is to hedge halfway between the model and the market delta. Then you’re never exactly hedged, but you’re never too far away either.

The inability to hedge perfectly continuously impacts your trading by introducing random risk. This risk decreases if you hedge more frequently, but only as fast as the square root. Therefore, if you want to halve your risk, you have to hedge four times as often.

This is a fantastic observation to give a sense of proportion:

Noise from hedging a one-year option on a daily basis instead of continuously is about the same as one volatility point. If you make one volatility point in expected profit and the standard deviation of your profit is one volatility point, then your Sharpe ratio is about one.

And remember…the risk from not hedging continuously can be diversified away.

His final point here echoes what I wrote in a misconception about harvesting volatility.

Which brings us to…

The Moontower Treatment

The original paraphrased question once again:

“I’m backtesting delta-hedged straddles and I’m worried the vol I use to compute my hedge delta is ‘wrong.’ Does the choice of hedge vol bias my P&L, and if so, how?”

My dead-leg-on-the-toilet response:

Here’s the quick answer…the vol that generates your delta introduces bias that you discover after the fact but you can understand how the bias is correlated to your p/l in different scenarios.

For example, if you are long vol and the stock trends, you will wish you hedged on whatever delta was the “lowest” of the reasonable options you could have chosen from…so if the option is ITM you will have wanted to hedge deltas on a high vol, but if it was OTM you will have wish you hedged on a low vol!

I’ve never done this, but you could create a little cheatsheet matrix with:

• option ITM or OTM
• market trends or chops
• preferred vol i wish i would have hedged on = “high” or “low”

By comparing that matrix to your strategy you can see which biases cause you to double down on your implicit exposure vs hedge it (for example, if you are long ITM options and vol expands in a trending market you will hedge on that desirable light delta…but you are already winning on vega so maybe this codependancy is too much “doubling” down which hurts extra if you were short that option)

Of course, I had to make the cheatsheet now that I got a moment to focus on the question. To start, I fed my response to Claude and it whipped something up. I did have to re-work some of its understanding.

[These are Gell-Mann amnesia moments, where it stumbles on things you know well, and wonder about what it tells you in domains you are less equipped to discern.]

Let’s begin with the cheatsheet, memorialized at https://delta-hedging.moontowermeta.com/:

The sheet is self-explanatory, but there are biases we can anticipate. It’s what I referred to as “the doubling-down” in my response to the reader.

Suppose you follow the rule:

“Hedge On Implied Delta”

IF:

[You buy an OTM option because you think IV < forecasted realized]

AND:

[Your vol signal is correct]

THEN:

[Your hedge ratios will be “light”…I buy OTM calls and sell too few shares]

THEREFORE:

If we trend, you will make “extra” p/l beyond the fact that you bought underpriced volatility. This is “doubling-down”.

If we chop, you will make less gamma scalping p/l than you would have with a heavier delta. The forgone p/l will be buffered by the fact that you were right on the vol being cheap.

In this case, hedging on the delta of the implied vol, is doubling down on your vol forecast in the event that we trend, and offsetting some p/l in the event that we chop.

💡Your choice of delta to hedge on begs you to wonder if a high realized vol forecast is more likely to coincide with trend or chop.

Most of the time, options embed a risk premium above the realized vol.

[The bridge between this idea and making money on selling options sways wildly and has a few missing planks. Many have died trying to find the treasure on the other side so take it easy Indiana Jones.]

That said, it’s understandable if you never want to buy an option. But sometimes you want an option like exposure, just like you might want an insurance policy. You want protection against a high-impact event even if you don’t think it will happen.

I discuss this in the Moontower community, where I prefer to hold BTC exposure as options rather than as a hard delta allocation (I actually use a blended approach, but the reasons aren’t germane to this post).

I pick my spots when I buy the options. My most recent call purchases feel validating because I thought the vol was cheap, so despite losing on direction, they were much better buys than the counterfactual of owning hard deltas.

[Welcome to vol trader cope. This is literally what life is like as a vol trader. I lost money but made the right decision. Yay. You only hope that your career lasts long enough to realize the sum of all the right decisions. The alternative of just guessing in a high-variance game and trying to get lucky is good too. If we focus on survivors. And we do. This is America after all.]

But what if you wanted to replicate the call exposure without actually buying the calls?

Replicating a Call When You Think It’s Overpriced

The closest neighbor to the term “portfolio insurance” in a database of vector embedding is “1987” (Did I put those fancy words in the right sequence? Who cares, you get the joke).

Don’t let that taint your mood going into this next section. You know that I know about that history. Calm down, we’ll extract the fruit from replication and point out the poison you can’t eat.

Step-by-step here.

You want BTC call exposure. You look at the options and think they’re overpriced. So you decide to skip the call and instead replicate it dynamically.

How?

You will be delta hedging in reverse. You’re assuming the posture of someone who sold a call and now needs to replicate it. An option market-maker who sells you a call must go out and manufacture it. If they can manufacture it for less than the price they sold it, they make a profit.

In this case, you are taking the role of call buyer, but instead of buying the call, you are going to try to manufacture it yourself, just like the market maker would have if you bought a call from them.

Mechanically, you’ll hold some BTC, intermittently rebalancing your position as spot moves, synthetically tracing the call’s payoff without paying the upfront premium.

How much is some?

You look up the delta of the call you would have bought, and you hold that much BTC.

How does intermittently rebalance work?

As BTC rises, delta increases, you buy more. As BTC falls, delta decreases, you sell some. You’re manufacturing the call’s convex payoff with a series of linear trades.

How often?

How often does a market-maker hedge? This is the question we’ve tackled many times. It’s a trade-off between the “noise” Maymin alludes to as you sample volatility. If you are a market-maker hedging a short option and the market trends, you’ll wish you hedged often (sampling a lower vol than experience from point-to-point).

If it chops, you’ll wish you hedged weekly, sampling a much lower vol than the daily ranges suggest. Both you and the market-maker face the same problem. You are both trying to manufacture an option whereby each time you trade you “sample” a realized volatility. The more you sample, the closer you get to the real vol. The less you sample, the more likely your replication strategy will differ from the real vol and you could get lucky or unlucky to the platonic (and non-existent) continuous vol.

The cost of this replication comes from the adjustments. To replicate a call, you buy more as the market rallies because the option for the strike you’re trying to mimic increases. You sell as the market falls. You are always buying high and selling low. The sum of those round-trips is your premium. You’re just paying it in installments instead of upfront. If you think these installments net of all transaction and slippage costs would exceed the call premium, you should just buy the call.

To feel good about this strategy, you’re rooting for the options to have been overpriced. If realized vol comes in lower than implied, your rebalancing costs less than the call premium would have. You built the same payoff for cheaper.

To determine how much stock you need to buy, you’re computing your delta at some vol, and that choice determines whether your delta is heavy or light. If you hedge at a high vol (say, the implied you think is too rich), you’re holding more BTC than you “should” — heavy delta. If you hedge at a lower vol (your realized estimate), you’re holding less — light delta.

The cheatsheet as an aid to your hedging strategy

The sheet has the posture of someone long an option, who by replicating is manufacturing an equivalent short option. They paid a premium upfront, but hope the sum of their gamma scalp stream exceeds the premium they paid. In other words, their replication posture is the opposite of yours. You are trying to replicate a long option because you think it will cost less than actually buying a call.

So you invert the logic of the sheet!

If BTC chops you want a light delta. Fewer round-trips means less friction eating into the savings you’re generating by not paying the full premium. If you are right about the IV being overpriced but you hedged using the implied delta, then you will suffer a bit because your delta will have been heavy. But this will partially offset the profitable decision to not buy the call outright. If you hedge on your “model” delta (ie the vol based on your realized forecast), then you are doubling down on your prediction that the vol is cheap in the event we chop.

Again, the idea of vol and its coincidence with trending or chopping is lurking beneath but now you are aware of it.

Restriking Your Synthetic Call

Say BTC has run from 70k to 90k. You’ve been replicating a 100k-strike call, but you want to “roll” it up, taking profit and starting fresh with a 130k-strike call.

You can just look up the 130k call at your chosen vol and adjust the delta to match. That will result in monetizing some of your BTC as the 130k call will have a lower delta than the 100k call.

Notice that if you don’t roll your 100k call is closer to ATM with the spot BTC now up to 90k. It has more gamma than your old deep ITM 90k call. More gamma means your rebalancing is more frequent and more costly. You’re “long” a more expensive option. There’s no free lunch. If you substitute your replicated call for a real call, that call’s theta will reflect the higher rebalancing costs you tried to avoid.

So….

What Makes You Wish You’d Just Bought The Call?

This question strikes at the heart of the Black-Scholes assumption of continuity.

Gaps.

The call buyer pays implied vol upfront and owns the path, for better or worse, for the duration of its life. If a stock gaps up 20% over the weekend, the call captures the full move. The gamma which you prepaid for, ensures your delta adjusts automatically.

The synthetic call you tried to manufacture missed buying deltas in the gap. You are not as long as you should be and to make it up you need to buy all your shortfall deltas up 12% as opposed to prices along the way.

Hard optionality is valuable and impossible to replicate. This is why Option Market Maker 101 class teaches you that the only way to hedge an OTM option is with another OTM option. Nobody knows what the SPX down 25% put is actually worth.* You can reason about a relatively tight put spread only because the error is bounded in proportion to the risk you know you are taking beforehand.

(Although we can reason that it commands a premium and likely trades for more than its actuarial value which is not really known. It’s all a bit circular. And you are still left to contend with the fact that the people, as a category, who buy those teenies know a lot more about vol trading than you. There is no non-vol trader buying that option. Also, this paragraph was written in invisible ink to reveal the VIX basis traders on the mailing list.)

Portfolio insurance failed because it was crowded thus blowing up the cost of put replication by feeding on itself. Meanwhile, the owners of the actual puts went on to start the trading firms you know of today.

The “Bridge of Asses”

📺Option Pricing Explained: No Arbitrage + Financial Mathematics from a Quant | 52 min watch

Doug Costa (SIG quant, former math professor, and the teacher I learned Black-Scholes from 25 years ago) builds no-arbitrage derivatives pricing from scratch using a binomial tree. No calculus, pure replication.

The thing I want to point you to is the profound role of the no-arbitrage axiom. It is the basis of derivatives replication and, by my assertion, represents the “bridge of asses” in investing education.

As a reminder, since nobody clicks links, Wikipedia says the pons asinorum or “bridge of asses” is:

used metaphorically for a problem or challenge which acts as a test of critical thinking, referring to the “ass’ bridge’s” ability to separate capable and incapable reasoners

The notion of replication is the pons asinorum of investing education because it is:

the conceptual rails of looking at a web of branching future payoffs, seeing how they could be replicated, and measuring the cost of that replicating portfolio today. It is the formalization of finance’s deepest truth — you cannot eradicate risk, but only change its shape.

You could make an even stronger claim that it lies at the core of decision-making itself, as it formalizes opportunity cost.

And I say this without being able to appreciate its deeper impact. Doug pauses for a moment in the video to marvel: when you add no-arbitrage condition to the standard axioms of mathematics, he says, the entire field of financial engineering “blossoms” out.

His colleague frames the no-arbitrage axiom joyfully:

Either we get a formula [so we win mathematically]. Or it’s violated and we make free money. Either way, we win.

Towards the end of the video, Doug discusses reflexive pushbacks he’s encountered after teaching this.

“One piece of pushback is typically, well, maybe it’s just that with stock prices, you don’t really know the probabilities. So it’s just a matter of knowing the right probabilities— if you could really discover somehow what the true probabilities were, then it would be better to use them [than the risk neutral probabilities].”

Doug’s rebuttal shows how you would still be arbed.

“I’m going to give you an example to debunk that idea. And I call this example the coin flip contract. So I’m going to postulate that there’s a company, a corporation, that finances itself, not by selling stock, but by selling what they call coin flip contracts. And the corporation has gone to great trouble and expense to manufacture a perfect coin, meaning a coin that is exactly 50% to be heads and 50% to be tails every time it’s flipped. So the probabilities are always 1 half and 1 half guaranteed…

You can watch the video, but I paraphrased it here as well. Here’s how it works.

A company issues coin-flip contracts based on a provably fair coin. The contract pays $150 on heads, $75 on tails. These trade in a secondary market at $100. Interest rate is 0%.

So we know everything. The probabilities aren’t hidden or estimated. They’re printed on the coin: p = ½.

Now: what’s the no-arbitrage price of a 110-strike call on this contract?

p̂ = (100 − 75) / (150 − 75) = ⅓

Call value = ⅓ × $40 + ⅔ × $0 = $13.33

Delta = (40 − 0) / (150 − 75) = 8/15 of a contract

Now suppose you say: I know better. The real probabilities are ½ and ½, and I’m not going to ignore them. Expected payoff is ½ × $40 = $20. So you buy the call from me at $20.

Here’s what I do next. I’m short the call. I immediately buy 8/15 of a contract to hedge.

Heads: My 8/15 position gains 8/15 × $50 = $26.67. Plus your $20 premium, I have $46.67. I owe you $40 (I have to buy the contract at $150 and sell it to you at $110). Net: +$6.67.

Tails: My 8/15 position loses 8/15 × $25 = $13.33. But I have your $20 premium. Net: +$6.67.

Every time. Both states. Guaranteed $6.67. I haven’t predicted anything. I don’t care what the coin does.

What did you get? Heads: gain $40 on the option, paid $20, net +$20. Tails: lose your $20 premium, net −$20. You’ve turned a coin flip into a coin flip — a $20 bet where you win or lose based on what the coin does.

If you try to hedge back? Doesn’t matter how you move delta. Win more on heads, lose more on tails. Move it down: vice versa. The best you can do is lock in a guaranteed $6.67 loss.

You had perfect information about the true probability….and you still got arbed buying the calls (you should have bought the contract!).

The market-maker doesn’t need a view on the coin, just the ability to trade the underlying and the derivative simultaneously. And acquiring the knowledge to cross the “bridge of asses.”

A random personal thought:

I suspect is kind of triggering for some people. It offends one’s sensibilities to think

that understanding derivative pricing ends up trumping knowledge about the true odds of things.

It’s like you spend all this time researching and learning and at the end of the day some market-maker knows just enough to not trade at the wrong price with you anyway. I’m overstating that reality, getting picked-off is real and market-makers are rightfully paranoid. But I guess that’s why I’m drawn to replication as a way of thinking. A trader is just looking for some free money when your bid or offer presents a contradiction. And that hunt makes all prices a little smarter, which, is a public good (but also a frustrating result for traders themselves, which is why the job is always uphill. A byproduct of your success is a smaller TAM).

Just to be thorough, this replication thing applies mostly to derivatives. The arb needs to be able to trade the derivative and the underlying and all advantage comes from the relationship between the two. The arb is useless without relative value.

Related learning:

🔗 Understanding Risk-Neutral Probability | Moontower

🖥️Moontower Presentation on Black Scholes “As a Trading Strategy” | Slides

• The slides for that presentation are based on this post: The Intuition Behind The Black-Scholes Equation

[UPDATED]

As I expected, the post how to get arbed with perfect info would trip people up. I didn’t expect confusion because I thought Professor Doug Costa, whose explanation is featured in that post, was itself confusing. But because the concept of replication is hard and feels like it violates the good. It’s triggering. It means you can know the truth and still get arbed. Again, this is why I call it the pons asinorum of finance.

A reader brought it up in our Discord so I’m going to share the discussion here as he felt like our back and forth helped.

Before getting to the conversation, let’s refresh the problem Doug set up:

A company issues contracts based on a provably fair coin. The contract pays $150 on heads, $75 on tails. It trades at $100. Interest rate is 0%.

You calculate the true expected value of the 110 call using the true probability of 50%.

It’s worth $20 because it has a 50% chance of being $40 in-the-money.

You pay $20 for it (but even if you paid a bit less for an unambiguously positive EV trade, this analysis will hold. I just want to stay with Professor’s example)

The dealer sells it to you, hedges with 8/15 of the underlying contract, and locks in $6.67 profit in both states. Pure arb.

You had perfect information about the true probability and you still got arbed. The dealer made money in all scenarios, trading the call at fair value with you.

Doug is showing how the real-world probability doesn’t matter to the derivatives trader if they can also trade the underlying. In this case, the underlying is mispriced, but the dealer doesn’t know that. All the dealer cares about is whether the relationship between the derivative price and the underlying price is mispriced. In this contrived example, the mispricing was more profitable than knowing the true probabilities.

And to add something Doug doesn’t mention…if the investor knew the stock was underpriced and bought that instead, they’d have a positive EV trade (the fair price of the stock is $112.50) but they are still worse off than the dealer who knows the relative value of the 2 securities is wrong and gets to make a profit in all scenarios.

This is a good place to insert the chat.

Reader: I see, so the main point is we can converge a spread by trading two things instead of betting on one.

Kris: In a world with no derivatives, you’re left with having to be good at guessing real-world probabilities, but derivatives are their own source of possible edge that doesn’t inherit from knowledge of the future but from relative mispricings between the derivative and the underlying.

It’s obvious that being able to handicap probabilities would be a source of edge, but it’s quite subtle that once you introduce derivatives and the idea of replication, there becomes a source of profit that doesn’t rely on such an ability.

Reader: Right, so it’s instructive in giving one more spread to look at. If you used a put in your example, then the dealer would lose because they’d be too short. Then a dealer that actually has no information and sells both sides ends up $0. This example is picking the (long) side where it wins.

Kris: Yeah, the underlying in this example is too cheap RELATIVE to the call option.

If the call option was $13.33, then from the vantage point of real-world probability, both the underlying and call are too cheap, but they are priced correctly with respect to each other.

Which makes the point — if a derivative and underlying are correctly priced to each other, then the real-world probability is not important to the dealer. The dealer only cares about the relative values.

You can just compute the Sharpe of buying one vs the other I suppose to see which is better (that’s one lens). The call is more underpriced in % terms, 3.33 when it’s worth 20. But it’s more volatile as it will lose all of its value when it loses.

I’d just stick them both in a Kelly calculator in Claude or something and whichever one it says bet more on is the better one lol.

There are some important implications here. And brain damage — investor brain and derivatives brain collision.

The goal is that derivativesbrainskill.md becomes something one calls as needed, like Neo downloading kung fu. But you don’t wanna get carried away with it and shoot it at everything in life. It’s this weird artificial thing that works in a replication context, but it’s also not artificial in that its violation presents hard cash arbitrage!

That’s the end of the chat, but let me add one more thing to make you feel better if it’s still foggy.

I’ve seen this subtlety trip up seasoned options traders where they take B-S pricing to mean that the forward for a stock is stock grown at the risk-free rate (RFR), but this is ONLY true in a world where you can trade the underlying AND the options. Outside the context of replication, you cannot make that assumption.

Struggling with this idea is entirely forgivable. I mean, the realization that you could use RFR as the discount rate was a revolutionary breakthrough. Bachelier figured out option pricing in the early 1900s, but he and his contemporaries were stumped by what rate to discount the payoffs.

Later academics wondered if you should use something like the required return from CAPM or something, but it was the whole idea that if you trade a derivative vs the underlying against one another, then you can have equivalent payoffs and therefore it’s riskless to go long one and short the other. If it’s riskless, then RFR is the appropriate discount rate.

Warren Buffett sees the necessity of agnostic dealers using the RFR to price options in arbitrage-free ways as an opportunity. He asserts that put options are overpriced because they use too low of a discount rate, but the dealers don’t care so long as they can trade the underlying, they can arb any other rate assumption. Again, so long as “they can trade the underlying.”

This single idea allows derivatives traders who know nothing about the fundamentals of securities to make money in a sea of people who do. It’s quite profound and not a small part behind why I think vol trading is easier than directional trading.

A final follow-up

A reader asked:

How did I get to the 8/15 hedge ratio?

It came from Prof Costa’s setup:

A stock is 50/50 to go to $150 (up) or $75 (down) from $100. What is the no-arbitrage price of a 110-strike call on a one-period binomial?

p̂ = (100 − 75) / (150 − 75) = 1/3

Call value = 1/3 × $40 + 2/3 × $0 = $13.33

Delta = (40 − 0) / (150 − 75) = 8/15 of a contract

Let’s take this apart.

Risk-neutral probability of stock going up

p̂ is the risk-neutral probability of the stock going up.

How do we get that intuitively?

Start with the payoffs:

• Up_payoff = 1.5x
• Dn_payoff = .75x

The risk-neutral probability is the one that makes the stock price fairly priced given the possible payoffs. In other words, if you buy the stock, the expected return is 0. It must satisfy this equation:

p̂ (Up_payoff) – (1-p̂ ) (Dn_payoff) = 0

Solve for p̂:

p̂ (Up_payoff) – Dn_payoff + p̂(Dn_payoff) = 0

p̂ (Up_payoff + Dn_payoff) = Dn_payoff

p̂ = Dn_payoff / (Up_payoff + Dn_payoff)

Concisely stated:

p̂ = d/(u + d)

p̂ = .75/(1.5+.75)

p̂ = 1/3

When in doubt, you can always set up the expected value equation and solve the algebra. You don’t have to memorize a formula.

But you can also put on your gambler goggles.

If something pays 2-1 odds like a money line of +200 or a prediction market trading at 33, then the implied risk-neutral probability is 1/3.

If you buy this stock, you risk $25 to make $50.

Just remember the intuitive odds to probability converter:

x-to-y odds = y / (x + y) probability

2 to 1 odds = 1/(2+1) = 1/3

This is just symbols for “If I get paid 2-1 when I win, then I must win 1 out of 3 times for this to be fair”

You can practice some more in these posts:

• the arbitrage reflex is more profitable than the opinion reflex
• Binary Straddle Example Based On The 2016 Election

The hedge ratio

Back to the original example…if you’re short one call, you buy 8/15 of a share to be delta-neutral. The numerator is the spread in the call’s payoff across the two states ($40 vs $0). The denominator is the spread in the stock’s payoff ($150 vs $75).

Delta is just option change over stock change. How much the derivative moves for a move in the underlying.

The formula in terms of return:

delta = (C_up – C-down) / S (Up_payoff – Dn_payoff)

delta of the 110 call= (40 – 0) / 100(1.5-.75) = 40/75 = 8/15 = .533

Relative value

To reinforce the main point of Prof Costa’s talk, risk-neutral probabilities are enough to make money if you can find an inconsistency between option prices and the underlying. The real-world probability doesn’t matter in a relative framework.

By understanding the distribution of the stock, we were able to compute a delta for any contract. That distribution implies some probability embedded in the underlying. This is not the same as the real-world probability, which is decreed to be 50/50.

Professor Costa showed that if you buy the stock which was underpriced (although you didn’t know that because you computed it must be fair with a 1/3 probability of going up) as a hedge on the call delta, then if someone paid anymore than the risk-neutral fair value of the call, even if they paid less than what the real-world implied probability price is, the dealer makes free money!

Work through the logic for a bunch of strikes and this is what you get if you sell the calls at the risk-neutral fair value (green) or real-world fair value (purple).

The dealer always wins against the real-world probability!

It is also true that the call buyer who pays some price between risk-neutral and real-world has positive expectancy but they don’t have an arbitrage.

So who loses?

The people selling the stock at $100 when the real-world probability is that it’s 50/50 to be 150 or 75. The stock should be $112.50.

Learn more:

🔗 Kellogg lecture notes that walks through exactly this kind of binomial pricing and hedging

Recall the levered silver flows post where we see the quick math of levered ETFs. For a fund to maintain its mandated exposure, the amount of $$ worth of reference asset they need to trade at the close of the business day is:

x(x - 1) * percent change in the reference asset * prior day AUM

where x = leverage factor

examples of x:
x=2 double long 
x=-1 inverse ETF
x= 3 triple long
x= -2 double inverse

This isn’t just a levered ETF thing. The -1 leverage factor is exactly the same as just a vanilla short position. It’s a sneaky reason why the shorting is mathematically challenged.

The easiest way to think of this as an individual investor is to imagine you have an account value of $100. The account is holding $100 in cash, but it’s the proceeds from shorting a $100 stock (assume you don’t need any excess margin to maintain the short). If the stock falls to $50, your account value is now $150 (your cash + $50 mark-to-market profit on the short). You earned a 50% return on a 50% drop in the stock.

Now what?

If the stock falls another 50%, you make $25.

$25/$150 = 16.7%

If you want to maintain the same exposure so that you make 50% on your account on that second 50% drop, you would have needed to short more shares at $50.

How many more dollars’ worth of stock?

-1 (-1 -1) x -50% x $100 = -$100

You needed to sell an additional $100 worth of stock or 2 more shares at $50. Then on that last leg down, you would have made $25 on 3 shares total or $75.

$75 profit /$150 account value = 50% return

Learn more:

🔗 The difficulty with shorting and inverse positions.

“Vega wounds, gamma kills” is an esoteric expression that’s still common enough that you can google it and return a bunch of hits. It’s a reasonable acknowledgement of realized vol p/l being quadratic with respect to how large a stock move is.

I’ve recently been cross-posting my writing on how this works on X since they’ve been pushing their Articles functionality.*

* A lot of people (and bots) are boosting these. I am treating these releases as a spaced repetition exercise for long-time readers. Analytics show very high engagement so X must be signal-boosting them. This is a 1-year chart. The recent spike is Articles:

A lot of people cry about the growth of Articles longform on X but twitter is a long way from the community it used to be anyway, so don’t really care as much if I’m burning the house for warmth in the eyes of diehards. Although I don’t think I am since the reason I came to twitter in the first place was to find stuff to read and learn not hot takes. It's different things to different people and when they suppressed Substack it shifted the appeal for me. This is some re-alignment, albeit on their terms. Fine. It's a reasonable negotiation. 

The Articles I’ve posted on the theme of non-linearity in options

• Moontower on Gamma
• using elementary school geometry to explain gamma
• Finding Vol Convexity

This last one is about the “gamma” of vega. For OTM options, the vega of the option, its sensitivity to changes in IV, itself changes. We call that second-order sensitivity volga. Volga is to vega as gamma is to delta.

I don’t have a dedicated post on vanna I’ll cover it briefly right now.

Vanna

The definition of vanna you are most familiar with is change in delta due to change vol. You have heard of this because of dealer flow discourse. For example, if dealers are long calls and hedged with short shares, as vol declines on a rally, their long option deltas shrink. If this happens faster than their long gamma increases their net delta, then they will have stock to buy to rebalance to neutral.

But vanna has an alternate definition. One that dominates our understanding of trading skew:

the change in vega for a change in underlying

If you are short puts on a risk reversal as the stock falls, you get shorter vol and vice versa. Your vega changes as the spot moves.

---

I suspect the “gamma kills” idea is popular because it’s a common experience. Option volume is dominated by near-dated expiries where gamma and theta dominate the p/l. Most people will simply never feel what it’s like to be wrecked or celebrated by volga or by a delta-hedged skew position. They might know what it’s like to get crushed to vega directly, but even that will be less familiar than realized vol-driven performance, given typical trade duration.

But I can tell you that my most memorable p/ls have all had vanna and volga at the scene. 2020 was especially dramatic in this regard as an explosion in vols led to position sizes exploding and finding myself sitting on a growing pile of vega that varied from “increasing in demand” to “panic bid”.

Qualitatively, the repricing of vega is significant because vega is illiquid. You can delta-hedge your way to a replication of a relatively short-dated option. In a sense, the volume in the underlying itself is a form of liquidity for options even if the options themselves are illiquid. But this idea extending to a long-dated option is only theoretical. In practice, if you are short a long-dated straddle that doubles in value, the mark and its accompanying hit to your capital may leave you in a forced position. You don’t have the luxury of manufacturing that vol via delta-hedges for a year.

This will be exacerbated if you were short, say 100k 1-year vega, but because of vol exploding you find that you are now short 200k vega. Maybe you can stomach the p/l hit due to vega, but you might not be able to hold the new position size. If Street Fighter’s Vega had Mortal Kombat finishing moves, they would be called vanna and volga.

The recent silver move has been so crazy that vega p/l has dominated realized p/l (realized p/l is the tug of war between gamma p/l from the equation at the opening of the post and theta). It’s an outstanding case study in how higher-order effects are fundamental to understanding options.

We’ll begin with a classic “trap” trade.

Imagine back on Dec 31st, with SLV at $64.44, you bought put and sold call on the 60/100 risk reversal delta neutral with the plan to hedge the delta at the close each day.

This position starts:

• Long vega
• Long gamma
• Paying theta (you laid out extrinsic option premium)
• The 60 put you buy is 59.6% IV, the 100 call you sell is 78.7% IV

The risk reversal would have cost you $2.89 of option premium since the put is much closer to at-the-money.

💡I used the Moontower Attribution Visualizer to compile data for this article

What happens between when you opened the trade and the snapshot I took this past Tuesday, 1/27/26, when the stock has risen to $97.09 and the options still have over 3 weeks to expiry?

This daily hedged risk reversal has lost $.82 net.

You are short gamma albeit less gamma than you were long when you initiated the trade because the ATM vol is so much higher!

More things to note:

The IV on your long strike: 59.6% → 99.6% or 40 vol points!

The IV on your short strike: 78.7% → 99.4% or 25 vol points.

You won on the vega spread between the options.

So why did you lose money? Was it the realized vol? That seems suspect, after all, you were long gamma at the start of a big move. You’re short gamma now, yes, but it’s not even that much.

The clue is right there in the table:

You went from long 5 cents of vega to short almost 14 cents of vega as your short strike is now at-the-money.

Yes, the vol on your short strike went up much less than the IV of your short strike, BUT it went up when the vega of that strike was much larger than the vega of the strike you were long.

In short, you were getting shorter vol as vol was ripping higher. The vega p/l totally swamps the realized p/l:

Here’s a snapshot from the interim on 1/13/2026, when the stock had rallied almost to the midpoint of the 60 and 100 strikes.

The 60 put you own has gone up over 7 points, and the 100 strike you are short barely budged from the elevated vol from the original skew. You are up $.37 on the hedged position…but your risk is changing quickly. You are now short vega, flat gamma, and collecting theta.

Wait, you are collecting theta without being short gamma.

Technically your gamma is very slightly short, but the point stands — in fact, if the 60 put IV was a bit lower you could even be long gamma and collecting theta. 

New option traders will brag about such a favorable greek profile. An experienced trader knows that the ratio is an indication that you are simply short a premium IV and premium IVs happen near the prices where hell breaks loose. As I’ve said many times…the skew just tells you where the bodies are:

In sum,

Despite these options not being “long-dated” their performance has been dominated by IV. In this case, mostly through vanna which is best seen at the interim.

• Despite the 60 put vol increasing 7 points, the vega of the option halved as it was now much further from ATM (it went from being a -33 delta put to -9 delta by 1/13/26)
• Meanwhile, the 100 call’s vega doubled due to it becoming closer to ATM (it went from a 9 delta call to 21 delta).
• Note that volga is not playing much of a role in 100 call vega doubling. The change in option vega can’t be due to IV increasing. Why? Because IV didn’t change on the 100 strike during the rally from $64.44 to $78.60!

From the vol convexity article, we know ATM options have no volga. In fact, ATM vega is insensitive to vol level and holding DTE constant, it only depends on the spot price!

But OTM options have a lot of vega to gain if IV increases since IV ripping higher makes all OTM options look closer to ATM as they are “less far away”. Their delta increases (vanna) and their vega increases (volga). In the above example, the 100 call IV did not rip higher by 1/13/26, so we couldn’t see volga in action. The vol only roofed on the strike once the option was close to ATM.

To give volga its due, we should zoom in on Monday when Feb SLV vol ripped higher on silver popping 10% (before giving back nearly half its gain).

We’ll look at a call nearly 14% OTM with less than a month til expiry.

The $1.33 of hedged option p/l for that call is only partially explained by the initial vega of .033 and a vol change of 26 points. The difference could be explained by the fact that the average vega of the call as vol (and stock) increased was probably closer to .05.

26 vol points x .05 vega = $1.30

Since the stock only rose by 6%, we can safely guess that the 50% increase in the vega of the option is mostly driven by volga.

Gamma is not the only killer. Any position that grows faster than the underlying changes contains risk that is not seen in a snapshot. That delta hedged vertical spread or risk reversal might look gamma, theta, and vega neutral today but that profile gets battered as soon the clock ticks and the waves start coming in. The snapshot neutrality is dangerous because it can easily lull you into thinking your risk is smaller than it really is.

Ask anyone who bought an SLV and nat gas 1×2 call spread because “the skew was fat” or because they are “long gamma, collecting theta” how that’s working out?