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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. 

By now y’all know option traders have the ATM straddle approximation burned into their retina:

straddle ≈ .8 Sσ √T

A useful approximation I did not explain in the interview is the similar-looking ATM gamma formula for a Black-Scholes straddle:

Γ ≈ .8 / (Sσ√T)

The three things that shrink gamma are in the denominator:

Higher S (price): The same $1 move is a smaller percentage move on a more expensive underlying.

Higher σ (vol): The option is already “priced for action.” The curvature of the price function gets spread over a wider range of expected outcomes. More vol → flatter curvature near the money → less gamma.

Higher T (time): Same logic as vol. More time spreads the curvature out. The more time to expiry the less a given move influences the delta of the option. The delta of 10-year option is not going to change much based on how the underlying changes day-to-day.

A couple of educational points:

• Take note of the scaling. Double the vol, gamma roughly halves. You need to quadruple DTE to get the same effect.
• As always, a good habit when trying to understand greek levers, is to take examples to extremes. If you raise DTE or vol to infinity, all options go to their maximum value. For calls, that’s the spot price itself. For puts, it’s their strike price. That means calls go to 100% delta since they move dollar-for-dollar with the spot. Puts go to 0 delta. It doesn’t matter where the spot price goes, the option is already at its max value. It doesn’t change. If a call is 100% delta and a put is 0% delta, the option has no gamma. Its delta doesn’t change with respect to the spot.

Going back to those formulas for a moment:

straddle ≈ .8 Sσ √T

Γ ≈ .8 / (Sσ√T)

The denominator of gamma = straddle/.8

Substituting:

Γ ≈ .8 /(straddle/.8)

Γ ≈ .8 /(straddle/.8)

Γ ≈ .64 /straddle

So when you want to do mental math you take “2/3 of the inverse of the straddle.”

This might sound obtuse, but taking inverse or “1 over” some number should be one of the fastest mental math operations anyone dealing with investing does. After all, when you see any ratio like P/E or P/FCF you are immediately flipping that to a yield where it can be compared with things like interest rates or cap rates.

If a straddle is $5, the gamma is 2/3 of $.20 or ~.13

And we know that doubling the straddle halves the gamma so you can just memorize that a $10 straddle has ~6.6 cents of gamma and linearly estimate gamma for any straddle price relative to that (ie $20 straddle is about 3.3 cents of gamma and $15 straddle is in the middle of 3.3 and 6.6).

And of course there’s time scaling. To find an option that has double the gamma you need to cut the DTE by 1/4.

Keep flipping this stuff over in your head, it’s satisfying, and it thickens the myelin around whatever brain cells you sacrifice to options damage.

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

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

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

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

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

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

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

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

Your equity option intuition is of no help here.

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

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

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

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

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

What to expect today:

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

Data study setup

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

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

Address the temptation head-on

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

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

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

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

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

Gamma revisited

A quick review is in order.

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

P/L = ½ · Γ · (ΔS)²

The ATM gamma formula for a Black-Scholes option:

Γ ≈ .4 / (S · σ · √T)

The three things that shrink gamma are in the denominator:

Higher S (price): The same $1 move is a smaller percentage move on a more expensive underlying.

Higher σ (vol): The option is already “priced for action.” The curvature of the price function gets spread over a wider range of expected outcomes. More vol → flatter curvature near the money → less gamma.

Higher T (time): Same logic as vol. More time spreads the curvature out. The more time to expiry the less a given move influences the delta of the option. The delta of 10-year option is not going to change much based on how the underlying changes day-to-day.

A couple of educational points:

• Take note of the scaling. Double the vol, gamma roughly halves. You need to quadruple DTE to get the same effect.
• As always, a good habit when trying to understand greek levers, is to take examples to extremes. If you raise DTE or vol to infinity, all options go to their maximum value. For calls, that’s the spot price itself. For puts, it’s their strike price. That means calls go to 100% delta since they move dollar-for-dollar with the spot. Puts go to 0 delta. It doesn’t matter where the spot price goes, the option is already at its max value. It doesn’t change. If a call is 100% delta and a put is 0% delta, the option has no gamma. Its delta doesn’t change with respect to the spot.

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

It will come back to that denominator in 2 ways:

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

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

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

Hedge Ratio Squared: Mapping M12 Gamma Into M1 Move Space

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

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

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

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

Review: From CAPM to Hedging

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

This chart pulls all of this together.

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

h is the hedge ratio for delta.

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

Gamma P/L

M1-Equivalent Gamma

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

Notice how:

• Delta scales with h
• Gamma scales with h²

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

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

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

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

The Raw Gamma Paradox

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

Look at the raw gamma numbers:

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

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

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

The longer-dated option has more curvature.

How?

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

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

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

The hedge ratio (h) is only .0215.

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

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

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

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

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

What does this do to your vega?

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

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

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

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

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

Spread Gamma

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

Real-life risk

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

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

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

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

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

Wrapping up

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

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

---

Appendix

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

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

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

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

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

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

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

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

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

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

Data Source:     IB API