If you buy an option, your exposure changes due to time passing alone. So that initial decision thrusts you into a series of future decisions, each one challenging you to tighten your thesis and face yet another bid/ask spread.

To demonstrate an appropriate use of options we stepped through a highly specific example during the lab and even executed the trade live!

I’m going to walk you through it here today.

Setup

The session was on Wednesday, September 30th. Nike (NKE) reports earnings after the close Thursday, October 1st.

Some context:

NKE is down ~75% since its lifetime peak which occurred 5 years ago. While falling short of the peak, it did stage a large rally from late 2022 into 2023, before resuming its downward March which accelerated this year with the stock down >40% YTD at the time of the session.

NKE typically pays about $1.40/yr in dividends or greater than 4% yield at recent stock prices. However, it can no longer fund the dividend from current cash flows. They could fund it from balance sheet assets (ie dip into their savings) to be able to pay it for several more years. I learned this from Brett Caughran’s tweet:

I used moontower.ai’s MCP (which is now included with individual plans

) to ask Claude if the options market was anticipating a cut. It turns out the implied dividend about 1 year out is 50% of what NKE has been paying. NKE’s woes are out in the open, the stock and options markets see them.

With such a backdrop, we expect this to be an important earnings call. After all, if a plane is nosediving, we expect the captain to say something. (We know the content of the words will be reassuring, but we’ll listen to the tone of his voice to infer how scared or optimistic we should be.)

Pulling up the option chain

With the above context in mind, we load the October 2nd option chain. This is clean as a market gets around an earnings move since the options expire 24 hours after the earnings call.

During the session, I reason aloud through what I’m looking at.

The stock is ~$36. The 36 straddle, which can be interpreted as the average expected move size, is ~$3 or 8% of the stock price. Earnings are a big deal.

I gracefully elide the question of whether you should be bullish or bearish. If I was privy to such information, I’d be a Market Wizard. But many people have views and wanna bet. I can help you reason through your choices given you have a view. So let’s get specific.

What are you paying for?

We arbitrarily side with the bulls, but the analysis that follows works for bears as well if they substitute different contracts.

If the 36 straddle is at-the-money, meaning the stock is also $36, then the 36 call is half the straddle or $1.50.

If the stock experiences the expected move to the upside those calls will double your money as they will be worth $3 of intrinsic value at expiry. In the language of odds, you get paid 1–1 on an 8% up move.

But if this is our specific bet, can we find an expression of the trade with more octane?

We load up the 38.5/39 call spread. It’s marked for $.12 and if NKE expires $39 or higher, it will be worth $.50.

You risk $.12 to make $.38 so you are getting just over 3–1 odds for the same $39 outcome.

Why does this sound so much better than getting 1–1 if we just buy the 36 call for $1.50?

Because the outright call continues to pay off if the stock climbs more than 8%. You are paying for the extra profit potential if NKE goes up 10, 15, 20 percent. Being vague leads you to pay for more margin of error. But if you’re specific, you can get extra leverage on being right. Of course, this is risky because if you’re wrong most scenarios lead to total loss.

Let’s deal with the risk

The setup of the session was that we have $5,000 of capital.

If we are wrong on the vertical spread and the stock doesn’t clear $39 then we likely lose our entire premium.

We get a 3-to-1 payoff when we are right.

This type of binary outcome is a perfect setup to use Kelly sizing. The nice thingabout Kelly is that it assumes you have a total loss when you are wrong, so the sizing itself is the risk management. You’ve already budgeted for the maximum risk.

We’ll get to the sizing in a second but just for building fluency in thinking probabilistically, let’s acknowledge a few heuristics:

1. We know from the derivation of the straddle approximation, that the straddle is 80% of 1 standard deviation. Therefore, it encompasses ~79% of the distribution assuming normality. Therefore the upper tail, where NKE goes up more than $3, is expected 21% of the time. To get 3–1 odds would require a minimum probability of 25% to be a positive EV bet. But, normality is doing a lot of lifting in that sentence. Earnings are more likely a binary outcome, and while it’s too simplistic to say there’s a 50/50 chance of the stock going up or down 8% (which would make this vertical spread seem irresistibly cheap), it’s not unreasonable to think the probability of that spread hitting is greater than 25% (or for that matter, for the symmetrical downside put spread for those with a bearish view).

In the name of walking you through sizing GIVEN you already have the bull view, we’ll presume you believe there’s a 40% chance the stock pops higher by the amount of the straddle or $3 to $39.

How much should we bet on the vertical spread if we are getting 3–1?

Kelly:

f* = p − q/b

where:

p = probability of winning

q = 1−p or probability of losing

b = the odds you’re getting → what you win divided by what you risk

Plug and chug:

f* = .40 — .60/3 = 20% or $1,000 of the allotted capital

Converting to contracts:

Each option contract is $.12 or $12 once we adjust for the 100 share multipler.

$1,000/$12 ~ 83 contracts

We don’t want to risk overbetting if we overestimate our p (i.e., probability of winning), so let’s size to about half Kelly and only buy 40 contracts.

We actually bought the vertical spreads live during the session.

Postscript

NKE ended up falling after earnings. The equivalent put spread on the downside would have been the 33.50/33 put spread, looking to get paid if NKE fell $3.

NKE closed Friday at $33.90, so the put spread would have expired worthless. At one point, the stock touched $32.09, so the spread would have been deep in-the-money, but because there was still some time value remaining, you wouldn’t have been able to capture its maximum value, although you probably could have flipped out of it and tripled your money, from $.12 to $.36

But it would have been hard to do since, with the stock sitting around $32 you have theta on your side, and you’d probably be thinking, “even if the stock rallies $1 or 3% intraday, I’ll have turned my $.12 into $.50.”

Alas, the stock ripped 6% intraday after bottoming, leaving the put spread worthless!

Notice how tricky this is for everyone. If you were short the spreads, you ended up collecting the full $.12 of premium, but the path would have greyed your hair as it looked like you’d face maximum loss or feel forced to cover only to later watch NKE stage a heroic rally.

Closing words

Options are surgical. I think of them like term life insurance. I have a specific thing I’m trying to mitigate or take advantage. If I buy them and lose, it may or may not have been a good outcome depending on what the counterfactual would have been. The term life analogy is an obvious demonstration of benchmarking to the right outcome.

The more tightly you can wrap an option expression around a terminal value thesis, the easier the risk management is because you can budget for total loss and let sizing do the heavy work.

Thinking probabilistically is a habit of mind. Reasoning between what’s in a price and your own confidence intervals around scenarios will develop not just discipline but it gives you metrics, even if they are finger-in-the-air estimates, to journal or track, allowing you to get a bit more calibrated with each rep.

If you don’t regularly read Money Angle (this is the finance section of my letter) maybe see if today works for you. It’s long, but that’s because we are going to drill the concept. If you succeed in walking away with a new mental tool to practice, then the ROI on the reading time will look like a steal.

My benchmark for success on this article is: did you actually learn something useful?

Let’s see…

One of the most important concepts in risk-taking is bet sizing. Which is unfortunate because people are quite bad at it, while the effort to be way above average is quite low.

A jarring and famous demonstration of this is the Haghani-Dewey Coin Flipping study, which showed how even college grads with business, economic, and technical backgrounds incinerated their capital or massively underperformed the expected profits presented to them by a game they knew was rigged in their favor.

You can read my synopsis in Bet Sizing Is Not Intuitive

For a binary wager (ie win or lose), if you know the payoffs and the probability of winning, both of which were known to the participants, the solution is to use the Kelly Criterion.

The tragedy is that it is incredibly simple to compute and applies to many conventional gambles and decisions (the examples in the quiz will span various life situations!).

If something is both easy and widely relevant, it should be common knowledge. So let’s fix that today. I’ll show you how easy it is to use, and you’ll forever be able to do it in your head.

First, a succinct definition:

Kelly is the bet size, as a fraction of bankroll, that maximizes the long-run compounded growth rate of your wealth. It’s a mathematical solution to bet size that doesn’t seek to maximize expected profit per trial, but the size that optimally balances compounding rate and survival.

If you want to go deep on this, see Moontowerquant’s Kelly Criterion Resources, but today’s focus is on getting straight to usability.

We will use this formulation of Kelly because it’s general:

f* = p − q/b

where:

p = probability of winning

q = 1−p or probability of losing

b = the odds you’re getting → what you win divided by what you risk

The easiest way to learn it is just jump right in with a few worked examples:

Fair coin wager (even odds style bet)

p =50%

q= 50%

b =1 (ie even money, for a $1 bet you either lose a $1 or make a $1 profit)

f* = 50% — 50% / 1 = 0 → bet nothing, zero edge

Coin biased in your favor (even odds style bet)

p =60%

q= 40%

b =1 (ie even money, for a $1 bet you either lose a $1 or make a $1 profit)

f* = 60% — 40% / 1 = .20 → bet 20% of your bankroll

Roll a 6 on a die (underdog bet where you get odds)

p =1/6

q= 5/6

b =8 (for a $1 bet, you either lose a $1 or make an $8 profit)

f* = 1/6 — (5/6) / 8

f* =8/48–5/48 = 3/48 → bet 6.25% of your bankroll

If f* is 0 or negative, you have no edge, so not betting is prescribed

Sports moneyline (betting as a favorite where you lay odds)

A −200 favorite. You risk $2 to win $1, and the line implies 2/3, but you think it’s closer to 3 in 4.

p = 75% q = 25% b = 0.5 (getting 50% return on the amount you risk)

f* = 75% − 25% / 0.5 = 75% − 50% = 25% → bet 25% of your bankroll

Wait a minute, these are large bets?!!

If these bet sizes seem surprisingly large for the given advantages, then your senses are well-tuned. For most people, “full” Kelly is too big!

Kelly maximizes long-run growth on the assumption your probability is correct. Well, it probably isn’t because the world is messy. We can inject some humility by using a fraction of Kelly:

• “Half Kelly” gives up about a quarter of the growth rate and roughly halves the drawdowns. If you invert that, you see that the Kelly scaling law means as you bet bigger, you get diminishing returns per unit of risk. Extrapolating that logic, betting more than “full Kelly” is incinerating compounded wealth even if the individual bet has positive EV.
• “Quarter Kelly” or less is far more common in practice.

Please don’t let the equation scare you, it’s intuitive and easy to remember

Look at the equation again:

f* = p − q/b

It’s just “how often you win” minus “how often you lose.” It’s just that the second term incorporates the payoff. The loss term gets divided by b, which represents the return you collect when you’re right.

• When b = 1, you’re getting even money. A 100% return. Dividing by 1 leaves q alone, and the equation collapses to pure hit rate: p − q. That’s the coin case where you bet $1 to make $1.
• When b > 1, you’re getting long odds. The division shrinks the loss term. This is why the die works. You lose 5 out of 6 rolls. Straight subtraction says you’re down 66 cents on the dollar and should never play, but you’re paid 8-to-1, so that 5/6 becomes 5/48, and suddenly the 1/6 win percentage is the bigger number. Long odds forgive a bad hit rate.
• When b < 1, you’re laying odds. Now division stretches the loss term. The moneyline: you only lose a quarter of the time, but at −200 each loss costs you two units to earn back one, so that 25% loss percentage behaves like 50%. Being right three times out of four barely clears the bar. Lay enough odds and even a very good record is a losing proposition.

The graphic shows how you can think of the odds (the denominator) as shrinking or inflating q, as you collapse your thinking to a comparison of p vs q.

A word on b

b trips people up because “odds” is loaded gambler jargon. A wider interpretation of b is that it’s a percent return.

It’s what you make divided by what you risk. Even money is b = 1: risk a dollar, make a dollar. That’s a 100% return on the amount at stake. 3-to-1 is b = 3, a 300% return. Laying −200 is b = 0.5 because if you risk two to make one, it’s a 50% return.

[Return is a profit, while multiples don’t subtract your initial risk. It’s the difference between “I 2x’d my money” vs “I made 100%” or “I 10x’d my money” vs “I made 900%”. The percent return is the multiple minus one because we subtract our initial risk.]

The reason it’s a return and not just “the odds” is that Kelly assumes a loss wipes out the whole stake. The denominator is always the same number: everything you put up. b is comparable across a coin, a die, and a moneyline because it’s the return on risk, always measured against a total loss.

b = 1 is a natural reference point. At 100% return, a win exactly cancels a loss, so you need to win more than half the time. The breakeven hit rate changes with b.

Set f* = 0 and you get p = 1/(1+b).

Read the table as a menu of the hit rates you’re allowed to have. At b = 24 you can be wrong 24 times out of 25 and still be flat. At b = 0.25 you can be right four out of five and still be flat

Applying to real life: when is Kelly the right tool

Kelly needs a few inputs: a bankroll, a payoff you know, and a probability estimate.

Which of these is a Kelly problem?

1. A prediction market contract trading at 30¢. You think it’s worth 45¢.
2. You’re all-in-or-fold on the river with a read that you’re good 40% of the time, getting 3-to-1 from the pot.
3. How much of your 401(k) to put in equities.
4. Writing checks as an angel investor across 30 startups.
5. Buying a weekly call on a biotech ahead of an FDA decision date.
6. Whether to take the new job.
7. Your buddy offers you 5-to-1 that it rains in Oakland tomorrow. The forecast says 30%.
8. Buying homeowners insurance. The premium is clearly more than the expected loss — that’s how the insurer stays in business.
9. Your neighbor doesn’t carry homeowners coverage. She banks the premium instead.
10. Your auto policy offers a $500 deductible or a $2,500 deductible, for a $340/yr discount on the premium.
11. You’ve got vested startup options. Exercising costs $40k out of pocket in strike, and you think there’s maybe a 15% chance the company gets somewhere that makes them worth $1M.
12. A merger arb spread. Target’s at $46, deal price is $50, and it trades back to $38 if the deal breaks. You think it closes 90% of the time.
13. Your agency spends 20 hours of unbilled time on a speculative pitch. You win about a quarter of them, and a win is worth 80 billable hours.

Solutions to Kelly Problems

1. Yes. Cleanest case there is. Binary, known payoff, and the price provides b directly. Risk 30¢ to make 70¢, so b = 2.33. f* = 45% − 55%/2.33 = 21%.
2. Yes. This is the canonical one. p = 40%, b = 3, f* = 40% − 60%/3 = 20% of your stack. The wrinkle is that in poker your stack isn’t really your bankroll. There’s a whole literature on pros using Kelly for bankroll management across sessions rather than for a single river decision.
3. No. Not this version of it. Stock returns aren’t generally binary so there’s no p, q, or b. There’s a continuous analog called Merton’s Share, which is similarly rooted in reward vs variance. What Gamblers Can Teach the Buy-and-Hold Crowd can get you started.
4. Sort of. The structure is right: repeated, roughly binary, long odds. The problem is that p is a guess and b is a bigger guess, and Kelly is violently sensitive to overestimating your edge. Garbage in, garbage out.
5. Approximately. If you treat it as approve/reject it’s binary enough to size with. If the expiry aligns with the date such that you are betting strictly on the terminal intrinsic value, the option piece will inherit the binary modeling you imposed on the stock.
6. No. The variables are too opaque.
7. Yes. p = 30%, b = 5, f* = 30% − 70%/5 = 16%. Note, you’ll lose this bet more than twice as often as you win it, so your most likely scenario is losing 16%. You can shrink the Kelly fraction if this makes you uncomfortable.
8. Wrong side of the equation. Run f* on this and you get a negative number, because you’re buying a negative-EV bet.
9. Yes. It’s the same policy, so notice that the bet only exists on the insurer’s side of it! Every year your neighbor doesn’t buy, she collects a premium and writes a tail. Rebuild cost $500k, premium $3,000, call it a 1-in-500 chance of a total loss. p = 99.8% q = 0.2% b = 3,000 / 500,000 = 0.006 (risk $500k to win $3,000) f* = 99.8% − 0.2%/0.006 = 99.8% − 33.3% = 66.5% Positive, as expected since insurers price premiums well above fair value. In this case, her bet size is the $500k house. If the house is most of her net worth, she’s at 100% on a bet capped at 66%. Rather than overbet, she should buy the policy. If she’s worth $5M, she’s betting 10% when she’s allowed 66%, which puts her near quarter Kelly (~14%), and she could skip the insurance. If she’s worth $1mm, it’s a 50% bet, which is more than half Kelly. I’d say take the insurance but I’m a wimp. There are other considerations (would she have the liquidity to rebuild the home or maybe taking the insurance with a high deductible is a better fit), but just doing this exercise gives you a sense of how risky or conservative your choices are relative to the bet share that maximizes long-term wealth.
10. Yes. Raising the deductible is like you writing a $2,000 policy and collecting $340 a year for it. You’re the insurer again, so work out what you need to believe. Take the high deductible and save $340. Have a claim, and you’re out $2,000 more, but you already banked the $340, so the loss is $1,660. b = 340 / 1,660 = 0.205 f* = p − q/0.205 = p − 4.88q Set that to zero, and you get p = 4.88q, which, with p + q = 1, means q = 17%. Your breakeven is a claim every 5.9 years. Anything less frequent and you’re the one with the edge. Let’s say real-world collision frequency is more like 6%. So p = 94% f* = 94% − 6%/0.205 = 94% − 29.3% = 64.7% Which says risk at most ~65% of your bankroll. The risk here is $1,660. That clears as long as you have about $2,600 in liquid savings, which is to say the sizing check is trivially satisfied for almost everyone so you should generally opt for the higher deductible. For quarter Kelly, we’d need savings of $1,660/(.647 * .25) = $10,262.
11. Approximately. It’s not truly binary, but if you frame it in a way where you are comfortable with the no consolation prize of a medium outcome, you can see it as paying $40k for a 15% shot at $1M. b = 24, so f* = 15% − 85%/24 = 11.5% of your liquid net worth. This is quite sensitive to your estimate of p of course.
12. Yes, a classic example of binary-type risk in markets. You risk $8 to make $4, so b = 0.5 — you’re laying odds, same as the moneyline. f* = 90% − 10%/0.5 = 70%. That number is only as good as the 90%. Revise p to 75% and f* is 25%.
13. Yes, in a subtle way! Your bankroll is capacity, not cash. b = 80/20 = 4, so f* = 25% − 75%/4 = 6.25%. Twenty hours has to be 6% of what you’re working with, which means you can’t run this pitch out of a 100-hour month. If you’re the manager, you can put it in dollar terms by converting to wages.

Finally, I strongly recommend William Poundstone’s book Fortune’s Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street

Description:

In 1956, two Bell Labs scientists discovered the scientific formula for getting rich. One was mathematician Claude Shannon, neurotic father of our digital age, whose genius is ranked with Einstein’s. The other was John L. Kelly Jr., a Texas-born, gun-toting physicist. Together they applied the science of information theory — the basis of computers and the Internet — to the problem of making as much money as possible, as fast as possible. Shannon and MIT mathematician Edward O. Thorp took the “Kelly formula” to Las Vegas. It worked. They realized that there was even more money to be made in the stock market. Thorp used the Kelly system with his phenomenally successful hedge fund, Princeton-Newport Partners. Shannon became a successful investor, too, topping even Warren Buffett’s rate of return. Fortune’s Formula traces how the Kelly formula sparked controversy even as it made fortunes at racetracks, casinos, and trading desks. It reveals the dark side of this alluring scheme, which is founded on exploiting an insider’s edge. Shannon believed it was possible for a smart investor to beat the market — and William Poundstone’s Fortune’s Formula will convince you that he was right.

And this is from my notes, Insights From Fortune’s Formula:

a gripping narrative full of 20th century trivia that ties together the birth of information theory, some of the greatest scientific minds of the 1900s, the rise of quantitative finance, and the role of organized crime. These topics come alive in a fresh, memorable way when discovered through the lens of its colorful characters.

It chronicles the history of the efficient market hypothesis (MIT, U Chicago, Paul Samuelson). You can organize its conclusion around this excerpt:

There is much truth in the efficient market hypothesis. The controversy has always been over just how far the claim can be pressed. Asking whether markets are efficient is like asking whether the world is round. The best way to answer depends on the expectations and sophistication of the questioner. If someone is asking whether the world is round or flat, as fifteenth-century Europeans might have asked, then “round” is a better answer. If someone knows that and is asking whether the earth is a geometrically perfect sphere, the answer is no.

Nor, when we consider the matter closely, can we say that all portions of the same commodity possess equal utility. Water, for instance, may be roughly described as the most useful of all substances. A quart of water per day has the high utility of saving a person from dying in a most distressing manner. Several gallons a day may possess much utility for such purposes as cooking and washing; but after an adequate supply is secured for these uses, any additional quantity is a matter of comparative indifference. All that we can say, then, is, that water, up to a certain quantity, is indispensable; that further quantities will have various degrees of utility; but that beyond a certain quantity the utility sinks gradually to zero; it may even become negative, that is to say, further supplies of the same substance may become inconvenient and hurtful…

The final degree of utility is that function upon which the Theory of Economics will be found to turn. Economists, generally speaking, have failed to discriminate between this function and the total utility, and from this confusion has arisen much perplexity. Many commodities which are most useful to us are esteemed and desired but little. We cannot live without water, and yet in ordinary circumstances we set no value on it. Why is this? Simply because we usually have so much of it that its final degree of utility is reduced nearly to zero.

Examples of the margin setting the price are all around us. The price of power is set by the last unit of demand, which necessitates the use of “peakers” or generation with the most expensive fuel inputs. The most optimistic bidder sets the price in a classic auction.

Heck, just imagine a neighborhood where everyone had to recertify the price of their home by being willing to pay the current market value. I live in CA. Most people living here would not and could not pay the current marginal price to live here. That many of those same won’t sell is not irrational; it’s called making a wide market. You can drive a truck through their bid/ask, but so what? The point is that marginal utility is setting the price, not the average. It makes sense. Real estate is an auction.

The idea that the margin sets the price is obvious when it’s pointed out. But it’s also obvious that people haven’t internalized it.

[This is an example of a general phenomenon in which deriving an answer is much harder than verifying whether it is correct. It’s harder to compute the cube root of 729 than to verify that 5 is wrong or 9 is right. This is the same principle behind crypto mining. It’s computationally expensive to find a specific nonce, but easy to confirm whether a solution satisfies the requirements.]

Your uncle, your cousin, your brother-in-law will, with a straight face, tell you they know more about a stock than the average person as if the average person’s opinion matters when it comes to making money. The consensus price is set by well-resourced, well-connected wiseguys. If you can beat them, you’re rich. If you can do it reliably and on large sums, you’re on your way to becoming one of them, but the standard for feeling like a winner at trading is nothing short of being able to consistently beat the marginal price. It’s your record against the spread.

Marginal thinking blind spot

People often take up my offer for paid calls because they want to bounce a pitch off me. They want a sparring partner, criticism, feedback, and tips. Nobody would say they were trying to persuade me, but it’s only human to assume that they would feel greatly encouraged if they did. But the best-case scenario on that front is a stalemate. The shape of everything that works, at least in public markets trading, is nuanced and requires handling many details at a high level. There’s no magic bean machine where someone is gonna make me say “take all my money”.

[If the idea is a massive layup, it’s resting on a special, likely fleeting access. I’d even shove high-barrier-to-entry infrastructure in this category even if the details vary. In all these cases, the provider of the strategy has a bargaining position that makes it a layup for them but not the capital provider whose results will be haircutted by the cost of the bargain.]

Many arriving with a pitch aren’t even getting to the stalemate. They are confusing their extensive knowledge, knowledge far better than average, with sufficient-to-be-competitive knowledge. To be able to trade, they must have a strong concept of marginal price, yet there’s a blind spot when the concept is applied to themselves.

You can’t go wrong by drilling the concept of value over replacement into your brain. It should be part of your mental OS to apply that lens to all of your major decisions. If you think of replacement value as a strike price, VOR prompts you to think of the moneyness of the strike and the possible upside above it. It goes hand-in-hand with the economic concept of opportunity cost as the strike of a put you’re giving up.

Ask yourself

Back to markets for a moment.

2 key questions to ask yourself:

Can the sharpest players here warehouse this entire risk?

If so, the price already reflects their view, and you have to beat them to make money.

If not, is it because they don’t know about it (maybe it’s very small capacity, but even then, you could reframe as “the sharpest players who would know about this”). Or maybe, it’s just because there’s a general “wrong way” risk premium. SP500 puts are overpriced since there’s no hedge to the economy actually imploding. All such hedges, at the limit, are constrained by general credit and solvency, making a short SP500 put trade an ever-present and therefore uninteresting trade, just as being long passive indices is underwriting an ever-present risk premia (notwithstanding any comparisons with implied forward returns with real “risk-free” rates).

Are you on par with the sharpest players who are setting the price?

This second question is a prompt to think strategically about your role in the system, not an invitation to consider your IQ (which in the range-restricted sample of “competitive investors” is uncorrelated with performance and perhaps even anti-correlated in a Berkson Paradox-style inversion). The prompt should make you consider who competes at your chosen horizon, what constraints you have versus them, and how incentives shape both the chosen horizon and constraints.

Finally, a meta-consideration that sits above these questions.

Notice how comparative trading is. It is inextricable from VOR, competition, and the zero-sumness of alpha.

The entire decision to care about this is a container that all your thoughts will need to live in. This isn’t to say that trading is uniquely competitive, but like sport it’s defined by it. Life is always competition in some abstract sense, but in trading you’re a crab in a bucket. An artist competes by being N of 1 and achieves that by competing with themselves. I’m not saying this is easier or harder, I have no such authority on which to even speculate, but it is different.

I’ll stop well short of lionizing infinite games over finite ones. We have both and I’d rather people just fit where they flourish rather than serve some high-minded ideal (the statement of “I’d rather” is itself some high-minded ideal, but I have no idea how to hit ctrl-break on the recursion so take it easy on me). I only make the distinction between game types in case it strikes a chord for someone debating what to focus on.

If you have $100,000 to invest, you can balance the risk by equal-weighting the holdings: $50k into each stock. You buy 500 shares of A and 5,000 shares of B.

But what if they aren’t the same volatility?

Equal-weighting means the most volatile stocks determine performance. If your $100,000 is split equally between the 2 stocks and A moves 10% per day while B moves 1% per day, you aren’t diversified. Stock A will mostly determine your returns.

We can achieve more balance via equal-risk-weighting, which adjusts how many dollars go into each stock based on its volatility.

Weight each stock by 1/vol:

• Stock A gets 1/10
• Stock B gets 1/1.

Divide by the total (1/10 + 1 = 1.1) and you get about 9% in A and 91% in B. Stock A is 10x as risky, so it gets about 1/10th the dollars.

On $100,000 that’s roughly $9k of A and $91k of B, corresponding to

• 91 shares of A
• 9,100 shares of B

Compared to the equal-dollar portfolio, the equal-risk portfolio requires you’d sell about 409 shares of A and buy about 4,100 shares of B, moving roughly 41% of your total portfolio value from the jumpy stock to the calm one.

No masochism for the kids but in case you’re interested…

Equal risk weighting is the starting point for so-called risk parity weighting. The difference is that instead of only considering the volatility a holding adds to the portfolio, the correlation is considered. A stock highly correlated with the rest of your portfolio contributes a lot of risk, while an anti-correlated one does a better job diversifying and reducing total portfolio risk. The effect can be so strong that even a highly volatile but anti-correlated stock can reduce total risk.

Computing a correlation-aware risk contribution requires a full covariance matrix and an optimizer — i.e., a guess-and-test calculator — to find the portfolio weights, since there’s no closed-form solution. If the kids can grok equal vol-weighting I feel like I’ve done my job, and they can discover risk parity on their own if they’re so inclined.)

Your own Portfolio HQ Spreadsheet

This workbook is designed to organize and monitor your first portfolio.

⏬ Download

The sheet is view-only. Select “duplicate” from the file menu to get your own copy.

In this video, I show you how to use the sheet and talk about the Investment “lab” we did this week.

https://x.com/KrisAbdelmessih/status/2070224748848742758?s=20

This gives users maximum flexibility. We are tuning our agent on a regular basis, but if you prefer your own tool stack and AI you have that choice now.

We use evals for automatically RLHF’ing Moontower Agent and I also have a manual process where I give the agent and the MCP (using Claude Code) the same prompt, and then judge them myself. Very old-fashioned. I’ll share more about what we’re learning from this in the future, but in the meantime, here’s a relevant article from the market-making firm Optiver:

Where AI Trading Models Work and Where They Still Fall Short (4 min read)

Optiver’s Applied AI team did a different kind of eval. They gave several leading large language models the same assessments they give human interns and junior traders.

The results indicate where LLMs excel…

• grasping trading theory
• calculating fair value
• recognizing risk

…and where they still stumble:

• multi-step reasoning
• updating beliefs on the fly
• maximizing expected value under pressure

Even before AI was dominating the conversation, traders have always been obsessed with learning from data. A common example is in transaction analysis. Looking at the trades you did filtered by counterparty, venue, method (ie voice/electronic) as you suss out where you are most likely to be adversely selected. This is a hard problem even with structured data. For example, it might be straightforward to filter by how you do against live option orders (as opposed to delta neutral packages), but there are so many possible permutations. Should I consider how the quote was framed before the order came in? Do I treat a resting order differently than if I’m hit or lifted? Does time of day matter?

But now consider the scope of the unstructured data problem. The counterfactual. The order a broker showed me, I passed on and proceeded to trade without my participation. You’d need to record every phone call (actually this is already done for compliance reasons. In fact, when I interned at a bank in 1995 one of my tasks was to change the giant reel of tape!). But you’d need to link the audio of what the order was to the print when it hit the tape. Or track the fact that it never even traded. It’s like tracking the p/l of a non-trade that could have been. With transcription so cheap, this is feasible now, but it wasn’t when I was thinking about it. You could have traders note when they passed on a trade, but this would be so tedious that it was always a non-starter on a high-volume market-making desk.

My guess is that some trading shops might be doing things like this now (if not, you’re welcome for the idea). But this Optiver article made me wonder when trading rooms will be mic’d up. Jarvis listening to all the conversations, meetings, and debates to cheaply turn unstructured data to structured data.

Your voice, its quiver, your cadence, your pauses, your keystrokes, your glances, your heart rate. Insofar as humans will still be trading, it’s hard to imagine the data obsession that’s already penetrated the MLB not make its way to desk talent.

You’ll know singularity is close when the employee handbook stipulates bathroom breaks as the only acceptable cause to remove your electrodes. Buy stock in Gillette. Every man on a W2 will need to shave their chest for a clean connection.

Related

Elm Wealth let AI compete with humans in their popular Crystal Ball Challenge. You can give it a try yourself:

https://crystal-ball.elmwealth.com/

Elm’s founder Victor Haghani:

A couple of weeks ago we let you loose on our Crystal Ball Challenge: tomorrow’s headlines, $1 million to trade in stocks and bonds, and four AI models to beat. Humans showed up in force, logging thousands of plays and adding over 1,500 entries on the leaderboard.

Here is how the AI models are doing against human players so far:

– Claude: winning 65% of the time
– ChatGPT: 50%, a coin flip
– ️ Grok: 43%
– Gemini: 40%

Both the Wall Street Journal and The Economist covered the experiment this month, and both keyed on the same finding: the AIs are great at reading market-moving news, but they struggle to size their bets appropriately. Knowing what to trade turns out to be the easy part. Knowing how much is what trips them up.

If you have not played yet, three of the four AIs are losing more than half their matchups. Pick your fight. If you have played but not lately, your spot on the leaderboard might no longer safe.

And finally, just before I scheduled this to send out I came across Dwarkesh’s:

The next big breakthrough will be AIs learning on the job

Subtitle: “Labs are throwing away the most valuable data”.

transcript

The series’ materials are here:

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

This is the specific material for class #4:

• Slides
• Spreadsheet Game (File:duplicate to have your own editable copy)

I also created a web version of the game:

☀️🌧️Sun/Rain Game

While I’ve been doing the series for kids, I think a lot of adults could even benefit. The overall arc of the presentation:

1. Last class’s game ended with a humbling but common result, hinting at a key pillar of investing.
2. We use a few facts to dispel the recency bias that all investors carry with them.
3. They learn what the fundamental nature of stocks predicts about their individual and group behavior.
4. We widen the meaning of diversification beyond stocks, which was extremely easy to do in light of March 2026.
5. We play a game that makes the implications for portfolios concrete.

While moontower readers span a wide range of investment experience (although overall quite interested in investing and money), here are a few ideas that I hope are presented in ways that might augment even your understanding or at least help you explain to learners in your life.

The most naive strategy is hard to beat

The kids spent Class 3 picking stocks based on a bunch of variables they could sift through, only for the equal-weight benchmark to beat everyone except the team that contrarily concentrated in the highest momentum company that is very much still an enigma to the market (TSLA).

The equal-weight strategy which I just called a monkey (although it’s not random, just dumb) beat 2/3 of the 15 individual stocks themselves.

The reason you shouldn’t be surprised that the naive strategy is hard to beat

Companies eventually die, but indexes shed them before they are in hospice.

Only 17% of the original S&P 500 companies from 1957 survived 50 years. The average company lifespan on the index was 33 years in 1964 — it’s now under 20. Kodak invented the digital camera in 1975 and buried it because of the innovator’s dilemma.

In a crash, stocks remember they’re all stocks.

Diversification works differently in good years than bad ones. In the class data, stocks spread widely in bull years. Then we looked at Jan 2022 to Jan 2023: 13 of 15 stocks fell together, the spread collapsed.

I didn’t want to lean into the word correlation, but I noticed a different way to convey the same idea. The inter-quartile range (IQR) of annual returns was smallest in the worst years. This chart is rich with insight. Notice the IQR’s visually but also how the equal-weight portfolio performed relative to the individual stock and median stock returns each year:

These observations are non-CAPM ways to arrive at the familiar language of diversifiable risk (company-specific stuff you can eliminate for free) and systematic risk (market-wide stuff you can’t diversify away but do get paid to carry). The crash revealed which was which.

If we zoom out from stocks alone, we see a race where the leaders change each year

The Novel Investor quilt shows 15 years of annual returns ranked best to worst across 9 asset classes. The diversified portfolio, that gray-ish bar, never wins a year nor comes in last. Note commodities, gold and BTC are absent from the series.

How do you think they would influence the gray portfolio?

The Sun/Rain Game

This leads to a game where we can build some intuition about the role of non-stock assets in a portfolio.

If you look at the sheet you can see how the kids actually did (I changed the kids names to letters):

• Spreadsheet Game

The game’s punchline is that owning the anti-correlated asset despite it having a worse expected return than the “good” asset leads to a better long-term portfolio.

But this is so unintuitive that I got a student’s question wrong during the discussion!

I’ll explain the mistake here.

A student asked if we played the game for 100 years instead of just 20 years, if owning the good asset ONLY would have led to the best return. I initially said no, then corrected myself and said yes because it has the higher expected return.

But I was right the first time. The answer is definitely NO.

It comes down to the fact that the good asset has an expected arithmetic return of +5%, BUT it has a negative expected CAGR or geometric return.

The math:

The company is 50/50 to return+40% or -30% in any given year.

.5 x 40% + .5 x -30% = +5%

But over 2 years, you expect 1 up, 1 down. Compounding math:

1.4 x .7 = 98%

You expect to lose 2% over a 2-year sequence of about 1% per year.

Formally, we compute the expected CAGR by multiplying (note how the arithmetic or single period return is added):

1.4^(1/2) * .7^(1/2) — 1 = .9899 -1 = -1%

[The exponents represent the probability of each outcome. If there were 3 outcomes, you’d have 3 terms and the exponents sum to 1.]

In the long run, the good asset destroys value. So you do not want to concentrate in it despite its superior expected arithmetic return.

The CAGR is being killed by volatility drag, which is the asymmetry of the fact that if you lose 30% you need to return 42.9% to get back to even, but the “up” years only return 40%. You are falling behind over time.

The bad asset returns -10% half the time and +8% half the time. It’s a “worse” asset, but it’s less volatile. Taking this quality to its extreme, isn’t this what cash is?

In arithmetic terms, our average return if we allocate to each asset equally is +2% (50% x 5% + 50% x -1%). But that portfolio is less volatile because one stock zigs when the other zags. The diversification cuts the volatility MORE than it cuts the expected return, leading to a better risk/reward!

If we rebalance each year back to an equal-weight portfolio, we “pull” the expected CAGR closer to the expected arithmetic return. It’s the only way we can get close to eating those expected arithmetic returns. Otherwise, they don’t really exist for you over time.

This table is worth staring at:

Here’s a message one of the dads sent me after the class:

Measure Your Own Diversification

I made you a tool to compute your portfolio vol and see how much the cross-correlations between your holdings have been reducing total vol from the vol that the individual assets contain. You can tinker by adding ETFs of other asset classes to your equities (ie GLD or USO or TLT etc) to see how they affect the volatility.

If you just want inspiration for an idea, use the tool to compare the Mag 10 index (MGTN) realized volatility with the average realized volatility of its holdings. The index is conveniently equal-weighted, 10% in each name.

Two ways to try this on your own portfolio:

To run in your browser

https://colab.research.google.com/github/Kris-SF/data-pipelines/blob/main/portfolio-vol/portfolio_analysis.ipynb

⚠️Just push through the warning it spits off

The output will includes metrics and charts:

To run locally

git clone <https://github.com/Kris-SF/data-pipelines.git>
cd data-pipelines/portfolio-vol
pip install -r requirements.txt
jupyter lab portfolio_analysis.ipynb

Either way, edit the WEIGHTS dict and the START / END dates, then Run All.

I’ll cut straight to the gold.

He’s been a repeat guest on Erik’s Outlier Trading podcast a few times. His writing and interviews mince no words. Despite never sugar-coating the reality of trading, I found his most recent interview even bolder. It’s this witch’s brew of insight that is somehow both timeless and underreported.

I’ll start with an idea from the interview he did from late 2025 that always bears repeating before moving to the more recent chat.

The most common misconception about trading volatility

In Euan’s 2025 chat with Erik, he was asked “What is the most common misconception about trading volatility?”

He zoomed in on the mistaken logic that because volatility is mean-reverting, selling it when it’s high assures a profit since it always comes back down.

There are several facets to the mistake. One is with respect to how volatility can cluster based on a market regime.

Saying volatility’s mean reverting is true, but the means also change, you know? So, if you sold volatility in, I don’t know, March 2020, right? Volatility didn’t go back to 15 for about a year, right? Volatility had a new normal. So just because something’s mean reverting doesn’t mean you’ll make money because it comes back.

Another flaw is in understanding that you are exposed to both realized and implied vol.

The other thing that can be wrong is that you’re not directly trading volatility. You’re trading the spread between implied and realized. And that doesn’t have to be mean-reverting and it doesn’t have to be negatively correlated with the level of volatility either. So just because the implied V comes down, that’s not necessarily going to help you if the realized vol still is higher than the implied.

[Kris: I remember suffering through a short option position in nat gas in the expiry right before I got married. The V09 option cycle (expiry date was late Sep 2009). Implied vol got up to 110% but it realized more than implied for our entire holding period. We chopped ourselves up on the short gamma even though we had “positive vega p/l” on some of the marks.]

It’s easier to understand this when you scrap the concept of “vol” for a moment:

Forget about volatility, right? Volatility is just a way of turning option prices into another thing. It relies on a model. Forget about all that, right? Let’s say you’ve got a straddle and the straddle’s trading at five bucks and that straddle normally is trading at two bucks. So you’re like, “Oh, that’s really high. I’ll sell it.” And then as soon as you sell it, it moves by $10. You know, the stock moves by 10 bucks. You’ve just got hammered, right? But you know, the straddle drops down to a dollar. So you were right about the price of the straddle, but you weren’t really just trading that.

Finally the VIX chart illusion:

That’s one of the problems people get wrong is selling volatility spikes isn’t as appealing as it looks as when you look at the VIX graph over time and you’re like, “Oh, the VIX these spikes. It always comes down again.” Yeah, but you that’s not what you’re trading when you’re trading options.

See this post from the Liberation Day period if this is not clear: you can’t trade spot VIX.

The rest of these excerpts are from Euan’s 4/1/26 appearance on Outlier. Emphasis mine.

Where to start: known effects

Erik asked Euan where retail traders should start. Euan’s answer is a tour through what he actually believes works, why it works, and how to think about learning from past data without fooling yourself.

If you’re prepared to have low enough expectations that they’re realistic, and then really work at this, um, and it doesn’t have to be a full-time job, but you can’t just think this is my extra income. You know, this isn’t like driving for Uber.

Erik: Thinking a little more deeply then about the kind of things retail traders should look at — I went through an interesting exercise on my own, because people would ask me for ideas on places they might want to start looking, and I always struggled to give a good answer. There’s a million variables. But I’ve recently started directing people toward really well-known market effects — stuff you could go look at research papers on SSRN in mass, well-researched stuff. Something like time-series or cross-sectional momentum as a general market effect.

The reason I’ve been going that route is that there are so many things you have to do right as a retail trader. Even if you’re doing all the right things but it’s centered on an effect that isn’t really there but you think it is, that’s massively detrimental long term. Even though the returns might not be — as you talk about — stuff that would give you Lambo money this week, you’re at least building the infrastructure around something we know exists.

Are there effects or markets that you think are better suited for those first few repetitions for a trader?

Euan: It’s not so much about finding a market where you have an edge. It’s not like, “Oh, you’ve got to trade shitcoins,” or “You’ve got to trade options on pharmaceuticals.” The actual instrument and sector are of lesser importance than the concept of what makes this thing work.

There are a few things in finance that we know broadly are real. Momentum is one. The other one I’d say, if you’re just starting out and you want to set up what I’d call a real trading operation, would be carry.

If you really understand the concept of carry, particularly as it applies to futures basis trading — in theory, we know that a future is going to coalesce to the spot price at expiration. But that doesn’t tell you how it’s going to do it. It doesn’t say the future’s going to stay there and the spot’s going to go up toward it. It doesn’t say the other way around. But what actually happens most of the time is that the future moves toward the spot price. Naively, that’s exactly what you wouldn’t think happens — you’d think the future was an expectation of the future or whatever. But no. It doesn’t matter what you naively think. That’s a very strong effect.

If you know about that effect, you’ve now got lots of places you can look for ways to apply it. And if you understand the effect, you’ll know places that are better than others. You look for something with a big basis. You look for something with high volatility, because that also gives you more of a basis. And you want something to move a lot, because if nothing moves, you can’t make any money.

That leads you to something like the VIX. The VIX futures have a very high basis to the cash usually. If you look at the difference and you annualize it, if that entire basis is realized, that can be 100% a year. Are you going to get 100% a year? No, because lots of other things are going to happen, and you’re not going to realize all of that, and there’s volatility. But that’s now an effect where you can start saying, “Okay, now I know this. What can I do to harvest it?”

Then you start saying, “Well, clearly I want to be short the future if it’s above the cash. That’s risky. What do I do to hedge that risk until I’m happy?” And maybe you’re like, “Okay, I’ll do a future spread. I’ll be short the front one, long the back one. What ratio? What futures?” Eventually you’re going to come up with — and there are papers written on this exact effect, I haven’t just pulled this one out of nothing.

A lot of the stuff’s out there. People tend to have this idea that no one’s going to tell you anything that works, whereas literally there are thousands of pages written on stuff that works. The universe isn’t going to give you money the way you want it to necessarily, but this carry effect — once you understand it in the VIX, then you’re like, “Holy shit, this also works on a ton of other commodities. It works in bonds.” And then, “Okay, if it works in bonds, does it work differently in treasuries, credit, corporate bonds?” And the answer is yes. So now you’ve got spreads. So relative value is the next thing you start looking at.

Pick a bunch of carry situations, learn to put them on, learn to manage them. They largely take care of themselves, but you have to adjust, you have to understand the risks. It’s like when you fly a plane — you’re learning on a Cessna, learning slowly. You don’t just get into a MiG and blast off. But this is that Cessna. It’ll get you money, and it’s a real trading operation.

Then you move on to relative value. These spreads move around. Maybe I can scalp those spreads. How do they move around? What’s the range? And then, “Well, sometimes they don’t move around.” Okay, and that’s going to lead you to momentum. That’s your next one.

Everyone should read that book by Antti Ilmanen, Expected Returns. It breaks down — it’s about 400 pages long, that’s how detailed it is — and it talks about the volatility premium for about four pages. It goes into hundreds of things like this that work. Carry is a huge unifying feature. Relative value. Momentum. The variance premium and options — again, that’s there. I wouldn’t recommend people start with it. It’s kind of slippery and a good way to lose a ton of money unless you’ve got everything else sorted out. But it’s another one.

That’s where I’d start. I’d start with one thing, move to the next, keep adding. Once you got to three or four things, that’s probably all you can handle. You’re not going to be able to have 10 different strategies and keep things together. That becomes a major logistical operations issue.

Where to expect edge: the hard leg is where the money is

[Kris: My biz partner would say the “hard leg” is where the money is.]

Typically in the world, you get paid for doing something that makes the world somehow better. You provide a service, and typically there’s something unpleasant about that. Otherwise people will just do it themselves.

If you’re selling flood insurance, that’s a tough business because you make money, make money, make money, and then you get absolutely blitzed. That’s a tough thing to live through. Everyone thinks they can, but in reality it’s a lot harder than you think. We’re very bad at figuring out how we’ll feel when something bad happens. Similarly, the guy in the flood who’s cleaning out the sewers, he gets paid.

If you’ve got a clear idea of, say, the carry trade — why am I getting paid to do the carry trade? Largely because I’m providing a source of risk insurance for other people. I’m short those futures that are going to go massively up when the market — like a couple of weeks ago — when the VIX spikes from 18 to 30, those short front-month futures are going to hurt you way more than the long back months. And they have to. Because if you hedged that, then there’s not going to be any premium for you to take out of the trade. All of these things are risk premia. In order to get risk premia, you have to accept that risk. Over time you’ll be fine, but you’ve got that unpleasant nature of the payoff.

Another good rule of thumb: anything where you think this is a bad idea because the risk looks unpalatable — there’s probably edge in there somewhere. If you as a professional trader are like, “Yeah, this looks dangerous,” you’ve got to accept that most of the rest of the world also thinks it looks dangerous, and you get paid to manage that fear. You are the bomb disposal guy. That’s why you’re getting paid.

So if it’s like, “Ooh, there’s no way I’d sell options over the weekend, that’s dangerous” — all right, what side do you think the edge is on? The guy who is prepared to sell options over the weekend, or the guy who wants to buy options over the weekend?

Walking into danger — I’m not saying you always have to, and it’s a risk judgment; you don’t have to take every risk that’s out there. But if it makes you nervous, there’s probably edge in there somewhere. If you can get it to a point where you’re comfortable with it, or can diversify, manage, or hedge it, that’s a good place to start making money. That’s why you get paid for selling options. You don’t get paid for buying options.

Again, difficult thing to say, because for the last three or four years, retail made a ton of money buying 20-delta calls in the indices. By the way, historically, what’s the worst option to be long? Index 20-delta calls. If you think that’s your edge — no. Take that money and good for you. But that’s not the way the world has typically worked. It might work like that for the rest of my life, who knows. You’ve got three years where that worked really well, and 100 years where it didn’t. Just as long as you know that.

Look for any situation that legitimately makes you uncomfortable, and there’s probably something in there.

Learning from the past

Erik: Two follow-ons. I’ll give you both, and you can pick. The first is how to use past information to inform future projections or predictions. The second is pricing of convexity and some of the internals behind that. You pick whichever one sings to you.

Euan: I think the first one is probably more suited for a podcast format.

Basically, the only thing we have to predict the future is the past. And there are two things you can say: the future will be somewhat like the past, or it won’t be like the past. Of those, clearly the better one is to say it will be somewhat like the past. That just makes sense, and usually that’s the way it’s been.

The problem, particularly in finance, is it’s an adaptive system with people on the other side of, and everyone is going through this thought process to a certain degree as well. Everyone’s looking at the same information. Everyone’s saying, “This is the way it’s behaved.” Everyone’s trying to predict on the same thing. Largely coming to a spread, but broadly the same conclusions.

So using the past to predict the future is always going to be murky. And there’s an unfortunate tendency we have now. Like, 30 years ago, no one did backtesting, because you couldn’t. There was no data, no computers. Even before Excel — you’d have to write a program in Fortran. It was hard. No one did it.

Now, everyone does backtesting. The problem is that means most people do it wrong. They see backtesting as a way to find patterns, and they’ll test stuff, do cross-validation, test walk-forward, go out of sample, and they’ll find things. The problem is they look at so many things. Of course they’re going to find something. That’s the big problem with using the past — you think you’re doing statistics correctly, but really all you’re doing is looking at stuff over and over again until you find it.

You’ve also got to remember — and this is an Aaron Brown thing as well — as soon as you’ve looked at some data once, you’re done. That data is in sample forever. You can’t say, “Oh, I’ll try my trend-following system on the S&P. That didn’t work. I’ll try a mean-reversion system on the S&P.” The only reason you’re trying that mean-reversion system and think it might work is because your trend-following one didn’t. So you’re already overfitting. It’s not so much overfitting to the numbers. It’s applying information you’ve learned by looking at it already, even if you’ve looked at it for a different thing.

At this point, there is practically nothing I can do to study the VIX that’s going to tell me anything, because I’ve looked at the VIX for so long that I know what’s in there already. I’m not actually making any sort of new judgment. All I’m doing is applying what I know has worked in the past because I’ve looked at it so many times.

The way to address that is to start with something you believe first. Do you believe in carry? Yes. Do you believe in momentum? Yes. Cross-sectional momentum? Yes. Risk premia? Yes. And then you come up with an idea, and then you say, “Does this work in the past?” And you test it. If it works, great, you can make it better. If it doesn’t, you give up.

We see this with options particularly. People are like, “Well, I tried selling strangles on Monday and holding them all week and it didn’t work. But then I found if I sold straddles on Tuesday and got out on Thursday, it did work.” Both of those are driven by the same effect — they’re both variance premium plays. If one of them didn’t work and the other one did work, all it means is you got lucky picking the entry point for one and not the other. As soon as you’ve looked at that second one, you’ve just completely overfit the whole thing.

That’s one of the big problems people make by using past data. You use past data to come up with the overall belief. Carry means something. Credit spreads mean something. We’ve got, I don’t know, 5,000 years of stories about credit. Credit’s older than money. It’s a real thing.

Looking at numbers should be the last thing you do, and only to confirm something you’re already pretty sure works. My risk premia thing — I’m sure it works. I can test does it work better in a one-month option or a one-year option, but I should not be just blasting combinations in until I find something that is the best.

One of the most dangerous words that’s come out in quant finance is certainly “alpha drift.” This isn’t a retail problem either. I see this in quant firms all the time. They optimize and think they’re doing it out of sample, but they’re not, because they’ve already — this is the fourth model they’ve run on the same data because they keep looking at the S&P 500 or something.

Optimization is a horrific thing. It’s one of the worst things to ever hit the world of finance — that concept that you can make something better, or perfect, or optimal.

The other one, by the way, is theta. I can’t think of a thing that’s cost any more money than theta — this idea that options decay over time. That is literally not what the Black-Scholes equation is telling you. The Black-Scholes equation is literally telling you they don’t. If everyone knew an option was decaying over time, no one’s going to pay for it now — they’ll buy it tomorrow, it’ll be worth less. The number of people who’ve fallen into that “I’m harvesting theta” thing — and there’s plenty of influencer types who are peddling that story, typically also owning a brokerage at the same time.

The difficulty of learning from the past is that people think they can learn too much from the past. You have to discount everything you know based on how much you actually believe the thing. Show me an effect you found in the market — I don’t know what it is. A few years ago, “gold goes up on Fridays” was a big thing. Everyone looked at the data — look, gold went up on Fridays. Okay, why would that happen? “Well, it’s because fund managers are scared of risk and they go into gold on Fridays.” All right, find me one of those people. Not you, or some other person who buys gold on Fridays. I mean someone who runs an appreciable amount of money — doesn’t have to be Pimco, but people who run hundreds of millions or billions. Show me one of those people. No one’s ever managed to do that.

Does it go up on Fridays? Well, it looked like it did. T-stats, sure it did. Now how much do you actually trust that? On a scale of zero, which is candlestick charts, to 10, which is carry is a real thing — I’ll give gold on Fridays a three. Could it be true? Sure. But I don’t see a compelling reason. You look at enough things, you’re going to find that.

That’s the problem people get. They look at the t-stats and go, “Look, statistics, man.” All right — now how many other things have you looked at that didn’t work? Because you’ve got to include those.

The philosophical problem of what you can learn from testing

Starts with sound reasoning:

I’d like to start with the idea that people buy bonds at the end of the month because of window dressing and rebalancing of 60/40 funds. There’s an idea. If the stocks go up more than bonds in a given month, people rebalancing have to sell the stocks and buy the bonds. That should have an effect. That’s your hypothesis.

So let’s look for situations where that hypothesis has a market effect. Do you see it? Do you see it in situations where your hypothesis would tell you you should see it? Equally, do you not see it in situations where your hypothesis tells you you should not see it? If this also happens on the second week of the month, maybe it’s not what you think it is.

I’d always like to start with a reason and then test the reason.

…But there’s thousands of years of philosophers who pointed out all the contradictions in this stuff. Nothing in the world is out of sample. The only person in the world who’s out of sample is a baby who doesn’t know anything. That’s the great curse — the more you know. With me and the VIX, it’s not that I’m the greatest VIX person in the world or even close. I’ve just looked at it, I know it, I can’t be surprised by anything it does. That’s the downside of experience: nothing is out of sample…

The best you can do is find a situation that looks like it’s somewhat constant over time. If you look at the statistics of the VIX, the distribution is pretty constant. If you look at what it did in the ’90s and in the 2000s, break it down in big blocks — it kind of looks the same. Whereas stocks tend not to. You can’t look at Tesla 10 years ago and Tesla two years ago. The business models were totally different, but in one case the stock was a dollar, and over the last two years it’s been between 200 and 400. These are different situations. Stocks are not stationary.

But really, one of the things you have to do is accept how little you know about the world. The good traders never say, “I’m right because of this, and this, and this.” The good traders are like, “I think I might be right because of this. But on the other hand, I could be wrong because of this and this and this.”

Aaron Brown, who’s probably thought more about this stuff than anyone I know, says that a bad trader is always saying “and furthermore,” and a good trader is saying “on the other hand.” A good trader is looking for holes in their argument. A bad trader is continually trying to find other reasons why they’re right.

David Hume — smart guy, all philosophers do is think about thinking — they were pointing out the problems of trying to learn from induction hundreds of years ago. There’s no answer. That’s the only information we have, but we can never really draw certain conclusions from it. So I always apply a big whacking discount to everything. And it’s not statistics. It’s a meta level above that. It’s like belief. I’m not just saying there’s a statistical rule for how you judge if this is good or bad. In fact, literally there isn’t. There’s an actual degree of belief that is independent from what the numbers are telling you.

Vibes vs quant is a false dichotomy — there’s vibes in everything

[The general construction of his portfolio isn’t] based on statistics. It’s based entirely on my degree of belief. Like that carry trade we talked about — how much do I believe that? A lot. Given what I know about its volatility — and I mean “things move around” volatility, not standard deviation of returns — given I know how much that is, given I know how scary it can be, but given I know the belief, I might give that 40%. Whereas there might be another trade where statistics are just as good, but I’m like, I don’t know, I don’t really believe in that one. That gets 20%.

That gives me my baseline, and then I go run all the statistics, and then I say, “Well, okay, I’m shrinking it to this vibes portfolio, because that’s the one I kind of wanted to be at in the first place.” There’s a lot of people who’d be like, “Well, that’s not quantitative. That’s not systematic.” Well, first of all, who gives a shit what they say? You don’t get a medal for being quantitative. You make money or you don’t. Being quantitative is seen as this kind of goal, and it’s not the goal. It’s a tool toward the goal.

Every time you make a portfolio or sizing decision, you are at some point applying that kind of thing. If you’re going to go through all the math, eventually you’re going to have a utility function, and that utility function is going to have a risk aversion parameter. So you’re going to go through all this math, and then at the end it’s going to say, “Oh by the way, what’s your risk aversion parameter?” You have to make that decision somewhere. I’m just doing it at the start. It’s much more of an exogenous thing that’s obvious.

It’s more a reflection of what I think I actually know than a mathematical statement. It’s not a mathematical statement. It’s an epistemological statement.

A lot of people get hung up on “insult found: the discretionary trader.” How is that an insult? Everything you do is discretionary at some level. Literally, you have chosen to do something. You have exercised your discretion to do something. Whether you do that in the math you choose, the problem you attack — you can’t go through life without exercising discretion. That’s actually called knowledge or experience. It’s a useful thing. If you don’t use that, you’re an idiot.

The cost of systematization

There’s certainly this whole idea that if you’re systematized, it’s emotionally easier to stay with things. I don’t think that’s true. I have operated completely systematic stuff when I’ve had jobs, and it loses money. It’s not easier. It just isn’t.

The reason I think you should systematize something is because you have to. Every time you’re removing yourself from interacting with the market, you’re removing yourself from an opportunity to learn. I think you should do things as manually as possible — and I mean it. If you can get away with writing stuff down on a legal pad, then that’s the way to go. If you need to use a spreadsheet, use a spreadsheet. But if you can do it on a spreadsheet, yet you choose to write some massive API call and do the whole thing in some language — you’ve done that because you wanted to, not because you needed to. That’s a mistake.

The places I’ve systematized things are because I wanted to avoid operational risk and key-man risk. If I’ve got a strategy I want to run for an ETF every day, it has to operate if I crash my car and go into a coma. That has to be systematized. But if I’m running something on my PA because I find it interesting and I’m still messing around with it — systematizing that, all I’m doing is removing the opportunity to learn.

It’s one of those things where retail look at institutional and go, “Well, it’s all systematized. I have to do that.” But they don’t think of the reason we systematize things. It’s not to make the strategy better. Honestly, it probably makes it worse, because you’ve locked it in. You’ve said, “This is it, I’m not going to be learning anything more about that now.” But it definitely removes operational risk. It makes it cheaper to run, because you can just let it run on its own. You don’t need a person doing it all the time.

People on the outside are making a judgment of how the things on the inside work, and they’re missing the point.

How do you avoid it? I don’t know. Hopefully if I pointed that out to you now, you’d be like, “Oh yeah, now I get it, this is the important thing.” Think about anything else you’ve ever learned. You’ve learned by doing, by the immersion in it. Trading now, the actual act of trading, is click, click, click, done. You don’t learn anything from that.

But what you do learn about — especially things like adverse selection and the ability to execute — like, “Oh, it looks like there’s plenty of volume. Oh, but every time I put a 100 lot in, all the bids disappear.” That’s the sort of thing you’ll learn so much about just doing it and interacting with it. You’ll never learn that if you automate the trades.

[Kris: To insert something here. I’m reminded by an interview with Agustin Lebron where he talks about getting ideas from just staring at an order book for ours. Seeing how it updates, how bids and offers change after trades, or in the absence of trades but while the rest of the market is doing things. He recommends doing this in medium or low-liquidity names where you can possibly see a story unfold in the order book. I agree with all of this. I mean I don’t think I never explained it quite like that because it’s just the water you swim in. If you trade something relatively illiquid, every movement matters. If some 20 delta put trades on the offer, you have to triangulate whether it’s skew expanding or volatility in general. See Mermaids, Fireflies, and the Bid-Ask Spread.

But there’s a corollary hidden in wisdom of learning from sparse order books: edge is inversely related to liquidity. The more liquid the market, the more whatever’s left looks like a risk premium. SPY is one of the most liquid markets in the world. If you had any edge in trading it, the edge would be so scalable your heirs would be set for an eon. Remind me to rant about this sometime.]

If you’re in a situation where you have to automate the trades, you’re probably playing in a game where you’re not going to win as a retail trader. You shouldn’t be trading, “Oh, I have to run this through the API so I can trade 15 times a day.” That’s one of those quant-envy things where you’re like, “I’m going to do this the way Citadel would do it.” Again, you’re playing basketball one-on-one with LeBron. Good luck. Don’t care how tall you are.

Inefficiency vs risk premia

There’s a split between inefficiencies and risk premia. With an inefficiency, price competition is going to drive it to zero. Nothing’s purely risk and nothing’s purely inefficiency, but market making in options has a lot of inefficiencies. It’s like, “I’m going to buy this option because I can sell this option. Those things should be worth the same. They’re off by a little bit.” That kind of thing. The act of providing liquidity is a risk premium, but the way you do it is looking at all these little inefficiencies.

If you look at the way equity market making has gone in the last 30 years or whatever, ever since the floors have gone and we’ve gone electronic, that has been driven to almost zero. It’s not at the point now where any sort of random person with an off-the-shelf piece of software is going to consistently make money making markets in equity options anymore. It’s gone from profit margins like Tiffany’s might have, and now it’s Walmart. Everything’s driven to zero.

Risk premia are different, though, because they’re there for legitimately a reason. It’s a different utility function. You’re doing something you know is wrong because you don’t want to deal with the risk. You’re buying insurance. You’re not buying insurance because you’re dumb, you’re buying insurance because you want insurance. And I’m selling it to you because either I can take the risk, or I can lay off the risk, or whatever reason — I’m doing it to maximize expected value. You’re doing it to maximize another utility function. We’re both happy.

That kind of thing has got more legs to it. If you look at the variance premium over time it’s been reasonably constant. Obviously it goes up and down. You’d think that after 2008 it would get wider; surprisingly, didn’t much. You’d also think that now, because of all these zero-DTE covered call funds, it would collapse. And it kind of has. But sooner or later people will realize a lot of these funds don’t make money, or some of them will blow up, and it will go back to where it was. All these risk premia are cyclical, but there seems to be a natural limit that doesn’t get driven to zero.

It helps to be able to say: is this an inefficiency that people can’t access, or is it genuinely a thing that everyone knows is there and people are doing it for different reasons? Because that’s the sort of thing you can build a trading life around. Inefficiency — if you find one, good for you, knock it out of the park as hard as you can. It’s not going to be there forever. If it is there forever, Citadel is going to come along and build a desk on it and crush you.

Adaptive markets, but not adaptive traders

Erik: The follow-on, though, even within something like risk premia that we know is there for a reason — an example you’ve given previously is there’s a $20 bill on a busy highway. If I’m standing on the sidewalk and there are five crackheads, and I say, “Hey, I’ll give you 10 bucks to go get that.” And then one of the other crackheads says, “I’ll do it for eight. I’ll do it for five.” And then at some point we get to a floor where the crackheads are like, “I am not going to risk my life for this anymore.” Do you think that with something like risk premia in index options, based on that consistency, we’re generally about at that level? Or do you think it’s likely to shift around more?

Euan: That’s difficult to say. But with your crackhead example — one crackhead might say, “I’ll do it for six bucks,” and he runs out there and he gets killed. And then the next crackhead’s going to come along and say, “Actually, I want seven now, because I saw what happened to this guy.” It’s not the same people all the time.

What we tend to see is a new cohort of traders come in and blow up doing the wrong thing, and then a bunch of other people come in and end up doing the same wrong thing. The market never completely adapts, because it’s never exactly the same group of people who learn. Markets learn, but traders as a whole don’t. For things like this, a lot of it’s propped up by new people coming along making the same mistakes as the old people. Adaptive markets, but not adaptive traders. In a lot of cases people think they’re the same thing. I don’t think they are. There were a ton of floor traders, good floor traders, and then everything went electronic, and they tried to do what they did on the floor on the screen. It just didn’t work. And I don’t know any of those guys who were like, “It’s not working, I’ve got to do this and this and this instead.” That didn’t happen. Instead, a whole bunch of other people — studied computer science at Cornell or whatever — came in. It was a whole different group of people who made money at that point.

(Matt Levine wrote about the fundamentals of the squeeze on 4/15)

So this was Thursday:

Also, note that the change in the basis per expiry increased beyond October, meaning the implied carry cost is no longer negative.

When a stock is hard-to-borrow, its options will imply a future price below the spot price, since a market-maker that is getting saddled with long calls, and short puts from the flow must short shares to hedge. The cost to borrow those shares is reflected in the synthetic futures (ie the option combo of long call/short put on the same strike).

The carry rate increased on Thursday, which means puts fell relative to calls on the same strike, as presumably the borrow will loosen as the squeeze subsides. This was another headwind for people who bought downside puts to bet on CAR coming back to earth. Far OTM December puts, like the 70 strike, actually declined in value on the sell-off. A classic example of what I call positive delta puts.

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

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.