Iron Condor Destroyer

Workflows are Moontower’s screeners organized around what you’re trying to do: sell premium (Income), buy protection (Defensive), finance that protection with your upside (Collars).

We just released a new one to help directional traders scan across the market for the best payoff for a given move: Verticals

About the Verticals Workflow

It starts from a simple need:

How do I compare payoffs across tickers in volatility-adjusted terms?

You don’t want to screen across names for the best bang-for-your-buck on a 10% rally because 10% means something very different in SPY vs MU or TSLA.

Instead we use standard deviations computed from the name’s own surface. Pick a move, say +1 SD by the November expiry, and ask the same question of every name: what’s the cheapest vertical that pays in full if the stock gets there?

The Verticals workflow returns the answer in a grid based on the watchlist you care about.

Let’s see how it works (and learn some option math in the process).

Two knobs

You set two things:

An expiry. The picker lists every expiry any name in your list carries, with a count of how many names list it. Every row in the grid is the same maturity, so you’re comparing like with like.
A signed move. ±0.5, ±1 or ±1.5 SD. Positive means call spreads, negative means put spreads.

For each name, the grid shows the tightest vertical that is fully in the money at that move. Tightest means adjacent listed strikes. Fully in the money means if the stock lands exactly on the target at expiry, you collect the whole strike width.

Finding the breakpoint

The target strike for a z-SD move is:

Then pick strikes:

Calls: the short strike is the closest listed strike at or below K_z. The long strike is the next one down.
Puts: the short strike is the closest listed strike at or above K_z. The long strike is the next one up.

Both rules keep the whole spread inside the target, so a stock that gets to the breakpoint maximizes the max spread value (ie it pays the width of the strikes.)

Marking the spread

The obvious price is the spread’s mid, but it’s far too noisy if each leg is 40 cents wide and the spread is worth 30 cents.

Collecting an accurate mark for a spread is a bit of an art, combining curve fitting and option pricing. In the app, the Price you see for the spread is not mid but a Moontower fair value which lives within the bid-ask.

Reading the grid

The headline number is Payout:

A 1-point spread priced at $0.20 pays 4 to 1 if the stock gets to the target.

Payout is the first column your eye goes to, but we include several columns to judge how much to trust a mark.

Filtering for confident markets

Spread bid / ask: the spread’s own market.
Mkt width: the legged market, both legs’ bid/ask widths added together. It’s what you’d pay to cross both legs.
Mkt / strike width: Mkt width as a share of the strike width. A 20-cent-wide market on a $1 vertical might be hard to execute nearfair value.
Leg width / vol: the wider leg’s bid/ask expressed in vol points, as a share of its IV. It puts a 20-cent market on a $400 stock and a 20-cent market on a $30 stock on the same footing.
Mid: Compare with Price. When these two sit close together, the market and the model agree. When they’re far apart, the Moontower fair value is probably more reliable.
Liq: the name’s liquidity tier, which we provide in all our filters. It’sa function of how wide the markets as a percent of vol allowing us to compare across names.

The grid also drops names whose markets are too wide to mark at all. They’re counted in the “N names excluded” line under the grid, so you can see what got left out.

A few screens to start with

Tight markets only. Mkt / strike width under 20%, Liq High, sorted by Payout. This is the honest version of the leaderboard.
Where the market and the model agree. Add the Mid column and keep rows where Price is within a few cents of it.
Cheap upside: +1 SD, Call skew %tile low, sorted by Payout.

Everything filters, groups and sorts like the other workflows, and your choices persist between visits.

This tool will let you look at the market and answer where the cheapest put spread or call spread to bet on a .5 or 1 standard deviation move by some expiry date.

You can also just reverse sort if you’re looking for spreads to sell. Knowing how much people love to sell options, I should probably rebrand this as the iron condor destroyer (an iron condor is a package of 2 OTM vertical spreads typically marketed as a less risky way to sell a strangle).

vibey vs discrete trades

At the Robinhood Summit last week, Mat Cashman and I hosted a “Trading Lab” emphasizing that options are surgical tools. You only want to reach for them when you have a specific view. While their payoff is highly levered to nailing a particular timing, they are unforgiving to vague forecasts like “I generally want to be long this stock”. If you buy the stock instead, you know your exposure. Your delta is 1.00 and doesn’t change.

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.

Taylor Expansion — Cheatsheet

The idea in one sentence

Stand at one spot on a curve, measure everything you can there, and use those measurements to guess the height somewhere else.

You’d do this when the curve is hard to compute everywhere but easy to measure at one point, or when you want to see what drives a change. A bond’s price is a familiar case: you know its yield and duration today and want to know what happens if rates move.

The guess is built in layers. Each layer uses one more thing you measured at the anchor. The first layer is a straight line. The second bends it. The third bends the bend. You stop when the layers stop mattering, or when you run out of measurements.

Worked example: guess x³ at x = 1.2 using only what you know at x = 1

The curve is y = x³. You are standing at x = 1. The true answer is 1.2³ = 1.728, but pretend you can’t compute it.

What you know at x = 1

Thing	How you get it	Value at x = 1
Height	x³	1
Slope	derivative, 3x²	3
How fast the slope changes	derivative of that, 6x	6
How fast that changes	derivative of that	6
Anything further	derivative of a constant	0

The walk. Destination minus start: 1.2 − 1 = 0.2. Call it h.

Layer 1: pretend the slope stays 3. Guess = height + slope × walk = 1 + 3 × 0.2 = 1.6. Off by 0.128.

Layer 2: the slope drifts. It goes up at 6 per unit, so over the walk it rises from 3 to 3 + 6 × 0.2 = 4.2. Use the average slope, 3.6, instead of 3. Guess = 1 + 3.6 × 0.2 = 1.72. Off by 0.008.

Written as a separate correction: the new piece is ½ × 6 × 0.2² = 0.12, added to the 1.6.

Layer 3: the drift rate drifts. Same move one level deeper. The correction is 6 × 0.2³ ÷ 6 = 0.008. Guess = 1.728. Off by exactly 0.

Layer 4 and beyond: the measurement is 0, so every further correction is 0.

The guess is now perfect at x = 1.2, and it’s perfect at every other x too. x³ only has three pieces of information in it. Use all three and you have rebuilt the function.

Try it at x = 3 (walk h = 2): 1 + 3(2) + 3(2)² + (2)³ = 1 + 6 + 12 + 8 = 27 = 3³. Still exact, even two units from the anchor.

Layer 1 is the tangent line. Layer 2 bends it into a parabola that hugs the curve near x = 1 but misses on both sides. Layer 3 (dashed) lies on top of the black x³ curve at every x, which is the whole point: for a polynomial, enough layers is exactly the function.

Why you divide by 2, then 6, then 24

Each layer’s measurement gets divided before it’s used: layer 1 by 1, layer 2 by 2, layer 3 by 6, layer 4 by 24. Those are 1!, 2!, 3!, 4!. Two ways to see why.

The averaging picture. Layer 2 used the average slope over the walk. The slope was a ramp going from 3 to 4.2, and the average of a ramp is halfway: that’s the ÷2. Layer 3 needs the average of something that grows like a parabola, and a parabola from zero spends most of the walk being small, so its average is only a third of its end value: another ÷3. Stack them: 2 × 3 = 6. One more level and the average of a cubic is a quarter: 2 × 3 × 4 = 24.

Check the parabola claim with numbers. Sample t² at t = 0.1, 0.2, …, 1.0: you get 0.01, 0.04, 0.09, 0.16, 0.25, 0.36, 0.49, 0.64, 0.81, 1.00. They add to 3.85, average 0.385, and with finer sampling it settles to ⅓.

The power-raising picture. Differentiating hⁿ gives n × hⁿ⁻¹: lowering a power by one multiplies by that power. So raising a power by one divides by it. To turn a constant measurement into a term in h³ you raise the power three times, paying ÷1, ÷2, ÷3 along the way. That product is 3! = 6.

Layer	Measurement (x³ at 1)	Divide by	Term
1	3	1	3h
2	6	2	3h²
3	6	6	h³
4	0	24	0

Worked example: ln x, where the corrections stop helping

Same game, anchor at x = 1. Height is ln 1 = 0. Slope is 1/x, so 1 at the anchor.

What you know at x = 1

Layer	Derivative	Value at 1	Divide by	Term
1	1/x	1	1	h
2	−1/x²	−1	2	−h²/2
3	2/x³	2	6	h³/3
4	−6/x⁴	−6	24	−h⁴/4
5	24/x⁵	24	120	h⁵/5
6	−120/x⁶	−120	720	−h⁶/6

The measurements never hit zero. They alternate sign and grow. So there is always another correction, forever. Whether the corrections help depends on how far you walk.

Short walk: x = 1.5, so h = 0.5. True value ln 1.5 = 0.4055.

Layers used	Guess	Gap
1	0.5	0.0945
2	0.375	0.0305
3	0.4167	0.0112
4	0.4010	0.0044
5	0.4073	0.0018
6	0.4047	0.0008

Each layer roughly halves the gap. Keep going and it heads to zero.

Long walk: x = 2.5, so h = 1.5. True value ln 2.5 = 0.9163.

Layers used	Guess	Gap
1	1.5	0.5837
2	0.375	0.5413
3	1.5	0.5837
4	0.2344	0.6819
5	1.7531	0.8368
6	−0.1453	1.0616

The gap gets worse. Each term is ±hᵏ/k, and with h = 1.5 the hᵏ grows faster than the k can shrink it: 1.5, 1.125, 1.125, 1.27, 1.52, 1.90, … The guess swings wider and wider around the truth.

Inside the band the colored curves pile onto the black one, each layer tighter than the last. Right of x = 2 they fan out: layer 3 shoots up, layer 4 dives, layer 5 shoots higher, layer 6 dives harder. Every extra layer swings further from ln x instead of closer.

The rule. For ln x about 1, the corrections help when |h| < 1 and hurt when |h| > 1. That distance, 1, is the radius of convergence. It’s set by where the function itself breaks: ln x blows up at x = 0, exactly one unit left of the anchor, and the series can’t reach further right than it can reach left.

x³ had no such limit because its corrections ran out before they could misbehave. That is the difference between a polynomial and everything else.

The formula, decoded

Everything above, written the standard way:

Pₙ(x) = Σₖ₌₀ⁿ f⁽ᵏ⁾(x₀) / k! · (x − x₀)ᵏ

Spelled out for the first few terms:

Pₙ(x) = f(x₀) + f′(x₀)(x − x₀) + f″(x₀)/2 · (x − x₀)² + f‴(x₀)/6 · (x − x₀)³ + ⋯

Every symbol, in the order it appears:

Symbol	Read it as	What it is	In the x³ example
f	“the function”	the curve you’re guessing; f(x) is its height at x	x³
x	“x”	the destination, any point on the horizontal axis	1.2
x₀	“x-nought”	the anchor, where you stood and took measurements	1
x − x₀	“the walk”	destination minus start, also written h	0.2
P	“the polynomial”	the guess; a polynomial because it’s a sum of powers of the walk	1 + 3h + 3h² + h³
n	“n”	how many layers you used; the highest power in the guess	3
Pₙ(x)	“P-n of x”	the guess using n layers, evaluated at x	P₃(1.2) = 1.728
k	“k”	the counter: which layer you’re on, running 0, 1, 2, … up to n	0, 1, 2, 3
Σₖ₌₀ⁿ	“sum from k = 0 to n”	add up the term for every k from 0 through n	four terms
f′, f″, f‴	“f-prime, double-prime, triple-prime”	first, second, third derivative: slope, rate of slope, rate of that	3x², 6x, 6
f⁽ᵏ⁾	“f-k”	the k-th derivative; f⁽⁰⁾ is f itself, f⁽¹⁾ is f′, and so on	f⁽²⁾ = 6x
f⁽ᵏ⁾(x₀)	“f-k at x-nought”	the k-th derivative evaluated at the anchor, a plain number	1, 3, 6, 6
k!	“k factorial”	1 × 2 × … × k, the divide-by column; 0! = 1	1, 1, 2, 6
(x − x₀)ᵏ	“the walk to the k”	the walk raised to the layer number	1, 0.2, 0.04, 0.008
f(x) − Pₙ(x)	“the gap”	true height minus guess	0
ξ	“xi” (Greek letter)	some unknown point between x₀ and x; used only in the error formula	somewhere in [1, 1.2]

So the k = 2 term of the sum is: take the second derivative (6x), evaluate at the anchor (6), divide by 2! (3), multiply by the walk squared (0.04). That’s 0.12, the layer-2 correction from the worked example.

How big is the gap? It’s controlled by the next measurement you didn’t use, taken somewhere along the walk:

f(x) − Pₙ(x) = f⁽ⁿ⁺¹⁾(ξ) / (n+1)! · (x − x₀)ⁿ⁺¹ for some ξ between x₀ and x

Read it as: the gap is small when the walk is short (the hⁿ⁺¹ is tiny), when the next derivative is tame, or when n is large enough that (n+1)! dominates. The gap is large when the walk is long and the higher derivatives are big, which is exactly what happened to ln x at x = 2.5.

Real-world uses

Four places you’ve met this without the name. Each is worked with numbers.

Bond prices: duration and convexity. A 10-year zero at a 4% yield is priced 100/1.04¹⁰ = 67.56. The anchor is 4%. The two measurements are duration (layer 1, slope) = 10/1.04 = 9.62 and convexity (layer 2) = 10 × 11/1.04² = 101.7.

Yield rises 1%, so the walk is 0.01:

Layers	Guess	True price	Gap
1 (duration only)	67.56 × (1 − 9.62 × 0.01) = 61.06	61.39	0.33
2 (add convexity)	61.06 + 67.56 × ½ × 101.7 × 0.01² = 61.40	61.39	0.01

Yield rises 3%, walk 0.03: duration alone says 48.07, convexity pulls it to 51.16, true is 50.83. The gap is 30 times larger than for the 1% move. That’s the long walk. Traders quote duration and convexity for exactly the reason your tool quotes delta and gamma.

How a calculator computes sin. There’s no sin key inside the chip; it sums the series about 0: x − x³/6 + x⁵/120 − x⁷/5040 + ….

sin(0.5): 0.5 − 0.0208 + 0.0003 = 0.4794. True value 0.4794. Three terms.

sin(3): 3 − 4.5 + 2.025 − 0.434 + 0.050 − 0.004 = 0.137. True value 0.141. Six terms and still off in the third decimal. The series converges everywhere, unlike ln x, but a long walk needs many more layers. Calculators dodge this by folding the input back to a small angle first, which is the “walk less” strategy.

Pendulum clocks. The textbook period T = 2π√(L/g) comes from replacing sin θ with θ, which is layer 1 of the sine series. The next layer says the true period is longer by a factor of about 1 + θ₀²/16, where θ₀ is the swing in radians.

Swing	θ₀ in radians	Correction	Period error if you ignore it
5°	0.087	1.0005	0.05%
20°	0.349	1.0076	0.8%
60°	1.047	1.069	7%

A clock built on layer 1 keeps time at small swings and drifts at big ones. Same story: the anchor is θ = 0 and the walk is the amplitude.

Compound growth. (1 + r)ⁿ about r = 0 is 1 + nr + n(n−1)r²/2 + …. For 5% over 10 years: layer 1 says 1.50, layer 2 says 1.50 + 45 × 0.0025 = 1.61, true is 1.63. Layer 1 alone is the “simple interest” mental shortcut, and the layer-2 term is exactly how much compounding beats it.

All four break the same way ln x did. Past some size of move the layers you kept stop describing the function, and the fix is either more layers, a shorter walk, or computing the real thing.

Quick reference

Recipe for any function about any anchor

Pick the anchor x₀. Compute the walk h = x − x₀.
Take derivatives of f until you have as many as you want, and evaluate each at x₀.
Divide the k-th one by k!, multiply by hᵏ.
Add them up. That’s the guess. The leftover is the gap.
If the derivatives hit zero, the guess becomes exact. If they don’t, check whether the terms are shrinking; if not, you walked too far.

The two examples side by side

	x³ about 1	ln x about 1
Derivatives at anchor	1, 3, 6, 6, 0, 0, …	0, 1, −1, 2, −6, 24, …
k-th term	3h, 3h², h³, then 0	(−1)ᵏ⁺¹ hᵏ / k
Exact after	3 layers	never
Works for	every x	0 < x < 2 only
Why	polynomial: information runs out	ln x breaks at 0, one unit from the anchor

Factorials

k	k!	Average of tᵏ over [0, h]
1	1	h/2
2	2	h²/3
3	6	h³/4
4	24	h⁴/5
5	120	h⁵/6

Common series about 0, for reference

Function	Series	Converges for
eˣ	1 + x + x²/2 + x³/6 + …	all x
sin x	x − x³/6 + x⁵/120 − …	all x
cos x	1 − x²/2 + x⁴/24 − …	all x
1/(1−x)	1 + x + x² + x³ + …	\|x\| < 1
ln(1+x)	x − x²/2 + x³/3 − …	−1 < x ≤ 1

The last two have a radius because the function breaks at x = 1 or x = −1. The first three never break, so the series works everywhere even though it never terminates.

Key Insights

What	Why it matters
Anchor and walk	Everything is measured at one point; the guess only ever knows about that point
Layers = derivatives	Each derivative at the anchor buys one more correction; that is all the information you have
Divide by k!	Raising a power costs a division each time; averaging a ramp, parabola, cubic costs ÷2, ÷3, ÷4
Polynomials terminate	Derivatives hit zero, so finitely many layers rebuild the function exactly, everywhere
Radius of convergence	For everything else, past the distance to the nearest breakdown the layers make the guess worse
The gap formula	The error is the first term you dropped, evaluated somewhere on the walk

Kelly Criterion — Cheatsheet Derivation

Step 1 One Period Expectancy

E = pB − q

where p = win probability, q = 1−p = loss probability, B = net odds (win B per unit staked, lose 1).

Why: Weighted average of outcomes. Win B with probability p, lose 1 with probability q.

Step 2 Per-Flip Wealth Multipliers

Bet fraction f of current wealth W:

Win: W₁ = W(1+Bf)
Lose: W₁ = W(1−f)

Why: You keep the unbet portion W(1−f) regardless. On a win you collect B times your stake Wf on top. On a loss your stake Wf is gone.

Step 3 Wealth After n Flips

After h wins and (n−h) losses:

Wₙ = W₀(1+Bf)^h(1−f)^n−h

Why: The flips compound — each one rescales whatever the previous left. That means multiply, not add. Order doesn’t matter, only h and n−h.

Step 4 Per-Flip Growth Rate G

Take the nth root of total growth to extract the per-period rate:

G = (Wₙ/W₀)^1/n = (1+Bf)^h/n · (1−f)^(n−h)/n

As n → ∞, law of large numbers: h/n → p, (n−h)/n → q.

G = (1+Bf)^p(1−f)^q

Why nth root: Same logic as extracting r from (1+r)ⁿ = total growth. Geometric mean, not arithmetic, because the process is multiplicative.

Step 5 Take ln Before Differentiating

Define g = ln(G). Since ln is monotonically increasing, maximizing g gives the same f* as maximizing G.

Apply two log rules:

ln(AB) = ln(A) + ln(B) → product becomes sum
ln(A^p) = p·ln(A) → exponent drops to coefficient

g = p·ln(1+Bf) + q·ln(1−f)

Why: Differentiating a product of powers is a mess. A sum of logs is trivial. Valid because ln is monotone — same maximum, easier math.

Step 6 Differentiate and Set to Zero

Rule: ddx[ln(x)] = 1x. Chain rule: multiply by derivative of the inside.

ddf[p·ln(1+Bf)] = pB1+Bf ← chain rule gives B from inside (1+Bf)
ddf[q·ln(1−f)] = −q1−f ← chain rule gives −1 from inside (1−f)

Set dg/df = 0:

pB1+Bf − q1−f = 0

Step 7 Solve for f*

Cross-multiply:

pB(1−f) = q(1+Bf)

Expand:

pB − pBf = q + qBf

Collect f terms:

pB − q = pBf + qBf = Bf(p+q)

Since p+q = 1:

f* = pB−qB = p − qB

The Answer

f* = pB − qB

Read as: edge / odds

Numerator pB−q is your expected profit per unit bet
Denominator B scales it by the odds

Special case B=1 (even money): f* = p−q

Your optimal bet equals your raw edge.

Key Insights

What	Why it matters
Multiplicative wealth function	One bad bet can’t be offset by other bets — sizing matters
Geometric mean not arithmetic	Compounding processes need per-period rates, not averages
ln transform	Turns product into sum without moving the maximum
Chain rule on ln(1−f)	The −1 derivative is what creates a finite optimum
p+q=1	The simplification that closes the algebra cleanly

“end behavior”

Alex is an options trader you should follow in case he ever tweets a lot. Because he doesn’t, when he posted the question below a year ago, it got few responses. I took the liberty of posting it myself this week.

This was fun because it led to a lot of discussion on the timeline and DMs. I was told it sparked a bunch of quant debate on one trader’s desk.

The most popular answer, which was still less than 1/3 of the responses, was the correct answer.

Why?

The maximum value of a put is the strike. The maximum value of a call is the stock price.

Straddle is C + P so $100+$100 = $200

Notice how this means all call spreads go to zero since the calls are worth the same — the stock price. All put spreads go to their max value— the distance between strikes because the puts themselves are worth the strikes.

Logic for delta:

Delta is the change in option price per change in stock.

But the put’s strike is fixed, so the value of the put doesn’t depend on the stock price. The put has zero delta. It’s always worth $100. Which means it has no gamma either

The call is $100 because the max value of the call is the stock price. The call value moves 1-to-1 with the stock, so it has a delta of 1 or 100%

The max value of a straddle is therefore the stock price plus the strike price.

If you sell the straddle or either option at max value and hedge on its delta one time (this is known as a static hedge in contrast to dynamic hedging where you would rebalance as your hedge ratio changes), you cannot lose. It is that simple fact of arbitrage that makes it the upper bound.

To address the second most popular response in the poll, those who said the straddle is $100 (wrong) and has a 1.00 delta (correct), we will demonstrate why this is incorrect.

What’s your p/l if you sell 1 straddle at $100 and buy 100 shares against, if the stock goes to $300?

The straddle will be worth $400, so you lose $300 but make $200 on your long share.

Hmm, maybe I’m just underhedged. Fine, what if I hedge on a 200 delta?

In that case, you actually make money; you win $400 on your 2 shares more than offsetting the $300 straddle loss. But what if the stock went to zero?

Your straddle p/l is unchanged, but you lost $200 on the long stock position. Arbitrage max value means you cannot lose if you sell at that price. Since we found a losing scenario, the price is not the maximum arbitrage bound. If you sell the straddle at $200 and buy a single share of stock, there’s no scenario where you lose. It is the lowest straddle value for which this no-lose scenario is true, thus it’s the arbitrage bound.

Of course, this is but a toy problem where the call and put go to their maximum values because it’s a degenerate case of infinite time or vol. But learning how a function (an option price is just a function) behaves by observing its boundaries is good for intuition. You did this in 9th grade. Khan Academy can jog your memory:

In the real world, you can fleetingly find options that trade beyond their arbitrage values:

Earlier in the week, @DeepDishEnjoyer aka p4 wrote a thread about a dividend mispricing.

It led to some back and forth with passersbys who use options but appear to have large gaps in the fundamentals.

Between the maximum value poll and p4’s dividend lesson, it’s worth saying it:

In a proper option education, you spend a lot of time on arbitrage relationships, cost of carry, and synthetics before you ever hear the word “volatility”.

I didn’t study formal math but I imagine there’s a lot in common with the process of proofs. Arbitrages rest heavily on assumptions. So to understand the relationships, you are forced into an intimate familiarity with the assumptions. And in the extremes of everything, it’s the failure to examine assumptions that leads to being blindsided. But also, when things get extreme, to go on the attack means asking yourself, “Who’s on autopilot? Is this price resting on a stale assumption?” The arbitrage relationships give you the highest conceptual ROI that derivatives offer, you never learn the most useful thing derivatives can teach…passage over the “bridge of asses”.

If you want to see more examples of why option basics are so key to understanding assumptions and opportunities when things get weird:

Variance & Covariance Cheat Sheet

Starting points (where every derivation begins)

Everything below is derived from these definitions. They’re the raw material — average squared deviation for variance, average product of deviations for covariance. When a derivation feels stuck, come back here and plug in.

Variance — average squared deviation from the mean

Var(X) = E[(X − μ)²] μ = E[X]

Covariance — average product of deviations

Cov(X, Y) = E[(X − μ_X)(Y − μ_Y)]

Sum of squared deviations (the un-averaged version)

SS = Σ (x_i − μ)² Var = SSn

Variance is just SS divided by n (or n−1 for a sample). Same object, before you average.

The move in every derivation: plug into one of these, expand the square or product (pure algebra), apply E using linearity, then recognize the Var/Cov patterns that fall out. The computational forms below (E[X²] − (E[X])², E[XY] − E[X]E[Y]) are results of doing this, not starting points.

The identities

Variance from the definition

Var(X) = E[(X − μ)²] = E[X²] − (E[X])²

Average of the squares minus the square of the average. Worth showing where that second form comes from, since every later grind reuses this exact collapse. Start from the deviation definition and expand the square:

1n Σ(x_i − x)² = 1n Σ(x_i² − 2x_ix + x²)

Average term by term. The key is that x is a constant (already computed), so it pulls out of the sums:

First term: (1/n)Σx_i² = E[X²]
Middle term: (1/n)Σ(−2x_ix) = −2x · (1/n)Σx_i = −2x · x = −2(E[X])²
Last term: (1/n)Σx² = x² = (E[X])² (averaging a constant returns the constant)

Put them together — and notice the last term carries a coefficient of 1, not 2:

E[X²] − 2(E[X])² + (E[X])² = E[X²] − (E[X])²

The −2 and +1 combine to −1. That collapse — middle and last terms both becoming (E[X])² and partially cancelling — is the same move behind every Var/Cov identity on this sheet.

Covariance from the definition

Cov(X, Y) = E[(X − μ_X)(Y − μ_Y)] = E[XY] − E[X]E[Y]

Average of the products minus the product of the averages.

Variance is covariance with itself

Cov(X, X) = Var(X)

Scaling rule for variance

Var(aX) = a² · Var(X)

Constants pull out as their square.

Scaling rule for covariance

Cov(aX, bY) = ab · Cov(X, Y)

Shifting rule for covariance

Cov(X + c, Y) = Cov(X, Y)

Adding a constant doesn’t change covariance.

Covariance with a constant is zero

Cov(X, c) = 0

Constants don’t co-vary.

Variance of a sum

Var(X + Y) = Var(X) + Var(Y) + 2 · Cov(X, Y)

Variance of a weighted sum (the workhorse)

Var(aX + bY) = a² Var(X) + b² Var(Y) + 2ab · Cov(X, Y)

Here a and b are the amounts held of each asset. They’re portfolio weights when they sum to 1. Var(X+Y) above is just this formula with a = b = 1 — one unit of each, no weighting lever. The weights are what turn a raw sum into a portfolio.

Worked example. Two assets: σ_X = 20%, σ_Y = 10%, ρ = 0.3. Equal weights a = b = 0.5.

Var(X) = 0.04, Var(Y) = 0.01
Cov(X, Y) = ρ · σ_X · σ_Y = 0.3 · 0.2 · 0.1 = 0.006
Var(P) = 0.25·0.04 + 0.25·0.01 + 2·0.5·0.5·0.006 = 0.01 + 0.0025 + 0.003 = 0.0155
σ_P = √0.0155 ≈ 12.4%

Compare to the naive weighted-average vol, 0.5·20% + 0.5·10% = 15%. The cross term (with ρ < 1) is what pulls portfolio vol below the average of the two vols. That gap is the diversification benefit.

Variance of a difference (spread variance / pair-trading formula)

Var(X − Y) = Var(X) + Var(Y) − 2 · Cov(X, Y)

Same as Var(X+Y) but cross term flips sign. When X and Y are highly correlated, spread variance is small — the math behind why pair trades work.

Bilinearity of covariance

Cov(X, A + B) = Cov(X, A) + Cov(X, B)

Same in the first slot by symmetry.

Where it’s used. This is the move that lets you compute an asset’s covariance with a whole portfolio without re-deriving anything. Say a portfolio P = 0.5A + 0.5B and you want how asset A co-moves with the portfolio it sits in:

Cov(A, P) = Cov(A, 0.5A + 0.5B) = 0.5 Var(A) + 0.5 Cov(A, B)

Distribute across the sum, pull the weights out. That number — an asset’s covariance with its own portfolio — is its marginal contribution to portfolio risk, and it’s exactly what you FOIL out when you expand Var(w₁X₁ + … + w_nX_n) into the full covariance matrix. Bilinearity is the engine under every portfolio-variance calculation.

Variance of a binomial

Var(H) = np(1−p) where H = Σ X_i

Derived in two steps, both from scratch.

Step 1 — variance of a single flip. One flip X is 1 with probability p, 0 with probability (1−p). Mean is E[X] = p. Plug into the squared-deviation definition — deviations are (1−p) for heads and (0−p) = −p for tails, each weighted by its probability:

Var(X) = p(1−p)² + (1−p)p²

Factor out p(1−p): the bracket is (1−p) + p = 1, so

Var(X) = p(1−p)

(Peaks at p = 0.5, value 0.25 — the fair coin is the most uncertain, most variance per flip.)

Step 2 — n flips. Write H as a sum of n independent single flips, H = X₁ + … + X_n. Variance of a sum adds the pairwise Cov terms, but independent flips have Cov(X_i, X_j) = 0, so every cross term drops. The n identical variances just add:

Var(H) = Σ Var(X_i) = n · p(1−p)

The np(1−p) isn’t handed to you — it falls out of one Bernoulli’s p(1−p) times n, because independence kills the covariances.

Standard deviation scaling

StDev(aX) = |a| · StDev(X)

Correlation definition

ρ = Cov(X, Y)σ_X · σ_Y

When ρ = 1: Cov(X, Y)² = Var(X) · Var(Y).

The derivation recipe

Every identity in this neighborhood comes out of the same five moves. When you see a Var or Cov of something built from linear combinations of random variables, this is the procedure.

Plug into the definition. Use Var(Z) = E[Z²] − (E[Z])² or Cov(X, Y) = E[XY] − E[X]E[Y] depending on what you’re computing.
Expand squares and products. Pure algebra on the random variables. FOIL out any binomials. No expectations yet.
Apply E using linearity. Distribute E across sums, pull constants out of expectations. This is the step that does the most work. Always handle linearity first when you have the chance — squaring is not linear, so you simplify E first and let the square wrap what’s left.
Group matching terms. Line up the things that share factors (a² terms together, ab terms together, b² terms together, etc.).
Factor and recognize. Pull out shared factors and spot the patterns: (E[X²] − (E[X])²) is Var(X), and (E[XY] − E[X]E[Y]) is Cov(X, Y).

The reason this recipe always closes: variances and covariances are quadratic in the underlying random variables, so expanding any square or product of linear combinations only generates more variances and covariances. Step 5 is recognition, not computation. There’s nowhere else for the algebra to land.

Two applications

Interview problem: E[H · T] for n coin flips

Flip a fair coin n = 100 times. H = heads, T = tails. Find E[H · T]. Worked slowly, because the one-line answer hides about six moves.

Step 1 — first reach, and why it fails. The instinct is E[H · T] = E[H] · E[T] = 50 · 50 = 2,500. But splitting a product of expectations like that is only legal when the two variables are independent. Check the precondition: H + T = 100, so knowing H pins down T exactly. Not independent. The naive split is off by a correction.

Step 2 — name the correction. That correction is what covariance is. Rearranging Cov(X, Y) = E[XY] − E[X]E[Y]:

E[H · T] = E[H] · E[T] + Cov(H, T)

Independent → Cov = 0 → naive split exact. Locked → Cov ≠ 0 → you need the term.

Step 3 — get the sign first. H + T = 100, so when H is above its mean, T is forced below. They move opposite, always. So Cov(H, T) is negative, and the true answer lands below 2,500.

Step 4 — compute Cov(H, T) via substitution. Since T = 100 − H, write Cov(H, T) = Cov(H, 100 − H) and split with bilinearity:

Cov(H, 100 − H) = Cov(H, 100) + Cov(H, −H)

First term is covariance with a constant → 0. Second term, pull out the −1 (scaling rule, ab = 1 · (−1) = −1):

= 0 − Cov(H, H) = −Var(H)

So Cov(H, T) = −Var(H). Now it’s earned, not asserted.

Step 5 — Var(H) is the binomial variance. H is the count of heads in n flips, so Var(H) = np(1−p) = 100 · 0.5 · 0.5 = 25.

Step 6 — land it.

E[H · T] = 2,500 − 25 = 2,475

Where n(n−1) comes from. Keep everything in symbols instead of plugging in. E[H] = np and E[T] = n(1−p), so E[H]·E[T] = n²p(1−p), and Var(H) = np(1−p). Then:

E[H · T] = n²p(1−p) − np(1−p) = p(1−p)[n² − n] = n(n−1)p(1−p)

The n² is the naive product, the −n is the variance shortfall, and factoring out p(1−p) leaves n(n−1). Sanity check: 100 · 99 · 0.25 = 2,475. ✓

The through-line: the answer falls short of E[H]·E[T] by exactly Var(H), because Cov(H, T) = −Var(H) whenever H and T sum to a constant.

Two-stock equal-weight portfolio with equal variances σ² and correlation ρ

Start from the workhorse with a = b = 0.5:

Var(P) = 0.25 Var(X) + 0.25 Var(Y) + 2(0.5)(0.5) Cov(X, Y)

Impose equal variances Var(X) = Var(Y) = σ², and write the cross term with correlation, Cov(X, Y) = ρσ² (since Cov = ρ · σ_X · σ_Y and both vols are σ):

Var(P) = 0.25σ² + 0.25σ² + 0.5ρσ² = 0.5σ² + 0.5ρσ²

Factor out 0.5σ²:

Var(P) = σ²(1 + ρ)2

Why this is the instructive form. Weights are fixed (50/50) and both vols are fixed (σ). The only thing left moving is ρ. So the entire diversification effect is carried by the single factor (1 + ρ)/2 — a clean dial from 0 to 1 that multiplies the single-name variance. Sweep ρ and read what correlation actually does:

ρ	Var(`P`)	σ_P (vol)	vs. one stock
+1	σ²	σ	no benefit — identical names
+0.5	0.75σ²	0.87σ	13% vol cut
0	0.5σ²	0.71σ	29% vol cut (the √½ case)
−0.5	0.25σ²	0.5σ	half the vol
−1	0	0	risk fully cancels

The variance scales linearly in ρ, but the thing you feel — vol, σ_P = σ√((1+ρ)/2) — scales as the square root, so the first chunk of decorrelation buys more than the last. Going from ρ = 1 to ρ = 0.5 already takes 13% off your vol. You do not need negative correlation to diversify; anything below +1 helps. Negative correlation is just the strong form, and ρ = −1 is the perfect hedge where the two positions cancel outright.

The whole two-name diversification story lives in that (1 + ρ)/2 factor. Same vols, same weights, and correlation alone moves you from “no benefit” to “risk gone.”

Two-stock unequal-weight portfolio with equal variances σ² and correlation ρ

Var(P) = σ² [1 − 2w(1−w)(1−ρ)]

Diversification benefit is the product of a weight piece (2w(1−w), maxed at w = 0.5) and a correlation piece (1−ρ). Need both to get benefit. With equal variances, equal weighting is optimal — any tilt from 50/50 sacrifices diversification.

Two-stock unequal-variance portfolio → inverse-variance weighting

Now drop the equal-variance assumption. Keep σ_X² and σ_Y² separate. Weights w and (1−w):

Var(P) = w²σ_X² + (1−w)²σ_Y² + 2w(1−w)ρσ_Xσ_Y

Minimize over w. Var(P) is an upward parabola in w (positive coefficient on w²), so the critical point is the min. Differentiate term by term and set to zero:

2wσ_X² − 2(1−w)σ_Y² + 2(1−2w)ρσ_Xσ_Y = 0

Divide by 2, expand, collect the w terms on the left and constants on the right, factor w out:

w(σ_X² + σ_Y² − 2ρσ_Xσ_Y) = σ_Y² − ρσ_Xσ_Y

w* = σ_Y² − ρσ_Xσ_Yσ_X² + σ_Y² − 2ρσ_Xσ_Y

The denominator is Var(X − Y) — the spread variance from earlier. The numerator is σ_Y² − Cov(X, Y).

The payoff — set ρ = 0 (independent names):

w* = σ_Y²σ_X² + σ_Y² = 1/σ_X²1/σ_X² + 1/σ_Y²

That’s inverse-variance weighting: each asset’s weight is its inverse variance over the sum of inverse variances. The quieter asset gets more money. This is the result behind Kalman filters, weighted least squares, and meta-analysis — anywhere you optimally combine noisy estimates, you weight by precision (1/variance). It’s also the “optimal” cousin of the inverse-vol risk-parity heuristic, which ignores correlations.

Intuition — hold σ_X fixed at 20% (σ_X² = 0.04), turn the σ_Y knob:

σ_Y	σ_Y²	`w`* on X
0	0	0
10%	0.01	0.20
20%	0.04	0.50
40%	0.16	0.80
∞	∞	→ 1

X’s weight is driven by Y’s variance, not its own. The noisier the alternative, the more you pile into X. Three anchors: σ_Y² = 0 → w* = 0 (Y is riskless, hold only Y); σ_Y² = σ_X² → w* = 0.5 (equal variances recover equal weighting); σ_Y² → ∞ → w* → 1 (Y is pure noise, flee into X). The cleanest limit: if σ_X² = 0, then w* = 1 — a riskless X takes the whole book. Precision is just quietness, and you trust the quiet estimate more.

Three-variable portfolio variance → why the matrix shows up

Same Form A grind, one more variable. Expand (aX + bY + cZ)², apply E, subtract the squared-mean term. Every squared term becomes a variance, every cross term a covariance:

Var(aX + bY + cZ) = a²Var(X) + b²Var(Y) + c²Var(Z) + 2ab Cov(X,Y) + 2ac Cov(X,Z) + 2bc Cov(Y,Z)

Counting the terms. For n assets you always get:

n variance terms — one per asset (the a_i² Var pieces)
_nC₂ = n(n−1)/2 covariance pairs — one per distinct pair

Total = n + _nC₂. For n = 3: 3 + 3 = 6. For n = 100: 100 variances + 4,950 covariance pairs. The cross terms grow as n², which is exactly why nobody writes portfolio variance longhand past n = 3 — you switch to the matrix form.

The double-sum / matrix form. Organize every term into a grid indexed by asset pairs. With weights w_i and returns r_i:

Var(P) = Σ_i Σ_j w_i w_j Cov(r_i, r_j) = w^⊤Σw

Reading the double sum: the outer Σ over i and the inner Σ over j together form every ordered pair (i, j). For each pair you drop in one term, w_iw_jCov(r_i, r_j), and add them all up. For n = 3 that’s 3 × 3 = 9 cells:

	`X` (`j`=1)	`Y` (`j`=2)	`Z` (`j`=3)
`X` (`i`=1)	`a`²Var(`X`)	`ab` Cov(`X`,`Y`)	`ac` Cov(`X`,`Z`)
`Y` (`i`=2)	`ab` Cov(`X`,`Y`)	`b`²Var(`Y`)	`bc` Cov(`Y`,`Z`)
`Z` (`i`=3)	`ac` Cov(`X`,`Z`)	`bc` Cov(`Y`,`Z`)	`c`²Var(`Z`)

Sum all nine cells and you get the six-term formula above. Two things to see:

Diagonal (i = j, shaded): Cov(r_i, r_i) = Var(r_i), so the diagonal is the n variance terms.
Off-diagonal (i ≠ j): each unordered pair appears twice — cell (X,Y) and cell (Y,X) are identical — and those two copies are exactly where the factor of 2 on each covariance comes from. You never write the 2 by hand; the grid double-counts it for you.

So Σ is the covariance matrix: variances down the diagonal, covariances off it. w^⊤Σw just says “sweep every cell of the grid, weight it, sum it.” The n + _nC₂ count is the matrix — diagonal plus (doubled) upper triangle. You’ve already discovered why the matrix form is inevitable; it’s just bookkeeping for the term explosion.

Figure — the double sum is the matrix: one sweep, three views

Figure — three forms of the same variance, and where the 2s go

Notation you’ll see in practice: w^⊤Σw. You’ll run into this constantly in risk models, optimizers, and quant papers. It’s the same portfolio variance, packaged as a matrix operation. Reading it piece by piece:

w — the weight vector, weights stacked in a column.
w^⊤ — “w transpose,” the same weights laid flat as a row. Transpose just tips a column over into a row.
Σ — the covariance matrix (the grid above). Watch out: this capital-sigma is the matrix, not a summation sign. Variances on the diagonal, covariances off it.

So w^⊤Σw is row-of-weights × matrix × column-of-weights, which multiplies out to a single number — the portfolio variance. It’s identical to the double sum, and in a spreadsheet it’s literally =MMULT(MMULT(TRANSPOSE(w), Σ), w).

Why bother, when the double sum already shows everything? Three practical reasons, none of them “it’s more correct.” It doesn’t grow — three symbols whether n is 2 or 2,000, where the double sum for 500 names is 250,000 terms. It’s how software actually computes portfolio variance (one fast matrix op). And optimization only speaks matrix: the minimum-variance weights come out as Σ⁻¹1 normalized, and the inverse Σ⁻¹ has no double-sum spelling. The two-asset inverse-variance weighting derived above is Σ⁻¹ for n = 2 — the matrix form is how that generalizes. For understanding, the double sum is enough; this is the notation for doing things with it.

The ladder

Each rung is built from the one before it — the definition first, then the algebra of scaling and adding, then portfolios, then the jump to the matrix. Nothing is assumed that wasn’t derived earlier.

1. Variance as average squared deviation

2. Var(X) = E[X²] − (E[X])² — from the definition

3. Cov(X, Y) = E[XY] − E[X]E[Y] — from the definition

4. Var(aX) = a²·Var(X)

5. Cov(aX, bY) = ab·Cov(X, Y)

6. Cov(X + c, Y) = Cov(X, Y) and Cov(X, c) = 0

7. Var(X + Y) = Var(X) + Var(Y) + 2·Cov(X, Y)

8. Var(aX + bY) — the full weighted-sum workhorse

9. Two-stock equal-weight portfolio variance → the (1+ρ)/2 diversification factor

10. Two-stock unequal-weight portfolio variance (equal variances)

11. Var(X − Y) — spread variance / pair-trading formula

12. Variance of a binomial = np(1−p)

13. Two-stock unequal-variance portfolio → inverse-variance weighting

14. Var(aX + bY + cZ) — three variables, and the n + _nC₂ term count

15. General n-asset portfolio variance — the w^⊤Σw matrix form

Is this one lesson in a math course?

No. This would be roughly half a semester of a first probability course, or a full chapter and a half of a more applied stats book.

Rough mapping to a standard curriculum:

Variance from the definition, E[X²] identity: one lecture, plus a problem set
Covariance and the product identity: one lecture
Scaling rules, bilinearity, variance of a sum: one to two lectures
Portfolio variance, weighted sums, correlation: one lecture in the probability course, or the opening week of a finance/portfolio-theory course
Binomial variance, applications: another lecture or two

So this is the equivalent of maybe four to six lectures of material, plus the problem sets that go with them. The reason it feels like a lot is that this sheet does the whole pipeline — derivation, intuition, numerical examples, applications — for each piece, instead of showing a formula and moving on.

The trade-off is real: this is slower but produces durable understanding. A typical math course shows you Var(aX + bY) on day one, leaves the “why it’s that and not something else” fuzzy, and you pattern-match for the rest of the semester. Done this way, when σ² · (1+ρ)/2 turns up in a textbook two years later, you see the bilinearity FOIL behind it instead of recognizing a memorized formula.

That’s the trade. Slower, but it sticks.

after this post you will be sizing bets in your head

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?

A prediction market contract trading at 30¢. You think it’s worth 45¢.
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.
How much of your 401(k) to put in equities.
Writing checks as an angel investor across 30 startups.
Buying a weekly call on a biotech ahead of an FDA decision date.
Whether to take the new job.
Your buddy offers you 5-to-1 that it rains in Oakland tomorrow. The forecast says 30%.
Buying homeowners insurance. The premium is clearly more than the expected loss — that’s how the insurer stays in business.
Your neighbor doesn’t carry homeowners coverage. She banks the premium instead.
Your auto policy offers a $500 deductible or a $2,500 deductible, for a $340/yr discount on the premium.
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.
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.
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

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%.
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.
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.
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.
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.
No. The variables are too opaque.
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.
Wrong side of the equation. Run f* on this and you get a negative number, because you’re buying a negative-EV bet.
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.
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.
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.
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%.
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.

stock-bond correlation

Return Stacked’s RSSB gives you a dollar of global equities and a dollar of Treasuries on the same dollar of capital. They just listed options on it this week. It will be a useful option market to watch if it gathers liquidity because its implied vol allows you to back out an implied stock-bond correlation since we know the vol of the legs.

Risk-parity funds and strategies are diversified, which gives them cover to use leverage but also means they are structurally short stock-bond correlation. In theory, they should be natural buyers of RSSB vol. If they actually did that, I’d expect the implied vol to trade at a healthy risk premium since it’s a one-way risk. There’s no real natural seller of that correlation.

I asked the Moontower Agent to work out the details and generate a deck (which is also a heat check on its abilities.) It produced a deck I could download, walking through the variance identity, a IEF-for-GOVT proxy swap, a scenario table for implied ρ across RSSB IV levels, and the noise associated with realized correlation.

The deck is short and educational:

Reading Implied Correlation Through RSSB Options

Confidence Intervals on Correlation

This made me look up a time series of the correlation between stocks and bonds using VTI and IEF as respective representatives of the asset classes. The 30-day came back +0.43, the 1-year at +0.28.

The agent volunteered the following statistical insight:

N=22 daily returns is a small sample. The 95% confidence interval on a correlation of +0.43 with N=22 is roughly [+0.02, +0.72] — wide. You cannot confidently distinguish +0.43 from +0.20 or +0.60 at that sample size. The 1Y number (N=251) is statistically much tighter — CI roughly [+0.16, +0.39].

Normally, when you compute a confidence interval, you effectively make a market by scaling the standard deviation to your desired confidence (so 1.96 or “2 sigma” for ~95% confidence)

But take notice of the first interval: +0.02 to +0.72 isn’t centered on +0.43. The upper tail runs 0.29 above, the lower tail 0.41 below. Weird. You wouldn’t see lopsided error bars if the math was something like “estimate ± 1.96 × standard error”.

Agent teach me what you did and why.

The problem is that the error bar’s width depends on the answer

Correlation lives on [−1, +1]. Bounded. If the true ρ is 0.9, your sample estimates can’t overshoot by much (the ceiling is 1.0) but they can undershoot plenty.

That boundedness shows up in the standard error of r itself, which runs about (1 − r²)/√(N − 1). Look at what that does. At r = 0.43 with N = 22 the SE is 0.177. At r = 0.9 with the same 22 observations it’s 0.042, four times tighter. Not because you learned more, but because you got squeezed against the wall.

The agent used something it called Fisher’s fix to move to a coordinate system where the standard error stops depending on r, do the easy symmetric thing there, and come back.

The recipe

1. Transform the point estimate: z = arctanh(r) = ½ · ln[(1 + r) / (1 − r)]

2. Take the standard error in z-space: SE = 1/√(N − 3). Note what’s missing. No r. Sample size is the only input.

3. Build the interval symmetrically, the boring way you already know: z ± 1.96 · SE

4. Recover each endpoint separately: tanh(z_low) and tanh(z_high)

Broadly educational bits to notice

Step 1 does almost nothing to the estimate.

0.4332 becomes 0.4635. The transform is only there to get the r out of the standard error in step 2.

Step 4 is where the lopsidedness comes from.

Both 30-day endpoints sit exactly 0.4497 away from the center in z-space. Perfectly symmetric. But tanh squashes hard once you’re out past 0.5 but does almost nothing near the origin, so the top end at 0.9131 gets crushed down to +0.72 while the bottom end at 0.0139 is nearly untouched at +0.014. What matters is each endpoint’s distance from zero, not its distance from your estimate.

Step 2 accounts for sample size

Ten times the sample and the standard error only comes down by a factor of 3.6. At large N the confidence grows by √N scaling, but at small N, subtracting 3 makes the confidence grow slower than square root scaling.

2 lessons

The object-level lesson: Correlations are volatile and because they are bounded from [-1,1] we need to recenter our point estimates before cuffing their ranges
The meta-lesson: LLMs are great math tutors so when they volunteer info of the “I don’t know what I don’t know” variety they can hold your hand. I’ve shared how I use LLMs as a tutor before in Socrates 2026: how to use highlights

collar shopping

At the end of July, Dean Curnutt tweeted:

The thread should sound familiar. Weeks earlier, Dean tweeted about SNDK vols presenting attractive collar pricing for hedgers.

From high implied vol can work for or against you?:

The high vol is a gift to the natural holders. Millennial employees can lock in their unborn grandkids’ inheritance.

A non-technical way to appreciate how high vol creates this opportunity in upside call vs downside put differentials: Imagine a stock starts at $100. It gets to $125. From $125 to $150 is only 20%. But if the stock fell to $75 the distance from $75 to $50 is 33%. Both $50 and $150 are 50% from the initial price, but in a compounding sense 150 is much “closer” to the starting value than $50. The higher the vol the less “distance” a fixed dollar move represents. As implied vol increases OTM calls grow faster in value than OTM puts. This is the source of the attractive pricing you see in the risk reversal (ie option collar).

That pricing falls out of the risk-neutral world that rests on “no-arbitrage” assumptions. In fact, the lognormal return process, described in the quote without ever using the word “lognormal”, is only one of many assumptions. The forward price of the stock is also assumed to be a function of the risk-free rate, which pushes the price that the options are based on higher than the spot price. This, of course, pushes up calls and puts down by their respective deltas. These assumptions conspire to make calls look quite expensive relative to puts to someone comparing the risk-reward of options intuitively.

Intuition would likely lead to very different prices, but present an arbitrage in the process because you live in the real world not the risk-neutral world.

A few examples from that post:

Warren Buffett sells the long-dated puts because he believes the no-arbitrage assumption of the risk-free rate underestimates real-world forward prices. The option trader he faces is quite content to buy those puts since they are looking for an easy flip in the vol market.
FX carry. The speculator holds the future for the risky but generous profit the risk-neutral price creates. The derivatives trader is content to make a penny of arbitrage profit.

This is an admittedly mind-bending state of affairs, but it creates disagreement because of different horizons. This is the basis for trade!

The collars (also known as risk reversals or fences depending on the trading floor you grew up on) are another example of risk-neutral assumptions presenting wacky prices to those who don’t require arbitrage to trade.

The arbitrageur prices the collar like a contractor bidding a job. The daily hedges are materials and labor. The replication price is a manufacturing cost. Call it $500 per square foot. But the client trading the collar is looking at the finished house represented by the payoff of where the stock lands. They are happy to pay $500/ sq ft if the final product is worth $800/ft. The derivatives trader is not underwriting the final value but operating a cost plus business. They have no view on the value of the final product. It’s literally none of their business.

If you want to tangle with this deeper, you are welcome to revisit how to get arbed with perfect information (again) but take heart that you aren’t alone if you struggle with the idea of no-arbitrage replication. I always say it’s the “bridge of asses” for finance.

Collar Shopping

Dean’s tweets highlighted stocks where the calls you overwrite can finance a fairly high strike put, creating an attractive hedge and therefore risk-reward to a long position.

Consider an example where you buy a $100 stock and a 1-year 20% out-of-the-money put while selling a 1-year 40% out-of-the-money call at the same price as the put. In other words, a “zero-cost” collar. The most you can lose is 20% in the case where the stock tanks. Your upside is capped at 40% before your shares would be called away. If you think the stock has a symmetrical distribution where it’s going up or going down are the same probability this looks like a good bet because you are getting 2-1 odds.

If the collar costs $2 or 2% of the stock price, we can roll that into the basis (so $102). Now you are risking 22% to make 38%, a risk reward (R:R) of 1.73 to 1.

This is a snapshot from the Moontower Collars Workflow on 8/11/26 for options with about 3 months to expiry.

Every row on the screen prices the package of buying the .25 delta put, selling the .25 delta call, and buying the stock.

You can restrict your universe to certain sectors and re-sort. You can filter for stocks a certain threshold above their moving average or stocks that are strongly correlated to each other to hunt for the best bang for your buck for a given beta, or just any number of screens to narrow your candidates.

A thread for the masochists

The strikes here are delta-defined, so the vol gap between names is already baked in. You could imagine beta-adjusting the strikes to go a step further (although you’d need to use the API/MCP).

Beta is correlation times the vol ratio. So if correlation < 1.0, beta-adjusting pulls the call strike in toward at-the-money. You’re short that call in a collar, so a nearer strike prints fatter making the risk-reward look better. But it’s a trade-off. The result comes from truncating idiosyncratic upside that is not captured in beta.

Defensive-minded investors should take note (as Dean clearly has). The multi-year highs in deferred interest rates (pushes up forward prices) and the fact that high-flying stocks that have made concentrated investors quite rich are also relatively high-volatility presents those of us in the real (not risk-neutral) world an attractive menu of hedges.

A final puzzle for the masochists

Collared stock has the same hockey stock diagram as owning a call spread of the same strikes but the total cost will vary by some amount. How much is the amount and why? The hint lies in put-call parity. Solving this should feel like a puzzle and is a step on the “bridge of asses”.

How I Teach Middle Schoolers To Build Stock Portfolios

If you construct a portfolio from 2 stocks and one is $100 and the other is $10, buying a share of each means the first will dominate your portfolio’s risk, assuming they have the same volatility.

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.