A Moontower reader asked a question that gets into one of the most confusing topics for option traders:

If I’m pricing options using calendar days(365), then I should even annualize realised volatility by multiplying 18.8(√256) instead of 16 (√256, approx trading days). In order to compare the VRP ratio on same scale, am I right?

I know firsthand from watching people wrestle with option models that this topic has put many brains in a blender. It’s worth a blog-post sized answer. My hope is that you will not only walk away clear-headed but bursting with ideas to explore.

A typical starting point

You compute close-to-close realized daily volatility for the past 252 trading days. Those days comprise the past year. You get an average daily vol of 1.875%. You annualize it by multiplying 16 to get 30% volatility.

observations:

• Before the addition of Juneteenth, a non-leap-year had 252 trading days.
• After Juneteenth was added to the holiday calendar, there are 251 trading days.
• A leap year will typically have 1 more trading day than a non-leap year.
• 2024 is a leap year that has 366 days and 252 trading days which is what we expect for a leap year.
• If the daily standard deviation is 1.875% you should annualize by multiplying by √252 but traders will typically just estimate by multiplying by 16 (ie √256). If you are building a model, don’t use the estimate, but a lot of trader workflow involves quick assessments so it’s worth noting where the mental math shortcuts are.

The central question is:

Can you put 30% annual volatility into a typical 365-day option model or should you have annualized by √365?

The answer is a satisfying mix of reasoning and arithmetic.

What’s even better is you will be able to appreciate a range of answers across a spectrum of complexity and be relieved that for 99% of you, the additional complexity is not worth the brain damage. But the insights can still lead to a flood of additional inspiration for anyone interested in volatility!

The key to the question: the meaning of 252 trading days

Straight to the heart of it:

Recognize that when you sampled 252 days of trading data you did in fact sample the volatility that transpired over a 365 day year.

Why?

Because the daily volatility that transpires from close-to-close is not just the volatility from the open to the close!

Close-to-close volatility = close-to-open volatility + open-to-close volatility

[Note: I’m using the word “volatility” in place of the technically correct term “variance”. Variance is volatility squared. Variance is additive across time so it’s the units you use to do the underlying math but the more colloquial “volatility” is  reader-friendly. You probably encounter the word “volatility” an order of magnitude or more than the term “variance” (does Zipf’s law apply to financial glossaries?) so why raise the cognitive load for the typical reader when the advanced reader’s burden of translation is quite low by virtue of them being, well, advanced.]

When you computed the realized vol from 252 days you included the volatility that occurs overnight and over the weekends/holidays. Although you only have 252 samples, it includes information about 365 days.

The core of the issue isn’t that you are missing information, it’s that you haven’t allocated it to the correct containers (ie time periods). You bluntly assigned it to trading days creating the illusion that the information only applies to 252 intervals whose boundaries are restricted to 6.5 market hours.

This is more clear if you compute your own ratios of close-to-open volatility divided by close-to-close volatility. You can start to answer questions like:

• What percentage of an asset’s volatility accumulates overnight?
• Do you think it would be higher for global assets like oil and gold or GME?
• How about weekends…how much of Friday close to Monday close volatility is captured from Friday close to Monday’s open?

All of this reduces to a comforting answer:

If you put your 252-day annualized realized volatility in a 365-day model it will generate a well-priced option assuming next year’s realized volatility is similar.

[Similarly had you annualized by √365 or ~ 19 you will overestimate the volatility and therefore the option price].

Where the brain damage begins: interpreting implied vols

The harder problem is interpreting what an IV means in the first place.

Calendar day (ie 365 day) option models

If we believe every calendar day is an equal contributor to that 30% vol then we are saying that volatility accumulates uniformly across every day, weekend or weekday. This will overstate weekend volatility and understate weekday volatility. In terms of options pricing, the straddle would experience the full theta for every hour from Friday’s close to Monday’s open. But I assure you it doesn’t (and if it looked like it did then IV actually fell — this will be more clear soon).

Business day (ie 252 day) option models

If we use a business day model we are saying no volatility transpires over the weekend. If that were true then the straddle wouldn’t decay at all over weekend.

The reality is somewhere in between. Volatility time doesn’t pass linearly. It passes slower over the weekend (so we experience some decay but not what the full theta predicts) and faster during the week. In other words that 30% vol gets a different weight depending on the day.

The difficulty in interpreting what an implied volatility in an option model is the flipside of the time vs volatility coin — different models disagree on how much time remains in the life of an option where the time remaining is measured as fraction of a model year.

To demonstrate this, imagine it’s the night of December 31st and you are looking at an option that expires on the evening of the following December 31. An option with 365-calendar days until expiry. At this moment both models agree that a full year remains until expiration.

• The 365-day model says there is 100% or 365 out of 365 days remaining.
• The 252-day model says there is 100% or 252 out of 252 days remaining.

Ok, January 1st comes and goes. It’s a holiday.

• The 365-day model says there is 99.7% or 364 out of 365 days remaining.
• The 252-day model says there is 100% or 252 out of 252 days remaining.

Let’s say the price of the option is unchanged.

[For some reason you can see the option price but the market’s closed. The fantasy actually doesn’t screw up the point. Also, if you have traded cotton you know the options market can be open while the underlying futures market is closed — this itself is a conclusive thought exercise on the Schrodinger’s question of does volatility time transpire when a market is closed.

What if Elon dropped dead on a Saturday, do you think TSLA’s share price is unchanged on Monday — if not then you have also answered the question does volatility transpire when a market is closed.  The fact that you can only measure its impact on Monday doesn’t mean it hasn’t transpired. Don’t confuse accounting challenges with reality.]

Both models are looking at the same option price, but the 365-day model thinks there is less time til expiry — it will therefore mechanically imply a higher volatility.

January 2nd is a business day. It comes and goes.

• The 365-day model says there is 99.45% or 363 out of 365 days remaining.
• The 252-day model says there is 99.60% or 251 out of 252 days remaining.

The gap in time remaining between the 2 models has narrowed .15% apart versus .30% apart but the 365-day model must still imply a slightly higher vol to account for “less time to expiry” relative to the 252-day model.

Let’s skip ahead a couple business days to the end of January 6th.

• The 365-day model says there is 98.36% or 359 out of 365 days remaining.
• The 252-day model says there is 98.02% or 247 out of 252 days remaining.

Now the 252-day model has less time until expiration. Again both models are fed the same option price but now the business day model implies a higher volatility!

Visual aids

Let’s pretend we are looking at a $100 stock and a call option struck at $100 (an at-the-money option) that expires in 365 days.

Assume the stock price never changes and the option price every day is the price that makes the IV 30% in a 365-day model (these models are the most common and usually the default when you find an online calculator or in your brokerage software).

I populated a table including the 2024 NYSE holiday schedule.

Earlier when we stepped through the first week of the year, you could sense a sawtooth tug-of-war between “DTE % remaining” between the 2 models.

• A business day rolling off impacts 1/252 of the second model but only 1/365 of the default model.
• A holiday or weekend day impacts 0/252 of the second model but still 1/365 of the default model.

Therefore, as the week progresses, more time comes off the business day model and pushes up the IV relative to the default 365-day model. Then on Monday, the business day IV falls because the option prices will have experienced some weekend erosion, but the business day model thinks no time has passed. The opposite happens with the calendar day model — the volatility falls throughout the week, but then pops up on Monday because the weekend doesn’t experience a full dose of decay.

If we use the time remaining in the default 365-day model as a baseline, we can compute the difference from the 252-day model. Likewise we can display the spread of the IVs between the 2 models. As the fraction of year remaining in the 252-day model falls relative to the 365-day model, the IV implied by the 252-day model increases relatively.

This is a plot of the option’s life where the time spread means the difference in time remaining from the 252-day model vs the 365-day model:

Note that the IV difference (orange line) for the first 10 months is less than 1/4 of a vol point. It’s not until you get into the last 60 days, that the IV differences get more significant and themselves volatile. This makes sense…when you have just a few weeks until expiration a business day rolling off has a larger impact on the “business-to-calendar days remaining ratio” (the self-loathing astute reader will have noticed that this ratio is exactly what drives the volatility difference. Technically, it’s the square root of that ratio — again the volatility vs variance thing).

Let’s zoom in on the IVs. Remember, we chose an option price that makes the 365-day model always imply 30%. We are seeing how the 252-day model IV bounces around relatively based on that very same option price. (You could have fixed the 252-day model as the default and saw how 365-day IV moves around).

This is the first 9 months. The IVs are fairly close.

The last 3 months:

The same option price is creating a 5 vol point difference in IVs between the 2 models.

[It makes sense. On the last day the default model says there is 1/365 days remaining and the business day model says there is 1/252 remaining. The square root of (252/365) is 83%. The business day model thinks there is much more time remaining than the calendar day model and therefore to generate the same option price it implies 83% of the 30% IV or ~ 25% IV]

Observations

1. When an option has 1 year until expiration, no matter what model you use you will see the same implied volatility.
2. The moment, the clock starts ticking, the way the day is categorized will change DTE when measured as a percentage of a year (which is how the t in an option model works — fraction of a year.)A different t yields a different IV for the same option price.

   This is a big reason why all IVs are “wrong”. There is no right IV. They always depend on the ruler we use to measure. When I was on the floor most traders used a 365-day model. When it got to Friday, traders might start “running Sunday’s sheets”…what that means is they push the days ahead in the model to fit the Friday option prices. This is a kluge so they don’t have to lower their model vols only to have to raise them again on Monday when the straddle doesn’t experience its full model theta. The sawtooth incarnate.

3. Bonus observation: It gets better. Vol time doesn’t even pass linearly intra-day either. The fist 30 minutes of the trading day is 1/13 of the business day but far more of the vol time has elapsed. Your “fraction of the year remaining” has passed faster than what the clock says.Intraday volatility decay schedules will look similar to the percentages prescribed by a VWAP algo — the first hour of the day might be 25% of the traded volume and 25% of the accumulated variance. Just like we decompose close-to-close volatility into close-to-open plus open-to-close we can decompose the open-to-close period into hours, 15-minute intervals, or even finer.

The endpoint of all this volatility accounting is a granular calendar which specifies weights to various periods. This framework can flex to accomodate earnings, economic releases, corporate events/conferences, rebalance dates, or whatever your creativity can imagine. The goal is minimize noisy changes in IV that are simply artifacts of lumpy, discrete decay schedules.

The practical takeaways

I have good news.

Unless you are in the business of trading for a fraction of a vol point, almost none of this matters. I was implementing volatility cleaning functions to trade cross asset >15 years ago. I used discrete methods like you see in the table above. Today, option firms are doing the same thing continuously. They imply IVs by integrating under the curve of a smooth, “event aware” voltime function.

For some it’s cute to know about this stuff if you want to explore further or add new friends to your idea sex orgy. But more importantly, there’s enough scaffolding here to walk away with actionable heuristics.

1) Your annualized realized volatilities (252 annualization factors) are acceptable to use in option models.

• Implied vols from option models apply that average volatility uniformly to a set of days. This can make them difficult to interpret without grounding it in assumptions of how the volatility is allocated to business days, overnights, and weekends/holidays. But if you are comparing options to one another much of that fog cancels out.
• And if you are hedging or speculating with options that are more than a few weeks out, the minor IV discrepancies between models are irrelevant.

2) Here’s the one that applies to most:

Don’t worry about small differences in absolute IV measures!!

Why?

• Again: the variation in IV between models is negligible with months until expiry!
• The difference in IV is probably swamped by the width of the spread in your long/short rates.
• If you trade 0dtes or weeklies you are probably better served to think in terms of straddles rather than IV anyway. This renders the IV noise due to “how much time does the model think remains” moot.[Although the topic of “how much volatility should be ascribed to the overnight” is definitely an area worth exploring if you trade short-dated options since those overnights are significant percentages of the variance time.]
• The typical option user is not doing vol arb for a few cents across asset classes (if you trade oil options that expire at 1pm on Wed vs USO options that expire at 4pm because they are relatively mispriced than you need to care about this stuff). Again, for must cases, the IV noise cancels out if you are trading listed options of the same asset type against one another.

Learn more:

⏳Understanding Variance Time (Moontower tutorial)

If you use options to hedge or invest, check out the moontower.ai option trading analytics platform

We started talking about Kelly criterion a couple weeks ago. As you play with the ideas yourself, I’ll point out 2 subtleties. One here and another below in the Masochism section.

Edge/Odds

I posted a couple ways to express the Kelly formula. Because it’s easy to remember, I prefer the simple expression edge/odds.

If you use this version too, let me offer some user notes.

1. It only works when there’s a possibility of lossThis is a technicality but consider the following bet:A stock is $100 and you believe it is 90% to worth $100 and 10% to be worth $300.

   The expected arithmetic return is therefore 20% (.90 x 100 + .10 x $300 minus your $100 investment)

   The odds or percent return when you win is 200%

   f* = Edge/odds = 20/200 = 10%

   With this version of the formula…

   

   …you get a divide by zero error. Which is nature’s way of saying “Bruh, you can’t lose with this proposition you should bet 100% why you asking a calculator.”

2. The second user note for using edge/odds is noticing a a counterintuitive idea:For a given level of edge, the optimal Kelly fraction to bet decreases as you get better odds (ie the denominator increases).Kelly has a preference for high win rates, an attribute that always arrives with negative skew.

   We’ll address this in the next section.

Bias towards negatively skewed bets

Consider 2 bets:

1. A 10% chance of getting paid 10-to-1, 90% chance of losing my betThe expectancy is straightforward. If you start with $10 and play 10x betting $1 on each trial you will lose $9, and your last dollar will get you paid $10 leaving you with $11 total. A 10% total return or 10% arithmetic expectancy.Using the spreadsheet:

   

   The prescribed Kelly fraction is to bet 1% of your capital on this proposition.

   This is a positively skewed bet. You lose most of the time, but win a large amount occasionally.

   Let’s look at a negatively skewed bet with the same 10% expectancy.

2. A 90% chance of getting paid 22.22%, 10% chance of losing my betAgain, we start with $10 and bet $1 each time. You will earn $.22 9x or $2 and lose a dollar on the 10th trial. Once again you’re net profit is $1 or a 10% expected return.But look what calculator spits out:

   

   The expectancy is the same but now Kelly wants you to bet nearly 1/2 your bankroll.

My intuition is that Kelly conclusions are loaded on volatility as opposed to higher order moments of a distribution. I’ve discussed this many times but to find the links I asked MoontowerGPT:

The first link of the responses is the most relevant (it’s embedded in the second link as well):

🔗Lessons From A Skewed Coin

Kelly’s bias towards negatively skewed bets is already understood:

And here you have Euan’s adjustment:

🔗The Kelly Criterion and Option Trading

[Euan needs no boost from me but I’ll add that his book Positional Option Trading was terrific. My notes here]

In real-life, almost nobody is aggressive enough to bet full Kelly (at least amongst those who would consider using Kelly in the first place). Half or quarter Kelly is more common and Euan’s adjustment will lower the prescribed full Kelly amount even further in the presence of strong negative skew.

This bit from Fortune’s Formula is instructive:

A Kelly’s bettor’s wealth is more volatile than the Dow or S&P 500 have historically been. In an infinite series of serial Kelly bets, the chance of your bankroll ever dipping down to half its original size is 50%.

A similar rule holds for any fraction 1/n. The chance of ever dipping to 1/3 of your original bankroll is 1/3. The chance of being reduced to 1% of your bankroll is 1%.

Any way you slice it the Kelly bettor spends a lot of time being less wealthy than he was.

A Kelly bettor has a 1/3 chance of halving the bankroll before doubling it. – The half Kelly bettor has only a 1/9 chance of halving before doubling.

The half Kelly bettor halves risk but cuts expected return by one 1/4.

• If you have gotten this far, you’ll probably enjoy these poll questions which strike at a lack of strict risk ordering and transitivity in comparing propositions.

• I’m done writing about Kelly and my current take is when faced with a bet whose properties lend themselves to the formula I’d like to see what it prescribes to get a ballpark for the upper bound of how much to bet. The ultimate choice of sizing would incorporate my instincts about the shape of the payoff and personal comfort.
• I’ve shared my summary of the Haghani bet sizing study and the overwhelming conclusion is people, including economists and grad students, instincts are quite poor on bet sizing. Just acquiring the knowledge that Kelly exists would help a reader recruit their “System 2 thinking” even if the details are foggy.

  This was a widely read post:

  🔗Bet Sizing Is Not Intuitive

If you use options to hedge or invest, check out the moontower.ai option trading analytics platform

In Sunday’s Getting Paid To Flip Million Dollar Coins, I mentioned that exponential functions such as investment compounding are best displayed on a semi-log chart. Let’s do another example of that step-by-step for anyone that wants to learn or anyone who has struggled to teach it to someone else.

Suppose your wealth grows according to this compounding formula:

where:

a = starting wealth

r= compounding rate (ie 10%)

t = time in years

For our examples we just use a = 1, so our charts are “growth of a dollar”.

For rule of 72 fans, you know that at a 10% growth rate wealth doubles every 7 years.

Wealth = (1.10)⁷ = 1.95

If you started with $10,000 after 30 years you’d have about $175k.

This chart is not necessarily hard on the eyes, but the fact that time is the exponential variable is a clue that over long stretches an exponential chart is going to become low resolution.

Here’s a 90 cumulative return history for the SP500

2 observations:

• The later years where you are compounding on a larger base of wealth stretch the chart so the earlier years’ changes are invisible.
• The resolution of the chart and the ‘larger base effect’ obscure what you probably care about — how the rate of return is changing.

Here’s the log chart:

The log chart now shows the resolution of zigs and zags in the early years by making the Y-axis distance between wealth levels of 10 and100 the same as 100 to 1,000 or 1,000 to 10,000.

To create our own log chart, we transform the wealth function:

Wealth = a(1+r)ᵗ

Log(Wealth) = Log(a) + Log(1+r)ᵗ

Log(Wealth) = Log(a) + t * Log(1+r)

This fits the form of a line:

Y = b + mX

Set “starting wealth” to a = $1.

That reduces the equation to:

Log(Wealth) = t * Log(1+r)

t, time, is our independent variable and Log(1+r) is a constant slope that depends on the rate of return.

Rule of 72 enjoyyyers know compounding at 10% for 7 years doubles wealth:

Wealth = (1.10)⁷ = 2

We take the log of both sides:

Log(Wealth) = Log(2) = 7 * Log(1.10)

You can just use a calculator to see that log(2) rounds to .29 and slope of the log chart will be Log(1.10) = .041

To interpret the log chart we observe, if:

• Log (Wealth) = .29 that represents a doubling of wealth
• Log (Wealth) = 1 that represents a 10x increase in wealth aka an order of magnitude increase

Let’s now chart the wealth function as Log(Wealth):

Note: each of the 10 series corresponds to a rate of return of 10%, 9%, 8% and so on. The middle series (purple) is 5% per year and the flattest line corresponds to 1% per year.

• If log (wealth) = .3, wealth has approximately doubled
• If log (wealth) = .48, wealth has tripled
• If log (wealth) = .7, wealth has 5x
• If log (wealth) = 1, wealth has 10x

Also note that at 10% growth per year we computed the slope of the log chart earlier to be Log(1.10) = .041.

And voila, it takes about 25 years (1/.04) to 10x your wealth, aka Log(Wealth) = 1, a whole order of magnitude.

Moving your eyes to the right along the line where Y=.3, to the light blue numbers. Those numbers represent a rate of return of 4%. You can see that it takes 4 extra years to get to the same level of wealth if you compound at 4% instead of 5%.

What you can generally observe is that earning 2% instead of 1%, is vastly more important than going from 9% ror to 10% ror. This idea is captured in the fact that 2% is double the rate of return of 1% and 10% is only 11% bigger than 9% but in practical terms it is a reminder that:

• a 1% difference in performance is a big deal
• taxes are a big deal
• fees are a big deal (“oh it’s just 1%”)
• inflation rates (and real returns) are a big deal
• but all these “big deals” matter more when the difference is a compounded rate of 2% vs 3% as opposed to 9% vs 10%

Here’s the chart zoomed in to holding periods of at least 10 years:

At 5% per year, you’ll double wealth in 14 years. At 3% it will take almost a decade longer.

If you use options to hedge or invest, check out the moontower.ai option trading analytics platform

A foolproof way to get engagement is post this thing on Twitter every couple months. Sometimes my mood is to hate on such dredging but in this case, screw it, let’s take this sucker apart and see how many things we can learn from it.

Let’s start with the obvious.

• The expected value of choosing green is $25mm
• Many people would choose red. Some of those people know the expected value of green is $25mm and choose red anyway.

There’s no dissonance here. The red button guarantees an entirely new life to most of the world’s population. The green button means they still might have to set an alarm for work tomorrow.

The joy of wealth has diminishing returns. I just found $40 in a pair of pants I hadn’t worn in a while (plus a covid mask). If that happened 25 years ago, it would have been a serious enough discovery that I’d hoof it to the local bank branch with a deposit slip.

Economists talk about the “utility” of wealth. They will demonstrate the concept with a sub-linear function to relate “utils” to the quantity of wealth. It’s typically a logarithmic or power function. The sub-linear part means “if your wealth doubles your happiness increases but not by 2x”. The empirical shape of the function is something academics will split hairs about.

I’m going to make one up in the spirit of Nick Maggiulli’s post Climbing the Wealth Ladder.

We will say your “utils”, the made-up satisfaction units, are equal to the cube root of wealth:

Let’s start with the simplest chart.

• As your wealth goes up by 100x from $10k to $1mm this function says you get “only” about 5x happier.
• As your wealth goes up by 2,500x from $10k to $25mm this function says you get “only” about 13x happier.

The function is reasonable — happiness increases at a slower rate but maintains that more wealth is always better than less (which I’d describe as a “no-arbitrage condition” — if it wasn’t you could just give money away).

But just as you want to look at long term investing returns on a log chart (compounding is an exponential function), we want to compress the chart for a more zoomed out view. Plus, there’s a non-negligible number of 🌙 readers with more than $25mm and we want to be inclusive around here, right?

Let’s transform the wealth axis to a log(wealth) axis by invoking 10^x (ie $1,000 = 10³)

The underlying table:

We use log charts to frame insights in a more functional way.

By using log (base-10) to transform the wealth axis, we can now see what cube root utility means:

For every order of magnitude increase in wealth, your happiness doubles.

Your wealth goes up by 10x, your happiness increases by approximately 2x.

But another fun learning moment is upon us.

When I look at that semi-log chart I’m bothered because it’s still exponential. Utility is growing by 2^x.

In the case of exponential functions (like compounded returns in an investing context), a semi-log chart creates a straight line.

But a cube-root function is a power function. To get a straight line, we must use a log-log chart instead of a semi-log chart!

Let’s do that and see why such a transformation aids interpretation. First the table:

It’s handy use log (base-2) for the utility axis because utility is growing by 2^x

Here’s the log-log chart:

Observations:

• The x-axis is log base-10(wealth) and the y-axis is log base-2 (utility) and we get a straight line — that leads to an easy inference: Every order of magnitude in wealth doubles our happiness.
• It’s obvious why many would choose a guarantee of $1mm over an expected value of $25mm — if you have $10 today your happiness doubles more than 6x (it increases more than 50x, 2 to 100) over 5 orders of magnitude. Happiness only increase about 3x (100 to 292) between $1mm and $25mm. Those $24mm are worth less than the very first $1mm.

Of course, this utility function needn’t describe any individual but is qualitatively inferred from the idea that your lifestyle looks pretty similar until you climb to a higher order magnitude of wealth. We can quibble over the actual rate but unless you are a megalomaniac it’s almost certainly sub-linear.

Next time you see the red/green button question you can appreciate how people’s answers are self-rational despite any EV-maxing wonkiness.

Addendum

This walk-through showed how to select log transformations to convert exponential charts into linear charts and maintain intuition by saying things like

• “Y increases by a fixed rate for order of magnitude increases in X (log base-10)”
• “Y increases by a fixed rate every time X doubles (log base-2)”

Deriving the linear transformations of semi-log and log charts:

1. Why exponential functions are linear on semi-log chartsStart by taking log of both sides of an exponential function:

   Y = a^X

   Log(Y) = X log(a)
   
   which looks like a line: Y = mX + b

   where:

   X log(a) corresponds to mX therefore slope or m= log(a)

2. Power functions are linear on log-log chartsDerivation by taking log of both sides of power function:

   Y = aX^b

   log(Y) = log(aX^b)

   log(Y) = log(a) + log(X^b)

   log(Y) = b log(X) + log(a)

   which looks like a line: Y = mX + b

   where intercept is log(a) and slope is the exponent b

Money Angle

Now if you have trader blood you look at the question above and say “I’ll just auction this red/green option off to the highest bidder.”

So what price do you think you’d get?

Let’s reason through this.

Someone that is truly risk-neutral is ambivalent between a certain $1mm and $1mm in expectancy.

The red button is worth $25mm so our risk-neutral friend Spock would not pay more than $24mm for the chance to push the button.

Proof of $1mm in expectancy if you pay $24mm:

.50 * -$24mm + .50 * $26mm = $1mm

Unfortunately, all we did was identify an upper-bound of $24mm that one might pay for this option.

But what do you think someone would actually pay?

🤔🤔🤔

Let’s make this more relatable and see if we can scale our logic up.

Imagine the green button guarantees just $1 and the red button is a 50% chance for $50.

Would you pay $24? Probably not unless you were risk-seeking but it’s not out of the question. I mean Robinhood has millions of users who trade for the lols and the E-trade babies were back in the Super Bowl ads.

Would you pay $23 to push the red button? $22? If you are unwilling to pay $20 please just close this tab right now.

What I’m getting at with this thought experiment is to have you feel that the answer to the question depends on:

1. your bankroll (gambling with $20 is feasible and acceptable, gambling with your net worth not so much)
2. your risk preferences

With this in mind we can move to the next section, where we’ll generate a concrete answer to the original question.

Money Angle For Masochists

$24mm to someone worth $100b is the same as $24 is to someone with $100k.

There’s 10 people in the world who can nonchalantly take this bet as easily as someone just gambles with $20.

But like finding the upper-bound of what someone might pay, this is barely a start.

This is actually a great place to use the Kelly Criterion. In short, the Kelly Criterion is a formula that prescribes the ideal percentage of your capital to wager. The prescribed fraction is the mathematical solution to “For a given amount of edge, how much should I bet to maximize my compounded growth rate?”

I created a collection for those who want to learn more (caveats, history, and much more):

🏇🏽Kelly Criterion Resources

…but for now we want to focus on our question.

The Kelly formula for what fraction of your bankroll to bet is simply:

f* = Edge / Odds

where

f* = bankroll fraction

Edge = expected return

Odds = percent profit when you win

If my original investment is $24mm and I expect to make $1mm then:

Edge = $1mm/$24mm = 4.17%

When I win I make $26mm for a $24mm bet:

Odds = 26/24 = 108.33%

f* = edge/odds = 4.17% / 108.33% = 3.85%

Kelly prescribes betting 3.85% of your capital on this proposition.

$24mm is 3.85% of a capital base of $624mm

The number of funds, trading firms, or even individuals who could reasonably take this bet is way larger than just the 10 richest people.

And remember this bet is a game — it’s uncorrelated with markets or economic growth. Trading firms diversify across bets like this all the time. As a market maker, I’d describe the business as “pay me $10,000 up front and I’ll flip a $1mm coin with you”.

If the coin is fair it’s worth $500k and I’m basically buying it for $490k or selling it for $510k. Either way I’m getting 2% edge.

My odds are $510k/$490k = 104.08%

The prescribed bet size is 2%/104.08% = 1.9% which is only half as good as the red button for $24mm! [Market-making biz in 1 sentence: Make a dime of edge on a $5 option a few dozen times a day, make sure the edge is real, and manage the risk.]

So yea, I expect this red button opportunity to trade for about $24mm by some large firm that is used to absorbing risk for a fee.

Byrne Hobart wrote a fantastic post recently in his educational Capital Gains letter that gets into related real-world messiness:

What’s The True Bankroll?

Matt Levine referenced it as well:

The Kelly criterion tells you what percentage of your money you should put on some favorable bet. If you work in financial markets, you want to make a bunch of bets where you think the odds are in your favor, and if you can estimate the odds then Kelly gives you a guide to how much of your money you should put on each bet. Kelly gives you an answer that is a percentage of your current bankroll. But what is your bankroll?

We talked a few times last year about a dumb story from Sam Bankman-Fried’s internship at Jane Street, where he kept making the maximum bet on slightly favorable coin flips, and I was like “well that’s not very Kelly is it.” But probably I was wrong. Jane Street interns were limited to losing $100 per day, so I sort of took $100 to be the size of his bankroll and thought he was aggressive to bet it all on a 51% coin flip. But readers pointed out, no, come on, his net worth at the time was not $100; $100 was nothing to him even though it was all he could bet that day. As a percentage of his actual bankroll that was a fine bet.

Anyway here is a fun post from Byrne Hobart titled “What’s the True Bankroll?” Sometimes the true bankroll is much bigger than the obvious bankroll: Sam Bankman-Fried’s $100 daily betting allowance was much smaller than his true bankroll, and Hobart points out that if you start your first job and have $1,000 to invest, your true bankroll is more like your lifetime expected savings than it is your current $1,000. Other times the true bankroll might be smaller than the obvious bankroll: If you are a portfolio manager at a multi-manager hedge fund, and you run a $500 million portfolio, you might think that your bankroll is $500 million. But if you know that you’ll get fired for a 10% decline in your portfolio, is your actual bankroll $50 million? No, but also maybe a little bit yes.

Learn more:

• Fortune’s Formula on The Kelly Criterion (Moontower)
• My notes on Kelly Criterion (Moontower)
• Understanding Risk-Neutral Probability (Moontower)
• Bet Sizing Is Not Intuitive (Moontower)

If you use options to hedge or invest, check out the moontower.ai option trading analytics platform