Category: Money Math

Index Investing: The Nature of the Proposition

When a friend asks me what I think of investing in the SP500 I have a standard answer:

It’s a broad basket diversified across industries including foreign revenue via multinationals. If the economy grows and companies make profits in the long run you should do fine.

The next question is inevitably “what percentage of my investable assets should I put in it?”

The quick answer is that custom is not a bad guide. If you don’t want to do any work 60/40 is a reasonable weighting or you can follow the glide path of a target-date fund. Ed Thorp admits he puts all his money in the index because that’s the recipe for the highest long term return. But he’s also admits that if he suffered a 90% drawdown he still has more money than or his heirs ever need. Most of us would be injured if the market swept the leg right before we retired.

So while 60/40 or age-based weight is a good starting point if you know nothing, most people who care enough to ask “how much should I put?” don’t know nothing. They know their life circumstances. They probably have goals. And concerns. For some goals loom larger than concerns in their minds. For others it’s the opposite. Both types of people find that the balance of these considerations makes them a candidate for a non-standard answer.

Of course you don’t just go full dominatrix and say “I heard what you need, now do THIS”. You let them own their decisions by laying out the proposition and letting them come to a conclusion. This is how I describe the proposition of investing in the SP500 or broad index:

It’s a box that has historically paid about 9% per year with a standard deviation of about 19%. The sharpe ratio has been close to .50. Under that hood, you have negative skew and fat-tails which means you are getting paid to hang on through some scary turbulence.

[A more granular answer would say something like “you earn the risk-free rate + 3 to 6%”]

They’ll ask if that performance will continue. I’ll die a little inside because the presumption is they think someone could know. Any answer is merely epistemic icing on a layer cake of conditional probabilities emanating from whether humanity will self-destruct. But ruining your Thursday was not on my to-do list so I play along:

“You see, it depends on whether you think there’s something persistent about the last 40 years that padded those hundred year stats. Have you wondered if the fact that companies stay private for longer matters for future returns? I mean it was kinda cool that Gates, Jobs, Bezos, Zuck, Brin, and Page had the public as their LPs during their massive growth phases. Not sure if they started their companies today if that would happen. On the other hand, the GFC marked the end of defined benefits for private companies in exchange for defined contribution. If the stock market is now the new social security I can see the political argument for not only a Fed put but a permanent federal jobs program.

You know, I have this saying that markets are biology not phys— um, hello…oh…at what point did I cut out, ahhh, sorry, go ahead…no, no, it’s ok, really go ahead, you should never be to a late cat seance”

You get the idea. Any discussion of the future has speculative errors bars that pale against the tyranny of circumstance. I’ve always appreciated the humility behind my buddy Nick Maggiulli’s observation that if you had invested from 1960-1980 and beaten the market by 5% each year, you would have made less money than if you had invested from 1980-2000 and underperformed the market by 5% a year. May your capital appreciation years coincide with economic growth where you live. Deciding to invest is a faith-based exercise before anything. Reminds me of a riddle:

If you have 3 of me, you have 3.

If you have 2, you have 2.

If you have 1, you have 0.

What am I? 1

Discussing the future with me is nothing but entertainment.

What I can help with is deconstruct the properties of historical proposition that the SP500 offered and pull out a bunch of interesting lessons. If the world carries on, they’ll be useful context for considering how much to invest. But I’d bet you’ll learn a lot more than that.

Let’s roll.

We will play show-and-tell with SP500 returns from Jan 1928 until April 15, 2024. By stepping through several exhibits with commentary you will come away with intuition for properties of the historical returns.

First, I computed non-overlapping logreturns (ie compounded) and volatilities for the following intervals:

  • 1-day (“daily” returns)
  • 5-day (“weekly”)
  • 21-day (“monthly” — note these don’t line up to calendar months, I’m just labeling a 21-day return as “monthly”)
  • 63-day (“quarterly”)
  • 126-day (“semi-annual”)
  • 252-day (“annual”)

For each partition of returns, we compute:

  • mean return (µ)
  • standard deviation (σ_raw: this is an unannualized measure of volatility)
  • mean absolute deviation (MAD: mean absolute move size — this is another measure of volatility. Unlike standard deviation the return data is not squared so large moves are not given extra weight.)
  • Sharpe ratio (SR: µ/σ_raw — this is a measure of return per risk)
  • St dev of MAD (This is a measure of volatility of volatility — the standard deviation of a typical change)
    • (St dev of MAD) / MAD normalizes this for the size of the typical move. The St dev of MAD will obviously be higher for 1-year moves than 1-day moves so we divide by the MAD of 1-year or 1-day respectively
  • MAD/SD (For a normal distribution MAD ~ .8 * volatility. Options folks will recognize the MAD as the straddle or expected move size. Observing MAD/SD ratios less than .80 is a quick way to recognize data is not normally distributed but has fat tails or skew)
  • annualized volatility [σ_annualized is σ_raw annualized by a factor of √(252/interval days…so monthly return are annualized by multiply the raw vol by √(252/21)]

Here’s the table:

What I notice:

1) Annualized vol is similar regardless of the sampling interval

Option traders split hairs over how to measure realized volatility because they are in the business of discerning 19% vol versus 20%. The general heuristic is more frequent sampling periods will converge on a sensible estimate of volatility faster. Having 252 daily data points is better than a single annual point. Option traders will often use more than just the closing price from a single day. There are methods to incorporate OHLC (open, high, low, close) or even use tick data to for even faster updating measure of realized volatility.

But the good news is — for general investing applications, this table says “just pick something reasonable”. It’s true that if the market sells off 3% and rallies back to unchanged intraday that using closing prices will mask the true volatility, but if a market chops around violently everyday but reverts back to a steady price the less frequent sampling might be fine for your context.

Picking a vol for portfolio weights doesn’t need the same scalpel you’d use to price a straddle.

2) The vol of vol declines with time

Look how (St Dev of the MAD/MAD) declines as you lengthen the sampling period. In other words, vol is more predictable over longer periods.

This is the same idea behind a popular option trader tool: the vol cone (explained in Understanding Vega Risk)

QQQ realized vol cone via moontower.ai

3) Sharpe ratio increases with time

We define SR as mean return / volatility

Daily SR = .04% / 1.19% = .03

Annual SR = 9.21% / 19.52% = .47

Why the disparity?

Friends — this is the entire basis of trading, the casino business, thinking in terms of repeated positive expectancy bets. It’s the gravity of investing that says a small positive edge ensures you get rich while a negative one means you go broke.

In a line: edge scales linearly, volatility scales by √(time)

*If you were a poker player you’d substitute “hands” for time.

Let’s try something with those daily numbers:

.04% * 252 = 10.08%

1.19% x √252 = 18.89%

SR estimated by annualizing from daily samples = .57

I’ve explained this before in Understanding Edge

4) The distribution, at least for periods less than 6 months, looks like it might be non-normal

Look at those MAD/SD ratios —> all below .80 with the daily ratio significantly lower at .64 (I’m going off memory, but I believe the price returns of oil future spreads which are definitely not normal have MAD/SD ratios in the .60 range). The ratios are a clue to look deeper for skew and/or fat-tails.

Fat tails and Skew

Let’s examine the data by filtering for larger moves. Let’s start with moves greater than 1 standard deviation for each sampling period.

note that up and dn thresholds are the mean return for the interval + or – 1 st dev

Note that we expect 31.73% of all returns to exceed 1 standard deviation if the distribution is normal.

We count how many moves exceed 1 st dev and divide that percentage of the sample by the expected 31.73%.

Whoa…the ratio is less than 1.

We observe that there are actually less moves of > 1 st dev than we expect!

But then we remember — if we filter for moves greater than 1 st dev we are also catching moves greater than 2 st devs, 3 st devs, etc. We need a finer look.

Here’s the full table with me highlighting points of interest. Observations follow.

Observations:

  • There are less > 1 st dev moves than we expect, but far more moves of > 3, 4, or even 5 standard deviations. This is a distribution with a high peak (more small moves than we expect) but also fatter tails (more extreme moves) than a Gaussian or normal curve predicts.
  • 4 and 5+ standard deviation moves occur several orders of magnitude more often than you expect but of course are still low probability. This isn’t shocking — the Black Monday crash of ‘87 was a single day freefall ~23%
  • Negative skew: the overwhelming majority of extreme (3 sd or more) moves are negative for all sampling periods. Extreme single day moves are the most balanced with a down move being a slight 6-5 favorite.
  • We have never seen a 5 standard deviation 1 year move (~90%) but during the early 1930’s there was a 70% drop over one year
  • [Be careful: this is not a drawdown study — it’s a survey of point-to-point non-overlapping total compounded return]

Let’s pause for pictures.

  • The peak of the distribution is centered around the mean monthly return of .77% reflecting the upward drift
  • The shoulder region between 1 and 2 st devs (sd = 5.64%) has less density than the normal curve predicts.
  • The tails are fatter with actual observations when we would expect those outcomes to almost never occur

Here’s the same picture zoomed in so the range cuts off tails beyond 20%:

The zoomed in pic makes the left skew, upward bias, and taller peaks around the mean more visible.

Relating to options

When SP500 put skew increases implying a higher probability of a far left tail move. But by making the far OTM puts more expensive, the ATM/OTM put spread gets cheaper which means the probability of going down at all is implied to be lower. This makes sense if option prices change while the spot price remains fixed…the implied densities are shifting around but the sumproduct of probability x outcome must still cash out at the same expected stock price.

This tends to make more sense when you use an extreme bimodal distribution — like a stock worth $100 because it’s 10% to be worth $1000 and 90% to be a zero. Such a stock will have:

  • Extremely expensive ATM volatility — in fact the straddle is worth $180 when the stock is worth $100! If you could buy the straddle for less than $180 you can construct a riskless arbitrage by shorting shares against the straddle. (I’ll leave it to inclined readers to construct a table of straddle prices vs share hedge ratios that ensure a profit).
  • Very expensive OTM call skew. The $900 strike call is worth $10 (but I’m sure there’s someone on Reddit selling at $5 “for income”)
  • Very expensive put spreads (and conversely cheap put skew!)

You have enough info to price calls and puts for any strike in this toy example to prove all this to yourself. If you need guidance see any of these:

✍🏽Practice Pricing Options By Hand

🐍The Snake Eyes Option

📏What We Can Learn From Vertical Spreads

Wrapping Up

Here are the main points of what we covered:

  • Annualized vol is similar regardless of the sampling interval (it just takes longer to get data that is sampled less frequently).
  • Vol of vol declines with time. Remember the “vol cones”.
  • Edge stacks linearly while volatility or risk scales sublinearly. This is the entire basis of repeated edge thinking.
  • SP500 historical returns exhibit negative skew, especially for large moves, and fat tails.

If history is a guide it’s reasonable to see:

  • a 15% selloff in a single month in a 4 to 5 year period
  • a 25% selloff in a single month in a 20 year period and to lose over 1/3 over the the course of a year

Large moves are rare so it’s hard to make guesses about them but when the next financial panic strikes remember such moves would be entirely precedented.

And to think we are just looking at the ultimate survivorship bias market — America and only in nominal terms at that! There’s no point in being humorless about this stuff. It is certain that future humans will remember that English was spoken in the United States the way we remember the Romans spoke Latin. May that not happen during the investing years of anyone you know.

(I have a strange urge to go watch the Enter Sandman video now…”if I die before I wake…”)

☮️

Stay groovy!

Related reading:

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

Growth rate = 70% * (doublings/years)

Friends,

I saw this chart on LinkedIn and the call of mental math immediately lured me onto the rocks.

Since 1972, the SP500 is up 250x.

So what’s the CAGR (compound annual growth rate)?

One can open the calculator on their phone and type 250^(1/52) – 1

I cannot. I am impelled to estimate the answer. By what? I don’t know, probably the same ghost that makes people play games like Wordle.

I figured I’d share how I did this because practicing mental math is fun.

Recipe:

  1. Immediately recognize that 250x return is about 8 doublings (2⁸ = 256)
  2. If we estimate a 10% log return per year, Rule of 72 predicts in 50 years you double 7 times since at 10% you double every 7 years.
  3. Log returns are handy because they are proportional to time. If you doubled 8/7 or 14% more cumulatively, then your return per year must have been 14% greater than 10% or 11.4% compounded.

This reasoning got us quite close to 250^(1/52) – 1 = 11.2%

Shortcut for estimating a 50-year compounded return: 10% (1 + doublings/7)

Compounding feels unnatural. If we compound at 11.2% instead of 10%, over 50 years we get an additional doubling!

  • At 10% CAGR wealth grows by 2^7 or ~250x
  • At 11.5% CAGR wealth grows by 2^8 or ~500x
  • At 13% CAGR, wealth grows by 2^9 or ~1000x

Rule of 72

Formally, the rule of 72 says that if you earn 10% per year, it will take 7.2 years for your money to double. A derivation of rule of 72 can be found here.

If you already know how it’s derived then you can guess that I prefer rule of 69.

[Let’s the Beavis chuckle pass]

If you double wealth, then your log return is ln(2) or .69

.69 / 7 years ~ 10% annual log return

The rule of 69 is very close to rule of 72 but is derived from continuous compounding instead of annual.

I prefer log returns because they are directly proportional to time.

From Using Log Returns And Volatility To Normalize Strike Distances:

The expression ex is a total quantity of growth. It’s actually assumed to be

e 1 * x where the 1 represents 100% continuously compounded growth and X represents a unit of time. The natural log or ln(ex) then solves for how much time (ie x) did it take to arrive at the total quantity of growth assuming 100% continuous compounding. 

A key insight is that we don’t need to assume a 100% rate and x to be time. We can simply think of x as the product of “rate multiplied by time”. This allows us to substitute any rate for the assumed rate of 100% to find the time. Once again we turn to BetterExplained:

If you review that section of the post a few times and make up a few examples for yourself, you’ll never get confused about e or ln again. You might even start thinking about all numbers in terms of their logs.

For any number X:

  • log (base-10) ~ “how many orders of magnitude to get to X?”
  • log (base-2) ~ “how many doublings to get to X?”
  • ln (aka log base e) ~ “compounding continuously at 100% how long will it take to get X”In this last example, it takes one unit of time to get to 2.718 compounding continuously at 100%. If our unit of time was a year, then 100% return compounded continuously would turn $1 into $2.718In the earlier examples, if you compounded continuously at 69% you’d double your money in 1-year. At 6.9% continuous compounding, it takes 10 years. At .69% continuous compounding it takes 100 years.

Generalizing

Continuously compounded growth rate = 10%* 7 log₂(wealth)/years

Compactly:

.7 log₂(wealth)/years

or

Compound growth rate = 70% * (doublings/years)

As long as you keep this in terms of doublings, ie log₂(wealth), then you can compute the compounded growth rate in your head.

Testing it remembering that the SP500 250x return was about 8 doublings in 50 years:

70% * (8/50)

70% * 16% = 11.2%

If you can count doublings then you can easily dazzle your friends with how fast you estimate a growth rate for any number of years.

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

Nah…you just ain’t seein’ the ball

When I hear someone mope “the market doesn’t care about fundamentals” I change the channel.

Market prices are a collection of point spreads. I don’t really understand the logic of such an argument.

If the relationship between fundamentals and returns were easy to understand in advance, then price would adjust to make the trade hard. A “cheap” company that stays cheap is implying a low future ROIC. If you buy shares and the company is able to find better opportunities to re-invest its capital than what was implied, you’ll win.

But it’s going to take time to find out. You can’t make a statement like the “market doesn’t care about fundamentals today”.

Moaning that NVDA is unmoored to fundamentals, besides being underperforming-the-index cope, is comparing something you can see (a price) to what you can’t (what happens later). The price itself is mostly driven by the future. You can’t evaluate if the price did a good job until the later happens.

And even then you can be fooled.

In my early SIG days I remember Jeff gave a talk arguing that the dot com boom then bust wasn’t irrational. You only needed to look at the options market to see why.

A price is just an expected value — if the underlying distribution is highly uncertain, exactly as you might expect the distribution to be in 1999 when the internet and fiber promised to change the world. The price of dot coms are bounded by zero so they have no choice but to go through the roof based on such world-changing mathematical expectation.

But option surfaces make higher resolution statements than the 2-D nature of a share price. [See the Market Innovation section]

AMZN used to be a 250 vol name. Stick 250 vol into a 1-year option price and look at the difference between the mean and median expected stock prices (median = geometric mean)

The options market said the stock was probably worth zero.

[Relevant reading: this post about the windowmaker is a lesson in what option butterflies mean]

And the options market was right for many of the dot-coms.

In fact the AMZN we are familiar with today is a delightful justification of the prevailing pricing back then — some company is going to reach breathtaking proportion if this tech is as important as we think. We just couldn’t predict which one. And in fact, owning the basket and following a index rebalance algorithm that sheds the losers (aka the Nasdaq index) worked out just fine because AMZN made up for Pets.com.

[Fun fact: iirc, AMZN did end up trading down to the implied mode, ie the most expensive butterfly, in the early 2000s washout.]

You can disagree with share prices. Your portfolio is how you disagree. And when the “later” happens you’ll find out if the future fundamentals were well anticipated by the today price.

I’ll be blunt. When I hear investors bitch about prices vs fundamentals I just hear a confession. “I can’t see the ball clearly anymore.”

Which is exactly what you should expect to happen in a competitive red queen domain unless your learning rate increases faster than the market’s lesson-internalization script runs.

Competing for provable alpha, the type that sits on many reps lending itself to statistical summary, means playing the trading game. Finding mispriced coins that will be flipped in the short term. Similar to arbitrage-inspired trading where futures and options expirations are a catalyst for convergence between prices and reality.

The competition for provable alpha is fierce. However the focus does open the door to alphas from having a long horizon. Long-term investors call this “time arbitrage”. That’s more of a clever marketing term than an investing phylum but it does hint at the reality of investing. (The fact that there is alpha in having a long-term horizon is also convenient cover to say “ignore our short term results”.)

You are unlikely to find the kind of provable alphas that are easy to raise money for. In fact, most investors who have such alphas don’t need or want your money. The “time arbitrage” people will happily take your money though. And you won’t be able to prove if they have alpha (otherwise you’d never hear from them), but this also means it’s the only chance you have of investing with someone who has an edge.

You just won’t know until later.

And even then you might still not know.

[Unless they boot you as an LP. Then you definitely know.

“Wait so if I pick the best horse my reward is being asked to redeem?”

Sorry, there’s no luxury more protected than a sure-fire compounding machine. You can buy your way into almost anything else from sex to a trip through space — but if you try to buy your way into a money copier the price adjusts until the ink runs out.

(Actually, as “strategic” investors know, you can buy your way in with other ways to add value. But nobody’s money is greener than another’s.)]

A couple pieces I’ve read recently suggest market prices are doing their job. And also enlightened me on a significant (and not what I expected!) reason for “value” underperforming.

[Emphasis in the excerpts is mine]

Total Shareholder Return Linking The Drivers of Total Returns to Fundamentals
Michael Mauboussin

A terrific paper by one of my favorite researchers.

From the conclusion:

Total shareholder return (TSR) is the capital accumulation rate for investors who reinvest dividends at no cost. TSR is a popular return measure despite the fact that few investors earn it for the stocks of companies that pay a dividend. Investors fail to earn the TSR because they do not reinvest the dividend or cannot reinvest the full amount due to taxes or other costs. Price appreciation and the capital accumulation rate are the same for the stocks of companies that do not pay a dividend, which includes about 65 percent of U.S. public companies.

We can break down TSR into drivers, including net income growth, change in shares outstanding, P/E multiple change, and the benefit of reinvesting dividends. We examine each of these drivers and link them to underlying economic principles.

It is useful to think about the value of a business in two parts: a steady-state value and the prospects for future value creation. The steady-state value assumes that the company can sustain its current earnings forever. Future value creation reflects the ability to earn a return on invested capital in excess of the cost of capital, the amount of investment the company can make while maintaining a positive spread, and the length of time a company can create value with its investments. Historically, about two-thirds of the value of the S&P 500 has derived from the steady-state value.

The market determines the appropriate P/E to apply to current earnings through an estimate of the cost of equity capital. The cost of equity is the opportunity cost of equity investors and is commonly estimated by adding an equity risk premium to the risk-free rate. Over the past 60 years, the steady-state P/E has gone from a low of 5.1 in 1981 to a high of 17.7 in 2020.

The multiples of companies with high P/Es tend to regress toward the steady-state P/E over time because the relative contribution of future value creation shrinks. This reflects market saturation and competition. Some companies can defy this downward drift by either sustaining a high return or by investing in new businesses.

Investors often consider dividends to be part of total returns. But for investors using TSR, price appreciation is the only source of investment return that contributes to accumulated capital. Indeed, dividends and share buybacks have the same result in a TSR calculation because they both increase the percentage ownership in a company. In the case of dividends, the investor uses the proceeds to buy more shares at the ex-dividend price. In the case of buybacks, the investor does not sell and hence winds up with a higher percentage of the company. These are equivalent, save for the reality that dividend reinvestment generally has a cost (see appendix A).

Value traps exist when a company appears statistically inexpensive, often based on the P/E as compared to the stock’s past or to the market overall, but the future drivers of TSR perform poorly. The two main culprits in the bad results are growth below the average and P/E multiple contraction that captures fleeting or elusive prospects for value creation.

We explain TSR through drivers including EPS growth and changes in the P/E multiple. We have noted the severe limitations of EPS and multiples to explain value. To compensate, we seek to link these concepts to underlying fundamental drivers. Doing so gives investors and executives a framework to understand the past and to anticipate the future.

We decompose the returns for value and growth stocks in recent years.

Excerpts of note:

  • Autocorrelation in net income growth is -0.10 for 1 year, -0.20 for 3 years, and -0.28 for 5 years. This is consistent with the academic literature that shows low persistence in net income growth. These results show that extrapolating past net income growth into the future is rarely a good idea.
  • The presumption that buybacks always increase EPS is incorrect. The common portrayal is that net income is divided by fewer shares, which automatically leads to a boost in EPS. This simplistic analysis neglects the fact that the company has to fund the buyback, either with excess cash or additional debt. Excess cash generates interest income, and debt comes with interest expense. As a result, buybacks affect net income as well as shares outstanding.
  • Appendix A: Dividend and Buyback Equivalence—Think Percentage Ownership
    • Dividends and buybacks have the same impact on corporate value. But executives think of them very differently. They consider dividends to be a commitment tantamount to capital expenditures and buybacks as a way to return excess cash after they have paid all of their bills and made of all of their investments.
    • In practice, dividends and buybacks are different because of tax consequences and the impact of gaps between stock price and value. But to understand TSR, the central distinction is between actively or passively increasing the percentage ownership in a company.
  • Decomposing TSR for Value and Growth Over the past 15 years or so, “value” stocks have provided substantially lower Total Shareholder Returns (TSRs) than “growth” stocks. Value stocks are those with low multiples of price to sales, earnings, and book value. Growth stocks are characterized by above-average sales growth, high Price-to-Earnings (P/E) ratios, and positive stock price momentum.
    The S&P 500 Value Index tracks the investment return of large-capitalization value stocks in the S&P 500. The index includes about 400 stocks that have an average market capitalization of $60 billion as of September 29, 2023. The S&P 500 Growth Index draws about 235 stocks from the S&P 500 that qualify as large-capitalization growth. The average market capitalization was $110 billion at the end of the third quarter of 2023. The stocks of numerous companies are in both indexes.
    Exhibit 16 shows that the annualized TSR from 2007 to 2021 was 7.6 percent for the Value Index and 13.3 percent for the Growth Index. We end in 2021 in order to use forward P/Es, but the value index still underperforms the growth index by 330 basis points annualized if we include the returns from 2022.

    We can compare the drivers to see why the results varied. Companies in the value index grew their net income at a 5.2 percent rate during the period, a little slower than the 5.9 percent rate for the members of the growth index. But the translation from net income to Earnings Per Share (EPS) was impeded by net equity issuance among the value index constituents and aided by equity retirement for the companies in the growth index. As a result, EPS rose 3.3 percent per year for the value index and 8.0 percent for the growth index.

    Both indexes enjoyed P/E multiple expansion, although the contribution was 1.6 percentage points for the value index and a larger 3.6 percentage points for the growth index. Interest rates dropped over this period and growth companies have longer implied durations, a measure of the weighted average time investors have to wait before they receive cash flows. Assets with long durations are more sensitive to changes in real interest rates than those with short durations. The growth index likely benefited more than the value index when rates dropped during this time.

    The sum of EPS growth and P/E multiple expansion led to price appreciation of 4.8 percent for the value index and 11.6 percent for the growth index. Those drivers explain most of the TSR difference between the indexes.

  • When governments tax dividends and capital gains at the same rate, as is the case in the U.S. in 2023, the decline in the stock price is roughly equivalent to the dividend. If taxes on dividends are higher than those on capital gains, the decline in stock price will be less than the dividend.

Pods, Passive Flows, and Punters
Drew Dickson

Excerpt:

I am as convinced as ever that, eventually, it is the fundamentals that matter. Eventually, the market is a weighing machine. If you want some evidence – even from some of the most iconic, well-followed, index-heavy, retail-engaged, pod-owned, successful companies, it is still, eventually, about the fundies.

Let’s take some of the winners as an example. And by winners, I mean game-changing, world-dominating winners.

You’ve surely noticed what has happen to Nvidia lately. We used to just call these winners FANGs, and then FAANGs and then FAMANGs, but Nvidia has insisted on joining the league table. It now has a $1.7 trillion market cap. And in the last five years, the stock is up about 1,700%. Guess what else is up about 1,700%?

Nvidia’s earnings estimates.

How about Facebook, aka Meta, which goes through periods of hatred and love with equal vigor? Well, over the past seven years it has bounced around a lot but still has generated nearly 260% returns. And forward earnings projections? They’re up 280%.

We can stretch things further back, and look at Google over the past 14 years (earnings up 885%, stock up 980%); or Amazon during the same period (earnings up nearly 2,500%, stock up about 2,800%).

Or we can go waaay back and analyze Microsoft over the past 22 years. Forward earnings projections have increased from $0.93 in February of 2002 to $11.57 today. That’s nearly 1,150%. The stock is up just over 1,200%.

And finally, from one of my favorite former-CEOs Reed Hastings, we have good old Netflix. About 18 years ago, analysts were forecasting that Netflix would generate 11 cents of earnings in the coming 2006 year. Here in 2024, they are forecasting a whopping $17 of earnings in the coming year. That is a whopping EPS increase of 14,889%.

And how about the stock? We’ll it is up a whopping 14,882%.

Fundamentals matter, sports fans. Fundamentals matter.

Admittedly, some of these examples above are very long-term, but even when we self-select with some of the biggest, most exciting, long-term winners out there, and ignore the losers (of which there are many), it is still clearly apparent that it is the fundamentals that matter most.

So basically, it probably isn’t terrible advice to ignore the rest of it. Ignore the noise. Ignore the talking heads on CNBC. Ignore prognostications of meme-stock sith lords. Ignore the volatility. Embrace it, actually. And just focus on the fundamentals. Get those right, and you will likely win.

Can you take advantage of it? Can you take advantage of the noise?

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

If you annualize volatility with 252 days can you use that number in a 365-day option model?

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

A Visual Primer For Understanding Options

Note:

This is a guest post by @KeyPaganRush. You can find the origin of this collaboration here.

 

If you’re a normal civilian like me, who at the very most took vector calculus as an undergraduate, you’d probably look at a differential equation like this and have the following reaction:

It doesn’t help that most people in finance who work in the volatility space are mathematically adept and don’t necessarily know how to simplify this for mathematical muggles.

I’m not that great at mathematics myself and although I use statistics and probability on a regular basis for my non-finance day-job, most of this is stuff covered in the 2nd year of an undergraduate course.

Concepts become more intuitive when I can visualize them, which has helped me scrape together just enough mathematical literacy to be decent at my job. Applying the same thinking to options, I decided to give up on trying to interpret options from the standpoint of differential equations, and instead lean into my already existing intuition of probability and statistics.

Conveniently, both approaches get you to the same answer.

I expect this approach can help other civilians finally make sense of the vol space.

An intuitive understanding of options

Probability distributions

When looking at the price of a stock, there are only 3 results that can occur next: Go up, go down or stay flat. If we make a big assumption that the chances for each are all the same, you can simulate a price chart as just being a long series of up, downs or flats.

The Galton Board shows that repeating the simulation many times over, you will find that the final price of a stock (each ball) forms a bell-like distribution, the normal distribution. Even if the probability is not equal, this is a starting point to model price movements.

Option prices as segments of distributions

You might have heard that options represent the full distribution of the market and are thus the real underlying. Sure you can argue, in the literal sense, that they are not the underlying, but that viewpoint is useless for making money, where the underlying stock price is a blunt representation of what the market expects. To illustrate, think of a stock as having a probability of having an ending price, represented by the graph below.

 

The market probabilities assigned to each of these prices is influenced by the buying/selling supply/demand of options. The peak of the distribution is typically the ATM. When you buy a call option at strike K, you are paying for the probability (The shaded green area) on the right side of K.

 

Conversely when buying a put option you are paying for the probability on the left side of K. This area of probability you buy is what you pay for an option.

 

When you sell your call, whatever probability is still existing to the right side of K is your payoff. Larger area of probability = more expensive the option is. So what can influence how much this option will end up costing?

When long a call option, if the price of the stock goes up, this shifts the entire distribution to the right. Each time the distribution moves to the right, because the stock goes up in price, you are gaining more area of probability and thus increasing the price of the option.

 

Delta and Gamma

When playing around in your mind with these graphs, you can normalise the amount of area  you have, as a ratio of the entire distribution.
The area that is moving past your strike, as a ratio of the entire distribution, is Delta2.

Notice however, that the change in delta gets bigger as the price of the stock gets closer to your strike. It then begins to scale down as you get past your strike and start moving further away from your strike. This means your delta is changing as a function of price. This ratio between the change in delta and the change in price, is gamma.

 

Vega

The area of that call option can get bigger (more expensive) even if the price of the stock stays completely still. Notice that if we just make the distribution wider, you gain more area of probability to the right side of your strike. Remember that the height of distribution at each price is influenced solely by buying/selling of options, or IV. Thus just by the market increasing the width of the entire distribution, the option can become more expensive. This ratio between the increased area for each increase in the width of the distribution is Vega.

As a bonus, you will probably notice that as a result of increased IV, the ratio between the area of the distribution per change in price, has also changed. This is vanna.

Theta

As time passes, the width of the distribution gets thinner. Why?

Imagine a $10 stock moves on average $1 a day. What chances would you give it of getting past $20 if you checked on it 250 days from now?

What about if you only gave it 1 day to do so?

See how the chances drop dramatically when there isn’t much time left for the price to move around? Thus as time passes, the distribution to the right side of your strike is moving inwards towards the ATM, reducing the area to the left of your strike over time ( and thus reducing the price of your option). This is theta.

 

As a bonus, you will probably notice that as a result of passing time, the ratio between the area of the distribution per change in price, has also changed. This is charm.

Central moments

The price of a stock is really a representation of only one thing: where will the peak of the distribution go, left or right? In fancy speak, we say it is the first central moment of the distribution, otherwise known as the mean/average of the distribution. It is just one aspect of the distribution.

When you introduce the process of delta-hedging, (see my video Gamma and Vanna Exposures) you are trying to prevent your PnL from being influenced by the shifting of the entire distribution, Ie. changes in the price of underlying.

This temporarily  “locks” your distribution in place, meaning that it can now only change in shape. Since the distribution cannot slide left or right, the only way the price of the option can change is for the shape of distribution to change.

So how can options become more expensive or cheaper now?

2nd Central Moment: Implied volatility

If implied volatility increases the distribution gets wider and the option becomes more expensive. This width of the distribution, in fancy speak, is the 2nd central moment, or the variance of the distribution. Volatility is just the square root of variance.

3rd central moment: Skew

We can get even fancier by thinking, the total variance of the distribution might not necessarily change, but that one side of the distribution will get wider but the other side gets thinner, that there will be a difference in the relative widths on either end (tail) of the distribution. This is achieved by going short vol on one side of the distribution and long vol on the other side of the distribution, delta neutral. The difference in relative areas of the tails on either side of the distribution is the 3rd central moment, or Skew.

 

4th central moment: Kurtosis

It is even possible to make a bet that mean, variance and skew don’t necessarily change, but instead bet that the width of the distribution might be thin at the middle, but wider near the ends. This is done being short vol near the middle of distribution and being long vol near the tails. The relative widths of the distribution near the middle vs the tails is the 4th central moment, which we call kurtosis.

 

By looking at the options market, we are able to gain a rich source of information and opportunities for expressing very specific views on what the market thinks the shape of the distribution is, that are independent of the entire distribution shifting left or right (Ie, price of the stock going up or down).

Appendix

Coming full circle to the original differential equation, we can now break down the meaning of this.

 

Using some basic algebra, we can re-arrange the equation (Move the right-most terms to the left side of the equation) to read like this:

This equation is simply telling you how the change in price of the option is influenced by the price of underlying, delta, theta, variance and gamma.
(For simplicity we are ignoring “r”, which represents the risk-free rate; although for those who are curious, the influence of the risk-free rate on the price of an option is known as “Rho”).

You may notice I did not highlight any vega term in the equation, to avoid getting too far into the weeds with the mathematics since Vega gets embedded inside the delta and gamma of the equation. This is because both are influenced by variance (The distribution getting wider). You can visualize this concept by changing the width of a distribution and observing how it affects delta and gamma.

Trying this out with some real numbers, lets see how this breaks down.

  • Take a call option on a stock which is currently trading at 100.
  • The strike is 110, with a 1 year expiration, interest rate at 0% and volatility of 10% annualized.
  • This stock does not pay any dividends.

An option calculator yields:

Our fundamental equation in terms of Greeks can be used to relate the value of the option to the size of the stock move. If the move size is the same as the volatility used to price the option then we’d expect the p/l to be zero. 

To see this we need to keep time units consistent. We transform annual parameters into daily ones.

  • Convert annual volatility to daily volatility by dividing it by the square root of 365
  • Annualized rates can be converted to a daily rate, simply by dividing by 365

Plug and chug:

The left and right hand sides of the equation equal each other! 

Assuming a 0% risk-free rate, if an option were priced perfectly (ie volatility is perfectly forecasted), then any gain made from movements of the price of the underlying should be offset by the theta that is going to be bled off.

More practice

What happens if one day were to pass, but the underlying did not move at all?

The purple term representing the “change in stock” will be 0 rendering the “gamma p/l” for that day to be zero.  


The option will be decay by its theta and there will be no offsetting gamma p/l. For that day, the option was “overpriced”. 

What happens if after buying the option, realized volatility were to increase from 10% annualized to 15% annualized?

We capture a gamma p/l as follows:

In this case the gamma p/l of .0082 was greater than the theta of .0036 so the owner of the option has won. The option was “underpriced” for that day because the annualized move of 15% exceeded the 10% annualized volatility the option was priced with. 

In daily terms, remember a 10% annual volatility corresponds to a 1-day change of .52% and a 15% annualized move corresponds to a 1-day change of .79%. 

The readers is welcome to discover how the p/l is a non-linear function of the difference between the realized and implied move sizes (gamma attribution is a squared term!)

Summary Tables

Conclusion

Our original equation reinforces the idea that the Black Scholes Equation is a non-arbitrage condition stating that if volatility were perfectly priced the value of the option is equal to the cost of the replicating portfolio.

Takeaways

  • Stocks only represent the first moment, a single point of the distribution but nothing about its shape or how it changes over time. 
  • Options will tell you about the entire shape of the distribution, which is why I submit that they are in a sense the true underlying distribution.

Further reading

Moontower on Gamma

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

Career Advice For Quants

I added an outstanding post to the top of the Moontowerquant Career section.

Version with my emphasis:

🔗Buy-Side Quant Job Advice

I read it a few times. It’s both amusing and practical.

Landscape

  • Every firm is a bit like Orwell’s “Animal Farm”: all employees are created equal, but some employees are more equal than others. In PEs and VCs, quants are not at the core of the business, and in a good portion of asset managers, pension funds, and family offices, quants are not working on the most exciting problems. You probably want to begin your career in a place where quants are first-class citizens and are using their brains. I will focus only on hedge funds and prop trading firms.
  • the top 20 hedge funds have generated 19% of the total profits (out of maybe 50,000 HFs). In the past three years, the top three hedge funds (Citadel, Millennium, DE Shaw) have generated 38% of the total PnL.

Recommended Reading

  • Subscribe to Matt Levine’s “Money Stuff” newsletter; read his past articles too. They are informative, funny, and have aged well. They are free. They are just too long.
  • Read a few entertaining books for fun and profit: “My Life As A Quant”, “Against the Gods”, “Red Blooded Finance”, “The Education of a Speculator”, “The Man Who Solved the Market”, “A Man for All Markets”, maybe a Taleb book (but don’t take it too seriously).
  • People ask brain teasers, and I can think for a couple of reasons. First, to probe basic modeling and math skills. Second, because it is a focal point: everyone knows they are a likely topic. So I am not testing your intrinsic ability to solve a puzzle, but your ability to learn about puzzles. And there is a pattern to puzzles, which can be learned. Work through all of Peter Winkler’s books. And various firms, including Jane, IBM, etc. have puzzle sites.
  • Applied probabilistic modeling and statistics are very important skills to have. Physics is still a good major to hire from, because it is a model-based discipline, as opposed to a technique-based one, and you will be exposed to many models. Take classes at the MS level. Read at least the following books:
    • “All of Statistics” (both volumes) by L.Wasserman
    • “Applied Probability Models” by S. Ross
    • “Convex Optimization” by S. Boyd and L. Vandenberghe
    • “Numerical Linear Algebra” by Trefethen and Bau
    • “Linear Algebra and Learning From Data” by G. Strang
    • “How to Solve It” by G. Polya NoteI don’t recommend any finance book. You’ll learn on the job.

Read the following three essays. They are short and full of useful advice.

  1. You and your research by R. Hamming This is the most practical of my recommended readings. Please read this over and over again. My favorite sentence is: “I started asking, ‘What are the important problems in your field?’ And after a week or so, ‘What important problems are you working on?’ And after some more time, I came in one day and said, ‘If what you are doing is not important, and if you don’t think it is going to lead to something important, why are you at Bell Labs working on it?’” If you have time, read “The Art of Doing Science and Engineering: Learning to Learn” by the same author
  2. Real-life mathematics by B. Beauzamy. By a mathematician actually doing applied mathematics. Favorite sentence: “Real-life mathematics does not require distinguished mathematicians. On the contrary, it requires barbarians: people willing to fight, to conquer, to build, to understand, with no predetermined idea about which tool should be used.”
  3. Ten lessons I wish I had been taught by G.C. Rota. Although this is a bit more academic, it is extremely useful. For example, the first item is on “lecturing”, but it’s really about communicating ideas effectively. Favorite lesson (from Feynman, actually): “You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps.”

Non-obvious points in the essay

  • non-alpha related jobs can be extremely intellectually satisfying. Thinking about data, execution cost measurement, optimization, risk–these are all very deep subjects and you can have a great and long career in any of those. The road to hell is paved with mediocre alpha researchers who did not achieve their goals and burned out by the early 30s. Maybe a life of purpose is not the first thing that comes to mind when working in finance but, as much as it is in your power, pursue it.
  • As a pet project, over the years I have asked many (many= 50-100) successful traders, algo developers and portfolio managers what makes a great analyst for their team. The answers have been remarkably consistent.
  1. Curiosity. People who read articles and scientific papers on their own, maybe during weekends, for the sheer pleasure of finding things out.
  1. Creativity. Like obscenity, hard to define but easy to tell it when you see it. I guess, something like this: looking at the same thing everybody can look at, but noticing something different, and proposing an original course of action. Most ideas do not survive scrutiny, but a few are brilliant.
  1. Humility. When something does not work, admit it early and openly, examine the reasons why, and move on. In practice, humility (as described to me) is both willingness to take responsibility and openness to experience.
  1. Integrity. Following the letter and the spirit of the rules– the team’s, the firm’s, the regulators’.

A few personal comments on this list. First, these qualities are highly correlated; their definitions even overlap. There’s a single trait that perhaps explains 85% of their occurrence. I can’t determine whether this trait is innate or cultural, but I’m fairly confident that by the time you join a firm as a researcher, you either have it or you don’t. Interestingly, not a single person highlighted “capability”, “mental throughput”, or “puzzle-solving” as a quality; yet, we partly select based on the ability to solve puzzles—go figure. In fact, many people I interviewed said that everyone can proficiently perform [task x] or work hard to execute instructions. Also, no one mentioned soft skills like empathy, communication skills, etc. Indeed, some of the very best investors I know, while being very good people at heart, have the social skills of a thermonuclear reactor. Finally, every manager I interviewed sees their employees as researchers, not soldiers or doers.

  • Scout MindsetMaybe this is a good time to recommend a book on this subject: “The Scout Mindset” by Julia Galef, which explores the differences between explorers and soldiers.[Kris: See A Few Blurbs From Slatestarcodex’s Review of Scout Mindset]
  • You can be successful (especially as an alpha researcher) in one of two ways.
    1. First one: You identify a completely new opportunity. Example: Gerry Bamberger at Morgan Stanley in the 80s developed statistical arbitrage. Also in the 80s: the early index rebalancing strategies, and convertible arbitrage.
    2. The second one: You apprentice in a team that has a successful strategy, learn the trade, and improve it marginally. Unsurprisingly, the overwhelming majority of successful traders belong to the second class. The lesson: try to join a team and a firm that has a habit of being successful. Don’t think you can make a huge difference, and don’t fall for the poetry of the underdog.
  • Don’t be paranoid. No one is going to steal your idea. The real risk is that they will not even listen to you.

He ends with a statement that I feel goes from insightful to cliche back to insightful for each decade you’re in the business until your 50. At which point only the sociopaths, alimony payers, and overly fertile still remain. (Calm down, I’m mostly kidding)

A final and non-strictly professional piece of advice: you will spend more time working with your colleagues than with your partner or spouse or family. If you have to suffer at work, try to suffer successfully by sharing a strong common purpose with your colleagues, then by pursuing it in the best possible manner. The accumulated wealth from having worked at several firms will not come from your W-2s, but from the relationships and friendships you will have developed along the way.

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

Kelly Math Weirdness

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:

tweet link

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

Getting Comfortable With Log Charts

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:

Wealth = a(1+r)ᵗ

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

https://www.macrotrends.net/2324/sp-500-historical-chart-data…

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

Getting Paid To Flip Million Dollar Coins

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:

�������=�����ℎ(1/3)\(utility = wealth ^{(1/3)}\)

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 10x (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 2x.

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 2x

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 = aX

    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 = aXb

    log(Y) = log(aXb)

    log(Y) = log(a) + log(Xb)

    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

Intraday Breakout Riddle

There was a famous futures trader on the NYMEX when I was there named Mark Fisher. I leased office space from him (his clearing firm had a large footprint) but I only spoke to him once or twice briefly in passing. We didn’t really know each other. Anyway, he wrote a book called Logical Trader and backed people to trade his system. A copy of it was laying around the office (it’s in my garage in boxes I haven’t unpacked since moving over 3 years ago). I read a chapter that happens to be available for free online.

The book was mentioned on Twitter and got me thinking a bit about its core observation:

If you subscribe to the “random walk” theory, which states that the market’s movements are random and totally unpredictable, then the opening range would not be any more important than any other price level during the trading day. Right? For example, crude oil trades from 9:45 a.m. Eastern time until 3:10 p.m. Eastern Time. If you divided that day into 10-minute intervals, you’d have 32 parcels of time (and five minutes left over). So, each 10-minute time interval would account for roughly 1/32 of the market activity. Using random walk theory, you’d expect that the opening range (established in the first 10 minutes of trading) would be the high 1/32 of the time, or it would be the low 1/32 of the time. Therefore, random walk theory would dictate that 1/16 of the time the opening range would be EITHER the high or the low. 16 Now, what if I told you that in volatile markets – not static, and not necessarily trending markets – the opening range tends to be the high or the low 17-23% of the time? Would that get your attention? Yes. Because this observation would tell you that the opening range being at the high of the low of the day roughly one-fifth of the time is what we call “statistically significant.” In complete layman’s terms, this means the opening range is not just another 10-minute interval out of 32 of them in the trading day. It has more weight than any other time interval.

Let’s take another example. Let’s say that you divide the trading day up into roughly 64, five-minute intervals. Random walk theory would state that the opening, five-minute range would be the high 1/64 of the time or the low 1/64 of the time. So it would be either of those extremes 1/32 of the time. However, in volatile markets, that five-minute opening range is actually the high or the low of the day about 15-18% of the time. So instead of about 3% of the time, as random walk theory would predict, the first five minutes of the trading day turns out to be the high or the low 15-18% of the time. Again, statistically significant. And, from a trader’s perspective, if you knew that something was going to market the high or the low 15% of the time, you’d want to know that.

In short, if the opening range is the high or low a disproportionate amount of the time Fisher concludes that the odds are in favor of an intraday breakout strategy. The gist of it is:

  • Once the opening range is established, if the futures breakout of the range by say some fraction of a standard deviation then bet that they will continue.
  • In the case of an upside breakout, set a stop near the bottom end of the opening range.

The book goes into sizing, money management, where to set levels and other details.

I’m not going to weigh in on the strategy’s merits because it’s not my wheelhouse. I have lots of questions and of course, come from a place of skepticism but that’s just a healthy reaction to any anomaly. It’s not yet the “work”.

But it did get me wondering about how likely you’d expect the opening price in a market to be the high or low.

First, I turned it into a simpler riddle that you could try to solve yourself or (give to some kids to noodle on).

 

You can dig into how I worked them out here:

Crossing Over Zero

 

There are a couple of neat ideas in the solutions. (You’ll also find the trick GPT taught me. Satisfying, clever and reusable.)

Based on tinkering with a simple random walk, it does seem that an opening price (or the zero crossover in my toy examples) being the high or low a disproportionate amount of time would not be random.

[Although that isn’t enough to suggest that there is a positive expectancy in the strategy. It’s possible the payoffs on the breakouts times their frequency don’t compensate you for the number of times you get stopped out. If any systematic traders reading this feel like being nerdsniped by researching it I’d love to see the conditional probabilities that surround the different scenarios].

Money Angle For Masochists

 

Let’s practice option intuition on this same problem.

The setup:

  • A 30% vol asset opens at $40
  • It rallies to $40.50
  • Half the trading day has elapsed

What’s the probability it crosses $40 again today?

If we assume a Black-Scholes lognormal distribution with no skew (not unreasonable for a single day) we can compute the probability by turning $40 into a Z-score.

 
explained simply here: https://moontowermeta.com/using-log-returns-and-volatility-to-normalize-strike-distances/

K = strike price

S = Spot price

σ = volatility

t = time (in years)

ln(K/S) is basically how much percent away $40 is from $40.50.

ln(K/S) = ln(40/40.50) = -1.24%

By dividing by σ√t we scale the 1.24% by standard deviation for the remaining time.

-1.24% / 30% * √(.5/365) = -1.12

$40 is 1.12 standard devs away.

The probability of the asset sliding at least 1.12 standard devs is 13%.

In Black-Scholes world, the probability of a strike expiring in-the-money is known as N(d2). But for short-dated options, delta is valid substitute for N(d2).

So we’d expect the delta of the $40 strike with half a trading day remaining and the asset at $40.50 to be 13%.

In the context of our earlier conversation, you might think that the probability of crossing zero (ie the opening price) is 13% but we need to make a key distinction based on using the option delta:

The delta is telling us the probability of expiring in-the-money…but our riddle is concerned with whether the price or random walk ever breaches zero even if it goes back up.

The riddle is not concerned with the probability of a vanilla option but a one-touch option.

Investopedia defines these exotic options:

A one-touch option pays a premium to the holder of the option if the spot rate reaches the strike price at any time before option expiration.

I’ve never priced one-touch options but I remember a quant trader telling me that their probability of being triggered was approximately 2x the delta of the vanilla option of the equivalent strike.

In this example, the probability of the asset touching a price less than $40 before the day ends is 13% x 2 = 26%

This is intuitive if we consider an at-the-money option that has a 50% delta. The asset is nearly 100% to touch prices on either side of the strike.

[It’s convenient and expected that option trader math gets reduced to rules-of-thumb (“straddle price is 80% of the vol scaled by time”, “multiplying the daily move by 16”, “implied correlation is ratio of index variance to avg stock variance”) since so much of flow trading is making quick decisions and on-the-fly comparisons or normalizations.]

If price changes were a random walk I wouldn’t expect the opening price to be the high or low more than 1% of the time. But the open price, while cannot be predicted, likely holds meaning once it’s established because it is a single clearing price of an auction that accumulated hours of overnight information.

[I spent almost 2 years on NYSE both as a broker in the “garage” if you are familiar with the place, and as a specialist in ETFs (in the “blue room”) . The open is the price that best clears the order book when considering the stack of market and limit orders on both sides. But consider this scenario —

  • At a price of $40.23 there’s an imbalance of 10,000 shares for sale
  • At a price of $40.22 there’s an imbalance of 75,000 shares to buy

You can expect the stock to open at $40.23 and for the specialist to buy the 10,000 shares for their own account and then to display a market with $40.22 bid for size to induce buyers. The opening price had information in it.]

Further reading:

 
  • Why We Use Logreturn in Finance (Quant Factory)
  • Understanding Variance Time (Moontower)This post ties in nicely with another riddle:

    In the option demonstration above I said there was half a day until expiration.

    What time is it?

    Hint: The answer is not the point. And you won’t get it anyway. I’d consider your response a success if you can just identify what the inputs the answer depends on. Godspeed.

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