Lessons From The .50 Delta Option

I was chatting with a quant friend who was bouncing an options idea off me. In the course of the conversation, he was surprised I did not assume the .50 delta option was the ATM (at-the-money) option. My friend is much smarter than me on finance stuff but options aren’t his native professional language. So if this idea had him tripped up I realized I had a reason to write a post.

If I do this correctly you will gain a better understanding of:

  1. Delta as a hedge ratio, not a probability
  2. How volatility affects the mean, median, and mode of these returns
  3. The relationship of arithmetic to geometric returns in option theory
  4. What these distributions mean for the value of popular option structures

Some housecleaning:

  • Option math is known for being calculus heavy. If you are a layperson, you are in luck, this tour guide likes to stick to the roads he knows. You won’t find complex equations here. If you are a quant, I suspect you can still benefit from an intuitive approach.
  • We are going to ignore the cost of carry (interests and dividends). While crucial to actual implementation it is just distracting to the intuition.

Delta Is A Hedge Ratio Not a Probability

Often delta and “probability of finishing ITM (in-the-money)” are indistinguishable. But they are not the same thing. The fact that they are not equivalent holds many insights.

Before we go there, let us revisit the most basic definition of delta.

Option delta is the change in option price per $1 change in the underlying

Consider the following example:

Stock is trading for $1. It’s a biotech and tomorrow there is a ruling:

    • 90% of the time the stock goes to zero
    • 10% of the time the stock goes to $10

First take note, the stock is correctly priced at $1 based on expected value (.90 x $0 + .10 x $10). So here are my questions.

  • What is the $5 call worth?

Back to the expected value:

    • 90% of the time the call expires worthless.
    • 10% of the time the call is worth $5

.9 x $0 + .10 x $5 = $.50

The call is worth $.50

  • Now, what is the delta of the $5 call?

$5 strike call =$.50

Delta = (change in option price) / (change in stock price)

    • In the down case, the call goes from $.50 to zero as the stock goes from $1 to zero.

Delta = $.50 / $1.00 = .50

    • In the up case, the call goes from $.50 to $5 while the stock goes from $1 to $10

Delta = $4.50 / $9.00 = .50

The call has a .50 delta

Using The Delta As a Hedge Ratio

Let’s suppose you sell the $5 call to a punter for $.50 and to hedge you buy 50 shares of stock. Each option contract corresponds to a 100 share deliverable.

  • Down scenario P/L:

Short Call P/L = $.50 x 100 = $50

Long Stock P/L = -$1.00 x 50 = -$50

Total P/L = $0

  • Up scenario P/L:

Short Call P/L = -$4.50 x 100 = -$450

Long Stock P/L = $9.00 x 50 = $450

Total P/L = $0

Eureka, it works! If you hedge your option position on a .50 delta your p/l in both cases is zero.

But if you recall, the probability of the $5 call finishing in the money was just 10%. It’s worth restating. In this binary example, the 400% OTM call has a 50% delta despite only having a 10% chance of finishing in the money.

I’ll leave it to you to repeat this example with a balanced distribution. Say a $5 stock that is equally likely to go to zero or $10. You will find the 50% delta call turns out to be ATM. Something you are used to seeing.

The key observation turns out to be:

The more positively skewed the distribution, the further OTM the 50% call will be. If a stock is able to go up 1000% and you sell a 400% OTM call on it you are going to need far more than a token amount of long stock to hedge.

The more positively skewed a distribution, the more the hedge ratio diverges from the “probability of finishing ITM”.


The Effect of Pure Volatility

Not to lead the witness too much, but an obvious feature of the binary example is the biotech stock is very volatile. That’s not a technical definition but a common-sense observation. “This thing is gonna move 100% or 900%!”.

Without math, consider how volatility alone affects a stock’s returns. If the stock price remains unchanged because we do not vary the expected value but instead inject more volatility what is happening?

  1. We are increasing the upside of possible payoffs.

In the biotech example, more volatility can mean the upside is not $10 but $20. 

  1. The counterbalance to the greater upside is a lower probability of rallying

If the stock is still worth $1 then the probability of the up scenario has just halved to 5% (95% x $0 + 5% * $20 = $1, the current price).

If we inject volatility into a price that is bounded by zero, the probability of the stock going down is necessarily increasing.

So volatility alone alters the shape of a stock’s distribution if you keep the stock price unchanged.

Let’s see how this works as we move from binary distributions to more common continuous scenarios.

How Volatility Affects Continuous Distributions

Let’s start with a simulation of a subjectively volatile stock.


  • The stock is $50
  • The annual standard deviation is 80%.

A basic presumption of option models is that returns are normally distributed but this leads to a lognormal distribution of stock prices 1.

Running the simulation:

  • I took a return chosen from a normal distribution with a mean of 0 1 and standard deviation of .80
  • I then ran that return through a simple log process to simulate continuous compounding. 2

S (T1) = S (T0) x e(random generated return)

  • I ran this 10,000 times.

Before we get to the chart note some key observations:

  • You get a positively skewed lognormal distribution bounded by zero. This is expected.
  • The median terminal stock price is $50 corresponding to a median return (aka the geometric mean) close to zero as expected.
  • The mean stock price is $68. corresponding to a mean return of 38%.
  • The modal stock price is $20 corresponding to a modal return of -60%.

Simulation Vs Theory

Let’s compare the simulation to what option theory predicts.

  • Median

As we stated earlier median expected return is 0 from theory and this lines up with the simulation.

  • Mode

The mode in the simulation lines up reasonably 3 with option theory which expects the mode to be:

S x e2 where σ is volatility

Note how volatility pulls your most likely outcomes lower. In this case, the most likely landing spot for the stock is $20 corresponding to a total return of -60%!

Average Arithmetic Returns

Look at the chart again. Note how the average arithmetic mean stock price is $68.89 in this sample. If the median return is 0, the positively skewed distribution has a mean arithmetic return of +37.8%! We don’t want to get excited about this since as investors we care about geometric returns which are zero here, but this 38% OTM strike is still very interesting.

It turns out it corresponds to the strike of the .50 delta option!

The equation for that strike:

S x e2/2)

That strike corresponds to 68.86 which is very close to the simulation result of 68.89.

This is the call that you must hedge with 50% of the underlyer.

The formula will look familiar if you remember that the geometric mean is pulled down from the arithmetic mean in proportion to the variance.

[This strike is special for option traders. This is the strike that has the maximum vega and gamma on the option surface. As implied vol changes the location of this strike can change, but it represents the maximum vega any strike can have for a given spot price. I’ll leave it to the reader to see how this relates to strategies that are convex in vol such as ratio’d vega neutral butterflies.]

Interesting Observations About Options

  • Even in a continuous distribution, the higher the volatility, the more positively skewed the distribution, the further OTM the 50d call strike lives.
  • The cheapest straddle will occur at the median outcome or the ATM4 strike. 
  • The most expensive butterfly will have its “body” near the theoretical mode. This makes sense since a butterfly which is just a spread of 2 vertical spreads is a pure bet on the distribution. If you chart the price of all the butterflies equidistantly across strikes you will have drawn the probability density function implied by the options market!

Enter Black Scholes

In a positively skewed distribution, the probability of finishing in the money for a call was lower than the delta. In the binary example where the stock had only a 10% chance of being worth $10, the probability of the $5 call was much lower than the delta of the $5 call.

What does this have to do with Black Scholes?

In Black Scholes:

  • The term for delta is N(d1).
  • The term for the probability of finishing in the money is N(d2).

What’s the relationship between d2 and d1?

  • d2 = d1 – σ√t

The math defines the relationship we figured out intuitively:

The higher the volatility 5 the more delta and probability will diverge!

Delta and probability are only similar when an option is near expiration or when it’s vol is “low”.

From Theory to the Real World

Markets compensate for Black Schole’s lognormal assumptions by implying a volatility skew. While a biotech stock might have a positive skew on steroids, a typical stock’s distribution looks more normal than positive. By pumping up the implied volatility of the downside puts and lowering the implied vols on the upside calls, the market:

  • Increases the value of all the call spreads.
  • Shifts the implied mode rightward.
  • Shifts the 50d call closer to ATM. Actually, it lowers all call deltas and raises all put deltas. This is important since deltas are the hedge ratios.
  • Fattens the left tail relative to a positive distribution and at least in index options even more than a normal distribution.

These adjustments reconcile the desirability of a simple, easy to compute model like Black Scholes which uses lognormal distributions with empirically consistent asset distributions that we observe in markets.



The next time you hear delta used as probability, remember this is only really useful when options are near-dated. Since most option activity occurs in the front end of the term structure the assumption is typically harmless.

Taking the time to understand why they differ turns out to be a great exercise in building an intuition of investment returns and their distributions. 



The “No Easy Trades” Principle

My “No Easy Trades” Principle

A recurring theme here is the wisdom of markets. If you find yourself constantly disagreeing with prices visible to anyone with a smartphone you probably get invited to lots of poker games. This doesn’t mean markets are always right. It just means nobody can claim there is easy money in slanging investments around. It means your guess is not consistently better than the market’s. I highlight that distinction because it seems to be lost on efficient market detractors.

So my definition of markets being smart is not academic. It’s actually a survival heuristic — there are no easy trades. Today I will discuss 3 things.

1. What this heuristic looks like in the options market
2. The puzzle I stumbled into when applying this logic to stocks.
3. How the puzzle got cleared up.

Before I mosey into this we need to agree on a stylized fact.

Investment returns are driven by:

  1. Actual earnings (returned to shareholders or re-invested)
  2. Expectations (risk discounted of course)

So a stock’s return is driven by actual earnings (I call that the “realized”) and changes in the multiple (the forward-looking growth prospects is the “implied”). Same with real estate. If the cap rate is zero in one location and 10% in another it’s usually investors factoring in different growth rates. The actual rent is the “realized”.

As time passes, the market will pay attention to the fundamentals on the ground to revise the growth rates. Is this investment cash-flowing more or less than expected? So the “realized” earnings interact with the expected or “implied future earnings”.

Option Returns Are No Different

First, some reminders about options:

  • Volatility is the standard deviation of a stock’s return. The amount of volatility that transpires until an option expires is a large driver of an option’s value.
  • Nobody knows how much volatility a stock will experience during an option’s holding period.
  • We know the other inputs into an option price. Since the other inputs are known, we say the price of an option implies a volatility.

For somebody managing a hedged options book, the p/l is driven by:

1. The realized volatility of the underlying

If the market bounces around 2% a day and you purchased the option implying 1% per day you will “capture” p/l in excess of the option’s time decay. (The p/l due to “gamma” exceeds the losses due to “theta”)

2. The implied volatility baked into the option

If the market suddenly believes the future will be more volatile, it will bid up option prices. This will lead to profits for someone who owns options. (The sensitivity of the option’s price to the market’s vote on volatility is known as “vega”)

Option Markets Are Smart

Without getting into the nature of realized and implied volatility it’s sufficient to say that they are mean-reverting. If a stock becomes more volatile, say moving 2% per when a longer history pegs it as a 1%-per-day type stock the options prices will increase to price the extra volatility. But as the realized day-to-day volatility reaches extreme levels, say 5% per day, something we may have seen in March, the option implied volatility will likely not rise by as much (I’m hand-waving term structure and more, it’s not necessary to the point). Why does the implied volatility not keep up?

Expectations. The market understands the shock is temporary. So there is no easy trade. People that want to sell or short expensive options will be disappointed to find that they will experience negative gamma p/l during the holding period because the realized volatility will exceed the implied volatility they had shorted.

And option longs who may be enjoying the positive gamma p/l (or “carry”) know that they have bought a high implied volatility that is eventually going to recede.

When option volatilities get very low the inverse dynamic occurs. There are period when the SP500 will realize less than 30 bps a day but the market never sells you options at such a low implied volatility.

It’s simple. The market will not let you have a position that simultaneously:

  1. Carries well
  2. And has a tailwind in the direction of the mean-reversion

Classic dilemma. You enjoy one while fighting the other. Market implied parameters reflect expectations. But expectations do not vary as widely as what actually happens because volatility is a mean-reverting quantity. Net result: no easy trades.

The Puzzle: Is The Stock Market Smart?

I presume the stock market is smart and must follow a similar principle of “no easy trades”. By analogy, I mapped implied volatility to P/E and realized volatility to actual earnings. So if profits (earnings) were high, I’d expect forward P/E ratios to discount the elevated earnings. Otherwise, it would seem like an easy trade to take the other side of a market that extrapolated unusually high earnings into the future. Just because LeBron drops 50 points one night does mean the “point futures” market is “49 bid” for the next game.

Now earning themselves are not mean-reverting. I wasn’t quite naive enough to apply my cargo-cult thinking to that metric. Enter fund manager John Hussman. He argued that profit-margins, which have been extremely high, are both mean-reverting AND being extrapolated into the future via fat multiples.

This was to be the rare set-up of an easy trade — a bear case with a double tailwind. Luckily I didn’t trade in my PA based on this. For all the $20 bills on the ground I miss out on, my disbelief in their existence has also saved me from being short stocks for the past 5 years.

So what’s the deal with the stock market? Can it possibly be extrapolating mean-reverting metrics with straight lines?

Carry On, Nothing To See Here

Well, it turns out there is no puzzle. Profit margins aren’t a metric that matters. It’s return-on-equity that matters. Get all the details from @jesse_livermore in his paper Profit Margins Don’t Matter (Link with my highlights). It’s several years old but the thinking that permeates through this paper is better than 99% of finance stuff you might read today. And if you are weak on accounting like I am you’ll want to bookmark this one.

Extra observations on the options vs stocks analogy:

  • When an option has lots of time until expiration the implied expectations dominate its price. Conversely, as an option approaches expiration, the realized volatility will dominate. This manifests as the option’s vega decreasing while its theta and gamma increase.
  • Unlike options which have fixed expirations, real estate and stocks are more like perpetuities making them highly sensitive to expectations.


Moontower Money Wiki (Link)

Latest updates:

  • Risk Is Unavoidable, Let’s Get To The Good News (Link)
  • TANSTAFL (Link)

WFH: Deus Ex Machina

Money is an important driver of what jobs people take. Especially at the level where basic needs are at stake. But at some point, flexibility in location, hours, and benefits gain relative importance. The more negotiable the mix the better chance we have of creating sustainable work situations for people. Sustainable means less burn-out, less disability, less dissatisfaction, and more productivity.

The growth of remote work shades in a fuller menu of points along the efficient job frontier. It creates more shots at sustainability. You can choose to work for less cash if it means a better overall life fit. (I know of a few people, moms to be precise, who were thrilled when employers allowed them to work 4 days a week for a 20% pay cut).

Why Is Sustainability So Important?

When I introduced the Moontower Retirement Model, I mentioned “how destructive I think the whole vision of retirement is as portrayed by Charles Schwab commercials. The fact that it takes a moment to figure out if it’s a financial commercial or a Viagra ad is enough of hint that drugs are being sold in both cases.” The entire notion of retirement is unsustainable.

The average American’s savings are inadequate to even cover an emergency. Forget having enough savings and social security to survive with dignity from 65 to 85 or 90. The presumption that investment returns will make up the shortfall even if everyone invested is fragile at best and quite likely nonsense.

The classic idea of retirement has outlived its usefulness. It is a vestige of a bygone era and was targeted to people who did back-breaking work. The cost of this charade is staggering. Underfunded pensions even after a 10-year bull market. Politicians needing to kick all cans lest they be forced to do arithmetic that doesn’t work in plain view. Too many lonely, poor senior citizens.

What Is The New Retirement Look Like?

More working years. But of a different type. Working years that don’t just feel like a thing to get-over-with. The future of work has the chance to be more sustainable. The subject is vast, beyond the scope of this post. But an emerging efficient frontier trade-off between salary and location offers a promise of better-centered lives. We should agree that matching ourselves to our environments is a public good. A sustainable balance means we can work more years. A double bonus. We save for more years and withdraw for fewer years. It is the only viable solution to the pension crisis.

If remote work means more sustainable working lives then WFH will be the pension crisis’ deus ex machina. And that’s a bigger cause for hope than the short-run sprint of getting paid SF wages while Zooming from a lakeside cabin.

To learn more:

  • I’ve summarized the roots of the pension crisis as told by Charley Ellis. (Link)
  • Once Facebook tells you how much they’d pay you if relocate to Boulder, Asheville, Nashville, Summerlin, or wherever you’d like to be, check Atmos to see what you can rent or build. Find your own efficient frontier. (Link)
  • Matt Mullenweg, founder of WordPress, explains Distributed Work’s 5 Levels of Autonomy. What does it mean to jump from remote to remote-first? How about from asynchronous to autonomy? (Link)

For remote-work to truly sustain us, we should, when practical, aspire to the pinnacle of the pyramid — autonomy. Daniel Pink identifies autonomy as 1 of the 3 pillars of human motivation (the other 2 are mastery and purpose). If we can grease the rails of motivation maybe we can scrap the retirement parties that in hindsight turn out to be death sentences.

Moontower Money Wiki Update

I’ve completed the section Evergreen Beliefs That Work Together. It has 4 parts:

  1. The Gift Of Market Efficiency
  2. Investing Is A “Loser’s Game”
  3. Time And Human Capital
  4. Towards A Prescription

While the wiki is ultimately about investing your savings, this particular section goes nicely with the topic of today’s letter which is really about a chance to live our lives better by maximizing the returns to our human capital not our financial capital.

Check the wiki updates. (Link)

Origin of the Pension Crisis

Investor Charley Ellis is deeply concerned about pensions being underfunded as we approach peak boomer retirement years. He discussed it with Ted Seides on the Capital Allocators podcast which I summarize in full here.

The roots of our pension crisis can be traced back to 19th-century European rail workers! Here’s a summary of the problem, how we got here, and what to do about it.

The Pension Crisis

Scope of the Problem

Public pension plans are impossibly underfunded

If you look at what are the biggest problems we as a nation have in the investments world, it’s pensions or retirement security. You can see it easily in the state and city funds that are seriously underfunded. They need 7.5% rate of return which they’re not going to get because they’ve got 25% in 2.5-3% bonds. They’re just not going to get it.

Households are underfunded

If you look at individuals, half the population does not have a retirement plan. For those that do 401k is increasingly dominant, taking over from defined benefit system. The average person approaching age 63.5 which is the retirement age in this country is thinking:

“I’ve got 165,000 smackers in my account. Why my wife and I are going to Florida to play some golf, some tennis, have some fun. We’re gonna have great years. We’ve earned it. It’s been a long long working run, but we’ve earned it it’s going to work out just fine.”


Anybody with any knowledge about investing knows right away — $165,000 if you take money out, from 63 years old to 85 or 90 is not enough. You’re not going to have anywhere near enough per year, cobbled together with social security to make anything like a decent connection.

Something over 65% of your life time health expenses are spent in your life six months. Well, that’s where half the bank for personal bankruptcies come from all kinds of trauma that goes with that as well assisted living expensive and dementia. So we’re going to have a real problem with old age, retirement security.

Political nightmare

So what are people gonna say?

“God damn it. I worked hard all my life I played by the game rules as everybody laid them out. And I was supposed to be able to retire at a decent age and enjoy retirement. That’s part of the deal.”

But the answer they will get back?

“Sorry, but nobody else understands that to be the part of the deal. And you’re on your own.”

So you will have a giant generation that is angry, focused, and motivated to do something about this false promise.

If you think we’ve had divisive politics in the past, imagine what it would be if you had millions of people and their relatives all saying “It isn’t fair. It isn’t right. These guys got screwed.” I think we’re going to have a terrible societal problem, political problem.

How Did We Get Here?

The retirement problem is rooted in an era of different needs and circumstances.

History of the retirement age

  • Age 65 came from Social security which dates back to 1935,

which came from:

  • Railroad Retirement act in 1923,

and even before that:

  • Churchill and Chamberlain jointly put forward in the United Kingdom retirement at 70, but people thought that was unfair because the Germans used 65.

And here’s where we get to the root…

  • German’s retirement age dates back to early 1880s

Baron Von Bismark tried to unify the German municipalities via technology namely the telegraph and the railroads. The telegraph combined with the post office allowed instantaneous communication anywhere in Germany.

We’re going to bring coal and iron ore from the rural and other areas to where the steel mills are and we’re going to build steel mills and have tremendous industry. And then railroads are going to be able to bring people from the cities out to the countryside for weekends, vacations  can be normal, and we will bring from the countryside, fresh fruit, fresh vegetables, all kinds of wonderful things that for people to eat, it’s going to make everything terrific. That’s great.

But where are you going to get the workers to work on the railroad?

Offer lifetime employment.

You get them to come out of the forest because they can get lifetime employment. That’s terrific. What do you call that? That’s guaranteed. This is a commitment. It’s the honor of Germany. Okay. Let’s go.

So what happened?

Well after a couple of years there were accidents on the railroads. Trains ran into each other, people were killed. Public outrage and scrutiny.

What’s going on?

Well, let’s send a study group and find out what the heck is going with these accidents. Well, we found out what the answer is in the work. Laying tiles, lifting heavy ties, brailles, shoveling coal, all kinds of heavy work. They’re saying to the older guys in their late 50s and 60s, your too old for this kind of work. You take the easy job. You’ll be in charge of the switches.

Then what happened?

So the switches are being manned by guys in their early 60s. A beautiful summer’s day and no trains coming in for the next couple of hours, why not take a little nap? And they’re just taking a nap, forget to wake up, and the accident happened. 

The solution?

Guarantees for life. Pay them not to work. To be cost effective find the min-max where it costs not too much to solve most of the problem. And the answer was 65. Most people don’t live to 65 in those days in Germany, but those who do are really doddering, so they will only last for another couple years after 65 anyway.

An obsolete model

We have inherited and retained a retirement model that is a poor fit for our post-industrial circumstances.

    • People live longer now. The ratio of non-working to working years has increased.
    • People are able to work longer as manual labor’s share of the economy has declined.

Dealing with the Crisis

Extend your savings

  • Take social security later…instead of 62 if you wait until 70.5 you make 76% more inflation-protected for the rest of your life. If you wait, you have fewer years in retirement, so they’re willing to give you a larger amount.
  • Continue funding your 401k in your 60s. These are the easiest years to save money. So you can ramp up your savings, dump it into the 401k as fast as you could. (also there are catch-up allowances)

Do all of these things and your chances of being in serious financial trouble in retirement go from awful to not too bad. So if we act soon, we could make a big, big difference in what could otherwise be one of the worst problems our society has ever faced.

Why has this been so challenging to solve?

The big problem is nobody’s paying attention to it. It’s too late. Congress is dealing with politically urgent issues. We need to agree to raise the retirement age to 70 but it’s easy to say that when you are not a ditch digger or coal miner.

Trying Too Hard

Excerpts from Dean Williams speech: Trying Too Hard

Effort to output in investing is weakly linked compared to other endeavors

“We probably are trying too hard at what we do. More than that, no matter how hard we try, we may not be as important to the results as we’d like to think we are.”

Worthless predictions

“One of the most consuming uses of our time, in fact, has been accumulating information to help us make forecasts of all those things we think we have to predict. Where’s the evidence that it works? I’ve been looking for it. Really…The
consolation prize is pretty consoling, actually. It’s that you can be a successful investor without being a perpetual forcaster.”


“Confidence in a forecast rises with the amount of information that goes into it. But the accuracy of the forecast stays the same. And when it comes to forecasting—as opposed to doing something—a lot of expertise is no better than a little expertise. And may even be worse.”

Related: Horse bettor example

Adam Robinson on the diminishing returns of data. Confirmation bias increases your confidence without increasing your accuracy

    • Study of horse handicappers found that the accuracy of their predictions did not improve from the original 5 variables they selected from a large menu of data. As they were given more variables there confidence went up (confirmation bias effect) although their accuracy did not!
    • In addition, the handicappers with only 5 variables were well-calibrated. They were close to 2x better than chance at predicting winner 20% vs 10% and they estimated their confidence as such. When they were given more variables their accuracy remained 20% but confidence grew to 30%!

Gather the right type of info — namely how to measure value (Me: this is how I trade)

We have security analysts. We get research reports from brokers. We get forecasts about the economy, interest rates, the stock market. We process that information and act on the basis of it. For all of that to make any sense, we all have to believe we can generate information which is unknown to the market as a whole. There is an approach which is simpler and probably stands a better chance of working. Spend your time measuring value instead of generating information. Don’t forecast. Buy what is cheap today. Let other people deal with the odds against predicting the future.

Sources of edge

“There are two ways we can try to gain an edge over the market. The one that most of us choose is to try to generate superior information. To know more than anyone else. The other choice is to be better at measuring value than others and not to care very much about what other investors think they know. To hold cheaper securities by today’s standards and to let the future speak for itself.”

Growth describes a phase not a category of company

“It is generally recognized that growth stocks produce a superior risk-adjusted rate of return. However, this is only true for stocks that are expected to grow in the future, and correlations between past growth and future growth are low”. There is no such thing as a growth stock. Only passing phases of growth in almost every company’s life. Phases whose beginning and end usually appear in disguise.”

Regression to the mean

The tendency toward average profitability is a fundamental, if not the fundamental principle of competitive markets. It is an inevitable force, pushing those profits and their valuation back to the average. It can be a powerful investment tool. It can, almost by itself, select cheap portfolios and avoid expensive ones. Its plain English equivalent is that something usually happens to keep both good news and bad news from going on forever.”

Riffs on Geo-Arbitrage

The Cul-De-Sac Reaches Out From The Past

Anytime you’re on a trip with friends, someone will always whip out the Zillow app and announce “hey look what you can get for $X here!” as you all think about that fat NYC or CA state tax your paying to live in a place that makes you feel like you are the villain. Places where the cover charge for a 2nd bathroom cost $1mm (clarification: half-bath). Alas you x out the Zillow app with a resignation that whispers “in my next life”.

Well, it’s starting to look like that next life might not require incarnation.

Twitter announced all employees can work from home forever. Many desk workers are finding out they can WFH. Elon Musk and Joe Rogan are the latest Californians contemplating moves to Texas. I saw that 400k people left NYC during Covid to hunker down elsewhere. The prospect of geo-arbitrage has many excited. Bigger house, more acres, lower costs, and less traffic while maintaining a big city salary. It’s the life I envision Chicago people have but without bone-chilling winds for most of the year.

An assortment of real estate riffs as they relate to geo-arbitrage:

On the high price-tag of real estate

• Is the price expensive or “high”?

CA has a high cost of living. However, I’d push back a bit on the idea of housing being expensive versus just high. Expensive implies an expected return is very poor going forward. People like to point to near-zero cap rates to claim CA homes are overpriced. That might be true but it’s hardly obvious. Real estate is priced according to supply which is pathologically constrained in SF and LA and demand which is dictated by job prospects. Low cap rates imply price appreciation not stupidity. Higher cash-flowing properties have lower or zero expectations of appreciating because of the supply/demand outlooks. This is basic finance. You are free to have an outlook on the supply/demand factors but zoomed out expensive is expensive for a reason and cheap is cheap for a reason. Markets are smart.

• Minimum Ticket Size Frustration

Real estate prices that are high are mostly annoying because they force you to put a lot of eggs in one basket. This can be true even if the prices are high but cheap (a house on Carbon Beach for $3mm is cheap even if the price is “high”). If you are worth $1.5mm and own a house worth $1mm it’s hard to diversify. Your home is 2/3 of your assets. Many homeowners are even more concentrated than that. The high price might not be a problem from an affordability point of view but it’s a problem from a risk or portfolio point of view. So when you live in a high cost of living area, the minimum acceptable house forces you to concentrate wealth more than you’d like to.

Here’s another way to look at it. Imagine if the lowest-priced stock in the world was $50,000 a share and there’s no way to buy fractional shares. We don’t need to make a statement about whether the stock is cheap or expensive (that depends on its earnings and how many shares outstanding there are) to be frustrated that the price is high even if it’s not “expensive”. We would just be frustrated that creating a diversified portfolio would be difficult if the minimum purchase prices were so high.

It’s important to differentiate “high” from “expensive”. I’d find Bay Area prices frustratingly high even if I was bullish.

• Location, location, location

When I lived back East I remember a trader friend commenting on how NJ suburb pricing was efficiently sorted according to commute times to NYC. I grew up in NJ, this sounds right. My family settled purposefully where the Academy bus line ran. People like to gawk at what $500k can buy you in the middle of nowhere as if to say “look what you can buy if you didn’t care where you lived”. As if the “where” wasn’t THE ENTIRE BASIS OF HOW REAL ESTATE IS PRICED! It’s all about location. Not the price of wood.

• Property Taxes

Here’s another difference in CA real estate. It’s a call on inflation because Prop 13 limits how quickly your property taxes can increase. Consider a state with high property taxes like NJ that get re-assessed as your property value increases. An apples-to-apples comparison would require amortizing the higher property tax value into a mortgage. So a $1,000,000 NJ house with a 2% property tax is equivalent to about a $1,220,000 CA house (An extra 10k per year in taxes corresponds to about a $220,000 mortgage). But as a CA property appreciated the tax rate falls as a proportion of the property. (The homeowner’s gain comes at the expense of everyone else but that’s another conversation).

The great-schools premium

I live in Lafayette, CA which is similar to neighboring Walnut Creek. But Lafayette homes command maybe a $250k premium because the schools are rated 10. When you sell your home you get the premium back in the sale price. So you can think of the cost as the additional interest costs on the $250k, as opposed to the full cost of $250k. So about $11k per year at current mortgage rates. The more kids you have the more value you get from the premium. Pretty obvious.

Beware, if the schools get worse the premium can erode by the time you sell.

How to spot a contractor shortage

If there is a large spread between newly finished homes and older homes of similar size, location and acreage you are looking at a market with a contractor shortage. That market is daring you to buy a fixer. It could also be a market like SF with a draconian permit process. Same thing. Go ahead and try to fix up a place. If you are especially handy and connected these make for good markets to be a flipper.

Migration patterns and doom loops

While I don’t have a high conviction view on CA real estate I can envision a nightmare scenario. Businesses leaving the state followed by well-paid employees. A shrinking tax base is asked to pay for the state’s bloat, leading to more fed up residents leaving, and so on. Doom loop. Imagine CA  with even higher income taxes or perhaps less services like police and fire (services that CA desperately needs even more of as things stand today). In such a downward spiral can you imagine state exit taxes? The US has one if you try to renounce your citizenship. Desperate times will call for desperate measures.

For all the Medium posts by techies thrilled to be leaving for Austin or Seattle, CA still sees a net inflow of people. But this will be an important flow to keep an eye on. Will the remote work momentum free people from the Bay or will the outperformance of tech during covid indicate even more magnetism to Silicon Valley?

For more on CA real estate and economics, I encourage you to check Byrne Hobart’s amazing post, Peak California (Link)

Shorting Bimodal Stocks


My friend and former colleague Jason took exception to the viral tweet I referenced last week about how shorting a bi-modal company is like an option. Not because it isn’t but because all equity is an option.

In short, the entire viral tweet is a tautology.

Jason joined Twitter to respond to it. Jason’s gripe was that all equity is effectively a call option struck at zero (you can argue that a positive book value sets the call strike higher but it doesn’t materially change the point).

Jason argues that if the viral tweet pretends it is saying anything beyond “being short stock is like being short an option”, you are mistaken. There are several reasons why, and I’ll use my intro to twitter to go thru them…(Jason’s reply)

Where do I stand?

I liked the original @HedgeDirty tweet because I am just a fan of presenting ideas in different ways. The flaws in the tweet are real and technical but it conveyed a correct impression even if it got there incorrectly. To Jason and @HedgeDirty the riskiness of shorting a bimodal company whose equity is probably worthless is obvious. I appreciated the narrative style which reinforced that point. Even if it’s self-evident to anyone who shorts stocks.

But I also see learning opportunities in deconstructing the flaws in the tweet.

You can read Jason’s reply to see the flaws he found. Especially resonant was the observation that the 0 strike call is a 100 delta call. It has no gamma or theta. The original tweet claimed otherwise.

What would I focus on?

1) Let’s do a little math to find the annualized vol. The first thing to note is how the bi-modality creates a very volatile stock. 125% per year standard deviation.

(Warning: it doesn’t make much sense to use standard deviation based on a normal looking return when we already stated it was bi-modal but the obvious takeaway is “damn, this thing is going to make large moves in return space.” If you know nothing else, you know going short this thing is like barebacking a bucking bull. Size appropriately.)

2) My biggest gripe with the example. The stock doesn’t trade for $185 because it’s hard to be short stocks.

If the stock is trading for $185 the market is implying different distribution. Either the 80% and 20% are not true, the stock’s upside is more than $250 (1.25B EV), or the recovery value is much higher.

It’s not trading for $185 with a theta that pushes towards zero or anything like that. We don’t need to invoke Greeks for an alleged $50 fair stock trading for $185. If it’s trading $185 the market doesn’t agree with the assumptions that compute a $50 fair value. Full stop.


Despite the flaws, I enjoyed the original tweet. You can read plenty of @HedgeDirty threads and see it’s a good account to follow. For example, the Why You Should Never Hold Levered ETFthread is great. I’ve written about the brutal math of levered, especially inverse levered funds before.

Shorting bi-modally distributed stocks is hairy. If it feels like it’s being short an option it’s only because the stock is volatile, equity is an option, and being short options is volatile. Nobody shorting stocks should have needed an education from that tweet. For everyone else, they should be aware of errors in mapping it to option theory even if I think overall the thread was net positively educational.

As for Jason who has been trading options since the late 90s, I imagine he felt that @HedgeDirty borrowed his bass and played it with a pick. Only Paul McCartney gets away with that.

Money and Inflation Musings

This is not my favorite topic but it’s impossible to watch the stimulus plans since 2008 and not think about the meaning of money and credit. This tweet-gif reminds us of what is happening. And this tweet-gif will set the appropriate mood before I continue. Ok, here we go. A friend sent me this post because it was formative for him in how to think about the nature of money from first principles. I agree it’s fantastic.

Enjoy Moneyness. (Link) (With my highlights)

Key points that resonated:

  • The distinction between credit (economy’s money) vs monetary base (bank’s money which credit references) and how expanding the base dilutes the credit which rests on it
  • Money is not wealth
  • The difference between accounting plane vs physical plane and his identity relating deficits to balance of payments
  • Gold is not money. It’s a tradeable asset that references money.
  • Hyperinflation is the catch-up period when prices zoom up to match the expansion of the monetary base which it typically lags in the beginning. Requires feedback loop of lack of confidence leading to expanding base more


Inflation Hedges 

 I have been thinking about hedges to inflation. Not because I expect it but because if it happens it will be painful. Same reason I buy homeowner’s insurance. I don’t expect a fire. In thinking about inflation I maintain skepticism of traditional hedges. Not for a specific reason but just because I’m paranoid about consensus solutions to any macro voodoo.

Let me give 2 examples of why I’m skeptical.

1. Elliott Management’s Q1 letter wonders about the conventional wisdom of commercial real estate being an inflation hedge. Yes, it’s a real asset. Yet they write:

Take real estate, for example. Of course, it is “real,” and you might think that it is a slam-dunk to preserve value in a serious inflation. But commercial real estate is a peculiar asset. It looks real because kicking it can break your toes, but it is generally highly leveraged and depends upon the relationship between rents and costs. If there are rent controls or moratoria, formal or forced by circumstances, and no controls on costs, commercial real estate can produce rapid insolvencies. A little thought will reveal many more examples of the complexities involved in a period of monetary destruction such as the one that is possible in the near future.

2. Since the Fed can control the short end of the yield curve it’s reasonable to think a curve steepener trade via options would be an effective inflation hedge under the dual premise that the Fed will be slow to raise rates and they can’t control the long rate anyway. But then I learned about the Fed Yield Curve Control Policy. That’s a mouthful. From an era without acronyms like TARP. I humbly offer FUCC. Not so much because it fits the words that well, but because what it would have done to a curve steepener.

You see, shortly after the US entered WWII, the Fed pinned the long rate at 2.5% to keep borrowing costs low. To an option trader, this means vol goes to zero. You think you have the right hedge on but forgot to read the rule that says the Fed can change the rules. Nothing like getting a prediction right and losing a ton of money on how it plays out.

Why You Don’t Get Paid For Diversifiable Risks?

Finance theory dictates that an investor does not get paid for “diversifiable” risk. You do not get paid for idiosyncratic risk, only systematic risk. I have not formally studied the CAPM pricing model and prefer more intuitive explanations anyway. So I went to #fintwit:

Twitter Roundup

You can click on the Tweet to see the responses. Here’s a roundup of the Tweets that most informed my understanding.

Here’s @Value_Quant:

Here’s @spreekaway:

My Take

The responses helped me consolidate my own non-technical understanding. I’ll walk through my own take and how I’ve seen the idea that you do not get paid for diversifiable risks in practice.

A Bidding Game

First, a quick game derived from this Bogleheads thread:

Imagine there are 2 boxes. One of them holds 100k and one is empty.

You have a net worth of $50,000. Multiple people are allowed to bid and none of them has more than $50,000.

How much would you bid for each box?

This is an extremely risky bet. Perhaps you bid $40,000 for one of the boxes, for an implied 25% return (you are buying an asset worth $50k for $40k)?

In this narrow example, a box will trade for a healthy discount to its fair value but at the price that the least risk-averse investor is willing to pay.

What happens if everyone in the world is invited to bid?

Jeff Bezos will bid $99,999.99 to buy both boxes and have a guaranteed profit. Even if he were only allowed to bid on a single box, he would bid $49,999.99 since he still has positive expected value and the potential loss is an invisibly small percentage of his net worth.

This Is A Good Model For How Markets Actually Work

Think of how market-making firms profit.

  • They are willing to trade for mere basis points of edge. Often it’s simply rebates.
  • They make thousands of trades a day that get thrown into a giant pool of positions.
  • The systematic risk or beta is hedged out so that they are left with a diversified, offsetting portfolio of idiosyncratic risks. Over time the law of large numbers crystallizes the expected edge into hard p/l dollars.
  • This model validates the idea that providing liquidity is a long-term positive expectancy.

The business model rests on two concepts.

  1. Bet an appropriate fraction of the bankroll for a given amount of edge.
  2. Diversification

It’s just like the casino business. The house doesn’t care what happens on any individual bet as long as the bet is a small fraction of its bankroll.

Competition Drives Efficiency

Competitive equilibrium will mean that the casinos who can bid the highest for the “customer” is the house that can:

a) source the most uncorrelated offsets to the wager


b) has the biggest bankroll

In the trading business, condition A is satisfied by the market makers with the best data/analytics and “see the most flow”. A firm entrenched in both equity markets and futures markets with licenses from both the SEC and CFTC is going to be more efficient at laying off the risks it acquires from serving tourists regardless of the venue they choose to play in.

A and B will create a virtuous loop. The best players will build larger bankrolls which allow them to outbid competitors further which earns them first look at the flow which improves their models and so forth.

Note the role of bankroll and diversification in the following examples:


Banks are able to trade options outside the bid/ask when they do bi-lateral deals with energy companies. The resources they pour into doing custom deals and loans earns them the extra spread. They create a high touch VIP area with fatter margins. They do this by being horizontally integrated across credit and derivatives markets 1


Bookies are able to take the other side of Mayweather’s sports bets because they have offsetting flow (phrased differently: they can make 100% negative correlated bets). They can take their vig between many offsetting bets providing an ample cushion to justify sweating the residual position they cannot expect to hedge…the actual outcome of the game.

Life Insurance

By pooling risk, insurance companies can underwrite policies at an affordable premium that still leave room for an actuarial profit. If you, as an individual, wanted to write an insurance policy for your neighbor you would not be able to offer a price that was simultaneously affordable to your neighbor and compensated you enough to tolerate the risk of their premature death. And even if you had enough money to insure a hundred people, you’d need the infrastructure to source them from multiple geographies so as to minimize the correlation between the deaths in the event of a natural disaster, terrorist attack, localized pandemic, or asteroid.

Stock Investing

You are faced with 2 stocks with the following attributes.

The portfolio math to construct the chart is described in my earlier post.

Interesting things to note:

  • Stock B is more volatile and a much worse risk/reward as proxied by its Sharpe ratio.
  • At positive correlations, the optimal weighting is to load up on Stock A. Not surprising. In fact, the more of A you have, the better the portfolio Sharpe ratio is.
  • But when correlations flip negative, optimal weights are now recommending significant weightings of Stock B.
  • As I tinkered with combinations of assets I found that assets with low or even negative Sharpe ratios improve a portfolio if they are both negatively correlated and highly volatile.

The insight for me is that negative correlations make assets outstanding diversifiers even if they have negative expected returns. And if the asset has a negative correlation, high volatility can even be attractive.

This is one more reason to suspect that volatile assets can be justifiably overpriced and not a source of excess return premium. In fact, they are valuable despite their unattractive stand-alone attributes.


In an efficient market, prices setters:

  • Are maximally-funded or have relatively low costs of capital
  • Have broadest perspective/market access

The emergent properties of markets will lead to idiosyncratic risks being held by the player most optimized to hold it. If the risk is borne by the most efficient holder, it is by definition priced so that there is no portfolio to which the risk is more valuable. What remains are systematic risks that no entity has discovered a hedge for. Those are the premiums you are allowed to pick up.

When there is no diversification left to hide behind, the systematic risk can only be compensated by a pure risk premium. At the end of the day, there may be no hedge for getting 2 to 1 on a coin flip. The long-run is going to have to do.

Further reading:

Quant and author Aaron Brown explains on Quora (Link)

My brief post on market efficiency: Dinosaur Markets (Link)

On Trading and Aptitude

I know there are many younger readers of this letter. I’m not a quant. I took Calculus BC in HS and one stats class in college (although I do want to take more math online — I’m not advocating ignorance). Many of you are very strong in math. There’s a perception that options are about graduate degree stochastic calculus and differential equations. There are research-oriented jobs for which this is true. These jobs require raw mental horsepower and lots of training to tackle technical problems.

On the trading side, don’t get discouraged by academic notation in option papers. Here reasoning and numeracy are the pen and hammer. The tools of the trade. I should add for college students looking to get into trading coding is now table stakes. You need to have something to give in exchange for learning. The business is harder than ever, fetching lunch is not enough. (I know what you are thinking. Every generation in trading always thinks “if I was just born 10 years earlier it would have been so much easier to rake in the bucks”. It’s as stupid as a 300 lb lineman who wishes he could have come up in a time when linemen only weighed 250. He’s committing a time travel fallacy where he gets to go to the past with knowledge of recent innovations in diet, drugs, and exercise). Continuing on. The ability to code is also self-reliance. My own ability is very limited and I’m sure a junior will look at me the way I used to look at older traders who struggled with Excel. Circle of life.

Perhaps more so than the pure quant roles, in trading there’s a lot of room for grit. The analogy is as simple as the fact that most poker players are not quants, but there’s no doubting their discipline, endurance, ability to focus, number-sense, and logic skills. Your liberal arts (and no economics and business degrees are not science) degree is not a life sentence in ops.

(To be clear, this is not an affront to ops…my wife went that way. In fact, there is a whole conversation to be had about why a career in ops can be a more lucrative route. But it’s a parallel route and if a person wants to trade and take risk, anything else will feel like they failed even if objective standards might say otherwise).

On the other hand, tying this all back to the Parable of the Talents essay — trading is not for everyone. It’s not even for many. You can do anything, except for what you can’t do.