Notes on Trading Volatility: Correlation, Term Structure, and Skew

Trading Volatility: Correlation, Term Structure, and Skew
Colin Bennett

The book is a broad reference on basic option theory, dispersion, and exotic options. It includes practical insight into managing a hedged book with a focus on correlation, term structure, and skew.

In addition its appendix includes the following topics and more:

  • a taxonomy of historical vol computations including and how they rank on “bias” and “efficiency”
  • shadow greeks
  • cap structure arbitrage theory

It’s an outstanding reference so I took notes. For public sharing I re-factored them by topic and tied some back to my own investment writing.

You can find these edited notes in my public Notion page. (Link)

Vol Premium [Partial] Justification

I’m about halfway through Colin Bennet’s terrific book Trading Volatility, Correlation, Term Structure and Skew (pdf).

Bennet is (or was) the Head of Quantitative Derivatives Strategy at Santandar. The book sits in a very sweet spot. It has lots of practical insights into managing vol portfolios and the mechanics of both vanilla and exotic options, var, and vol swaps. I’ll likely do a full post summarizing the takeaways I appreciated most, but in the meantime I thought to share this blurb about the oft-referred VRP (vol risk premium).

Just because implied vols trade over realized does not mean they are mispriced:

[To be fair the author asserts they still are. More importantly, you should read ch. 3 of the book to see how he decomposes the premium to systematic risk and pure vol demand premia.]

I wrote something similar a few weeks back:

Index options should be “overpriced”.

The question is how much premium do they deserve. If stocks warrant a risk premium over the RFR it’s because their systematic risk cannot be hedged. Index options must conceptually inherit this premium otherwise there would an arb in portfolio allocation.

An index option, held delta neutral, gets paid as correlations in the marketplace increase. It literally makes money when systematic risk embodies.

A standard for deciding if puts are expensive: Its price should have enough premium in it that by buying a put, if delta hedged, that you would actually have basis risk. In other words, it’s premium should make it uncertain that you would actually make money in a sell-off. If your argument is that it’s expensive in a vacuum (perhaps as a comparison to realized vol) then what if it was only 1% premium to realized? That sounds like a bargain for something that hedges the risk that, like, the whole world has. This isn’t news to most investors or anyone who understands portfolio construction and the beauty of neg correlations. It’s just another instance of my sun/rain example.

Let Chess Help Kids

Two years ago my wife Yinh started her podcast Growth From Failure. Her second guest was Berkeley Chess School founder Elizabeth Shaughnessy. It is one of my favorite interviews ever. We have referenced wisdom from it on many occasions since. Yinh texts with her from time to time and always comes away so invigorated. This past week I was stoked to meet the 83-years-young chess whiz. My expectations were high.

It turns out I still underestimated how special she is.

We went to lunch at Cafenated Coffee in Berkeley and 5 minutes into the conversation I immediately regretted not having a notebook. Elizabeth is bursting with passion for her mission and practical insights for teaching, life, and of course chess.

I did a full write-up that I’d love for you to check out: Lunch With The Amazing Founder Of Berkeley Chess: Elizabeth Shaughnessy (Link)

Here I’ll give a brief version of why it’s so special but the full article gets into ideas you can literally apply today in your life.

The Mission of Berkeley Chess School

BCS is a true Robinhood organization. As a non-profit, they are funded by donations and fees they receive for after-school programs around the Bay Area and private lessons (our son and his friends do group private lessons with BCS instructors). This supports their mission to provide free or low-cost chess instruction to students at poorly sourced Title 1 schools. In the past 40 years, BCS has taught over 250,000 kids.

But when you sit with Elizabeth you realize this is about far more than a game. Today, with Covid decimating enrollment, the school has re-purposed its building to teach disadvantaged kids. These are kids from low-income sections of Oakland, Richmond and Berkeley who are struggling with distance learning. These kids have no internet or computers at home. Without intervention, these kids, already struggling academically before the pandemic hit, may suffer an irreparable learning loss that could affect their health and financial well-being far into their adult lives.

From her experience, Elizabeth is convinced there is hope.

How Does Chess Help?

As a fan of games and games in learning, I like to believe that the skills acquired in play “transfer” to other domains. This is something I’ve wondered aloud about on Twitter. It is rooted in causality. I specifically asked Elizabeth if she thought a joy of chess was simply a symptom of a more general aptitude or if chess was imparting a more generalized skill that could be applied to other fields.

Elizabeth is a big believer that there is transference.

  • Chess asks kids to slow down and be methodical.

    Count how many pieces are threatening your pieces. Do this for every piece, on every turn, to find the strengths and weaknesses on the board.  Then look at all the checks you can deliver, then the captures, then the attacks. When all this is done, then make your move.

  • Consequences matter and compound.

    Chess teaches you that consequences matter. Make a rash move and you get penalized by your opponent.  Mistakes are expensive in chess and life. What scenarios can unfold if you always skip math class? How will this serve your long term objective of being a Wall Street wizard if you’re unable to calculate risk or odds?

  • Chess sharpens your focus.

    She has repeatedly seen firsthand the power of chess to harness kids’ attention. It’s an effective tool to settle kids so they can get into a better headspace for learning. Kids who start out resistant often do not want to go home after school.

Chess can show kids they are smart. It teaches them to believe in their own abilities. Many of the kids BCS teaches face long odds in life but chess can offer lessons in foresight, creativity, problem solving, and self-control.

Helping BCS

Children heatseek that which provides immediate benefit or stimulation. BCS has figured out how to stimulate children that have been written off. Any witness to that transformation will see one thing — the longest lever we have as a society to improve a child’s well-being today and into adulthood. When I listen to Elizabeth, I can feel what she has seen.

If you are looking for high impact ways to give back I encourage you to check out my full post or if you prefer you can simply head over to BCS site to learn more. (Berkeley Chess School)

Tips and Insights

Elizabeth cannot help but spill insights all over the place when talking. Check out the full post to get:

  • practical tips for learning chess today
  • how to play chess with children and why
  • insights into teaching girls specifically
  • the role of genius
  • the pros and cons of being a good loser

And if you are wondering her view on Netflix’s Queen’s Gambit — she thought it was too long but the beginning and end were fantastic. Ultimately, she thought it deserved high marks for making chess so compelling.

Wrapping Up

My 7-yr-old has been taking lessons with BCS intermittently since he was 5. Even our copycat 4-yr-old is into it. It took him all week of multiple games per night to learn how the knight (he’ll correct you if you call it a “horse”) moves. I better start learning more, they are hot on my heels. I’m MoontowerMeta on if you want to add me. I’m a beginner. I’m still beating the 7-yr-old but it’s getting tougher.

This is one of Elizabeth’s sons teaching chess at our pod a few weeks ago.

The ‘Volatility Is A Risk’ Strawman

In my short post Is Volatility A Risk?, I urged that any definition of risk:

should be evaluated by its usefulness. Any single definition is incomplete and insufficient for making an investment decision.

Here’s a specific case.

  • How The Sharpe Ratio Broke Investors’ Brains (Link)
    Institutional Investor

    This is a good overview the shortcomings of Sharpe ratio, most of which should be well-understood by anyone who has computed a standard deviation.

    I’ll expand on some of the less obvious points:

    • If you annualize Sharpes from monthlies you risk overstating it if the strategy returns are serially correlated.

      Why? Because you are understating the vol which you can no longer assume scales at the square root of time. This is a complicated issue because auto-correlation, while easy to compute, is itself subject to variation.

    • Pardon my yawn, but apparently option sellers game the Sharpe ratio fetish by selling nickels in front of a steam roller. If the image of straw allocators investing on the basis of a single measure keeps you up at night then, sure, sound the alarm. Skewness can hide within vol.

      A quick demo:

      a) Bet $1 on a fair coin
      b) Bet $.33 on heads on a coin that costs 9-1 if tails but has 90% of coming up heads (still a fair coin).

      These bets have the same vol ($.33 creates risk or vol parity weighting) but the payoff shape is materially different.

    • There are popular alt ratios like Sortino, Calmar, and Omega which try to correct for skewness by penalizing drawdowns and giving hall passes to upside volatility. These are not panaceas since they correlate strongly to Sharpes. This reinforces the idea that you can’t compress the nature of any strategy into a single number. (I feel the tendency to pretend that anybody evaluates investments so naively is a straw man drubbing of allocators signaling no deeper handle of the problem than an influencer who read Taleb on a cross-country flight. Like do you even know an allocator?)
    • A point the article didn’t mention: you can have high Sharpe strategies that cannot generate high returns. Like investing in T-bills. If the cost of levering the strategies is prohibitive then Sharpe would yet again not be the only number you can look at.

Ok, I’m done suspending my disbelief that anyone uses a a single metric in isolation to decide anything of importance. The post is worthy reading for new investors who just discovered Sharpe before they run out and impale themselves on it. I hope my additions made it a touch more interesting for the initiated.

California Love [Hate?]

Rewind nearly 3 months to August 20.

It was handy that I had a KN95 mask lying around since it was 2020. I needed it that day and for the next several weeks. We were trapped inside due to smoke and the AQI in my garage office would make your eyes tear.

This was the 3rd year in row that you needed to set your browser home page to if you live in CA. And it wasn’t even the heart of fire season yet, when we could look forward to planned PGE blackouts. Grrr. We already live on a faultline, I didn’t need more Old Testament-style risks on my land.

Sprinkle in the talk of yet higher CA taxes (the state pension situation is not improving despite being the epicenter of a 10 year bull market), word that some major insurance  companies wouldn’t underwrite home policies in the Bay Area, and the fact that WFH meant we could live anywhere and I felt a bit of an emotional tidal wave…

Why do I live here?!

Before I take one step forward let me caveat this. I love living in CA. I won’t list the reasons but I often liken it to a hot girl — you wouldn’t tolerate her behavior if she was less than a 10.

(The less crass version is a paradox of plenty — companies in super profitable industries can afford to be poorly run. Oil-rich nations use resources as crutch to mask imbalances. Byrne Hobart likes to say “never lend money to a country with good cuisine”, No matter how bad CA treats you, San Diego is always 72 degrees and sunny.)

So if I’m not in some smoky, reactionary mood I get why we live here. If it was any less awesome, we’d be long gone. Everyone I know who lives here is aware of the downsides. Every Californian has a wandering eye. Gee, that Incline Village sure has a sexy zero state tax.

But the wandering mood lingered longer that morning. That my entire team just high-tailed it to Denver didn’t help. At dinner that evening, I raised my concerns to Yinh. Her family is here, her mother lives with us. Her perspective always talks me off the ledge.

Not this time. She not only agreed, but having a hair-trigger bias towards action, Yinh texted a realtor friend to come by the next day.

A month later our house was on the market. It sold in a weekend (the city exodus meant a feeding frenzy in the burbs where prices ripped 10-15% in 3 months. A pandemic-spread-via-proximity broke the home price to GDP correlation).

So last week I didn’t write because we were busy moving to a rental in the same town. We don’t know where we will end up. Between distance learning, WFH, and the chaos of 2020 we decided we wanted max flexibility and optionality. The inertia of our script was stirred this year. We didn’t want to waste the chance to embrace the chaos. The chance to reconsider and test our values.

Owning a home requires mindshare and can anchor decision-making. I won’t say that’s true for everyone, but it did for us. We spent a lot of money and effort improving our home over the past 5 years. And yet, I couldn’t manufacture a shred of emotion when we sold it. No nostalgia. Staring at the home my second son was born in, I expected to feel something. Looking at the yard where we threw big summer birthday pool parties or watched Zak learned to ride a bike, I expected more. But alas there was nothing. When I canceled the home policy and signed up for renter’s insurance they asked me how much coverage I wanted. And it occurred to me, I don’t care. There’s almost nothing I own that I actually care about or couldn’t just replace. Everything that matters is either made of carbon or megabytes.

We moved from a cool, modern ranch house to a Boogie Nights-esque 60s style home that has not been updated.  Mirrored walls, black toilet, shag carpeted bathrooms, no closet space and a setup that required us to have the kids share the master while Yinh and I use a small-sinked, hallway bathroom that reminds us of the modest homes we grew up in. We haven’t seen a shower curtain in 15 years. On paper, a major downgrade.

But the ranch has a cabin feel bookended by fireplaces, bordered by old beautiful redwoods and Japanese Maples. It’s so California you want to take a psychedelic dip in the hot tub while White Rabbit plays in the background. I actually love it. And I really do love CA.

I guess I just have commitment issues.

Working for Free

This is the first year in nearly 20 that I haven’t played fantasy football. I blame Covid but it’s actually a blessing. [If you work in asset management I suggest you look away and skip to the next paragraph]. The bots level the playing field so any extra alpha would be work that pays like $2 an hour.

Despite FF being dead to me, I ended up reading every post by FantasyLabs co-founder Jonathan Bales. There’s no point in describing him. His writing style is addictingly personal. It’s smart stuff that goes down like a pint of of Salt & Straw.

I’ll get you started with one I have thought about from similar angles. [Special thanks to Adam who put Bales on my radar.]

Should You Work for Free? (link)
Sometimes, working for free is the most valuable thing you can do. Here’s how to figure out when it’s smart.

Why did I pick this essay in particular?

I tend to think “free” is misunderstood because free refers to price. Price is a narrow concept which gets all the attention. It’s the most legible. It’s easy to label. Really big companies love price. If they can get you to focus myopically on that single number they can ruthlessly optimize for delivering you a solution that they are best suited to supply. Big food wins when they commoditize. If you accept subconsciously all beef is the same regardless of where it came from then price is the only thing left to compete on.

So “free” is an idea that focuses you on a single aspect of a transaction. It’s short-term oriented. Free is not free if we imagine the repetition of that transaction. It has downstream effects that you are aggressively discounting in the present. (This stands in very sharp relief to what is happening in asset markets today — cash flows ever further out in the future are being inhaled into present prices).

Consider an obvious example: health. The cumulative effect of low quality food is an invisible debt that accumulates. Subsequent chronic illness is the equivalent of a debt service that crowds out future growth. Acute illness is a bankruptcy re-structuring (if you don’t test the limits of financial metaphors, are you even trying?).

A personal example is this newsletter. I do this for less than free. It takes about 7 hours a week to deliver these Pulitzer masterpieces every Sunday. That the rewards are invisible doesn’t make them any less real.

In business, I always enjoy the Costco example. Charlie Munger has written:

“When other companies find ways to save money, they turn it into profit. [Costco] passes it on to customers. It’s almost a religious duty. [They] sacrifice short-term profits for long-term success”.

It’s not as hokey as it sounds. Think of it this way. They are hiding profits in the customer’s own pockets. They will be return customers. That profit is hidden from competitors’ wandering eyes and the IRS. The strategy commits Costco to keeping the customers happy because the profit is realized over the long-term. It’s simple but requires rare discipline.

Bales on time horizon:

  • Reading, sleeping well, and working out. All stupid uses of time if your goal is to optimize your day…but if the question is “How can I create the most value for myself (happiness, money, however you want to define ‘value’) in, say, 2024, then you should probably create a long-term foundation for success, with reading, working out, and getting rest being among the most +EV things you could possibly do.

    Which cuts to the heart of the matter for working for a low or zero price.

    We’re all trying to strike a balance between maximizing money/value/happiness right now versus creating a sustainable foundation for long-term value generation. At one end, working for free makes no sense. At the other end, you should work for free all the time because it provides value to the maximum number of people.

Another aspect to delivering more than you get back to today is how goodwill accrues to you. It’s equity that cannot be taken away from you. Just because it doesn’t show up on the balance sheet doesn’t mean it’s not there. (Fans of gangster movies will note the more nefarious version — an asset you might call “favor receivables”).

Bales on entrepreneur’s mindset:

  • Do you know who works for free all the time? Entrepreneurs. Do you know who never works for free and gets paid for every hour they put in? Employees.

    Being an employee can be great. You can typically work only during set hours, get weekends off, don’t need to worry about problems that arise outside your expertise, etc.

    But, when you work for someone else, you (mathematically, at least in an efficient market) must take less money than you’re worth. And usually, you don’t get to participate in the upside if you (and your company) do an awesome job.

    To be clear, I’m talking about the typical mindset (and pay) of your average employee/entrepreneur; you can be an employee with an entrepreneur’s mindset, or vice versa. Some business owners are total shmucks and would be better off working for someone else. Most, probably. But many people are sharp enough to absolutely crush it on their own and just aren’t going out and doing it. Today, and tomorrow, more than ever, it’s easy to go get it for yourself.

If you actually believe in yourself you should imprint this point on your mind:

When you start to think about the value you can generate for yourself—again it can be happiness, freedom, money, whatever you want—you realize that getting paid for your time is -EV if you have awesome skills and can better people’s lives.

Bales is a realist. You should not let yourself be exploited.

  • You must be able to sort out those who are just looking for interns they never plan to hire from those from whom, at a minimum, you can really learn. And, to be clear, you should not be working for free for most people.

    And this next point is a brilliant tactic for reducing the adverse selection that would come from being chosen to work for free.

    It all comes back to providing value to the right people, meaning you’re the person who identifies them, and not vice versa. If someone asks you to do work for free, that might not be the best opportunity; your job is to spot the situation that’s going to improve your upside the most long-term—likely with someone who isn’t even necessarily looking for help—and then convince them you can improve their life…by actually doing it.

In sum, the article says:

  • Doing things that are positive EV in the long-run requires indexing your options by more criteria than just price.
  • Although working for free can be a hugely positive risk/reward, it’s still risky. Be discerning who you work for. (I’d lean heavily to those who have a lot to teach me, especially if the lessons are scarce in the wild).

I think you will enjoy the whole piece and think many of you will move right into his other posts. (Link)


40% Of Your Assets In…OTM Options?

The treasury issues EE Bonds that yield 3.5% guaranteed if held for 20 years. In the interim, they are totally illiquid.

Meanwhile 20-year US treasuries yield 1.5% if held to maturity. However these are liquid, so you can sell them anytime.

Is it worth giving up 2% per year for the liquidity?

In How Much Extra Return Should You Demand For Illiquidity I explore this question and what it depends on. There are other examples of how other investments are priced based on their liquidity. I provide 2 frameworks to consider as you try to price liquidity.
Applying the logic to the current environment
Putting your money in a lockbox for 20 years to earn 3.5% per year might sound attractive if you decide liquidity isn’t worth much to you. Especially when the equivalent liquid treasury only yields 1.5%.

But as @econompic shows, there is no period in the last 75 years that if you looked back 20 years at stocks did you only earn 3.5% per year.

It’s reasonable to point out that stocks are not bonds so the comparison is made of straw. But the counter to the counter is that if you are putting the money in a box and throwing away the key for 20 years, then the comparison is not crazy. A significant benefit of bonds comes from the ability to rebalance. But with a 3.5% bond trapped in a box you lose the option to rebalance.

So we are stuck with that 1.5% bond. It’s nearly cash. Let’s not sugarcoat this. Bonds at current pricing are just an option on deflation. And the premium is all extrinsic value since they have negative real returns. Since they are now an option that you pay for in real terms, they are no longer an investment but an insurance policy. Once you see it like that, you have to wonder if their appropriate allocation size should be more commensurate with that line of thinking. Would you put 40% of your portfolio in option hedges? I didn’t think so.

Is anyone still putting 40% of their portfolio in bonds? Asking for an industry.

Snowflakes vs Lemmings

My professional training and experience give me tremendous respect for the “wisdom of crowds”. In a prior post, Dinosaur Markets, I defend using the “outside view” as a surrogate for your own.

Obviously you don’t want to follow this logic right off a pixelated cliff. The tension is in knowing when the consensus is wrong.

You can fail in 2 ways.

Snowflake Error: contrarian when you shouldn’t be
Lemming Error: consensus when you shouldn’t be

If the consensus is correct most of the time, you’ll make Snowflake errors more often than Lemming errors. This is counterbalanced by the fact that Lemming errors are more costly.

When I hear “first principles” thinking my mind sees snowfall. Lots of snowflakes. Since I’m a default efficient market mindset, my first instinct is Chesterton’s Fence:

Don’t ever take a fence down until you know the reason why it was put up.

If you think something that exists is wrong, you need to be able to explain why the conditions for its existence are no longer valid. This idea is often invoked when people describe the difference between rigid conservatives and eager reformers. Well, ok. I suppose there will always be some people who put ketchup on steak.

The idea is more powerful than that. It’s more broadly about epistemic humility. When someone thinks they are “thinking from first principles”, my instinct is to ask “are you the first person to think from first principles on this problem?” It’s firsts all the way down. Chesterton’s fence is more generally about who has the burden of proof. A current example: Does the urbanist need to prove that cities will continue to be humans’ preferred means to self-organize or is it the dissenter’s duty to show otherwise?

In sum, nobody’s life rule is “copy what other people do”. But the opposite binary, “always think from first principles”, which is somehow more acceptable to say, is just as ridiculous. It’s almost like the phrase “reinvent the wheel” hired a PR firm.

The subject of when you should actually think from first principles vs listen to markets is obviously complicated. If you are interested in when to diverge from consensus, then check out the free book Inadequate Equilibria: Where and How Civilizations Get Stuck by Eliezer Yudkowsky. I have plugged it before and plan to re-read it soon enough. (Link)

How Much Extra Return Should You Demand For Illiquidity?

In some corners of asset management, marketers are offering to lock up your money to “save you from yourself”. These Samaritans don’t want you to succumb to behavioral biases and overtrading. I’m fine if private funds want to argue that the best opportunities are illiquid (I don’t have to believe them but I’m ok with them making this argument). But don’t tell me your lockups are doing me a favor. Don’t act like you shouldn’t be giving me a discount for tying up my money.

Should You Care About Liquidity Even If You Are Talking About Money You Don’t Need For A Long Time

Perhaps you are one of these people who doesn’t want to put their hands on the wheel. You are self-aware enough to know you’d chop yourself to pieces in the market. First, I commend this level of self-awareness but you still deserve a discount all else equal (I know it never is).


First of all, your needs or preferences don’t set the marginal price. Don’t be so vain, not everything is about you 🎶. The price of illiquid investments are set by those who do care about liquidity even if you don’t. You inherit that discount the same way you get power windows for free nowadays. You get that even if you think you’d be better off with the physical exercise of cranking your own windows.

The second reason why illiquidity deserves a discount or liquidity deserves a premium is liquidity itself is an option. Any argument that says liquidity is bad, whether for behavioral or any other reason, needs to address the value of that option.

Uh oh. Are we going to need to price some abstract option?

Fortunately, no. We can build the intuition from option theory to demonstrate that liquidity is not only valuable, but quantifiably so. It gets better. We can also point to another approach that demonstrates the measurable value of liquidity without pricing options. The best part is its driven by the same underlying logic that makes the option approach work.

Before we start thinking about the value of liquidity, let me start with why I started thinking about this question.

3 things I’ve come across recently have made wonder about how big a liquidity premium is warranted.

1. “Networks of confidence

I was listening to @Jesse_Livermore on the Invest Like The Best podcast.

I’ll paraphrase:

High valuations are increasingly dependent on liquidity or what he terms “networks of confidence”. He refers back to prior work that shows how you’d need a healthy discount to intrinsic to buy an asset you couldn’t sell.

On Twitter, he later posed a cool thought experiment where you price an asset that has no fundamental risk but unpredictable, perhaps zero liquidity in the future. The only thing you can rely on is its unchanging (even in real terms) dividend.

Look through the thread and you will not be able to unsee how little thought we put into pricing liquidity.

2. Illiquid vs Liquid Bonds

I came across this article about US Government EE bonds which showed how a feature of these treasury-issued bonds is, if held for 20 years, you are guaranteed 2x your money back. That means your worst case is you earn a 3.5% CAGR nominally. The catch: they don’t trade in an open market. Compare that to liquid 20 year US treasuries at about 1.5% yield. [Let’s set aside the fact that one can only buy $10k worth of EE bonds per year.]

There is a 2% per year difference in yield if you held both to maturity! That sounds big. But is it? This comparison is a perfect example of why we’d really like to be able to quantify the value of liquidity.

3. Insurance products

I know someone who is considering jamming a bunch of savings into an insurance product that “guarantees” around 3.5% per year if held for about 25 years. I don’t want to turn this into a post about insurance, I have enough brain damage from the email threads I’m privy to. The larger point is there are products where you can earn more yield for sacrificing liquidity even after after adjusting for the credit risk and the actuarial features of these products. (PSA: If you are interested in getting technical about insurance my buddy @RajivRebello covers it from the institutional side. In other words, he understands the math and levers in ways you cannot pry out of retail brokers.)

Hopefully I have convinced you that a) liquidity is worth a premium and b) we are faced with real-life comparisons that beg us to price it.

How big should the liquidity premium be?

As I alluded earlier, there are 2 frameworks in which I have started to think about this. Let’s start with the approach I found personally more intuitive (although I suspect most of you will find the second approach more natural).

The Liquidity-Is-An-Option Replication Approach

First, what’s the obvious advantage of liquidity?

You can cut risk.

The fact that a market is willing to show you a bid for your investment at all times has a real theoretical value. That may sound abstract but the entire options market is actually built upon that idea. Let’s see how.

Go back to the untradeable EE bond vs the 20 year treasury. The nominal EE bond has a nominally guaranteed CAGR of 3.5% if held to maturity. But it’s real return is not guaranteed. In real terms, you can technically lose 100%. In contrast, the liquid treasury bond can be sold. If you placed a stop order on it, you can create one of those hockey stick payoff diagrams where the most you can lose is your stop price.

So…you have created an option. This was the entire basis of portfolio insurance.

[reader recoils 🤮]

I know, I know.

1987 ruined the term portfolio insurance. But the reality is some version of it is done every time a market maker sells an option and delta hedges it. The market maker is trying to dynamically replicate the option they have sold. They are “manufacturing” a long option. The market maker hopes the accumulation of losses due to negative gamma (buying high and selling low) is less than the premium they collected up front for the option.

The key here is to recognize that the ability to “manufacture” an option by trading is possible because of liquidity.

Sure, there are caveats. The “manufactured” option fails in the presence of gaps. It’s not as valuable as “hard” or contractual optionality. 1987, in fact, makes my point…the constraint on theory is liquidity. Liquidity is valuable in itself because it sustains options. And options are good.

[aside: options are valuable because they allow you to fine tune risk. Slice & dice the expressions of your desired exposure or lack of exposure. Equity is an option. Capital structures allocate options according to what shape of risk people are willing to take. Some investors require insurance. Some investors only equity, while others may prefer debt. And there are some debt holders who want CDS, another type of option that can be relative-value arbitraged against vanilla options]

Back to the bonds…

So we can think of a liquid bond as having an option to sell that the EE bond does not.

Liquid bond = EE bond + Option

What’s that option worth? Pricing the option (if we assume the market is continuous) will be an exercise in portfolio insurance-esque replication.

The recipe will look something like this:

1. Pick some theoretical strike price (ie maybe a desired stop price)

2. Estimate what option is worth

3. Add it to the cost of an EE bond that guarantees 3.5% for 20 yrs.

Compare the portfolio comprising a 3.5% EE bond + this theoretical option  to a portfolio which simply holds the 1.5% treasury and you are taking a big step toward quantifying the value of liquidity!

That identity one more time:

Liquid bond = EE bond + Option

What is the main driver of the option’s value?


The premium we are willing to pay for liquidity depends on volatility. The higher the volatility the more the liquidity option is worth and the larger the gap should be between a liquid and illiquid price. 

It’s interesting to consider in light of recent valuations. The more volatile the future is, the bigger the discount we should ascribe to illiquid assets. Today, with implied vols relatively elevated, private investing should be worth less if all else is equal. (I expect private managers to claim that all the alpha is in private markets which is an argument they are entitled to)

To restate the main point of the liquidity-as-an-option replication approach:

Increased volatility raises the value of liquidity because it raises the value of the option embedded in the ability to trade.

I mentioned there’s a second framework for valuing the premium we can ascribe to liquidity.

Rebalancing premium

The ability to rebalance your portfolio is valuable.

Here’s an intuitive demonstration:

Markets take a dive. Pretend 1/2 your wealth was in stocks and 1/2 in your home. If homes were down more than stocks, you could sell stocks & upgrade your home while restoring a 50/50 allocation.

In Lessons From Coin Flip Investing, I showed how rebalancing between a coin flipping investment and a positive expectancy investment enhances performance. Specifically, rebalancing pushes your geometric return up towards the expected arithmetic return (remember geometric returns are lower than arithmetic because of volatility drain). You earn a “premium” for rebalancing.

There are 2 main drivers of the rebalancing premium.

  • Volatility 

    The size of  the premium is a direct function of volatility since the drain is half the variance. This should be satisfying — the option replication framework also said that volatility increases the value of liquidity.

  • Correlation

    A full explanation would be out of scope here. Instead, I direct you to
    @breakingthemark and his blog. His recent post, The Great Age of Rebalancing Begins, shows how lower fees/spreads provide unprecedented opportunity to capture “Shannon’s Demon” — the underlying concept behind rebalancing premium first identified by Claude Shannon. 

Key Takeaways

  • Liquidity-As-An-Option and rebalancing premium are 2 ways to price the value of liquidity
  • Both methods agree — the greater the volatility, the more liquidity is worth.
  • You should get a better deal for accepting less liquidity

Path: How Compounding Alters Return Distributions

Compounded returns experience “variance drain”. This idea captures the fact that typical result of compounded returns is lower than if you compute arithmetic returns even though the expected value is the same. We mostly care about compounded returns. This describes the situation in which your bet size or allocation is a fixed percent of your wealth, savings, or bankroll.

This is in contrast to keeping your bet size fixed (ie if you invested $10,000 in the stock market every year regardless of your wealth).

The distinction is critical because as humans we experience the path of our investments so we care about the distribution of returns in addition to the expected value.

Let’s back up for moment.

Recapping Intuition

  • What land are we in?
    • Compounding Land

      If you bet 1% of your wealth on a coin flip and win then lose, you are net down money. This is symmetrical. If you lose, then win, still down money.

      1.01 * .99 = .99 * 1.01

    • Additive Land

      In additive or non-compounding land we bet a fixed dollar amount regardless of wealth.

      So if I start with $100 and win a flip, then bet $1 again and lose the flip I’m back to $100. The obvious reason is the $1 I bet when my bankroll increased to $101 is less than 1% of my bankroll.

  • The order of win then lose, or lose then win leaves you in the same place in both worlds.

    The order does not matter if we are consistent about how we size the bet (so long as we are consistent to the style whether it’s fixed dollar or fixed percentage).

So is fixed percentage somehow “bad” in that it opens you up to volatility or variance “drag”? 

Well in the last example we used an alternating paths. Win then lose or vice versa. Let’s look at the case where instead of alternating wins and losses, we trend. Win-win or lose-lose.

  • In the additive case, we are either up 2% or down 2%
  • In the compounded case we are up 2.01% or down 1.99%

Wait a minute. In the compounded case, we are better off both ways! So the compounded case is not always worse.

The compounded case is better when we trend and worse when we “chop”.

If bet a fixed percent of our bankroll fair coin toss game we are in compound return land.

Compounding is not “bad”, it just alters the distribution of our terminal wealth

Your net compounded return in the coin-flipping game is negative more often than it’s positive, even though the game has zero expectancy.

So why is the median outcome negative?

It goes back to the trend vs the chop. Compounding likes trending and hates chopping as we saw earlier.

  •  Chopping happens more 𝐨𝐟𝐭𝐞𝐧 so you get a negative median
  • …but this is balanced by a larger trending bonus due to compounding.

Let’s illustrate.

2 Coin Flips

There’s 4 actual scenarios:

2u (trend)
1u, 1d (chop)
1d, 1u (chop)
2d (trend)

Zoom in on “compounding bonus/drag”:


  • Chop and trend happen equally.
  • The magnitude of the boost/drag is also equal.

3 Coin Flips

There’s 8 total outcomes, but again order doesn’t matter. So there’s really just 4 outcomes.

The “chops” are bolded. They represent compounding “drag”


  • You drag 75% of the time!
  • The larger positive boost magnitudes make up for the frequency.

Now that you have the gist, let’s do 10 flips.

10 Coin Flips

  • 65% of the results are chop giving you compounding drag.
  • The times you trend though crush your performance if you only bet fixed dollar!

Visualizing “The Chop”

Let’s take a look visually at paths where N=10  to see the “chop”.

Pascal’s Triangle is a quick way to to get the coefficients of a binomial tree. The coefficients represent combinations which are weighted by the probabilities in the binomial expansion.

I enclosed the “chop” or drag paths

100 Coin Flips

  • The negative median now becomes very apparent in the “cumulative probability” column.
  • The chop occurs in 68% of paths. The median return is -.50% after 100 flips though the expectancy is still zero.
  • In additive world if you win 50 $1 bets and lose 50 $1 bets your p/l is zero.
  • In compounding world, where you bet 1% each time you are down 50 bps in that scenario.
  • The negative median associated with compounding is balanced by better outcomes in the extremes.

Both the maximum and minimum returns in simulations are better than the fixed bet case. This simulation by Justin Czyszczewski (thread) shows just how substantial the improvement is in those less probably trending cases:

Lessons From Compounding Coin Flips

  • Your overall expectancy is zero because the common chop balances the rare but heavily compounding trends.
  • Paths affect distribution of p/l even if they don’t affect expectancy.

Since we actually experience “path” and all its attendant emotions, it pays to think about the composition of expectancy and returns.