A Former Market Maker’s Perception of PFOF

It feels like payment for order flow controversies flare up every few years. When I see some of the takes I know how marine biologists felt after Jaws hit the cinemas in 1975.

Except they didn’t have Twitter to scream into.

I’m going to assume you already know what payment for order flow is.

If you need the basics, A16’s Alex Rampell and Scott Kupor have you covered. (Link)
If you want the GOAT of high finance’s version, here is the Matt Levine post I shared last week. (Link)

Now if you stop at Levine’s post I’d forgive you. There’s really no following that guy. But now that I’ve said that, you own the downside of reading further and if I say anything useful here I’m in-the-money.

I think my experience qualifies me to hopefully add some perspective to the discussion. I have been trading options for 21 years with the first half of those years on the floor. Even though I’ve been trading prop for the past decade I’m a dyed-in-the-wool market-maker. You can take the dog off the floor, but you can’t take the floor out of the dog. (Full disclosure: I used to work for SIG who was an early payer for order flow, but I had no insight into that side of their business).

An Image Problem

Payment for order flow sounds terrible. It sounds like payola. Greasing the radio DJ to get your record played on-air. That’s a bribe to the regional gatekeeper. There’s widespread misconception that when Citadel pays for flow it’s attempting to use the info to front-run the order. This is a dizzying misconception.

No trader thinks front-running random retail flow makes any sense.

Write that on a chalkboard 50x please.

The Nature of Adverse Selection

Drive it home: no trader thinks front-running random retail flow makes any sense.

In fact, the opposite is true. The entire basis of trading against retail flow is that it is a random mix of buys and sells and not autocorrelated. You want to trade against your drunk uncle Sal who has a good feeling about the Jets this Sunday. We call this “dumb” flow. Sorry, but that’s what it’s called.

On the other hand, we refer to institutional flow as “smart flow”. Not because it knows which direction the stock is going to go, although this can be the case as anyone who has been contra to SAC flow back in the day can attest. The reason we don’t want to trade against the flow is that it’s autocorrelated. 1,000 shares is the tip of an iceberg. Nobody eats just one chip just as nobody buys just 1,000 shares.

The options equivalent is putting someone up on a trade, only to have them reload 5 minutes later. This past fall, Softbank string-raised tech calls every day for a couple weeks. Masa-son is not smart paper, but he has a big stack. Truthfully, the threshold to be an undesirable counterparty is surprisingly low. I remember hearing a SIG trader at a conference a few years after I left. He mentioned that their studies had shown that the adverse selection of an options trade went up dramatically once it was greater than 16 lots.

Let’s understand this. Consider a pro-rata exchange where your limit bid is on equal standing with other limit bids but your fill is proportional to your size. So let’s say you are bidding $1.25 for 100 option contracts and the total bid quantity is 1000. If a retail sized order sells the bid for 10 contracts, you get filled on 1 because your size was 10% of the total displayed size. The pro-rata system (vs maker-taker which is a queue based on speed) incentivizes traders to show far more liquidity than they really want to. They don’t want to get their whole bid hit but they need to show size to be entitled to any reasonable percentage of the incoming orders. When an order sweeps the book, banging out the displayed size on the bid, the market makers are instantly sad. They know they are on the wrong side of a “smart” order.

The possibility that the flow you trade against is adverse, smart, institutional – whatever you want to call it – has a deep implication. You make a wider market than you would have if you could just tell the difference between the adverse flow and the random retail flow.


The brokers have realized they can segment the market between orders that can be facilitated on tighter spreads and those that require wider quotes. Liquidity has a price. Without PFOF, spreads need to be cushioned by the probability that an order is institutional. Instead, PFOF creates a tiered market where the cost of liquidity is proportionally aligned with the risk on a per trade basis. Retail traders get better fills. There’s less deadweight loss.

Institutional traders might complain, but its an illusion that they should have gotten the price that a retail trader should get. The risk business is not the widget business. You don’t get volume discounts.

“The opportunity to trade against random flow” as a source of revenue is a bit abstract. You are already familiar with price discrimination in other domains.

  • Casino’s attracting whales.

    Casinos don’t like card counters, they want customers that have positive LTV in the long run. They like whales and the type of people who buy books titled “The Fool-Proof System To Beating Roulette”. Casinos are paying for order flow when they offer complimentary suites and blacked out SUVs to and from McCarran.

  • Ad tech

    What is the internet but reams of data on customers being sold to the highest bidder so platforms (the brokers in our analogy) and in turn vendors (the Citadels) more can more efficiently convert sales (trades)?

  • Financial products

    Good driver discounts on auto policies. Life insurance physicals. Credit checks for loans. Price discrimination based on risk is the norm not the exception.

  • Retail

    As a broke 20 year old I used to frequently buy and return products at GNC. Yes, you can return a half-used tub of creatine. GNC started keeping tabs as a policy. I get it. The Ponderosa wishes it could turn away Joey Chestnut.


The discourse around PFOF has an air of monopoly sentiment. Maybe not in the Standard Oil sense of the world. There’s more firms than Citadel. You have Virtu, G1 (SIG), Two Sigma, Wolverine. It looks more like OPEC.

But there’s a big difference. These are not natural monopolies or crony handouts. Contrast the dynamic with payola. Payola was a scam that worked because the value of the bribe to the briber (the record label) was very low compared to the payoff of getting radio exposure. Meanwhile the value of the bribe was substantial to the receiving DJ who was paid a conventional salary despite being the caretaker of a government monopoly — airwaves.

I don’t think it’s surprising that high fixed cost industries settle into oligopoly-type hierarchies. The competitive forces are so strong that they double as high barriers to entry. The HFT-firms here are not defending natural monopolies. They are the survivors of the trading game who invested heavily in technology early. @hidenotslide explains in his recent post about another storied traded firm, DRW:

This brings me to my first point – firms who embraced HFT early in its evolution are today’s kings. Of the 10-20 firms that make up the bulk of high frequency trading profits, a large majority were launched before the 2008 financial crisis and many even prior to 2000. Because superior technology leads to direct competitive advantages in HFT, barriers to entry have become insurmountable over the last decade as companies have invested in ever faster exchange connections & market data feeds. A 2017 paper from researchers at Cornell & Penn argues this exact point – newer, smaller entrants that engage in HFT can survive, but they don’t get anywhere near the share of profits that larger, more established firms enjoy.

What’s absent from the narrative is how tall the pile of bodies these firms stand atop. I should know. I used to be able to work five hours a day (NYMEX alum holla) and make a lawyer’s wage. And in some years, a law partner’s carry too. Well, if you were smart you saved your money and realized it wasn’t going to last. The days of “locals” (ie wildcat market-makers) is long gone.

Many of the small firms, who saw the writing on wall and had an appetite for the long game, plowed money back into massive technology capex. Most of them just earned the right to say they lost to the best. In some cases they found small, profitable niches where they play the role of suckerfish. Respect to them, even this was not easy.

How about the remaining firms? The private giants the media likes to call “shadowy”. They were the ones who were most adept at assembling teams of software and hardware engineers working with game-theory geniuses to devise algos in a cat-and-mouse battle with competitors. The ones who stayed step-for-step with the exchanges who themselves were experimenting with matching engine rules, data, product listings and connectivity in their own battles for market share.

The truth is progress is cutthroat.

I remember the days before decimalization where you could make $5 wide verticals 3/8 wide. Today that same vertical is a choice market and the market maker gets paid the equivalent of an inter-dealer broker commission or about 25 cents. On a 3/8 wide market the market maker used to earn nearly $18.75 (or 50% of 3/8)! My business partner and I always marvel at the innovation and how little vig a trader is willing to accept to flip million dollar coins. It’s such a flex for capitalism. So much so that how good these firms are is chalked up to monopoly and not that fact that they are the survivors of the capitalism’s most brutal tournament.

How Survivorship Bias Makes Firms Look Like Monopolies

Perhaps I should not be surprised at the monopoly sentiment. Some of you will nod. “How can they make money every day?” First, I’m not sure they do, but even if they did that’s hardly a red flag. Casinos might make money every day so long as they can open. They’re not monopolies. Worrying that financial firms make money everyday is conflating market makers with investment managers because they traffic in the same products. But one of them is a customer and the other is a supermarket. With tiny supermarket margins per trade. And high fixed costs. If volumes dried up, the losses would show up even if the margins stayed flat.

A stronger, but still naïve argument, that they were monopolies would come from noticing that these shops came of age at the same time as the giant tech firms. This is a hint of how much they have in common. The difference is the size of the relative opportunities, but the tactics are similiar.

It started with skill and luck. The early big bets on talent and technology meant they were bringing guns to a knife fight. SIG wasn’t know as the “evil empire” on the Amex just because of the black jackets we wore. They understood the risk-reward was completely outsized to what it should be 25 years ago. They were amongst the first to tighten markets to steal market share. They accepted slightly worse risk-reward per trade but for way more absolute dollars. They then used the cash to scale more broadly. This allowed them to “get a look on everything”. Which means you can price and hedge even tighter. Which means you can re-invest at a yet faster rate. Now you are blowing away less coordinated competitors who were quite content to earn their hundreds of percent a year and retire early once the markets got too tight for them to compete.

SIG was playing the long game. The parallels to big tech write themselves. A few firms who bet big on the right markets start printing cash. This kicks off the flywheel:

Provide better product –> increase market share –> harvest proprietary data. Circle back to start.

The lead over your competitors compounds. Competitors die off. They call you a monopoly.


Thus far I’ve only pushed back against the idea that PFOF is somehow nefarious. It is a form of price discrimination. The price discrimination is economically sensible when we price liquidity. There is a cost to having someone trade with you at the exact moment you want to trade. If you are a retail trader, that cost is tiny and we can thank technology and the competitive drive of very smart people to undercut one another so they can be the best bid for your business.

If you are an institutional trader that cost is higher. And it should be. Your cost to trade should be compared to your historical cost to trade. Not against what a retail trader’s costs are. I’d be shocked if an apples-to-apples TCA showed that this cost has increased over time. My null is the cost to trade for everyone has collapsed but probably more for retail.

I don’t have any strong opinions as to whether PFOF is the best equilibrium. One could argue we should have a single central order book, but then the exchange would have a monopoly. Plus it’s not obvious to me that the centralization of liquidity serves the heterogenous interests of all economic stakeholders across countries, regulatory regimes, strategies, time zones, and instruments.

We could entertain more incremental tweaks to the current architecture. For example an auction every minute or shorter trading hours to centralize liquidity in time but not venue. There’s probably some efficient frontier of tradeoffs. Nothing about PFOF looks villainous from my understanding of markets so if it lies along that frontier I would not be surprised.

And perhaps now you won’t be either.

How Options Confuse Directional Traders

2017 was a historically low-vol year, rewarding options sellers despite selling lower option premiums as the year progressed. Like they found a broken slot machine at the Cosmo. It wasn’t until the Feb 2018 “volmageddon” in exchange-traded VIX products, that retail discovered the dangers of selling options.

In the past year, retail, led by r/WSB, is back in the deep end of the options pool. This time they brought swimmies — they are only buying options. This limits their losses to the premiums.

As opposed to professional vol traders, most people use options as a way to bet on direction. You buy a put to bet on a stock going down and you buy a call to bet on a rally (or an “up” — a term coined by my doctor friend who always used to make fun of us finbros who talked about “puts and ups” all the time when we were in training). I tend to dissuade people for messing with options unless they have a very specific risk to hedge or if their speculative thesis is well-defined. Since options expire you need to be right not just on direction but timing as well. There’s a lot of ways to lose, get lured into trading more, and generally chop yourself up.

I’m going to demonstrate how you can lose money despite being very “right”. For good measure, we’ll extend the conversation to how hedging can actually increase your risk. Let’s jump in.

An Option Lesson

The recent action in GME justified the cigarette warning label I put on options. If the option user doesn’t appreciate the role implied volatility plays in an option price, then Benn’s tweet is mystifying:

Put options increased in value as the stock went up.

Then, with the stock on the way down @mark_dow tweets:

Put options lost value as the stock collapsed.

My kids would chalk this up to “opposite day” (apparently a modern holiday where kids wear pajamas to school). Alas, there is a more boring explanation:

Implied volatility increased as the stock price increased and fell as the stock price fell.

Pro Version

If you are eager, we can drill down a bit.

  • The option’s “vega” dominated its delta in both cases. The vega tells us how much the option’s price will change as the volatility rises or falls.
  • “Vanna” represents the sensitivity of the option’s delta to volatility — a second order effect. As the vol increased, the OTM option deltas increased. This is notable because it is a positive feedback loop. As the stock and vol both increase on the way up, market makers have to buy more stock to hedge. On the way down, it is stabilizing as the vol decreasing means the option delta decreases and market makers need to be “less short” to hedge the puts…it’s stabilizing because this offsets the negative gamma effect from being short puts in the first place. This one is tricky because the vanna effect is dampening the vanilla gamma effect.
  • Then there’s “volga” which is how the option’s vega changes with respect to vol. This is yet another second order effect of vol (I’ve written about that here). It feeds right back into vanna and acts as a reinforcer on the way up and a stabilizer on the way down (since spot and vol are positively correlated. We’ll get to this correlation later).
  • There are higher order Greeks than “vanna” and “volga”. Ironically, they are only known by the French. Don’t ask.

Vol traders care about these cross-currents because of how they accelerate or dampen the price of options. These effects alter hedging flows which change buying and selling pressures. Outputs become inputs so each sub-cycle in the process looks like a foreshock to something bigger, or the aftershock to something dissipating.

This might sound theoretical or academic but it’s the nuts and bolts of managing volatility portfolios. An option book with many names, maturities, and strikes looks like an amorphous blob until you use these concepts to give it shape. Once it has shape you can recognize what kind of animal it is. You can predict how it might respond to different scenarios. The measurable risk is how it will react to the market’s movements. The stock is going to do stuff. That’s a given. You are not allowed to be surprised by that fact.

The real concern is if the portfolio, this animal under your care, acts outside your range of expected behavior.

Normie Version

Rest easy. That was utter overkill for investors or even casual option punters. To understand why puts got cheaper on a selloff, you just need this picture:

It is a beautiful and simple visual intuition constructed by @therobotjames.

  • The purple bell curve is the distribution of GME stock when it’s trading for 600% vol and $200.
  • The green bell curve is the distribution of GME stock when it’s trading for 400% vol and $90.


Despite the higher stock price, the purple curve imputes a higher probability of the stock going below $20 because the distribution is much wider at 600% vol than at 400% vol. The impact of the vol totally dominates the moneyness, or distance, the stock is away from the strike. Another way to say this is “the $20 strike is closer to $200 than it is to $90” if the volatility is that much higher when the stock is $200. This is easier to understand if we simply make the volatility disparity wider. Imagine a govt bond that trades for $100 par and a stock that trades for $200. Nobody would be shocked if the 50 strike put for the stock was worth more than the 50 strike put for the bond.


I can see you scratching your head. In GME, we are talking about the same exact asset at 2 points in time with a contradicting proposition: namely that the probability of the stock dropping below $20 when the stock is $200 is higher than when the stock is $90!

This paradox is an illusion that happens whenever you have the benefit of hindsight. You don’t know which of these prices is the true odds. You can only trade with the information you had at the time. You cannot arbitrage the relative pricing between the 2 states of the world that we have the luxury of seeing in the rearview. Looking back you can say that the stock’s chance of going below $20 was underpriced when it was trading $90 or that it was overpriced when it was trading $400 but you couldn’t make those claims at the time. They only seem paradoxical when compared to each other.

At this point, I suspect retail traders, curious as to why they won to buying puts on the rally and lost to buying puts on the selloff, developed some understanding of vol dynamics.

Hopefully the tuition wasn’t too steep. Not all lessons are as cheap as a defined option premium.

The Expensive Option Lesson Pros Learn

Professional option traders adjust option greeks for spot-vol correlation. In the GME-case the correlation is positive just as it is in agricultural commodities. As the price increases, the vol increases. Most markets have a negative spot-vol correlation. The VIX falls when the SPX rallies. This is also true in the oil market. A supply of options hits the market during rallies as large hedgers overwrite calls.

To adjust for this, option traders will model a negative spot-vol correlation or “vol beta”. For example, suppose your ATM call option typically has a 55% Black-Scholes delta. you might model a 50% delta only, knowing that if the future goes up $1, your call option probably won’t increase by $.55 since implied vol will fall. (In fact, one of the ways to know if the counterparty you are quoting was a bank or not was by the delta the broker wanted to use on delta-neutral structures. Banks often quoted with Black-Scholes deltas while prop shops used deltas which incorporated vol betas, effectively lowering all call deltas).

When you model vol beta you are usually making a trade-off between hedging local behavior of common moves versus more unusual sized moves which will break the spot-vol correlation, in turn upending calibrated deltas. If a skirmish broke out in the Strait of Hormuz and oil ripped 10% higher I would not expect volatility to fall. Therefore, you also need to consider a matrix of outcomes.

(This is a hypothetical picture which tells us if oil rallied 10% and vol increased 50% we would lose money. Note that if vol fell in accordance with a vol beta we would have made money).

Even with respect for local and jump spot-vol correlations, you can still be caught off-guard. In nat gas, I’ve underestimated just how GME-like its vol surface can change. I’ve seen put prices not budge despite a 20% selloff in an underlying. If you are running a hedged book and have any long futures against the puts you enjoyed the full drawdown in futures without any offset from the puts. Enough to make a burly man cry.

The idea that “you only risk your premium” when you buy options is only true if you do not hedge. It’s diabolical to get crushed on a supposedly neutral position. Why? Because, you thought you were hedged. This tricked you into buying more puts than you would have if you didn’t hedge.

(All basis trades have this dangerous property. The illusion of being hedged induces you trade bigger or use leverage to push a small edge.)

Qualitative Appreciation For Spot-Vol Correlation

The GME put holder who lost money on a sell-off now understands how the change in implied volatility explains the loss. Regrettably, this is like being told you missed a flight because you were late. It’s just a mechanical explanation. What you really want to know is why did volatility come in as much as it did? In option trader terms, “why did the vol beta outperform or underperform in the first place?”

The beta itself will have quite a bit of variance since a price can follow many paths to a destination. Those paths will each be a sample of unique realized volatility. Did the price grind to X or did it gap to X? The realized beta will vary from your projected one depending on the path and the market’s interpretation of that path. If the stock gaps down due to a specific bit of news (for example news that a big short is done covering or the company issuing more shares) the gap can actually be vol-reducing as the market interprets the news as “stabilizing”. If the gap comes with no explanation, then the market might interpret this data point as another mystery piled on an already burning heap of confusion. The market will presume that the crazy stock might just rip back up again. In this case, the vol might hold up better on a sell-off that occurs without a reason in contrast to the the prior case where the reason had the narrative effect of curtailing the upside.

So in the GME case, most of the reasons the price can go down are stabilizing. We expect options to be sold in response to a sell-off, and for the vol to decline. But “most of the reasons” does not mean 100% of the reasons so there is a probabilistic distribution to what the realized spot-vol correlation could be. And that’s why we still have surprises.

The Beauty Of Options

Ultimately, options help to “complete” a market. A simple stock price is just the expected value of a stock (equity risk premia and arbitrage pricing theorists are welcome to have a cage match over that statement. I’ll be out back selling beer). By imputing more information than a one-dimensional expected value, option surfaces give us a richer picture of expectations. What’s considered stabilizing, and what’s considered unthinkable are encoded in options markets.

There’s a silver lining to the WSB obsession with options. Some of these people who showed up for a thrill will stick around to learn how to listen to how a vol surface whispers.

Prices Are More Than Expectations

Prices Are More Than Expectations

This week I came across this tweet by @NewRiverInvest:

Breakevens are not inflation expectations (thread)

It explains how implied breakevens from TIPs are not the same as inflation expectations. One of the reasons is because it’s a small market that is easily distorted.

The second reason is more technical. TIPs are actually implicit options. The face value increases with inflation but is floored at par. So if we experienced deflation you’d get your now-even-more valuable USD back. We should presume the TIPs price reflects not just inflation expectations but a premium for the option value.

This is a familiar lesson. Think about volatility risk premiums. Since convexity improves portfolio CAGRs the expected value of owning an option should actually be negative in arithmetic terms. This is why quants will fancily say “implied vol is a biased estimator of realized vol.” It’s overpriced on average but it’s correlation and convexity attributes suggest it is not overpriced in a repeated, compounding framework.

Another demonstration of “price does not equal expectation” is found in correlation itself. Correlation swaps trade at cheaper levels than implied correlation because being short correlation is a concave (ie negatively convex) position. A correlation seller will require an additional risk premia to be short it. A further explanation can be found in my notes.

So whenever you imply an expectation from a price you need to strip out any additional risk premia or preference that is embedded in the price.


This ties in well with Why You Don’t Get Paid For Diversifiable Risks (MoontowerMeta)

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.

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.

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.

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.

Making Property Taxes Apples to Apples

You will be working from home more often. Not all of you but many of you. That means browser tabs devoted to Zillow searches in Austin, Nashville, Vegas, Denver, and Miami. Geo-arbitrage won’t be as dramatic as software devs had hoped since the big companies will cut your pay when you leave, but in some of these places you could sustain a 20% pay cut and still be better off (at least if you’re leaving SF).

One of the biggest inputs into cost-of-living comparisons are so-called SALT (state and local) taxes. Since 2018, SALT deductions are limited to $10,000. They were previously uncapped. This has created even larger disparities in cost-of-living between states. CA, IL, NJ, and NY have income taxes that get a bit handsy with their residents.

Beyond state income taxes, one needs to consider property taxes for a more complete picture. Texans enjoy zero state income tax but hefty property taxes. NJ residents are assaulted from both ends — above average state income taxes and punitive property taxes. How about CA? The state income tax, gas tax and the cost of renewing a vehicle registration are nothing short of sunny weather ransoms.

But what about CA property taxes? The answer to this is sneaky and can be used to understand the impact of property taxes in general. But I’d go further and say that if you have not walked through the math the way we are about to, then you may be walking around with some very mistaken impressions about the cost of housing.

Property Taxes: Apples to Apples

The effect of property taxes depends on 2 core variables. The property tax rate and the assessed value. If you are weighing a house in CA to a house in NJ you want to make an apples-to-apples comparison. How do you do that when the rates are different and the methods of assessing value are different?

Let’s isolate each effect.

[Obviously the cost to buy a home has many factors that can mostly be tucked under the headings of supply and demand. Yet the effect of property taxes is significant so it’s worth isolating. It’s also worth noting that since a primary residence is most people’s largest asset, a property tax is a defacto, albeit incomplete, wealth tax. Economically it’s passed-thru to renters so it hits everyone]

Assessed Value Effect

Property taxes are waged on assessed value. In NJ, assessed value resets whenever a home trades. So if you buy a $1,000,000 home and the property tax rate is 1% you owe $10,000 per year in property tax. As the estimated market value of your home changes, your assessed value changes. So if your market value jumps 15% in one year you can expect a big increase in your tax bill. It may lag the full market return but the idea is the assessed value tracks the value of the home. Downturns in prices require homeowners to plead their case that the home’s value has declined if they want relief on their taxes.

Like NJ, CA assessed value resets to the purchase price after a transaction. But then CA diverges from other states. A month before I was born, in June 1978, CA passed Prop 13, a ballot proposition that has created distortions in wealth that few could have foreseen. Prop 13 froze assessed values at 1976 levels for homes which have not since traded. It also limits increases in assessed value to a cap of 2% per year.

Combined with a NIMBY attitude to permitting new construction, CA features a lopsided sight to behold — multi-million dollar homes with single-digit thousand tax bills. Nice for those owners but not socially desirable.


  • The flipside of having seniors be able to stay in their homes is that it limits worker mobility by poorly allocating big homes to people who don’t need them. It basically keeps rooms off the market. If you are a senior citizen on a fixed income you are not going to sell the home you’ve outgrown to buy a condo with much higher property tax than the big house you leave behind. And that’s after you pay a huge cap gains bill.
  • Prop 13 starves the state of tax revenue that needs to come from somewhere. So the state income tax can be seen as a wealth transfer from young, working Californians to older, entrenched Californians.

In a state that has seen generational wealth built on a loop of buying real estate, and cash-out refis it’s easy to see how Prop 13 has contributed to the party. Let’s pretend you buy a home in CA and NJ.


  • Each home costs $1,000,000
  • Each has a property tax of 2.5%. We are isolating the assessed value effect so need to hold the tax rate constant.
  • Each home has a real (inflation-adjusted) return of 2% per year.
  • The only difference is the CA home is assessed only when you buy it, but the NJ home is assessed each year.

The CA home’s IRR will be .14% after-tax while the NJ home’s IRR is -.52%. The CA home outperformed the NJ home by .66% per year over 30 years. On a $1mm home that’s over $275,000 simply because the NJ home is re-assessed every year.

It gets crazier. The effect actually explodes with higher appreciation rates. If we double the appreciation rate to 4% per year, the CA homes nets you $700,000 more than the NJ home. Remember that the tax rates are the same! We are just isolating the impact of fixing the assessed value at the purchase price.

The main takeaway is Prop 13 is a call option on inflation. Your home is much less of an inflation hedge than you think if its assessed value increases in-step with the market value.

[This year Prop 15 is on the CA ballot. Prop 15 would repeal Prop 13 for commercial properties only. Based on the examples above, it’s obviously something RE investors are highly concerned about.]

Rate Effect

What if you wanted to compare the price of homes in 2 places with different property tax rates? Let’s pretend CA no longer had Prop 13. Like NJ, it’s property taxes were re-assessed annually. This allows us to simply isolate the impact of differing tax rates.

Let’s assume:

  • Each home costs $1,000,000
  • CA tax rate is 1%
  • NJ tax rate is 2.5%.
  • The homes do not appreciate over 30 years (just to keep it simple)

Let’s explore 2 methods of comparison:

The Mortgage Method

If the homes do not appreciate then their assessed value remains fixed at $1mm. This makes it easy — the CA home owes $10,000/yr in taxes and the NJ home owes $25,000. On a monthly basis, the NJ home costs an extra $1,250. If mortgage rates are 3% we can find that a $300,000 30-year mortgage corresponds to a $1,250 monthly payment. So we can say that a $1mm house in CA costs the same as a $700,000 house in NJ since the $700,000 plus an additional $300,000 mortgage would equate to the cost of the CA home.

The IRR Method

The IRR on your home’s value will approximately differ by the spread in the tax rates. In the table below, we see that the CA home returns 1.44% more (close to 1.50%) over 30 years. If we use an inflation rate of 3% to keep consistent with what I chose as a mortgage rate, we find that the NJ home costs you $300,000 more over the 30 year holding period than the CA home, matching the result from the mortgage method.

Combining Effects

To compare the price of a home in CA to a home in NJ you need to account for both the difference in property taxes and how assessed values are treated. Let’s combine the results in one model with more realistic numbers:

  • A 4% annual home appreciation in both markets
  • A 2% inflation rate
  • CA tax rate is 1%
  • NJ tax rate is 2.5%
  • CA assessed values do not increase, NJ is re-assessed annually

CA, due to Prop 13 and a lower property tax rate, has an almost 2% edge in annual return (3.29% vs 1.34%). Since these are nominal returns and inflation is 2% per year we see that the NJ home end up actually losing value in real terms. The fact that the home is re-assessed every year means that even though the home’s value is growing faster than inflation the taxes are also growing very quickly.

I don’t want to have you miss the point — these CA and NJ homes were assumed to grow at the same rate of 4% per year and yet the CA home earned you an extra $900k in present value vs the NJ home. This is strictly due to lower property taxes and Prop 13.

We know that home appreciation in CA has been faster than NJ (my family considered moving to CA in late 70s, early 80s so we are very sensitive to the comparison). The difference in property tax policies has a staggering delta in terminal wealth when applied to CA real estate boom over the past 50 years.

Wrapping Up

Having grown up in NJ and now lived in CA, I have noticed a massive divide in how people have earned their money and wealth. You cannot live here and not notice the wealth built in real estate and not think about how policy has enabled it. When you start comparing apples-to-apples, the headline prices of CA homes are not as relatively expensive as they appear. Don’t hate on Californians though. Those SALT taxes are still burying all of us who still work for a living.

In sum:

  • Prop 13 allows homes to be a call option on home appreciation/inflation
  • High property taxes on homes that are re-assessed require rapid appreciation to not render the home ‘dead money’
  • Compare homes with different property taxes by amortizing the difference in monthly payments into a mortgage

Sending a thanks to @econompic who I discussed these topics with. As another NJ to East Bay transplant he has given these ideas plenty of thought as well. And on the math side, he gave me the idea to use IRRs instead of CAGRs. CAGRs are simpler because they are compounded returns which require no more than a start value, ending value, and time period. They are commonly used when calculating a return for a stock or fund that you buy and hold.

In this case, IRRs or NPVs are preferable since there are many cashflows.