Bubbles: Knowing You’re In One Is Not Even Half The Battle

Select excerpts from Aaron Brown and Richard Dewey’s paper:

Toil and Trouble, Don’t Get Burned Shorting Bubbles (SSRN)

It was not a mystery that there was a bubble in subprime from 2005-2008. That did not mean shorting it was an easy trade. With the benefit of hindsight, we can learn about the risks of shorting frothy assets that may even be a bubble.


From the abstract:

Bubbles are among the most puzzling and controversial phenomena of financial markets. Although rare, their cumulative impact on both investor returns and the broader economy can be great. One particular question that has motivated research is why shrewd short sellers don’t prevent excessive price increases. The “limits to arbitrage” idea argues that correcting inefficient market prices is neither easy, cheap nor riskless. The “rational bubble” literature identifies situations in which being long the bubble is a better trade than being short, even if investors know for certain the bubble will pop.

We examine the “short subprime” trade from 2005 to 2008 to evaluate these and other explanations. We argue that the short subprime trades had more risk than is commonly appreciated. We discuss how the opaque and illiquid nature of subprime mortgages deterred some investors from purchasing CDS contracts and note that other investors assessed the risk of counterparty failure, government intervention and unknown time horizon to be sufficient enough not to purchase CDS contracts.

Talking to investors who saw the bubble and passed on shorting it, instead opting for alternative strategies:

We interviewed and analyzed the internal research of several investors who evaluated the short subprime mortgage trade and decided not to purchase CDS contracts and present some of their reasoning below.
    • The Basis Trade: Magnetar Capital in Chicago.

      Magnetar did not cooperate with the media, so their story has not been widely told. Magnetar reportedly employed a strategy whereby they purchased the riskiest equity tranche in many CDOs which often offered double-digit returns. They used this positive carry to pay for protection on the AAA tranches that most investors assumed were safe. Magnetar appreciated that the correlation between the safest AAA tranche and the lowest quality equity tranche would be close to one in a crisis due to the way these securities were constructed.

    • Picking-up-the pieces trade: Soros and Tepper

      Soros

      Perhaps the safest way to profit from the subprime mortgage meltdown was the time-honored method of picking up the pieces at the bottom. George Soros and his Chief Investment Officer Keith Anderson smelled opportunity and hired two ex-Salomon Brothersmortgage experts, Mason Haupt and Howie Rubin.

      Tepper

      David Tepper purchased shares of Citi and Bank of America near the bottom, helping his Appaloosa fund return 120% in 2009 on $12 billion in capital.

    • Convex-listed hedges correlated with a downturn: Talpins and Dalio

      Ray Dalio at Bridgewater and Jeff Talpins at Element Capital are rumored to have purchased futures or options on government bonds that would rise in value if the Fed aggressively cut interest rates. Many on Wall Street believe that Element purchased Eurodollar options in 2007 that helped his firm generate returns of 26.4% in 2007 and 34.9% in 20083. Talpins has posted annualized returns north of 20%, without a single losing year in the decade that followed. And simply being long volatility in equity markets, fixed income or currency markets paid off nicely for many traders. The key to these trades is that they removed some of the unattractive aspects of the subprime trade, by using more liquid instruments, waiting for the crisis to materialize or constructing more nuanced expressions.

The collection of people that did these trades: George Soros, David Tepper and the team at PIMCO are investors with long-term track records. They made their money quietly, in sensible trades over several years, and were also able to put large amounts of capital to work.

The sobering difficulty of the short subprime trade:

Betting against subprime mortgages worked, but it was somewhat of a Goldilocks trade: it required default rates to get high enough to generate profits on your insurance, but low enough that the banking system survived to pay you and that the government didn’t help out borrowers at your expense.

Current backdrop:

As we write this analysis in the first quarter of 2021, financial market pundits are calling bubbles in everything from cryptocurrencies and TSLA to SPACs, high-end real estate and, most recently, stocks hyped on Reddit.

My takeaways:

    • Shorting subprime looked like a hero trade but the path was painful and uncertain. You needed to weather the negative carry and margin calls on bilateral trades with banks for years. And you needed to bet against on institutions that were not bailed out.
    • Taking a bubble on straight ahead looks like foolish risk reward, especially if the bubble is “rational”1 and has no clear correcting catalyst.
    • Need to think about the risks and incentives of the system. For example, if you believe bonds are currently overpriced and shorting them, even with their low-yield and therefore relatively small negative carry, you cannot ignore the possibility that the rules can be tampered with in the name of the system. Just as banks were bailed out, it’s not impossible to imagine a yield curve control policy (YCC) similar to post-WWII would cap any upside on a short Treasury trade.
    • The paper describes what you are up against eloquently:

      The reality is that structuring good trades is often every bit as difficult as forecasting. This is particularly true if a trade is contingent on a crisis materializing, when pricing is less reliable, liquidity dries up and contractual obligations are sometimes not honored. In these instances, trade construction is everything. Even if an asset price bubble can be confidently identified ex-ante (no easy task), making money from the bubble is perhaps equally challenging….

      This separates opinion havers from risk-takers. Getting the right odds on the right contingent payoff in the state of the world that matched that payoff. And then actually being able to collect. Having an opinion on markets is like have a business idea but no ability to execute.


Understanding Vega Risk

In a chat with an options novice, they told me they didn’t want to take vol (vega) risk so they only traded short-dated options. This post will explain why that logic doesn’t work.

Here’s the gist:

It’s true that the near-term option’s vega is not large. That is counterbalanced by the fact that near-term implied vols move faster (ie are more volatile) than longer-term vols.

The goal of this post is to:

  • demonstrate that near-term vols are more volatile both intuitively and with napkin math
  • show the practical implications for measuring risk

Near Term Vols Are More Volatile

An Intuitive Understanding

Think of the standard deviation of returns that a stock can realize over the course of a week. If there is a holiday in that week the realized volatility will likely be dampened since there are 4 days of trading instead of 5. If Independence Day falls on Friday, Thursday might see even lower volatility than a typical trading day as fund managers chopper to the Hamptons early. On the other extreme, if a stock misses earnings and drops 25%, then we have a Lenin-esque week where a year happens. The range of realized volatilities is extremely wide. This requires the range of implied volatilities to be similarly wide for a 1-week option. Those large single-day moves are diluted when they are part of a computation for 1-year realized volatility (there are 253 trading days in a year).

This concept is easily shown with a “volatility cone” (credit: OptionsUniversity)

Here we can see the standard deviation of realized volatility itself declines as the sampling period lengthens.

The Napkin Math Understanding

The intuition for why the range of short-dated volatility is wider than long-dated volatility is easy to grasp. To cement the intuition let’s look at a numerical example.

Consider:

A weekly option [5 days til expiry]

Assume the stock’s daily vol is expected to be 1% per day. The fair implied vol can be computed as follows:

IV = sqrt(.01² x 5 days x 52 weeks) = 16.1%1

Remember variances are additive not standard deviations so we must square daily vols before annualizing them. We take a square root of the expression to bring it back into vols or standard deviation terms.

Ok say 1 of those days is an earnings day and is expected to be 3% daily vol.

IV = sqrt([.01² x 4 days + .03² x 1 day] x 52 weeks) = 26%

Look what happened.

The single-day expected vol jumping from 1% to 3% means there is more variance in that single day than the remaining 4 days!

.01² x 4 days < .03²

How did this earnings day affect the fair IV of a longer-dated options?

A 2-week option [10 days til expiry]

 IV =  sqrt([.01² x 9 days + .03² x 1 day] x 26 bi-weeks) = 21.6%

A 1-month option [21 days til expiry]

IV = sqrt([.01² x 21 days + .03² x 1 day] x 12 months) = 19%

The increased vol from a single day is clearly diluted as we extend the time til expiry. When we inserted a single day of 3% vol:

  • The 1-week option vol went from 16% to 26%. 10 vol point increase.
  • The 1-month option went from 16% to 19%. 3 vol point increase.

To understand why this matters look at the effect on P/L:

Remember, the vega of the 1-month straddle is 2x the vega of the 1-week option.

    • The 1-week straddle increased by 10 vol points x the vega.
    • The 1-month straddle increased by 3 vol points x 2 x the vega of the 1-week straddle

      10x > 6x

      The 1-week straddle increased in price 10/6 (ie 66%) more than the 1-month straddle!

      (This is why event pricing is so important. The astute novice’s head will now explode as they realize how this works in reverse. You cannot know what a clean implied vol even is unless you can back out the market’s event pricing)

Practical Implications For Measuring Vega Risk

Comparing Risk

So while a 1- month ATM option has 1/2 the vega of a 4-month option2, if the 1 month IV is twice as volatile it’s the same vega risk in practice. You need to consider both the vega and the vol of vol!

In practice, if I tell you that I’m long 100k vega, that means if volatility increases [decreases] 1 point my position makes [loses] $100k. But this risk doesn’t mean much without context. A 100k vega position means something very different in a 1-week option versus a 1-year option. Looking at a vol cone, we might see that 1-week implied vol has an inter-quartile range of 30 points while 1-year vol might only have a 3 point range. You have 10x the risk if the vega is in the weekly vs the yearly!

Another way of thinking about this is how many contracts you would need to have to hold 100k vega. Since vega scales by sqrt(time) we know that a 1-year option has √52x or 7.2x as much vega. So to have the equivalent amount of vega in a 1-week option as a 1-year option you must be holding 7x as many contracts in the near-dated.

Normalizing Vegas

It’s common for traders and risk managers to normalize vega risk to a specific tenor. The assumption embedded in this summary is that volatility changes are proportional to root(time). So if 1-week volatility increased by 7 points, we expect 1-year vol to increase by 1 point.

This is an example of normalizing risk to a 6-month tenor:

Observations:

  • Your headline raw vega is long, but normalized vega is short
  • Your 2,000 vega in a weekly option is more vol risk than your 10,000 vega in the 6-month
  • You want the belly of the curve to decline faster than the long end. This is a flattening of the curve in a rising vol environment and a steepening in a declining vol environment.
  • If the entire vol curve were to parallel shift lower, you’d lose as you are net-long raw vega.
  • If we choose to normalize to a different tenor than 180 days, we would end up with a different normalized vega. The longer the tenor we choose, the shorter our normalized vega becomes (test for yourself).

Critically, we must remember that this summary of net vega while likely better than a simple sum of raw vega is embedding an assumption of sqrt(time). If you presume that vol changes across the curve move in proportion to 1/sqrt(t), the value of calendar straddle spreads stays constant. At this point, you should be able to test that for yourself using the straddle approximation in the footnotes. This would imply that as long as your total normalized vega is 0, you are truly vega neutral (your p/l is not sensitive to changes in implied vol).

As you might expect, that assumption of sqrt(time) vol changes across the curve is just a useful summary assumption, not gospel. In fact, on any given day you can expect the curve changes would deviate from that model. As we saw above, the bottoms-up approach of adding/subtracting volatility with a calendar has uneven effects that won’t match up to sqrt(time) rule. Your actual p/l attributed to changes in volatility will depend on how the curve shifts and twists. Perhaps the decay rate in a vol cone could provide a basis for a more accurate scaling factor. It does require more work plus scaling to time allows us to normalize across assets and securities more understandably rather than using some empirical or idiosyncratic functions.

Conclusion

Just because the vega of a longer-dated option is larger doesn’t necessarily mean it has more vol risk.

  • We need to consider how wide the vol range is per tenor. We looked at realized vol cones, but implied vol cones can also be used to approximate vol risk.
  • We need to recognize that a steepening or flattening of vol curves means the price of straddle spreads is changing. That means a vega-neutral position can still generate volatility profits and losses.
  • Changing straddle spreads, by definition, means that vol changes are not happening at the simple rate of sqrt(time).
  • Measuring and normalizing vols (or any parameter really) always presents trade-offs between ease, legibility/intuition, and accuracy.

A Quick Thought On Hedging Tail Options

These are copy/paste replies to a friend who asked my opinion on options.

  1.  What do you think about “hard” vs “soft” deltas? I.e treating your otm deltas differently than your atm or itm deltas. Specifically not delta hedging when you buy otm wings?My response:

    Depending on the name and how the skew behaves I’d say my experience in normal conditions is OTM options have lower deltas than the model predicts. The prices are “sticky”. So if I’m running a book long lots of OTM options I’m going to lean my delta in that direction.

  2. What about if you get long vs short those wings? Would you treat the delta differently depending on that?My response:

    You can’t hedge a wing short with anything but other options so it would depend on the option hedge. In general, I don’t open selling wings just because they are “high”. I will sell expensive wings closing. Also, there’s asymmetry with respect to how you ended up with a wing position. Typically you can get long them at reasonable levels because they are the leg of a spread that’s popular in some market. For example, in oil markets, when producers buy put spreads they are handing you the wings. Yummy.

    In general, the only way I end up short wings outright is if the underlying makes a huge move through [for example] a short call spread, blowing thru both call options. Well, now I’m synthetically long a put spread and therefore short the left tail.

Shorting In The Time Of ShitCos

HTZG, GME, now HWIN.  The more slandered or shorted or ridiculous the name is the more bullish it seems to be for the stock. Just imagine explaining this to an alien.

“I bought a deli for $100mm. It’s an investment.

A deli? Well… it’s a place where people from the surrounding neighborhood go midday for some protein stuffed into wheat…umm, no, not every person in the neighborhood. Just like some of them. Why not everyone? There are other delis I guess. And a McDonald’s. Oh, you have those too? Yea I love the fries myself. Ah, yes back to the deli. Right, so the deli actually has to buy the ingredients. Correct, it doesn’t grow them. Slaves? What? No, no, no. Those people are called “employees”. I have to pay them. And yes, that guy needs to be paid too. IRS. We call him IRS.

Did I mention it has the best dills?”

The entire shorting business model appears to broken. In a period where concentrated shorts are getting lit up, in a period where diamond hands combined with brick brains, shorting just looks like return-free risk. Or at least the style where you try to recruit support after establishing the short.

I think @Mephisto731 is correct. Probably super correct. The best time to sell insurance is after the earthquake blows out your competitors.

You’re sneering. Fine, I’ll play along.

Common Objections To Shorting

It’s common for shorting detractors to mock the strategy as negative EV for 2 reasons. I’m just going to annihilate them now so we can get to a more productive discussion.

  1. Stocks have positive drift (aka “stonks only go up”)

    I get it, you are fighting the most fundamental risk premia. The “equity risk premia”. First of all, that’s debatable. After, all most stocks go to zero. Stock indices have risen over time thanks to rebalancing. But more clinically, the negative drift, can be offset by just offsetting the beta. You can short the target and get long a basket to sterilize the drift. So, in practice, and possibly in theory, this positive drift objection can be put to rest.

  2. Stocks have unbounded upside but limited downside

    This has no bearing on the EV of shorting. Anyone familiar with options understands that individual stocks have positive skew. If a stock is $100 despite everyone knowing that it is bounded by zero and infinity then the odds of it going down are the counterbalance. And the fact that most stocks go to zero is in keeping with that understanding. So, stop citing the unbounded upside as a reason why shorting is negative EV. Remember EV is a sumproduct of terminal prices and probability.

That said, shorting is no stroll in the park. We just don’t need to fabricate objections like the ones above to show that.

The Real Reasons Why Shorting Is Difficult

  • No limit to arbitrage on the short side

    First, think of the long side. I’ll paraphrase Sam Bankman-Fried’s explanation from his recent Odd Lots interview:

If AAPL stock price went to $1 tomorrow, Warren Buffet or whoever would just buy the whole company. It makes billions of dollars in earnings and you could just buy all the earnings for less than the stock price if it got low enough. But on the short side, there is no mechanism to moor the stock to reality (although as we learned from the Archegos saga, a secondary to feed the ducks, has consequences).

This lack of limit to arbitrage doesn’t change the EV of the stock which is already balanced by probabilities, but it does change the path behavior. You need to borrow shares to be short, and any share borrowed means a future buy order. So inflows of cash can cascade into forced covering since the short-seller is effectively levered.

  • The negative gamma effect

    I’ve explained this before with respect to shorting, but I’ll re-hash it simply. When a fund sizes a short it does so as a percentage of its AUM. Say the short is 10% of its AUM. You can think of the AUM as the denominator and the dollar-weighted short as the numerator. This ratio starts at 10/100.

    What happens if the fund wins on the trade because the stock drops 50%?

    Well, now the fund has made 50% on a 10% position, so its new equity is 105. Yet, the size of the short shrank with the stock halved. So now the numerator is 5, not 10 units. So the short is now 5/105 or 4.7%. The fund needs to more than double the size of the short to maintain constant exposure as a percentage of AUM. Both the numerator and denominator moved in a way that reduced the position.

    This looks just like short gamma. You need to sell more as the stock falls!

    When the stock rallies, the size of the short (numerator) increases, while the fund’s equity (denominator) gets hammered. Both forces conspire to force short-covering. Or buying, in a rallying market. Negative gamma. And to think, you often pay to borrow stocks, so you get the indignity of paying theta to play this game.

The Options Approach

Let’s address the ways we can use options to be short.

  • Synthetic shorts

    If you want to implement the short in the most similar way to a short stock position, then you will want to structure a “synthetic short”. Just like a stock position, it has 100 delta and no Greeks except exposure to cost of carry. But you faced that risk from the prime you borrow shares from anyway.  In this case, the borrow cost is embedded in the options but the clearing rate for that cost will be inherited from the arbitrageurs with the best funding rates.

    How to implement a synthetic short

    You buy a put and short a call on the same strike in the same expiry. To prove to yourself that it is the exact same exposure as a short stock position work through this example:

    Stock is $100
    You buy the 1 year 100 put for $10 and sell the 1 year 100 call at $10.

    The stock drops to $80 by expiration. What’s your p/l?
    What if the stock ripped to $120?

The synthetic short will have the same path risks as an actual short so let’s move on to option strategies that mitigate the path risk.

  • Outright puts

    If short-selling seems like it has negative gamma, you could always substitute your trade expression with long options. At least, you get something for the theta.  So while you will be paying to borrow, it might actually be at a better rate than you can borrow from your broker. And the moment you buy the put, the funding rate is capped at the implied cost you traded at. If the borrow gets more expensive from that point forward, your put will actually appreciate in step with its rho.

    The risks of buying puts are familiar. You can be wrong on timing, vol, how far the stock actually falls.  You can get middled. Your thesis can be right but not right enough.

    The benefit is you cannot lose more than the premium (unless you dynamically hedge…but if you are using the puts directionally then you shouldn’t be doing that anyway). This simple fact turns your strong hand into a weak hand. You always reserve the right to roll your puts down as you take profits or up to chase the rising stock. But the basic position, while risky, is path-resistant. And path is why shorting is so hard.

  • Put spreads

    Buying a put vertical (buy 1 put, sell a lower strike put, same expiry) sterilizes many of the Greeks since you buy and sell an option, including some of the borrow costs.  The tighter the strikes the more the bet looks like a pure probability play. If the strikes are wide, your further OTM will not offset the Greeks of the near put as much (if you think about it, an outright put position is the same thing as a put spread where the further OTM strike is the zero strike).

    If the stock has a lot of negative sentiment around it, depending which put spreads you choose, it’s possible you are getting a bargain if the put skew is especially fat.

Options and the “Write Down Your Thoughts” Effect

I’m not shilling for options here. I’m just pointing out that in a market that is scaring vanilla short sellers away, there are trade expressions that allow you to stay in the game at the time when you probably want to the most. Even if you decide not to use options, there is a benefit from walking through the trade construction process — it will tighten up your thinking. It’s like journaling.

Before choosing an option implementation, you should write down your answers. I’d be surprised if the answers to these questions didn’t impact how you might frame a vanilla short.

Let’s walk through questions you must answer before buying a put spread.

  • Edge: if the put spread I’m looking at pays 6-1 what do I think the true odds are? 4-1? 3-1? The amount of edge AND the fact that we are talking about a bet with a sub 25% hit ratio will dictate my risk budget.
  • Risk budget: How much am I willing to lose in premium?
  • Should I spread my risk budget over several months or is there a specific catalyst or expiring lockup that favors concentrating the bet in a single month?
  • Which put spread should I buy? Would I rather buy $1,000,000 worth of the 85%-80% put spread or the 70%-65% if $1,000,000 buys me 2x as many of the further OTM spreads. Or maybe I prefer a higher delta trade, that pays off more often but pays smaller odds. This forces me to think about price targets and the market’s relative implied pricing of those targets. It directs your attention to the meatiness or winginess of your thesis.
  • Does the winginess or meatiness of my thesis correlate to any other forces in the market or is it a purely idiosyncratic idea? For example, if you were interested in owning put spreads on a portion of the ARKK basket, then you could concentrate your put spreads on the subset of the basket that offered the best implied odds. Your thesis wasn’t specific to a single stock but more of a general liquidity trade.
  • How much dry powder do you want in reserve to roll your put spread up when the stock rallies? What thresholds would trigger rollups? Likewise, if the stock sells off, will you roll spreads down? How about down and out into a further calendar month? Will you roll down on a 1-to-1 basis (taking profits) or aggro win-big-or-go-home style where you use 100% of the collected premium to buy a boatload of further OTM put spreads?

Working through these questions refines your thinking and creates a plan for different scenarios. I find that the granularity of options and layers of relative pricing force me to “write down my thoughts” in a way that delta 1 trading can easily gloss over.

Conclusion

Short-selling is hard. Not because it’s negative EV, but because limits to arbitrage and the reality of levered return math create perilous paths. Whether the bruises from the recent mania will usher in a “golden age of short-selling” remains to be seen. But removing an entire direction of returns from your arsenal seems short-sighted. It’s a surrender to the current moment just when you should be thinking hardest about profiting from names that on a long enough time frame will have prices that match their ShitCo status. Options provide a more path-hardy set of trade expressions and may become table stakes for investors (ie hedge funds) whose mandates should not allow them to ignore the short side.


Related:

The difficulty with shorting and inverse positions

Shorting Bimodal Stocks

A Thought Exercise For Outsourcing Liquidity Risk

Understanding Edge

In my indoctrination into trading, the term “edge” was equated to the bookie’s “vig” or a casino’s “house edge”. This makes sense since I started in this business as a market maker. The interview questions I faced were focused on mathematical expectation or expected value. For example, if someone offered you a game that pays you the number that comes up on a single die, what would you pay to play? The weighted average payout of the game is $3.50. So if you can pay $3 to play, you’d make $.50 in theoretical profit. Of course, you could still lose if you roll a 1 or 2, but if you could do this every day, you’d earn 14% ($.50/$3.50) in the long run.

The basic premise of the market-making business is 2-fold: capture edge and manage risk so you can survive to actually see that long run.

  1. The edge comes from identifying the fair price.
  2. The primary risk management levers are diversification and sizing.

If you can price accurately and manage risk competently, you can crystallize the edge as surely as the Wynn prints money.

In this post, I will share:

  • the nature of edge in both trading and investing contexts
  • unbehaved edge in the real world
  • intuitions you can take with you

The Nature Of Edge in Trading And Investing

First, let’s define fair value. I will decompose it into 2 concepts.

  1. Expectation

    This can be a price that is ultimately an arbitrage. The die game from the intro or a casino game can be squeezed into this since the asset’s expectancy can be computed. With a large enough bankroll or sufficiently small bet size, it’s practically impossible to lose in the long run. Cash/futures arbitrage and creating/redeeming ETFs trading away from NAV are market examples.

  2. The liquid price

    In the market maker pasture, I was raised in, we’d call any price that was transparently and liquidly trading “fair value”. If the market for an option was “choice” or “pick’em” with deep-pocketed players on both sides then it was “fair”. We might say “fair value is $5, Goldman Sachs by JP Morgan”. In other words, a GS client was $5 bid and a JP Morgan client was offered at $5, it was trading, and there was enough size available for anyone else to basically participate. It’s a fleeting concept, but useful. We could use that price as a benchmark to compare less liquid derivatives as we looked for relative value.

With the idea of fair value established, we can begin exploring the nature of edge with a familiar toy model — the coin flip.

The Power Of Small Edges

Imagine a coin flip game. Call the toss correctly, make $1, otherwise, lose $1. Let’s pretend you could predict the coin flip with 50.5% accuracy. Sweet.

  • What’s your edge?

The expected value of playing the game is 1% because your payoff is equal to .505 * $1 – .495 *$1

  • What’s the standard deviation?

    From the binomial distribution, we know the standard dev or vol is √(.505 * .495) or 50%

  • What’s your risk/reward (Sharpe ratio)?

    I’m going to use the term “Sharpe ratio” in a specific context, as the ratio of edge to volatility. This is intuitively important since edge doesn’t mean much without a measure of variance. For this single toss, the Sharpe ratio is a measly .02 (1%/50%).

1% edge on this coin flip doesn’t seem like much. The .02 Sharpe ratio is a laughable signal to noise ratio. But as we increase N from 1 flip to many, the binomial distribution can be closely approximated by the familiar Gaussian curve [Taleb, spare my window, I’ll address reality later].

Look closely. The Sharpe ratio increases with N. Specifically, it increases at the rate of √N.

Why? Because the edge or numerator grows linearly with N while the denominator, or vol, only increases at √N. This property of edge is the foundation of trading and gambling. With enough trials, victory is nearly guaranteed. With a 1% edge on a coin flip, you are 90% certain you will be up money after 4,000 trades. So if you have 10 traders making 20 trades each business day, in one month you are more than 90% certain you are winning. In one year, you can’t lose.

Getting A Feel For Edges

Let’s look at the math in reverse. In Excel, we can use Norm.INV() to find what return corresponds to a desired probability for a given EV and vol. Let’s say we want to be 95% certain we make money. In math language, we are interested in the point where the 5th percentile return of the CDF is equal to 0.

We want to ask Excel:

How many trials do I need to have so that my Sharpe ratio sets my 5th-percentile return to zero?

To do this let’s standardize the vol to 1. The equation we need to solve is:

NORM.INV(5%, EV, 1) = 0

To solve for EV we use Excel’s goalseek function. We find EV = 1.645

Since we standardized the vol to 1, then we have discovered that at a Sharpe ratio of 1.645 (again Sharpe is EV/vol), the 5th percentile return is 0. That is the Sharpe ratio we need to be 95% certain we make money.

Remember that having 1% edge on a single coin flip only has a Sharpe of .02

But as we increase N, the Sharpe increases by √N :

SR of 1 trial x N/√N = SRN
.02 x N/√N = 1.645
N = 6,764

If we flip the coin 6,764 times, we are 95% sure we will make money even though we have a tiny edge on a volatile bet.

Let’s recap in English what we did here:

  1. Compute the risk/reward or Sharpe for a single bet
  2. Figured out the risk/reward needed to be 95% certain we will make money on a series of bets
  3. Computed how many times we need to play to achieve that risk/reward

Let’s look at the relationship between a single bet Sharpe to how many trials we need to be 95% certain we win.

  • If we have .02 Sharpe per bet, we need to do 25 trades per day for a year to be 95% certain of making money.
  • If we have .10 Sharpe per bet, then 1 trade per day will help us realize the same risk/reward over the course of a year.

This table highlights another important point: by increasing the Sharpe per bet by an order of magnitude (ie from 1% to 10%) we cut the required number of trials by 2 orders of magnitude (27,055 to 271).

Think about that. The improvement in Sharpe leads to a quadratic reduction in trials needed to maintain the same risk/reward for the series of bets.

Inverting the logic:

If the risk/reward of your bet is halved, you need to bet 4x as many times for the strategy to maintain the same overall risk/reward.

From Trading To Investing

The domain of many individual bets fits more under the umbrella of trading. For investing, we tend to think of the annual Sharpe ratios of investing styles or asset classes. Without looking this up, I’d guess that the SP500 has a long-term Sharpe ratio of about .40. I’m estimating an 8% annual return divided by 20% vol.

We can use the same math we did above to see how many years we’d need to invest to be 95% certain we did not lose money in nominal terms. Turns out the answer is 17 years. The table below finds the number of years for other combinations of expected return and volatility.

Years Required to Be 95% Sure of Profit

The Real World

Bell curves are great to build intuition but they are not reality. We can’t really be 95% sure we’ll make money by holding stocks for a generation because the historically sampled returns and volatilities are just that — sampled. We don’t know what the actual distributions are. Fat tails, skew, other moments I don’t even know about. 

We can use a highly skewed bet to demonstrate how volatility can distort our impression of risk. This renders the Sharpe ratio useless in highly skewed scenarios.

Consider 2 stocks, both are fairly priced at $100. We’ll call them Balanced Corp and Skewed Corp.

Balanced Corp is 50% to go up or down $10.

Skewed Corp has a 90% chance of going up $3.33 and a 10% chance of dropping $30.

Using the bimodal distribution we find that the stocks have the same volatility. However, they would have different straddle prices if there were options listed on them.

(It’s a good exercise for the reader to use what we know about expected value to manually compute the call and put prices).

So here we have 2 stocks with the same true volatility but different straddle prices if we compute them via expected value. Of course, we would not use B-S for a stock that was discontinuous and was going to magically open at one of 2 prices in a year. But this does show how the effect of a strong skew would suppress the value of a straddle for a given level of volatility. 

This is actually more intuitive than it appears. FX carry is a highly skewed trade that might exhibit minimal vol on a daily basis. The volatility imputed by the straddle understates the risk because it derives most of its value from the behavior of daily moves, where the risk of a jump will be better reflected in the cost of OTM options. In the above case, the Balanced Corp 90 put is worthless while the 90 put on Skewed Corp is worth $2 (10% of the time it finishes $20 in-the-money).

So if you use straddle prices to impute volatilities which are then used to calibrate Sharpe ratios, you may be understating the risk of highly skewed assets. Your risk/reward ratio is actually overstated which means it will take far more trials to realize your edge, assuming you actually have any. And remember how diabolical the math is…if your Sharpe ratio is overstated by 2x (let’s say you think it’s .8 and it’s actually .4), then you need 4x the number of trades to maintain the same assumptions about making or losing money. How would you feel if you found at the long-run for your given strategy wasn’t 10 years, but 40?

Takeaways About Edge

Self-aware investors and traders are always questioning their edge. Evaluating a track record or doing post-mortems on your own strategies requires being able to handicap the true distribution of your trades. The more Gaussian they look (for example if you play limit poker instead of no-limit) the easier it is to ascertain the strength of your edge statistically. You can tell the difference between bad run vs a change in the quality of your edge. Some runs would be almost impossible if your edge was real.

Edge is scarce. When we prospect for it, we should expect to mostly find fool’s gold. There are many reasons for this.

On skew

While both high volatility or high skew make it harder to determine if you have an edge statistically, skew is especially tricky. It is hard to see without liquid option surfaces. Here’s an intuitive way to see how skew distorts reality. Imagine finding a video poker machine that didn’t show its payoff table. Under the hood, it gives slightly worse payoffs on a pair of Jacks or better, but offered a billion to one on the Royal Flush. You could play that machine for days or even weeks and never realize you had massively positive EV.

On sample size

  • Having a small edge or number of trials makes it hard to verify an edge. Remember that when evaluating anyone trading highly volatile assets (ie crypto), engaging in highly skewed trades (carry, staking tokens for yield, option selling), or making a few concentrated bets per year (much of discretionary fundamental investors).

  • Remember the phrase “to think in N not T”. If there is a flow that shows up every day for a month do you have a sample of 30 or just 1 bit of behavior spread over 30 days? It’s the philosophical version of how auto-correlation artificially inflates N.

On luck vs skill

  • If you have negative edge, trade less. Short-term variance may turn up a friend named “Luck”. In the long run, she’s lost your number. 

  • In chess, a difference in ELO can be used to handicap a match between 2 players. Chess has no element of randomness. The signal is extremely strong. Backgammon has randomness, so the predictive strength of the ELO spread increases with match length. This comment in a chess forum cements this:

    While Magnus Carlsen would stand virtually no chance against the top chess programs, the Elo rating difference between Extreme Gammon, (the best bot) and the top humans is more like 75 points, so XG would be something like a 2-1 favorite in a 25-point match against the top human player.

The importance of edge

  • When I was a market-maker we were always on the lookout for a new source of edge (perhaps a new name to trade or spotting a new flow to trade against). Edge is pure gold. Its scaling properties are amazing if it’s genuine. We were encouraged to not worry about risk if we could find a legit edge. The firm would find a way to hedge some portion of the risk if the edge was worthwhile, and you could always use sizing to manage the risk. Finding edges was top priority. It’s what you build businesses around.
  • A 1% edge in a stock or ETF is enormous. Imagine buying a stock that was trading “fair” for $50 for $49.50. This is an order of magnitude more edge than HFTs earn. Hold my beer now as we do options. If the fair price for a call or put is $.50 and the bid/ask is $.49-.$51, you are giving up 2% edge every time you hit or lift. Before fees! Option prices themselves are more volatile than the underlying stock so from the market-maker’s perspective the Sharpe of the trade might be pretty small (getting 2% edge on a security that might have a 100% vol for example). But think of the second-order effect…the optical tightness of the market and high volatility of option prices means it can take many trades before the option tourist realizes just how much the deck is stacked against them. For independent market-makers, like I was 10 years ago, the tight markets made our business worse because our risk and capital limits did not allow us to keep pace with the volume scaling required to make up for the smaller edge per trade. But the large market-makers welcomed the increased transparency and liquidity because they could leverage their infrastructure effectively. 

  • If you make a 50/50 bet with a bookie but need to pay them 105 to 100 you are giving up 2.5% per bet (imagine you win one and lose one…you are down 5% after 2 bets). Now think of a vertical spread or risk reversal in the options market. Pay up a nickel on a $2 spread? Might as well have a bookie on speed dial.

Edge in the real world is nebulous

Firms with provable edges don’t try to raise money. If it’s provable it does not need more eyeballs on it. The epistemological status of edges that are trying to raise money is unknown. Many will never get the sample size to prove it. Asset management is the vitamin industry. It sells noise as signal. It sells placebos.  There will always be one edge that never goes out of style — marketing.

True mathematical edge is hard to find.


Related:

  • Nick Maggiulli’s Why You Shouldn’t Pick Individual Stocks: On The Existential Dilemma Of Stock Picking (Link)

  • Moontower Money Wiki: Time And Human Capital (Link)

Real Talk On Options Trading

This is a cleaned up version of a “real talk” thread I did based on questions I get in my Twitter DMs about option trades. Novice traders/learners might find them discouraging but I think in the long-run knowing this now will help you plan your learning more efficiently.

There’s 3 large categories of using options. Each has a very different starting point. Be very aware which category your strategy targets.

  1. Options as expressions of directional axes

    If this is your strategy, then 95% of the work is upstream of your option implementation. Your fundamental work may uncover a forecasted distribution that disagrees with the option surface. Most DMs fall in this category and don’t realize it. Questions like should I sell the 5% put to buy the 10% put or whatever.

    I ask “what fundamental work have you done that suggests the surface is wrong?”

    For more depth, see Structuring Directional Option Trades (Link)

  2. Relative value volatility trading

    If you are doing this at home, may your god be with you. Correlation trading, IV vs RV, cross-sectional relative value. Do you understand funding, rates, and dividends? This is a trapdoor thru the Earth’s crust into a steaming pit of hot magma. And this just gets you to the R term in Black-Scholes which we hand-wave as a given so we can get to implied vol. Speaking of implied vol, are your IV computations even correct? Do you know how to clean vols for time and events?

    A clue to get you started:

    It’s Monday and the Friday straddle is $3.
    It’s Friday and the Thursday straddle is $3
    Both have 5 days to expiry. Your off-the-shelf model spits out the same implied vol in both cases.

    Uh oh.

    Welcome to the vol time vs wall time.

    This is the option trader’s version of accounting. It’s the foundation for measurement and cleaning data.

    The dashboard required to see what QVR’s Benn Eifert calls “disturbances in the force” requires expensive data and infrastructure.  And even then, you are trading for edges smaller than a bookie. Is your bankroll appropriately sized?  There is a good reason why most vol traders I know who go out on their own don’t do “volatility trading”. Instead they focus on more discrete bet types like special situations. SPAC trading is a recent example. You can also check out Kid Dynamite’s archive to see how an ex-institutional trader approaches markets as an independent.

  3. Using vol flows to generate directional alpha

    This is all the vanna stuff. It’s relatively new as the option market “tail wags dog” effect has amplified in the past few years. My own anecdotal experience is flows absolutely matter. Especially if options are priced too tightly ultimately providing more liquidity than the underlying. I’ve seen it in commods which is mostly the vol markets I trade. My own incorporation of it was understanding who held large chunks of OI and trying to anticipate players’ behavior on how they might manage around the greeks. Large hedging flows occur in oil and ags for example. Understanding their rhythm and triggers is important. Understanding how certain areas of the surface become “infected” is critical for survival. Although I never systematized my analysis, all discretionary vol traders always have a mental framework around “who’s holding the risk”. Accounts like @nope_its_lily@jam_croissant,  @SqueezeMetrics, and @HauVolatility are being methodical and public about how they do it.

    I link to them in the Moontower Volatility Wiki: Flow Tracking

The major takeaway from all this is know the source of alpha you are using options to access. If you ask me about a single stock option trade, I’m just going to ask you about your fundamental research. That’s the hard part. If it’s done well, the option part is comparatively trivial.

If you have an opinion on the vol, it better not be as naive as “well the realized vol is X or skew is Y”. You are pointing to info any donkey can easily see. Instead, you need a composite view which has seams nobody else can see.
So #1 isn’t really about options. #2 is a game very, very few playing from home can play (@darjohn25 and a few in his sphere are people to follow). And for  #3, you know who to follow.


You can think of my view of option trading as a sub-category of the taxonomy @therobotjames outlines:

Broadly, there are 3 types of systematic trading strategy that can “work”.

In order of increasing turnover:

1. Risk premia harvesting
2. Economically-sensible, statistically-quantifiable slow-converging inefficiencies
3. Trading fast-converging supply/demand imbalances

His full thread.

Twitter Reminds Me Of The Trading Pits

Floor trading and fintwit share an overlapping dynamic: “cooperative competition”. I’ll lay out the floor trading ecosystem so you can spot the analogies.

The Players

Let’s classify the traders on the floor:

  • “Locals”

“Locals” are independent traders. They trade their own money and secure the right to trade on the floor by owning or leasing one of the limited seats which represented an ownership stake by the exchange’s “members” (this was before demutualization). Since they are not as well funded, locals trade smaller and focus on extremely high Sharpe (although they wouldn’t use this lingo), low capacity opportunities. They tend to be seasoned and run tight risk. Many were independent-minded misfits, allergic to jobs and W2s.

  • Prop firms

Some of the big prop shops included Group One, Timber Hill (IBKR), Jane Street, CTC, DRW, Cooper Neff, O’Connor, Cutler, Wolverine, and my alma mater SIG.
Traders representing these firms usually have a salary and bonus. The bonus could be discretionary, formulaic (ie 20% of your p/l), or a hybrid. Firm traders tend to be more pedigreed, younger and less autonomous. They are well-resourced, well-capitalized, and can trade big often answering to the “mother ship”.

[A little digression. There’s a 3rd class of trader which is a “backed” trader which blurs local and firm. These firms were usually tied to clearing firms and offered economies of scale in risk management, software, data, clearing rates. I left SIG in 2008 to get backed. From 2009-2012, I put up escrow with a firm whose business was to bankroll traders. In my case I got 70% of my p/l, and I could sleep knowing I couldn’t lose more than escrow. Giving up 30% was worth it to me. My backers were well-capitalized and allowed us to trade as big as firm traders (but our escrows ranged from 6 to 7 figures). We answered to risk managers as well. But for the most part we were left to build teams and businesses as we saw fit.]

Floor Dynamics

Now that we know the players, let’s examine the interactions.
  • The floor is competitive.

    If a broker lifts the pit’s offer for 500 contracts, market makers fight for “recognition”. If 10 people scream “sold!” the broker gets final say on who they “heard”. Everyone wants an allocation. The power of the broker’s discretion means politics, justice, and game dynamics all come in to play as traders scrape for market share.

  • The floor is also cooperative.

    It’s an emergent trust system since you are moving large sums of money by the sound of your voice. Even a den of thieves has a code of ethics. Word is bond. Welch on a broker and you might as well be invisible to them from that point forward. This trust system is further reinforced by incentives. Yes, you are competing with each other, but the broker flow is the ultimate source of compensation. It’s the life blood of the pits. Everyone on your exchange wants the flow to keep coming. Traders, brokers, and the exchange itself. There’s an informal favor system or quid pro quo that ultimately serves the broker’s clients not having errors. Nobody wants clients taking their ball and going home.

  • The floor is made of humans

    You cannot help but make friends with many of the people you stand shoulder to shoulder with all day. After all, you share common interests. You want to know each other’s comp deals…I joined my backer because of a friend I made on the floor (thanks Mike!). You exchange trade ideas with the people you are close with. It was even common to share trades. If I like selling December vol and manage to get a big allocation on the trade, I might give some up to my like-minded buddy who was busy yelling at his clerk outside the pit. He would do the same for me.

  • Every day is an interview

    You are watching each other, sizing each other up, looking to hire or maybe get hired. You are watching and learning how the best traders were interacting with the broker as well as each other. Who’s smart? Who’s clever? Who is someone weak, raw, or exploitable? Whose someone you wouldn’t want to physically fight for their spot in the pit? Physical real estate in the pit was not assigned but earned through an invisible consensus of respect or even physical force. Go ahead, try to move me. Standing close to the brokers on the top step of the ring…that’s Central Park West. Down in the soup with the newcomers…might as well be in NJ.

  • The information game

    What can I share that isn’t too valuable but sounded valuable? There’s the constant sandbagging about what pits had edge. There could be 50 stinky dudes crammed into a pit the size of an airport Starbucks and if you asked them how the trading was in that pit they’d say gravely, “it sucks in here, nobody makes money”.

    Here’s a veteran move — sniffing out a disgruntled market maker at a large firm. Perhaps a young trader upset with their bonus. They might trade info for acceptance. They are dreaming you’d hire them (somehow oblivious that a willingness to divulge secrets is not a quality you advertise to future employers).

The dynamics of competitive cooperation always reminded me of the old wolf and sheepdog cartoons.
Sure you might get in a nose-to-nose screaming match for an allocation to the big order that came thru the pit that morning. But later you’d grab a beer at Suspenders.

Faster Learning

The proximity to your competitors in the pit was an opportunity to learn faster. For example, if I hear the Citibank traders talking to their Clearport trader on the headset and then aggressively hitting a bid in the pit, it’s safe to assume they got an exclusive look at a deal that implies vols are lower. (This is why traders usually cover their mouths when speaking into their headsets. Just like NFL coaches do on the sideline when talking to their upstairs booths.) In fact, seeing your competitors with headsets or “hoots” in the first place tipped you off to the possibility of inter-market arbitrage.

Another strategic example. Suppose there is a SIG trader in your pit. You will watch what they do because you know they see all options flow globally. If they are suddenly the best offer in your options pit you can adjust your opinion of fair value lower, both leaning on their offer and knowing their broad access means they can see moments into the future. Go deeper. If SIG was a seller of BBH vol you might ping your market maker camped in the DNA (Genentech) crowd to see if Susq was trying to leg correlation trades.

On the floor everyone loved the Timber Hill traders. Timber used their tentacles in every market to feed a brilliant cross-sectional vol model (the “box”) that supplied each of their traders with a fair value for every option. In doing so, they also removed discretion from their traders. So every other market maker in the pit came to see Timber’s undisguised behavior as valuable market intel to calibrate against their own opinions. You could “lean” on their markets to “free roll” on their transparency.

[The Timber model optimized for holistic edge/risk targets but ignored the role of negotiation tactics to maximize individual fills. It was a top-down tradeoff that was the antithesis of the SIG poker style. This is not a knock against it. They systematized a different strategy with clinical precision. In fact, seeing how there was many ways to win was one of the beauties of being on the floor. To demonstrate the philosophical difference consider the following scenario:

If an option was worth $1 the Timber trader would always make a balanced market say .95 bid, offered at $1.05. While a trader with discretion, might in anticipation of a sell order make the market .90 bid and offered at $1.00. Why? Expected value…if the order is any more than 50% to be a sell order than you will make more money on a .90-1.00 market than a fair market of .95-1.05.]

The Twitter Pit

Learning from watching your competitors is common in business especially public businesses. Same in sports. Maybe not at first but everyone eventually gets smarter when they watch Belichick go for it on 4th and 2 from his own 40. Everyone got smarter when the first baseball manager shifted the defense based on the batter’s tendencies.

The public has the opportunity to learn from these strategies that cannot be hidden. (The Patriots practiced the pass defense that was disguised as a goaline stand for the entirety of the 2014 season only to use it once — on the last play of the Super Bowl, leading to Malcolm Butler’s interception of Russell Wilson to seal a Pats victory. They intentionally hid the play all year understanding the power of its deception should not be wasted until the stakes are sufficiently high.) I miss the multitude of learning opportunities that were strewn all over the trading pits. I miss hearing and seeing the competition with my own senses.

Luckily we have Twitter. Twitter gives me the same sense of a cooperative competition. You can spend parts of your day shooting the bull with people tied together by the same callings of opportunity and learning. You are watching the tactics people use to build their online businesses. You are seeing how accounts posture and market. You can watch debates on mutuals’ timelines.  If you think about the floor dynamics I presented above, see if you can spot the analogies as you browse Twitter. They are everywhere. Who’s subtly or not-so-subtly angling for their next job? Whose bartering commoditized info that sounds good in exchange for valuable followers? Who relies on brute force? Who is the equivalent of the dishonest brokers playing the brazen quid pro quo game?

It took me awhile to realize it, but Twitter fills the void I left on the floor. There must have been a time when the end of the floors seemed unthinkable. That’s where we are with Twitter today. We’ll see what happens, but in the meantime…enjoy it.

Jesse Livermore Bangers

  • Upside Down Markets (Interview with @JesseLivermore)
    Invest Like The Best Podcast

    The blogger behind the Jesse Livermore pseudonym did a second public interview. He talks about his latest blog post. Here’s the description from Invest Like The Best:

    My guest today is Jesse Livermore. I’ve worked with Jesse as part of our research partners program at O’Shaughnessy Asset Management for years now. Whenever there is a huge, important, and complex issue to be studied, I believe he’s among the best minds in the world to tackle it. He did that recently on the topic of what he calls “upside down markets,” which is the topic of this conversation. We seek to answer the simple question: against a horrible economic backdrop, how can the stock market be near all-time highs? Jesse explains in detail the impact that fiscal policy has had on the market and may have in the future. Please enjoy this master class in upside down markets.

    His work is dense but he’s gifted at breaking things down into small steps to impart his findings. It’s a very rewarding experience to read his work because there is a meta-learning experience built in — how to think from first principles. The joy of building an understanding from accounting identities is a fresh break from the pseudoscience feel of macro theories. It seems fitting that this exceptional work would come from someone outside of finance.

    Go further:

    • The full Upside Down Markets post (link)
    • He mentions how exceedingly high valuations are increasingly dependent on liquidity or what he terms “networks of confidence”. I wrote a thread musing about liquidity (thread)
    • Jesse’s liquidity thought experiment (thread)
    • Jesse: Market-Cap Weighting under Policy Dominance (Fiscal nGDP Targeting): The disadvantages of market-cap (mcap) weighting are well known. Here, I want to elaborate on a benefit that it might offer, which I mentioned in recent ILTB podcast. (thread)
  • Profit Margin Mean Reversion Is Not Relevant (link)
    @JesseLivermore

    This post is one of my favorite older Jesse posts because he uses his first principles approach to dissect Hussman’s contention that the mean-reverting qualities of profits would be a tailwind to stocks. The upshot: ROE not profit margins are what matter. There’s a lot to learn from this deconstruction.

    A tangential takeaway for me was how stock market wealth is fungible with household savings at the aggregate level, even if the dynamic is regressive. It’s sustainable so long as labor continues to maintain positive savings rates. Unfortunately this say nothing of the risks to social fabric if equality widens in step.

    The post is 6 years old and it’s interesting to see that the recent embrace of fiscal expansion could help to narrow equality, as least at first glance.

A Thought Exercise For Outsourcing Liquidity Risk

A portfolio manager shorting a stock will size a position not just based on price target and conviction but based on the risk relative to bankroll and relative to the liquidity. There are conventional ways to do this. Volume, days-to-cover, variance, 90s nostalgia factor (kidding on that one…worrying about that going forward is like closing the barn door after the horse escaped). While these inputs inform a sizing decision, the bottleneck in the risk management process is not in the sizing. It’s in the remedy when your sizing turns out to be wrong. Eventually it will be. Your plan needs to tolerate that eventuality.

Most plans are to cut risk by buying back some percentage of your shorts. This is like trading with a stop. You are constructing an option since you cover as the stock goes against you and you add as the stock goes in your favor (remember if you short a stock and it falls, to maintain exposure as a percentage of AUM you need to short even more).

The problem with this plan is it is soft optionality. It’s not the same as buying a hard option like a deep ITM put, or buying an OTM call to hedge your short position. Hard options protect you from gap risk. You know, that thing that happens when a stock is halted. Or when the US goes to sleep. Or when a gamma squeeze creates a massive imbalance.

To improve risk management, managers should at least entertain the question: “if I wanted to buy X amount of deep ITM puts instead of shorting shares how much would it cost?”

The answer to that question comes from market-makers who sit in the middle of the marketplace. They hoover up market intel to synthesize a price so you can know exactly how much it costs to express your view with a hard option. Sure, that price embeds a consultation fee in the form of a vig, but at least they are on the hook for mispricing. Not you.

A market maker’s job is to price the spread between soft optionality and hard optionality by gauging the liquidity required to dynamically hedge. Market makers are also in highly competitive, low margin businesses. If you pass on their price for the hard optionality you must ask yourself…”is my assessment of the liquidity/gap risk that much better than theirs OR are their margins excessive?”

At the very least, you can consider this reasoning a sanity check before you size up a big short.

Interview Questions A Market Maker Gave Me in 1999

SIG is well known for asking probability questions to filter trainees. This is not surprising. They view option theory as a pillar of decision-making in general. Thinking in probabilities takes practice which is why they like to look for talent amongst gamers who make many probabilistic decisions and need to interpret feedback in the context of uncertainty. They require many hours of poker during  “class”. In this 3 month period, junior traders live and breathe options in lovely suburban Philly after apprenticing (“clerking”) on a trading desk for about a year.

Here’s some of the questions I remember from my interviews in 1999.

  1. You flip a single die and will paid $1 times the number that comes up. How much would you pay to play?
    • Suppose I let you take a mulligan on the roll. Now how much would you pay (you are pricing an option now btw)?
  2. My batting avg is higher than yours for the first half of the season. It’s also higher than your for the second half of the season.

    Is it possible your avg for the full season is higher than mine?

    (Hint: Simpsons paradox)

  3. You are mid game that you have a wager on. Opponent offers to double the stakes or you automatically lose. (Like the doubling cube in backgammon)

    What’s the min probability of winning you need to continue playing?

  4. You’re down by 2 with seconds left in regulation basketball game and have a 50/50 chance of winning a game if it goes to overtime. You have a 50% 2-pt shooter and a 33% 3-pt shooter.

    Who do you give the ball to?

    (simple EV question)

  5. You are given $1,000,000 for free but there’s a catch. You must put all of it into play on roulette.

    What do you do?

  6. There’s a 30% chance of raining Saturday. 30% chance of raining Sunday.

    What’s the probability it rains at least one day?

To encourage you to try before looking up the answers, I’ll make it annoying…the answers are somewhere in this thread.

I wrapped that thread with a short post on Trading And Aptitude (Link)