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.
- The edge comes from identifying the fair price.
- 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.
- 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. - 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 √(1-(2*.505 -1)²) or .9999
- 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 .01 (1%/99.99%).
1% edge on this coin flip doesn’t seem like much. The .01 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 16,000 trades. So if you have 20 traders making 40 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 .01
But as we increase N, the Sharpe increases by √N :
SR of 1 trial x N/√N = SRN
.01 x N/√N = 1.645
N = 27,060
If we flip the coin 27,060 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:
- Compute the risk/reward or Sharpe for a single bet
- Figured out the risk/reward needed to be 95% certain we will make money on a series of bets
- 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.
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