There was a famous futures trader on the NYMEX when I was there named Mark Fisher. I leased office space from him (his clearing firm had a large footprint) but I only spoke to him once or twice briefly in passing. We didn’t really know each other. Anyway, he wrote a book called Logical Trader and backed people to trade his system. A copy of it was laying around the office (it’s in my garage in boxes I haven’t unpacked since moving over 3 years ago). I read a chapter that happens to be available for free online.
The book was mentioned on Twitter and got me thinking a bit about its core observation:
If you subscribe to the “random walk” theory, which states that the market’s movements are random and totally unpredictable, then the opening range would not be any more important than any other price level during the trading day. Right? For example, crude oil trades from 9:45 a.m. Eastern time until 3:10 p.m. Eastern Time. If you divided that day into 10-minute intervals, you’d have 32 parcels of time (and five minutes left over). So, each 10-minute time interval would account for roughly 1/32 of the market activity. Using random walk theory, you’d expect that the opening range (established in the first 10 minutes of trading) would be the high 1/32 of the time, or it would be the low 1/32 of the time. Therefore, random walk theory would dictate that 1/16 of the time the opening range would be EITHER the high or the low. 16 Now, what if I told you that in volatile markets – not static, and not necessarily trending markets – the opening range tends to be the high or the low 17-23% of the time? Would that get your attention? Yes. Because this observation would tell you that the opening range being at the high of the low of the day roughly one-fifth of the time is what we call “statistically significant.” In complete layman’s terms, this means the opening range is not just another 10-minute interval out of 32 of them in the trading day. It has more weight than any other time interval.
Let’s take another example. Let’s say that you divide the trading day up into roughly 64, five-minute intervals. Random walk theory would state that the opening, five-minute range would be the high 1/64 of the time or the low 1/64 of the time. So it would be either of those extremes 1/32 of the time. However, in volatile markets, that five-minute opening range is actually the high or the low of the day about 15-18% of the time. So instead of about 3% of the time, as random walk theory would predict, the first five minutes of the trading day turns out to be the high or the low 15-18% of the time. Again, statistically significant. And, from a trader’s perspective, if you knew that something was going to market the high or the low 15% of the time, you’d want to know that.
In short, if the opening range is the high or low a disproportionate amount of the time Fisher concludes that the odds are in favor of an intraday breakout strategy. The gist of it is:
- Once the opening range is established, if the futures breakout of the range by say some fraction of a standard deviation then bet that they will continue.
- In the case of an upside breakout, set a stop near the bottom end of the opening range.
The book goes into sizing, money management, where to set levels and other details.
I’m not going to weigh in on the strategy’s merits because it’s not my wheelhouse. I have lots of questions and of course, come from a place of skepticism but that’s just a healthy reaction to any anomaly. It’s not yet the “work”.
But it did get me wondering about how likely you’d expect the opening price in a market to be the high or low.
First, I turned it into a simpler riddle that you could try to solve yourself or (give to some kids to noodle on).
You can dig into how I worked them out here:
There are a couple of neat ideas in the solutions. (You’ll also find the trick GPT taught me. Satisfying, clever and reusable.)
Based on tinkering with a simple random walk, it does seem that an opening price (or the zero crossover in my toy examples) being the high or low a disproportionate amount of time would not be random.
[Although that isn’t enough to suggest that there is a positive expectancy in the strategy. It’s possible the payoffs on the breakouts times their frequency don’t compensate you for the number of times you get stopped out. If any systematic traders reading this feel like being nerdsniped by researching it I’d love to see the conditional probabilities that surround the different scenarios].
Money Angle For Masochists
Let’s practice option intuition on this same problem.
The setup:
- A 30% vol asset opens at $40
- It rallies to $40.50
- Half the trading day has elapsed
What’s the probability it crosses $40 again today?
If we assume a Black-Scholes lognormal distribution with no skew (not unreasonable for a single day) we can compute the probability by turning $40 into a Z-score.
K = strike price
S = Spot price
σ = volatility
t = time (in years)
ln(K/S) is basically how much percent away $40 is from $40.50.
ln(K/S) = ln(40/40.50) = -1.24%
By dividing by σ√t we scale the 1.24% by standard deviation for the remaining time.
-1.24% / 30% * √(.5/365) = -1.12
$40 is 1.12 standard devs away.
The probability of the asset sliding at least 1.12 standard devs is 13%.
In Black-Scholes world, the probability of a strike expiring in-the-money is known as N(d2). But for short-dated options, delta is valid substitute for N(d2).
So we’d expect the delta of the $40 strike with half a trading day remaining and the asset at $40.50 to be 13%.
In the context of our earlier conversation, you might think that the probability of crossing zero (ie the opening price) is 13% but we need to make a key distinction based on using the option delta:
The delta is telling us the probability of expiring in-the-money…but our riddle is concerned with whether the price or random walk ever breaches zero even if it goes back up.
The riddle is not concerned with the probability of a vanilla option but a one-touch option.
Investopedia defines these exotic options:
A one-touch option pays a premium to the holder of the option if the spot rate reaches the strike price at any time before option expiration.
I’ve never priced one-touch options but I remember a quant trader telling me that their probability of being triggered was approximately 2x the delta of the vanilla option of the equivalent strike.
In this example, the probability of the asset touching a price less than $40 before the day ends is 13% x 2 = 26%
This is intuitive if we consider an at-the-money option that has a 50% delta. The asset is nearly 100% to touch prices on either side of the strike.
[It’s convenient and expected that option trader math gets reduced to rules-of-thumb (“straddle price is 80% of the vol scaled by time”, “multiplying the daily move by 16”, “implied correlation is ratio of index variance to avg stock variance”) since so much of flow trading is making quick decisions and on-the-fly comparisons or normalizations.]
If price changes were a random walk I wouldn’t expect the opening price to be the high or low more than 1% of the time. But the open price, while cannot be predicted, likely holds meaning once it’s established because it is a single clearing price of an auction that accumulated hours of overnight information.
[I spent almost 2 years on NYSE both as a broker in the “garage” if you are familiar with the place, and as a specialist in ETFs (in the “blue room”) . The open is the price that best clears the order book when considering the stack of market and limit orders on both sides. But consider this scenario —
- At a price of $40.23 there’s an imbalance of 10,000 shares for sale
- At a price of $40.22 there’s an imbalance of 75,000 shares to buy
You can expect the stock to open at $40.23 and for the specialist to buy the 10,000 shares for their own account and then to display a market with $40.22 bid for size to induce buyers. The opening price had information in it.]
Further reading:
- Why We Use Logreturn in Finance (Quant Factory)
- Understanding Variance Time (Moontower)This post ties in nicely with another riddle:
In the option demonstration above I said there was half a day until expiration.
What time is it?
Hint: The answer is not the point. And you won’t get it anyway. I’d consider your response a success if you can just identify what the inputs the answer depends on. Godspeed.
If you use options to hedge or invest, check out the moontower.ai option trading analytics platform