One notable point I forgot to add to the post: G. I. Taylor was not merely a random British physicist reading a magazine with the pictures. He not only worked on the Manhattan Project, but was one of the ~10 scientists actually present at the Trinity test itself. So while this is presented as him being clever and working things out from first principles, in fact the "knew" the answer already, and then figured out how to back into it from open sources to publicize what the yield was without breaking classification.
...I guess that does undermine the point somewhat, oh well.
For exploring dimensional analysis in physics, here is my "periodic table" for physics, arranging 50 types of the most common units by their factors of length and time, with different tables having different charge and mass factors: https://substack.com/@enonh/p-148207296
Using charge instead of current as a base unit made the relationships much more clear. The major thermodynamic unit types are also listed, but not geometrically arranged. It also has most of the most useful constants. The unit type names come from my favorite software, Alan Eliasen's physically-typed language / calculator / Swiss-Army chainsaw, Frink (frinklang.org).
Δ = sqrt(2Q/λ). - i still don't fully understand this λ term in this, if the orderbook ask is lets say 100Q @ $100, 150 @ $110 and 210 @ $120. What is the λ term for the price impact of 1,000Q units, is it correct to say its sqrt(2*1000/210) = 3.08? what am i doing wrong?
Your lambda is [change is shares offered] / [change in price], so in your instance you added 110 shares as you moved $20, so every $1 rise in price is expected to result in an additional 5.5 shares offered.
Just clarifying here, if 100 are offered @ $100, and sequentially 150 are offered @ $110, and 210 @ $120, then sequentially change in shares offered is +150/$10, and +210/$20?
This would make lamba = 15, and sequentially 21. Another way to define lambda would simply to be slope (or average slope if you ignore dSlope), in this case 18.
And so to definitively answer Aces, the impact of a 1000 units would be = sqrt(2*1000/18) = 10.5
...10.5 what? 10.5% is obviously not it. and 10.5x is not it either.
could assume here the 5.5 shares pre dollar and just divide 1000/5.5 = 181.8 or $181.8 higher than where we started which was $100.
I'm still a little square not circled on this one unfortunately
Any good physicist will tell you that the speed of light in a vacuum is c = 1
This was a great read! Brings back fond memories; I recall thinking this was magick when I first sat through a lecture in which the professor applied dimensional analysis to another one of our favourite gadgets, the harmonic oscillator.
It’s not clear to me that % is a unit that can be dimensional-analysised in the equation; a priori I would have said it’s a dimensionless quantity. Perhaps it’s a stand-in for “return, cost or risk per invested capital” but this is still units [$/$]: dimensionless.
It's a slight abuse of notation. If you prefer, we could recast volatility, spread, and impact as having the dimensions of $, and then divide everything through by the mid price when we finished.
The way the G. I. Taylor story was told to our class was that, such was his accuracy in estimating the power of the device, he was suspected of being a security risk and counterintelligence officials were sent to find him in Cambridge, ironically when some of the Cambridge Five spy ring were nearby at the time. (I have no idea if the years even match up - the story has likely been embellished over the years.)
Sorry to post so late. I was revisiting this wonderful post and realized something was bothering me about the dimensional analysis…
Vol should be % / t^0.5, I think. Just as how expected volume V grows with the time unit of measurement, so does Vol (also like miles per minute vs miles per hour). That would leave the final units of the t-cost calculation as %/t.
When trying to figure out what I was missing, I realized the unit analysis of expected t-cost being %/t might be truly correct…
Repeated price impact has to be linear (otherwise you could arb by buying slowly and selling very fast). This dimensional analysis might actually be saying that, for your given time period and other trade parameters, here is your % t-cost. But if you repeat that same trade 3x in a row (for 3 periods of t), you should expect to pay 3x as much (not sqrt 3 like the formula might naively imply).
In practitioner models I expect this could be solved by varying the scalars per unit of time expected for the trades.
One notable point I forgot to add to the post: G. I. Taylor was not merely a random British physicist reading a magazine with the pictures. He not only worked on the Manhattan Project, but was one of the ~10 scientists actually present at the Trinity test itself. So while this is presented as him being clever and working things out from first principles, in fact the "knew" the answer already, and then figured out how to back into it from open sources to publicize what the yield was without breaking classification.
...I guess that does undermine the point somewhat, oh well.
Great post, brings back memories of physics classes!
What are the good papers on the impact to follow further pls?
I’m interested in learning more about K!
Is “K” at all related to kurtosis?
Do you just estimate it via regressing to the price impact of previous orders given the other factors?
Is it stable over time?
For exploring dimensional analysis in physics, here is my "periodic table" for physics, arranging 50 types of the most common units by their factors of length and time, with different tables having different charge and mass factors: https://substack.com/@enonh/p-148207296
Using charge instead of current as a base unit made the relationships much more clear. The major thermodynamic unit types are also listed, but not geometrically arranged. It also has most of the most useful constants. The unit type names come from my favorite software, Alan Eliasen's physically-typed language / calculator / Swiss-Army chainsaw, Frink (frinklang.org).
Δ = sqrt(2Q/λ). - i still don't fully understand this λ term in this, if the orderbook ask is lets say 100Q @ $100, 150 @ $110 and 210 @ $120. What is the λ term for the price impact of 1,000Q units, is it correct to say its sqrt(2*1000/210) = 3.08? what am i doing wrong?
Your lambda is [change is shares offered] / [change in price], so in your instance you added 110 shares as you moved $20, so every $1 rise in price is expected to result in an additional 5.5 shares offered.
You're a king, loved the article. Thank you.
Just clarifying here, if 100 are offered @ $100, and sequentially 150 are offered @ $110, and 210 @ $120, then sequentially change in shares offered is +150/$10, and +210/$20?
This would make lamba = 15, and sequentially 21. Another way to define lambda would simply to be slope (or average slope if you ignore dSlope), in this case 18.
And so to definitively answer Aces, the impact of a 1000 units would be = sqrt(2*1000/18) = 10.5
...10.5 what? 10.5% is obviously not it. and 10.5x is not it either.
could assume here the 5.5 shares pre dollar and just divide 1000/5.5 = 181.8 or $181.8 higher than where we started which was $100.
I'm still a little square not circled on this one unfortunately
Any good physicist will tell you that the speed of light in a vacuum is c = 1
This was a great read! Brings back fond memories; I recall thinking this was magick when I first sat through a lecture in which the professor applied dimensional analysis to another one of our favourite gadgets, the harmonic oscillator.
It’s not clear to me that % is a unit that can be dimensional-analysised in the equation; a priori I would have said it’s a dimensionless quantity. Perhaps it’s a stand-in for “return, cost or risk per invested capital” but this is still units [$/$]: dimensionless.
It's a slight abuse of notation. If you prefer, we could recast volatility, spread, and impact as having the dimensions of $, and then divide everything through by the mid price when we finished.
The way the G. I. Taylor story was told to our class was that, such was his accuracy in estimating the power of the device, he was suspected of being a security risk and counterintelligence officials were sent to find him in Cambridge, ironically when some of the Cambridge Five spy ring were nearby at the time. (I have no idea if the years even match up - the story has likely been embellished over the years.)
Sorry to post so late. I was revisiting this wonderful post and realized something was bothering me about the dimensional analysis…
Vol should be % / t^0.5, I think. Just as how expected volume V grows with the time unit of measurement, so does Vol (also like miles per minute vs miles per hour). That would leave the final units of the t-cost calculation as %/t.
When trying to figure out what I was missing, I realized the unit analysis of expected t-cost being %/t might be truly correct…
Repeated price impact has to be linear (otherwise you could arb by buying slowly and selling very fast). This dimensional analysis might actually be saying that, for your given time period and other trade parameters, here is your % t-cost. But if you repeat that same trade 3x in a row (for 3 periods of t), you should expect to pay 3x as much (not sqrt 3 like the formula might naively imply).
In practitioner models I expect this could be solved by varying the scalars per unit of time expected for the trades.
Am I still missing something deeper here?