You may be interested to know that Kelly's work was instrumental in a company called Axcom in the 60s. Elwyn Berlekamp, previously an assistant to Kelly at Bell Labs, implemented Kelly et al's work in early financial trading at Axcom, which was later turned into the Medallion Fund at Renaissance Technologies. Wikipedia [1] has some info on this, but I also highly recommend "The Man Who Solved The Market" (Zuckerman, 2019) for more history.

[1] https://en.wikipedia.org/wiki/John_Larry_Kelly_Jr.

You may be interested to know that Ed Thorps - Princeton Newport Partners/ the Santa Fe school work lives on at an even better performing fund called TGS Management based in Irvine.

> I used to use the principles of kelly betting back when I designed systematic HFT strategies.

possibly a dumb question, but how did this work exactly? the kelly criterion assumes you know the amount by which the coin is weighted, how would you know the equivalent for the stock market in the very near term?

You make a conservative guess. The Kelly criterion is somewhat forgiving about guessing it wrong.

Your question is not dumb: you figured out exactly what's hard about this stuff.

There's a bit of a discussion here: https://news.ycombinator.com/item?id=18484631

How did you apply Kelly to a HFT strategy? Usually those strats don't have a binary outcome so standard Kelly wouldn't fit.

Kelly goes beyond binary outcomes. The underlying principle is the same, though: you maximise expected logarithmic wealth.

To do that you need the joint distribution of outcomes (what are the possible future scenarios and how likely are they?) Estimating this well is the trick to successful application of the Kelly criterion.

Suppose we have 100 sequential bets with distribution U(-1,1.1) on each. How would we apply Kelly here?

You wouldn't unless you could vary your exposure to such a sequential bet.

Suppose you can though. For simplicity, suppose you can expose yourself to 0.4U(-1, 1.1), 40U(-1, 1.1), or any other fractional amount F U(-1, 1.1) you might like. Kelly is a technique for choosing F (maybe you had some other idea in mind like that you have to buy into a bet on U(0, 2.1) -- if so, that's nearly equivalent other than putting bounds on F -- the idea of maximizing expected logarithm will carry through to other bet structures).

Going through the motions, suppose you're starting with a bankroll B then you want to choose some ratio F=rB maximizing the expected logarithm of the bet. The distribution of your outcome is another uniform distribution U(B-rB, B+1.1rB), and you want to choose r maximizing the expected logarithm of that distribution. The details of that are probably beyond the scope of a HN comment, but you wind up with r approximately equal to 0.13624.

If you'd like you could plot the result of many instances of 100 such sequential bets with r varying. You'll find that those with r around 0.13624 will usually be much larger than for other choices of r.

For continuous payoffs, Kelly sizing reduces to the square of Sharpe ratio.

Kind of. Most simple models for continuous payoffs will assign a nonzero probability to losing all your wealth or your wealth going negative. The Kelly bet size for any thing with a nonzero chance of "ruin" is zero.

Sharpe is typically calculated on log returns. Price going to zero would weigh as negative infinity in log return space. Therefore Sharpe would also prescribe zero bet on finite chance of ruin.

A proper Sharpe ratio is calculated with arithmetic returns.

Where did you see this?

the binary outcome formulation you see everywhere is just "real" kelly boiled down. the real thing, which is contained fully in the first paragraph ("The Kelly bet size is found by maximizing the expected value of the logarithm of wealth"), has no such restrictions.

How do you maximize the E(log(wealth)) when applied to a HFT strategy? In such a strategy we have N sequential bets, each bet has a roughly normal distribution outcome with mean just above zero.

The example on Wikipedia supposes we are investing in a geometric Brownian motion and a risk free asset.

in the U(-1.0, 1.1) case you mentioned, kelly says not to bet.

optimize the value of the bet size over the expected value of the log of bankroll + betsize*outcome. you can do that for any probability distribution of outcomes.

if you can't write that in 5 minutes, then i already did half your homework for you.

> each bet has a roughly normal distribution outcome

hahaha.

Right so just do a simulation, no closed form solution.

that's not simulation.

for that trivial case, there's going to be a closed form solution. your nearest copy of mathematica can derive it for you.

not that having a closed form solution is relevant to anything. the answer is still the answer.

Not sure if it's how they did it, but there's this: https://en.wikipedia.org/wiki/Kelly_criterion#Multiple_outco...

Hi I work at a small hft firm and would love to discuss this more in detail, please contact me if you have the time.

Thank you