5 ms·

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

dcolkitt 6y ago
For continuous payoffs, Kelly sizing reduces to the square of Sharpe ratio.
- howlin 6y ago
  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.
  - dcolkitt 6y ago
    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.
    - kgwgk 6y ago
      A proper Sharpe ratio is calculated with arithmetic returns.
- fighterpilot 6y ago
  Where did you see this?
potatoman22 6y ago
Not sure if it's how they did it, but there's this: https://en.wikipedia.org/wiki/Kelly_criterion#Multiple_outcomes https://en.wikipedia.org/wiki/Kelly_criterion#Multiple_outco...
kqr 6y ago
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.
- fighterpilot 6y ago
  Suppose we have 100 sequential bets with distribution U(-1,1.1) on each. How would we apply Kelly here?
  - deleted 6y ago
    [deleted]
  - hansvm 6y ago
    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.
sigstoat 6y ago
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.
- fighterpilot 6y ago
  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.
  - sigstoat 6y ago
    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.
    - fighterpilot 6y ago
      Right so just do a simulation, no closed form solution.
      - sigstoat 6y ago
        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.
        deleted 6y ago
        [deleted]

dcolkitt 6y ago

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

howlin 6y ago
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.
- dcolkitt 6y ago
  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.
  - kgwgk 6y ago
    A proper Sharpe ratio is calculated with arithmetic returns.
fighterpilot 6y ago
Where did you see this?

potatoman22 6y ago

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

kqr 6y ago

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.

fighterpilot 6y ago
Suppose we have 100 sequential bets with distribution U(-1,1.1) on each. How would we apply Kelly here?
- deleted 6y ago
  [deleted]
- hansvm 6y ago
  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.

sigstoat 6y ago

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.

fighterpilot 6y ago
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.
- sigstoat 6y ago
  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.
  - fighterpilot 6y ago
    Right so just do a simulation, no closed form solution.
    - sigstoat 6y ago
      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.
      - deleted 6y ago
        [deleted]