I disagree. First most Kelly blowups happen precisely because of bad formulae. I didn't write the Bad Kelly thread and the other threads for my health.
You're stuck on the fiction that there is some return probability distribution out there that you can reach out and touch, if only you could calculate its parameters with some certainty. I reject this fiction. My formulas prove that all you need are accurate recordings of the trade returns. The mean tells you if you have positive expectation. Beyond that, the standard deviation, the skew, etc. matter not a wit. You don't need any of that artificial stuff to figure out your betting fraction.
There is no uncertainty in the trade returns. There is only uncertainty in the phantom distribution you imagine they adhere to.
I agree.
There is no uncertainty in trade returns.... in the past. You know exactly what they are. You can calculate what the right Kelly should have been in the past precisely.
This is no use in determing the right fraction in the future. I assume that's what we're trying to do once we've moved to the real world and away from the fascinating but financially unrenumerative world of problem solving.
So how do we determine the right fraction in the future? We can assume that the future will be
exactly like the past. But in another post you agreed that this was silly. These are financial markets. If the market falls by 5% every 500 days, that doesn't mean it will do the same thing again in the future.
There
is uncertainty in the future.
Thus we need some kind of model of how to project the returns of the past into the future, accounting for uncertainty.
You're stuck on the fiction that there is some return probability distribution out there that you can reach out and touch, if only you could calculate its parameters with some certainty. I reject this fiction.
Fiction eh? Well let me tell a story then. The model I use* is something like this:
* and, you know, all of finance
Treat the markets as a card game, in which each hand is a new return. We're being dealt from an infinitely large shoe of multiple decks. We can see all the cards that have been dealt so far (no need for card counting, just go to yahoo finance and get it for free). Suppose we see twice as many jacks as queens. Does that mean I should assume I'm exactly twice as likely to get a jack than a queen?
Actually I can never say that with certainty. But what I can do is look at how many cards I have been dealt so far. I can look at how variable the jack:queen ratio is. I can chop the stacks of cards dealt up randomly and measure the ratio inside each stack (bootstrapping).
If I do that it will tell me that on average there are twice as many jacks as queens. But it will also tell me that there is a 10% chance of being in a situation with twice as many queens as jacks. If I'm sensible I shouldn't play the game in such a way that I assume I will definitely get twice as many jacks as queens on every hand.
You're stuck on the fiction that there is some return probability distribution out there that you can reach out and touch, if only you could calculate its parameters with some certainty. I reject this fiction. My formulas prove that all you need are accurate recordings of the trade returns. The mean tells you if you have positive expectation. Beyond that, the standard deviation, the skew, etc. matter not a wit. You don't need any of that artificial stuff to figure out your betting fraction.
This is semantics. A set of past returns
always forms a distribution. It doesn't have to be parametric (i.e. with a cool name like Gaussian, and with a finite number of parameters). You can still do this kind of exercise with a non parametric distribution - just the stream of returns without summarising it by fitting it to a set of parameters.
You don't need the distribution to work out your betting fraction. But you do need to understand that the fraction is based on some estimates, and those estimates are uncertain. You need to have some model of that uncertainty or you're kidding yourself that you can predict the future. This isn't a casino.
And it doesn't matter if:
a) you estimate the mean, standard decviation and other moments and use them in a closed form Kelly solution.
b) You use the actual returns and use them in whatever you're doing, or in the numerical solution we've discussed.
It happens to be easier to do (a) which is why I used it in the examples
In both cases you're using some data, and you need to think about what the uncertainty is in that data.
Look your formulas are clever. I like them. They're neat. But I don't want anyone reading this to think that it's possible to know your correct position size to 3 decimal places in a real trading enviroment.
GAT
