Deriving the Kelly Criterion to Maximise Profits

The Kelly criterion is almost never used as-is because it is very sensitive to probability of success, which is hard to know accurately and in many cases, dynamically changing. This is easy to see in an Excel spreadsheet. Changing the probability by even 0.01 percent can vastly shift the results. The article calls this out in the last paragraph.

The article mentions fractional Kelly is a hedge. But what fraction is optimal to use? That is also unknowable.

Finance folks, correct me if I’m wrong, but the Kelly Criterion is rarely used in financial models but is more a rule of thumb that says roughly if you have x $ and probability p, in a perfect world you should only bet y amount. But in reality y cannot be determined accurately because p is always changing or hard to measure.

I am not sure what you mean by "never used as is."

The Kelly criterion is an optimization of capital growth (its logarithm) method/guide. Not using it doesn't change its correctness.

But yes you need to know the advantage/the edge you have. Like with pricing methods eg for European options for Black Scholes you need to know the volatility and there is no way to know it, you estimate. This is where all the adjusting for bias and ML comes in.

But do you calibrate p (say through estimation) and then apply the Kelly criterion in your portfolio?

I don’t think it is used in this way. It swings too much with a given p.

You calibrate for a reasonable distribution of p and use that to estimate (Monte Carlo, etc.) expected gain, optimizing your investment based on that. With this technique your estimate will probably end up somewhere around the common heuristics.

Yeah, but I think this misses the point a bit. The fact that your true edge isn’t knowable wouldn’t be so bad except that if you’re betting full-Kelly and overestimate your edge even a little bit, your probability of ruin in the long run goes to 1. Whereas if you underbet, you’ll compound wealth at a little lower rate but won’t risk ruin.

> Changing the probability by even 0.01 percent can vastly shift the results.

No, not generally. Since it's a quadratic function we're optimising, it's surprisingly flat at the top. Sure, there are some bets where the edge is tiny and 0.01 percent is a large proportion of that, but that doesn't invalidate the Kelly criterion – by what other criterion would you determine the appropriate bet size?

> is more a rule of thumb that says roughly if you have x $ and probability p, in a perfect world you should only bet y amount.

It applies far more broadly than to binary bets. It tells you how to allocate your spending optimally across any number of opportunities, based on joint probability of outcomes.

Both of your misconceptions are common, and they are addressed in the article linked in the submission: https://entropicthoughts.com/the-misunderstood-kelly-criteri...