If I had an investment that with 50% likelihood quadruples your investment on a given day and you lose it all also with 50% likelihood, what percent of your money should you invest each day to maximize your median return? I believe from memory the answer is 25%, but I'd like to understand the math behind this and believe the answer may be related to the Sharpe ratio.
I've asked a related question over on the money stack exchange and I got back the nonsensical (to my thinking) answer of infinite leverage, so I'd like to offer some math to better address this question more objectively: