Yup that's what I got.
For each turn, you would bet the following fraction
of your stack. In particular, whenever the number of red cards and black cards remaining are equal, you pass (bet $0).
Then one can calculate that the money you will have left at the very end will be:
Another way to write this formula when starting with n black and n red cards is:
For asymptotics, you can use Stirling's formula to get that the amount of money you will have at the end asymptotic to:
Math - It's in you to give.