Here's the second draft of the paper, revamped considerably:
The main changes:
- Added local and average jump volatility. Take and cash points depend only on the average volatility since on a pass/take decision you're comparing post-double equity with -1. The post-double equity depends on the jump volatility at the cash point; but should the game turn and head back up there it'll probably be in quite a different state than it's in near the take point. So using the average jump volatility seems more reasonable. Local jump volatility impacts the cubeful equity at the current cube level, which in turn impacts doubling decisions.
- I note that if I assume a constant (single) jump volatility, it maps very nicely onto Janowski's model with one x. But the local + average jump volatility version does not map as nicely onto his model extension to a cube life index for each player, x1 and x2.
- I changed the definition of jump volatility to be the expected absolute jump size instead of jump standard deviation. That's what the equity calcs care about. In fact, I only need to assume the jump distribution is symmetric for positive and negative jumps; if that is true then I don't need to know anything more about the jump distribution than its expected absolute jump size. Pretty nice! I got caught up for a while playing around with different distributional assumptions before I cottoned onto that.
- I rewrote Appendix 3, the one that works through the details of the jump model, to try to make it more clear when you should use local vs average volatility. That I suspect is the subtlest part of this. I think I understand this properly now, but I went through a couple of iterations myself before I got here, so it could be that I'm still getting something wrong.
This is on the back of a very rewarding conversation with Rick Janowski, who pointed out how I needed to rethink this in terms of something more like his model where he has a different cube life index for both players.