Bet Sizing When Anticipating Line Move

[BigCap]

Simple scenario:

Assume you have a 2% edge on an even money bet, with a 3 cent line (-103/+100) 8 hours before close. When you make the bet, you anticipate the closing line will distribute across -109.3/106.3, and 114.7/-117.7, with equal probability for each of the 25 possible incremental closing lines in this range. This means that the median expected closing line is 102.7/-105.7, which is consistent with your 49/51 projected edge.

How much should you bet on the +100 line being offered 8 hours before close, as a percentage of your bankroll? Assume you can bet on either side at close, the total of all bets cannot exceed your bankroll, and you will not encounter any house limits.

[Ganchrow]

Assuming the win probability of the +100 side remains constant at 51% irrespective of line move, and that the only 2 decision points are now (8 hours prior to the close at +100) and at the close where the 25 different possible closing prices on the opposing side, all appearing with probability 1 25 , are:

1. -109.3
2. -108.3
3. -107.3
4. -106.3
5. -105.3
6. -104.3
7. -103.3
8. -102.3
9. -101.3
10. -100.3
11. +100.7
12. +101.7
13. +102.7
14. +103.7
15. +104.7
16. +105.7
17. +106.7
18. +107.7
19. +108.7
20. +109.7
21. +110.7
22. +111.7
23. +112.7
24. +113.7
25. +114.7

then at full Kelly I show an optimal initial bet size of ~50.560%.

[BigCap]

You may assume that the closing no-vig line represents the true win probability, i.e. is an "efficient" line. I apologize for leaving this out of the original post.

[Ganchrow]

You may assume that the closing no-vig line represents the true win probability, i.e. is an "efficient" line. I apologize for leaving this out of the original post.

In that case at full Kelly I show an optimal initial bet size of ~50.539%.

[BigCap]

No guess on the bet at close?

BTW, I get 50.9789% for the initial bet +100, and 49.0211% on the other side at close, regardless of the actual closing number. This yields an expected bankroll growth of 0.6182%.

[Ganchrow]

No guess on the bet at close?

Obviously it would depend on the closing line:Which of course is not a "guess".

[Ganchrow]

BTW, I get 50.9789% for the initial bet +100, and 49.0211% on the other side at close, regardless of the actual closing number. This yields an expected bankroll growth of 0.6182%.

Yes it does, but one could do better.

Obviously it would depend on the closing line:Which of course is not a "guess".

This, OTOH, yields expected bankroll growth of 0.6237%.

[BigCap]

No offense intended by using the term "guess"; your numbers do in fact yield a slightly better expected growth. My model was simpler in that I was not changing the close bet size, but some improvement (although not significant) can be made as you calculated.

With this in mind, and assuming that the bettor has a bona fide 2% edge, would it be reasonable to assume that such a closing line distribution would occur, and the bettor should significantly increase the size of his initial bet (compared to the 2% kelly edge calculation) because of the opportunity afforded with the closing hedge?

[Ganchrow]

would it be reasonable to assume that such a closing line distribution would occur

Such a distribution could certainly be used as a first order approximation.

the bettor should significantly increase the size of his initial bet (compared to the 2% kelly edge calculation) because of the opportunity afforded with the closing hedge?

Most definitely. This is especially integral to exchange market making.

[BigCap]

So if one has a bona fide edge on a bet, and it can be reasonably assumed that the closing line will move toward the line equal to the bettor's edge (with some distribution), then the bettor should wager significantly MORE than the kelly ratio calculated by that edge (e.g. 2% in the above scenario).

If this is true, why would one bet at kelly ratio for the original bet, when the opportunities to hedge can be reasonably expected?

[Ganchrow]

So if one has a bona fide edge on a bet, and it can be reasonably assumed that the closing line will move toward the line equal to the bettor's edge (with some distribution), then the bettor should wager significantly MORE than the kelly ratio calculated by that edge (e.g. 2% in the above scenario).

If this is true, why would one bet at kelly ratio for the original bet, when the opportunities to hedge can be reasonably expected?

Sure, it's all about limits, spread, opportunity cost, and the anticipated line movement distribution.