Statistical Arbitrage Challenge, Prize Offered (Hard)

**louisvillekid** · 12-24-07, 12:22 PM

well since you said "hard", i assume the previous puzzles were easy, and since i didn't have a clue on those, i guess i ain't winning.

**robzilla** · 12-24-07, 01:27 PM

Will the New England Patriots lose in the 2007/8 Super Bowl? (NE must play for bet to have action.)
Yes +437

Will the New England Patriots win the 2007/8 Super Bowl?
Yes +118

- Odds:5.3700 - Bet:$115.50 - Ret:$620.24 - Profit:$220.24 - Profit:55.06%
- Odds:2.1800 - Bet:$284.50 - Ret:$620.21 - Profit:$220.21 - Profit:55.05%

64.49% - Profit: 35.51% Do i win something for this?

**Ganchrow** · 12-24-07, 01:38 PM

Originally posted by robzilla

Do i win something for this?

There are two problems with your answer:

As per the bet description, for bet# 3 to have action the Pats must actually play in the Super Bowl.
The solution needs is supposed to be half-Kelly optimal, not just zero-risk.

**Wheell** · 12-24-07, 02:06 PM

Rob, as per your bets if the Patriots were to lose in the Divisional round or the AFC championship game you would lose $284.5. If Ganch says it is hard, it ain't gonna be that simple.

**Wheell** · 12-24-07, 02:09 PM

Ganch, for the record, unless I am making a math mistake, you left a false clue in your title.

**Ganchrow** · 12-24-07, 02:17 PM

Originally posted by Wheell

Ganch, for the record, unless I am making a math mistake, you left a false clue in your title.

Hmm, I can't say I know to what clue you're referring...

**Wheell** · 12-24-07, 02:23 PM

Arbitrage. It is indeed statistical, it is indeed a challenge, the prize being offered is not a clue, and yes, this does rank as hard. Your half-Kelly bettor likes the Patriots at all times, particularly against an NFC team. The real question is how much to risk at any given time. It is a great puzzle.

**Ganchrow** · 12-24-07, 02:27 PM

Originally posted by Wheell

Arbitrage. It is indeed statistical, it is indeed a challenge, the prize being offered is not a clue, and yes, this does rank as hard. Your half-Kelly bettor likes the Patriots at all times, particularly against an NFC team. The real question is how much to risk at any given time. It is a great puzzle.

OK. I gotcha.

The title refers to the fact that this represents a so-called "statistical arbitrage" in contrast to a deterministic or risk-free arbitrage.

**Wheell** · 12-24-07, 02:39 PM

The term "Statistical Arbitrage" is positively Orwellian. July-August 2007 was justice being served. Positive EV is one thing, it does not deserve to be called arbitrage. I understand why the term is used, but the concept misleads financial investors.

**Ganchrow** · 12-24-07, 02:44 PM

Originally posted by Wheell

The term "Statistical Arbitrage" is positively Orwellian. July-August 2007 was justice being served. Positive EV is one thing, it does not deserve to be called arbitrage. I understand why the term is used, but the concept misleads financial investors.

I hear you, but whether you personally agree with the term or not, it's nevertheless a part of the vernacular.

I worked in US Equity StatArb for many years and I'm not about to go back and modify my old business cards just because of a two month draw down.

**Wheell** · 12-24-07, 02:50 PM

Alright, I suspect my answer is wrong, but hey, I admitted it was hard.

6570 to win 1095 (rounded up from 6568 to win 1094.67) on the Patriots to win the divisional round.

8432 to win 9950 (9949.76) on the Patriots to win the SB.

No other bets as at any given time the player is already risking enough of his bankroll.

**Ganchrow** · 12-24-07, 02:59 PM

Originally posted by Wheell

Alright, I suspect my answer is wrong, but hey, I admitted it was hard.

6570 to win 1095 (rounded up from 6568 to win 1094.67) on the Patriots to win the divisional round.

8432 to win 9950 (9949.76) on the Patriots to win the SB.

No other bets as at any given time the player is already risking enough of his bankroll.

Sorry.

But keep trying.

**Wheell** · 12-24-07, 03:07 PM

Apology accepted. I think I realized my mistake. Getting the right numbers might be tricky though.

**Art Vandeleigh** · 12-24-07, 03:11 PM

A question from the under-informed.

From a previous post introducing Kelly -

"Kelly Stake as percentage of bankroll = Edge / (Odds – 1) for Edge ≥ 0

Put in terms of win probability:

Kelly Stake as percentage of bankroll = (Prob * Odds – 1) / (Odds – 1) for Probability * Odds ≥ 1"

Is a half Kelly just these same equations, with the denominator multiplied by 2? Or is that just too obvious.

**Ganchrow** · 12-24-07, 03:29 PM

Originally posted by Art Vandeleigh

A question from the under-informed.

From a previous post introducing Kelly -

"Kelly Stake as percentage of bankroll = Edge / (Odds – 1) for Edge ≥ 0

Put in terms of win probability:

Kelly Stake as percentage of bankroll = (Prob * Odds – 1) / (Odds – 1) for Probability * Odds ≥ 1"

Is a half Kelly just these same equations, with the denominator multiplied by 2? Or is that just too obvious.

That generally works as a reasonable approximation of single-bet n-Kelly (although it will be significantly off for full-Kelly stakes close to 1).

The exact formula for single-bet n-Kelly however (0 < n < ∞) given decimal odds of d, and a win probability of p would be:

((d-1)*p)ⁿ - (1-p)ⁿ
----------------------------
((d-1)*p)ⁿ + (d-1) * (1 - p)ⁿ

More relevant to this problem however would be the n-Kelly utility function:

U(X;n≠1) = ^{X^1-¹/_n} / _{(1-¹/_n)}

U(X;n=1) = ln(X)

So for double-Kelly you'd be looking at U(X|p,d) = 2X^½.

**HedgeHog** · 12-24-07, 03:38 PM

Full Kelly vs 1/2 Kelly Win/ Lose

NE not lose div game 19.88% 9.94% $9940/$1657
NE not lose Conf game 5.15% 2.58% $2580 /$1290
NE not lose SB game 16.62% 8.31% $8310/$1662

Does not guarantee a profit, but follows the Kelly principle.

**Ganchrow** · 12-24-07, 03:46 PM

Originally posted by HedgeHog

Full Kelly vs 1/2 Kelly Win/ Lose

NE not lose div game 19.88% 9.94% $9940/$1657
NE not lose Conf game 5.15% 2.58% $2580 /$1290
NE not lose SB game 16.62% 8.31% $8310/$1662

Does not guarantee a profit, but follows the Kelly principle.

Nope, not there yet.

**HedgeHog** · 12-24-07, 03:48 PM

In above example, NE has a 90% chance of winning its 1st round game, but the line is a real bargain at -600. The Kelly calculator says bet 19.88%...so half Kelly would be a bet of 9.94% of bankroll?

**Ganchrow** · 12-24-07, 03:59 PM

Originally posted by HedgeHog

In above example, NE has a 90% chance of winning its 1st round game, but the line is a real bargain at -600. The Kelly calculator says bet 19.88%...so half Kelly would be a bet of 9.94% of bankroll?

If it were the only bet you were considering, then according to calculator the half-Kelly stake on a -600 bet with 90% win probability would be 16.1523% of bankroll.

But a word of warning ... in complicated situations such as these the Kelly calculator won't be of very much use.

**Wheell** · 12-24-07, 04:58 PM

Originally posted by Ganchrow

But a word of warning ... in complicated situations such as these the Kelly calculator won't be of very much use.

I've noticed. When Ganch says it is hard, it is hard.

**tomcowley** · 12-24-07, 05:57 PM

Are parlays allowed (except for bet 4?)? If so, assume true odds parlays?

**Wheell** · 12-24-07, 05:59 PM

What is a parlay?

GROAN!

**Art Vandeleigh** · 12-24-07, 06:09 PM

Well, I'm not going to be the one to answer this puzzle, but it's interesting and relevant to gambling and I'd at least like to understand the problem at the top level.

It seems to me that it's the "yes" side to all 4 questions that's being asked for.

If you bet $1000 on question 4 "Yes" and the Patriots win the Super Bowl, you'd win $1180.

So the hedging begins with question 1. If you bet, let's say $200 that yes, the Pats will lose, and they do lose in the first round, you'd get $1032 back, minus the $1000 you bet on question 4, giving you a $32 profit. End of story.

If NE wins in the first round, and you bet say $700 on them to lose the AFC championship at +183, and they did lose, you'd make $1281 on that bet, minus the $1000 you lost on question 4 and the $200 you lost on question 1, for a profit of $81.

Unfortunately, if you proceed to question 3, there is no amount you can bet to now be profitable.

In fact, after doing a rough estimation, it appears that with the numbers given, the best combination for all 4 questions still loses about $25-$30 for every $1000 bet on question 4, regardless of whatever NE does in the playoffs.

So, if I understand things, what balances things out is that the estimated probabilty of question 4 occuring is 55%, but the payoff is as though it is a 46% chance of occurring, giving you appx a 20% edge. So in actuality, $1000 wagered in question 4 effectively returns around $1400(??), not $1180.

Similar adjustments need to be made in questions 1-3 for assessed probabilities vs. actual odds, to determine optimal amount.

Is anything I've written here on the right track ?(I'm not really interested in claiming a prize, just trying to organize this somewhat confusing puzzle to an arbitrage novice, i.e. me)

**Ganchrow** · 12-24-07, 06:19 PM

Originally posted by tomcowley

Are parlays allowed (except for bet 4?)? If so, assume true odds parlays?

As long as they don't include #4, assume you can make true-odds IF-bets without any restrictions on size as long as they proceed forward in time. (In other words, we're assuming the conditional odds won't change following a victory.)

**tomcowley** · 12-24-07, 06:31 PM

So $1 on NE to win divisional, IF WIN, $1000 on NE to win AFC is a legal bet?

**Ganchrow** · 12-24-07, 06:33 PM

Originally posted by Art Vandeleigh

Well, I'm not going to be the one to answer this puzzle, but it's interesting and relevant to gambling and I'd at least like to understand the problem at the top level.

-snip-

You're reasoning is sound with respect to designing a perfect hedge, but realize that from an economic perspective assuming n-Kelly preferences, there's nothing sacred about a zero-variance solution. In fact, in this problem there is no feasible positive-return zero-variance solution.

The problem is sufficiently complicated such that it wold be to tough to simply intuit an answer. Probably the most direct to way proceed is to set up the expected half-Kelly utility function with the available bets as variables and then maximize. I couldn't imagine doing this without a spreadsheet, mathematical package, or other such software.

**Ganchrow** · 12-24-07, 06:43 PM

Originally posted by tomcowley

So $1 on NE to win divisional, IF WIN, $1000 on NE to win AFC is a legal bet?

Yes that's fine. $1 to win divisional and if that wins then $1,000 to win AFC.

But you couldn't bet that the other way around -- so betting $1 on NE to win AFC and if that wins $1,000 on NE to win divisional would not be legit.

In other words ... no funny business.

**Wheell** · 12-24-07, 07:39 PM

ok, I'll bet 11,845 on the Patriots to win the Super Bowl. I'll bet 3,155 on the Patriots to win round 1 and then lose round 2. And, because I'm afraid of betting too much against the Colts, I'll put 59.6 on the Patriots to lose in the AFC Championship game.

Anyway, Merry Xmas to all and to all a Merry XMas.

**tacomax** · 12-24-07, 07:40 PM

My money's on curious to win the prize.

**tomcowley** · 12-24-07, 07:52 PM

I think I've got it, expected 3.7% bankroll growth, but I can't win

**Wheell** · 12-24-07, 07:52 PM

I'd take no winner over the field. I know I don't like my chances.

**Wheell** · 12-24-07, 07:53 PM

Tom, you can't win a prize, but you can win the respect of countless forum readers.

**Ganchrow** · 12-24-07, 08:14 PM

Originally posted by tomcowley

I think I've got it, expected 3.7% bankroll growth, but I can't win

According to the letter of the original post, a player would only win IF he were the first to post the correct solution AND he were prize eligible.

So, if Tom were the first to post the correct answer in this thread then one of two things would happen: A) no one would win the prize, or B) I'd cave and award Tom the prize anyway.

<Insert clever Ganch/Grinch pun here.>

**Ganchrow** · 12-24-07, 08:16 PM

Originally posted by tacomax

My money's on curious to win the prize.

Smart money says he threatens to beat some one up first.