[Ganchrow]

Just in time for the holidays and taking the cue from Justin's original Arbitrage Challenge, I present my own statistical arbitrage challenge. The first correct answer receives a choice of several books from Amazon.com¹. A special cash bonus will be awarded if the answer is given prior to midnight of Christmas Day (Eastern time)².

---

The following bets are available:

1. Will the New England Patriots lose in the 2007/8 divisional round of the playoffs?
   Yes +516
   No -600
   
   
2. Will the New England Patriots lose in the 2007/8 AFC Championship? (NE must play for bet to have action.)
   Yes +183
   No -200
   
   
3. Will the New England Patriots lose in the 2007/8 Super Bowl? (NE must play for bet to have action.)
   Yes +437
   No -500
   
   
4. Will the New England Patriots win the 2007/8 Super Bowl?
   Yes +118
   No -128

Your unbiased estimates of the associated 2007/8 probabilities are:

1. NE wins divisional round: 90%
2. NE wins AFC Championship: 63%
3. NE wins Super Bowl: 55%

If Bet#4 (NE to win Super Bowl) can't be included in a compound wager what would be your optimal bet combination?

Assume you're a half-Kelly bettor with a $100K bankroll and that all bets must be placed immediately. You may ignore cost-of-capital.

---

¹Players ineligible for prizes include SBR employees, members with fewer than 15 posts at the time of this posting, previous Ganchrow puzzle prize recipients, and RickySteve.
²Cash bonus only awarded to eligible prize recipients and depending upon logistics may be awarded in a cash-equivalent form.

[louisvillekid]

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]

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]

Originally Posted by robzilla

Do i win something for this?

There are two problems with your answer:

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

[Wheell]

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]

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

[Ganchrow]

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]

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]

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]

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]

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]

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]

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]

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

[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.