How do those of you who bet futures tend to size your bets?
I've put together a modified version of Kelly (what else?) to account for timevalueof money, but am interested in knowing if anyone else has any ideas.
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How do those of you who bet futures tend to size your bets?
I've put together a modified version of Kelly (what else?) to account for timevalueof money, but am interested in knowing if anyone else has any ideas.
SBR Founder Join Date: 8/28/2005
I usually bet the limit on futures.
SBR Founder Join Date: 8/10/2005
Most books offer rather large limits on season totals futures, which may not always be a reasonable bet quantity, especially when betting at longer odds.
Back in October of 2005 (can't get there from here) I had posted what I had considered (and in retrospect still do consider) a longterm value play. The bet was at the Greek and the wording was "Independent candidate to win White House in 2008." This bet was offered at +50,000 which I believed to be quite generous. InTrade was at the time offering the same bet at +4,445 for and 5,782 against.
So even if we assumed that all the market vig on the InTrade bet is on the the dog side (resulting in a breakeven fave bet), this would imply a win probability of about 1.7001%, correspondsing to an incredible edge of a little over 750%! For a quarterKelly bettor, this would imply a stake of 0.3759% of bankroll (assuming no hedge). Considering that the maximum bet was $1,000 (I believe  although memory could be failing me again), this means that those quarterKelly bettors with bankrolls of approximately $266,000 or more (which, somewhat coincidentally, I'd expect to roughly represent the median bankroll among pro and semipro sportsbettors) would bet the max, and those with smaller bankrolls would bet less than the max.
So this has all been straight forward Kelly up to this point. But here's the question ... let's assume for the moment that you could bet the InTrade market at zero slippage (meaning that one could transact infinitely large quantities without impacting the market price  a rather specious assumption to be sure). In this situation, how much would one bet on the +50,000 instrument and how much would one bet on the hedge instrument?
It should quickly become apparent that Kelly is effectively useless on this situation. Because the market contains a riskless arbitrage, the sum of bets across the two instruments, assuming the max bet size constraint on neither leg binds, is 100%.
To best see why this probably won't accurately reflect most bettor's preferences, we should look at the specifics. The quarterKelly stake on this market would theoretically be 0.5939% on the primary bet (Independent wins) and 99.4061% on the hedge (Independent does NOT win). This means that if the Independent candidate won (a 1.7001% likelihood event) the player's bankroll would increase by 197.5%, and if the Independent candidate lost (a 98.2999% likelihood event) the player's bankroll would increase by 1.125%, an expected return of 4.4643% over 3 years, and an expected annualized return of 1.4665%.
So does this seem attractive? Would you really want to tie up ALL your money for 3 years just to earn an expected return of 1.4665% per year? (This corresponds to a certaintyequivalent of 2.0944% for 3 years and 0.6933% annualized. Pretty poor  especially when you consider credit rating uncertainty for the Greek). I know I'd never make this bet.
So this is the general problem with long odds futures bets ... you can't really bet all that much on the underdog as you're very likely going to lose, and if you hedge, you could be tying up a lot of money for a long period of time, with a greatly diminished expectation.
Conversely, on a short odds future, while the hedge would be very cheap, the expected return on the primary bet would be quite low. Furthermore for sufficiently short odds bets, chances are that even for the smallest bettors the max bet size would constrain. (This is part of the reason why short odds futures bets often trade at low or no vig. Justin had pointed out this phenomenon a month or so ago regarding Super Bowl futures. And if this were the case for 2 or 3day futures, imagine how much more likely it would be to hold for 2 or 3month or longer futures  this could be quite relevant to bettors sufficiently with low hurdle rates.).
So what's the solution? How does one optimize in the face of a pure arbitrage and a relatively long period to contract expiry? What about if you were able to buy on margin?
Does anyone have any thoughts on this? I'll post what I consider to be the most direct solution if people show interest.
SBR Founder Join Date: 8/28/2005
Very true, indeed.
I think it often can be fairly easy to identify prop bets that are likely to be profitable using a Poisson valuation methodology, an issue discussed by Stanford Wong in Sharp Sports Betting.
A fact that certainly hold true for the recreational gambler.
SBR Founder Join Date: 8/28/2005
No book of which I'm aware, although it certainly wouldn't surprise me if this were available in the UK.
In theory, there's no reason why books shouldn't provide this service. Margin rates would be another way for books to make money. Betting credits are essentially free to books and so the only real issue is risk.
On reconsideration, I believe that Tradesports has or had something like this.
SBR Founder Join Date: 8/28/2005
Here's where Justin talked about Super Bowl props. It's a good read, and Justin was righton in his analysis.
Super Bowl props. while technically only veryshortterm futures, may have more in common with longerterm futures than one might think at first glance. This would be the case insofar as the cost of capital may be considerably higher around Super Bowl time.
SBR Founder Join Date: 8/28/2005
Right. That's the whole point. While the "solution" I gave is indeed Kellyoptimal, no one is really going to play. The question is, what is the correct way to optimize?
Possibly. Although there's still an opportunity cost issue when playing at a one of a bettor's regular credit shops.
An interesting question would be how to optimize if on top of your $X 0% interest you could also borrow from the book at some specified interest rate. For example, if you could borrow at 1% per annum (remember that the book isn't really floating cash, but rather only betting credits hence it could afford to offer very low rate) and can make 1.5% per annum guaranteed on your futures position, then that's pure arbitrage profit (credit risk aside).
Last edited by Ganchrow; 030907 at 09:27 AM.
SBR Founder Join Date: 8/28/2005
This will only hold in the case of a riskneutral player. In the case of a riskaverse player, it's actually a bit more involved.
Assuming Kelly preferences, and
o = odds
p = win prob
r = riskfree interest rate over relevant time period
k = Kelly multiplier (1 => fullKelly, ½ => halfKelly, ¼ => quarterKelly)
the optimal stake, S* is given by
So to give an example, assuming a quarterKelly bettor, payout odds of +500, a win probability of 20%, and a riskfree interest rate of 4%, the correct stake would be (6×20%4%1) / (64%1) × (0.25+4%) / (1+4%) = .675%. (Using the linear method would yield 0.600%, which to be fair is only off by about 12½%)Code:(opr1) (k+r) S* =  ×  (or1) (1+r) (proof available upon request)
But this is all fairly trivial. What we're really looking to do is determine how to jointly optimize the primary and hedge positions subject to both individual and aggregate budget constraints.
The issue at hand is that of hedging a very large underdog. In such a case the opportunity cost could be substantial (and as mentioned earlier, this could be an especially powerful effect when liquidity is at a premium  say right around Super Bowl or NCAA Tourney time).
A case in point would be the 2008 election example I gave above. This is made even more explicit when one considers the hedge. On a $1,000 bet, hedging completely would require a $493,482.49 bet on the fave. Freezing up this credit line over a 3year period may well be substantial. Now one might certainly respond that the expectation on the bet would only be $7,517.51  certainly nothing to write home about for a bettor with a halfmillion dollar line of credit, but possibly not completely irrelevant either when considering it's a pure arbitrage.
So the issue returns by commodius vicus of recirculation to opportunity cost and environs.
Last edited by Ganchrow; 030907 at 06:39 PM. Reason: reformatted equation spacing
SBR Founder Join Date: 8/28/2005