Kelly Casino isn’t doing too hot as they’re struggling to attract new clients as many young Americans have opted to punt GME calls over heading to the craps table. Kelly realizes that to get such regarded1 investors through their doors they’re going to need a simple game with an enticing enough promotion.
Ultimately Kelly decided to opt for a good old-fashioned roulette table, but modified to be on the simple side; there are no numbers, just colors that the ball can fall into divided into 20 sections- of which, 9 are red, 9 are black, and 2 are green. Players can only wager on black or red colors and they pay out with 1:1 odds if the ball lands on the color wagered on.
“Deposit $1,000 with the casino and we’ll give you $1,200 dollars worth of chips.” Seems pretty good right? Just cash out instantly and you’ve got yourself $200 dollars just like that. Well not quite because there’s some fine print to pay attention to- “Customers must wager at least $2,000 in order to cash out any casino chips.”
Ah well now the offer doesn’t sound as good because in expectancy you probably lose the free chips given, and maybe if your impulses get the best of you, you actually lose more than the free chips Kelly decides to give you. One might calculate the expectancy of a single spin of the wheel as:
and from that naively use linearity of expectations to arrive at the conclusion that over the course of wagering $2,000, $200 dollars will be lost and hence the promotion is a wash. Without putting any real thought into this, this may come across as ~reasonable~. After all, the main job of a casino is to make sure that they have edge. But then again, even casinos can make mistakes so let’s dive a little deeper into the math and see if Kelly made any invalid assumptions.
First and foremost, the most glaring mistake with the above equation is that the loss in expectancy over the course of wagering $2,000 will also be realized in expectancy. Okay let’s actually unpack this thought a little more now. Suppose we go ‘all-in’ and bet our entire stack of $1,200 chips. One of two things will happen: we win and our stack doubles to $2,400, or we lose and our stack goes to zero, zilch, nothing, and we go home right then and there. How much did we actually loose in the second scenario? We wagered $1,200, but remember only $1,000 of that was our original capital. We can now see that on the first bet we make the payout is not 1:1, but rather 1:1.4 and this first roll actually has positive expectancy, from the viewpoint of our original starting capital, of $80.
This boils down to the fact that we do not personally incur the negative variance of Kelly’s chips, but reap the benefit of the upside variance.
We can now see that our goal should be to maximize the variance of our chips in order to come out on top (in expectancy). BUT, remember that this game is negative EV without the incentive, so we shouldn’t wager more than we need to in order to reach the $2,000 withdraw point. Therefor we will go ‘all-in’ on the first spin, and then, if and only if we do not loose our stack, we place one final wager of $800. This results in 3 potential outcomes: win, win; win, loose; loose.
It’s now clear that this promotion actually has a positive expectancy of $44 by levering the fact that we can place negative variance onto the casino.
Lowering Variance, Same EV
We can clearly see that maximizing the variance of the casino’s money benefits us, but is it still the casino’s money after the first roll? No. We can therefore place bets in as small of a denomination as possible after our original $1,200 wager to finish betting the remainder of the $2,000 needed. This doesn’t change our final expectancy of $44 at all, but because of LLN (law of large numbers), we will have less variance on our final PnL.
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So what’s the takeaway? When someone else takes on the downside risk, max out variance to maximize your own expectancy- even negative expected value plays can become personally plus EV as result. This is also known as the O’Hare Play2. Nota bene: I don’t actually condone the O’Hare Play.
Distributions with caps on one end, see a shift in the mean with a change of variance. Take log normal distributions for example. The mean of a log normal distribution is given by:
We can clearly see from this that when variance increases, so too does the expectancy, which is why the delta of call options approaches 1 as the implied volatility approaches infinity3. This is nothing new of course and I would hope that any seasoned options trader should be able to spot this, but then again, even Natenberg4 said that the delta of a call option approaches 0.5 as implied vol approaches infinity, so maybe truly understanding probability distributions is relatively rare.
Slick_Wick324. (2023, September 11). Origin of the regard. Reddit. https://www.reddit.com/r/wallstreetbets/comments/16fm03k/origin_of_the_regard/
Pisani, B. (2019, January 2). You won't believe the 'O'Hare Play' trade done by Chicago traders during their eighties heyday. CNBC. https://www.cnbc.com/2019/01/02/-you-wont-believe-the-ohare-play-trade-done-by-chicago-traders-during-their-eighties-heyday-.html
AKdemy. (2021, July 10). [Answer to "Effect of Implied volatility on option delta"]. Quant Stack Exchange. https://quant.stackexchange.com/a/65960/74652
Natenberg, S. (1994). Option Volatility & Pricing: Advanced Trading Strategies and Techniques. McGraw-Hill