The season finale of Last Week Tonight with John Oliver had a great main story: the many fallacies associated with lotteries. Among them:

- The probability of winning the ‘big one’ in the US, MegaMillions, is roughly the same as being stuck by lightning…while being eaten by a shark. I’m not quite sure this is right, as the odds I see on the internet are the following:
- MegaMillions jackpot: 1 in 258,890,850
- Attacked (not necessarily killed) by shark: 1 in 3,700,000
- Hit (not necessarily killed) by lightning: 1 in 300,000
- Assuming independence (probably reasonable…?), odds of both at the same time are 1 in (3.7m * 0.3m) = 1.1 trillion. So off by a factor of about 4,000 or so
- Anyway, good TV

- Lotteries may not be so good for charity/education as one might think. The program focuses on the replacement of education funds by lottery money; sometimes the shifted state funds are used elsewhere, and sometimes they’re just wasted.

OK, back on point. The program got me thinking about the humble lottery syndicate: suppose a group of 5 or so co-workers pool $1 each, on a weekly basis, to buy MegaMillions tickets. The jackpot would be shared equally. I’m sure many folks just do this as a way to build camaraderie, but there are indeed many people who honestly believe the lottery will help fund retirement. So let’s discuss a very slight improvement for this group of folks.

**Objective:**a better payout ratio than MegaMillions, while keeping similar ideas of a lottery (e.g. small investment with big potential payout). So I would like a higher expectancy, which is defined as- Prob(win) * Win$ – Prob(not win) * Cost$
- For MegaMillions, the average jackpot is $105 million. So the expectancy is:
- (1/259 million) * ($105 million) – (1- 1/259 million) * ($1) = $(0.60). I expect to lose $0.60 of my $1 invested each week, on average.
**NB:**I don’t include the other prizes, to simplify analysis. Take a look here for more rigorous computations.

**Strategy:**purchase of deep out-of-the-money put options on the S&P500. Similar to lottery tickets, they will almost always expire worthless, but will sometimes have some value and*very infrequently*have a*very large*value.**Method:**choose a near-dated, highest-strike put option on the S&P500 (SPX) which can be purchased for $5 – the ‘nickel option’. To give perspective, here are the best value nickel puts on the SPX (current value = 2040) this morning:- 2 days to expiry (DTE): 1795 strike. The S&P would need to ‘crash’ 12% over the next couple days for this to close in the money.
**I choose this one.** - 8 DTE: 1605 strike. A 20% crash needed.
- 16 DTE: 1450 strike. A 29% crash needed.

- 2 days to expiry (DTE): 1795 strike. The S&P would need to ‘crash’ 12% over the next couple days for this to close in the money.
**Scenarios:**- Under most conditions, the 2 DTE 1795 put will expire worthless. The syndicate has lost its $5 this week.
- The probability of reaching 1795 in two days can be
*very roughly*modelled as the probability of a 12% move given the S&P’s current 13% annualised volatility. That is a roughly 11-standard deviation move, meaning 0% probability given a normal distribution of returns. So conventional stats don’t really help us. - Going to our friend Quandl, we can calculate all the 2-day (overlapping) returns for the S&P 500 going back to 1950. It turns out our 12% OTM Put would have ‘come good’ a few times in that period:
- 20 Nov 2008 (-12.4%)
- 20 Oct 1987 (-16.2%)
- 19 Oct 1987 (-24.6%)

**So:**out of 16,318 overlapping observations, we’ve found 3 that would result in a win for the nickel option. That’s a historical probability of 3/16318, or around 1 in 5,500.**What do you win?**That’s a good question. With only 3 observations where this has occurred, the expected move*given*the 12% loss is difficult to estimate. Just averaging the 3 gives -17.7%, or today’s S&P crashing to around 1678. Our option would then be worth $11,700 at expiration.**Extension: what about volatility?**If we are indeed down 17% on the S&P, we can’t expect implied volatility to remain 13% annualised. Though the VIX wasn’t around in 1987, estimates of implied volatility around Black Monday were about 170% annualised. Suppose we assumed we bought the nickel option today, then the crash happened; suppose further the volatility expanded to 170% annualised. Our option would now be worth about $17,500 with the extra vol. If we’re playing for this scenario, we might actually prefer the options with greater DTE…but I’m getting too complicated.

**Our expectancy, then:**(1/5500) * ($11,700) – (1 – (1/5500)) * ($5) = $(2.87), or $(0.57) for each $1 invested. I expect to lose*slightly less*than the lottery.

**In sum, **this fun little exercise has taught me a few things:

- Lotteries are still terrible ways to make money. The expectation is
*horrible.* - Buying deep OTM options is also a pretty terrible way to make money. The expectation is nearly as bad.
- Statistics are fun when the numbers involved are ludicrous. I reckon almost every one of the stats mentioned above is dubious in one way or another. So fun for a thought experiment, but
**never**for creating an actual investment strategy.

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