### Sizing bets on FTX claims + a Monte Carlo simulation

#### January 17, 2023

There is a market for FTX claims, trading at around 15¢ on the dollar, up from 10¢ last week.

Since SBF is allergic to thinking through bet sizes[1][2] (a fact that once struck me as discordant with his Bayesian gestalt) — or maybe because I’ve been indulging in a slow read through Red-Blooded Risk — the Kelly Criterion has been on my mind, just like everyone else’s.

Last week, claims trading at 10¢ seemed undervalued in light of the seizure of 3.5bn by the Bahamas regulator. But of course the 3.5bn turned out to be largely fabricated. A solid number requires liquid assets, which is both obvious and funny. If the assets are not liquid, they are not solid.

This week’s recrudescence is the purported recovery of 5bn, announced in a US bankruptcy court. Entertain me in some hand waving towards a 45¢ recovery if true, 10¢ recovery if false. At 15¢ per claim, the implied odds of the 5b figure’s veracity are 1:71.

That seems low. I’d bet (but how much?) that it’s closer to a coin toss, in which case, the Kelly bet would be 125%2, a figure which feels impossibly large. Though the rule of thumb is to cut it in half, and if the claims trade up to 20¢, it diminishes to a comparatively sober 60%.

I also noticed that it’s possible to achieve similar changes in bet size by assuming, for example, a 5¢ rather than 10¢ recovery in the pessimistic scenario, without changing the expected value by more than a few cents.

Anyhow, I made a Monte Carlo simulator for a three-outcome version of this problem, more to develop intuition around how ternary outcomes and small parameter changes affect optimal bet size than for any ideas around FTX claim betting, per se, since the edge for this particular trade is probably dominated by information asymmetry.

Instructions: adjust the probabilities and claim prices. Click “Optimize bet size” to find the bet size with the highest median return3 among the simulated portfolios. Note that increasing the simulated number of bets increases the time horizon that the strategy plays out over, and affects the proportion of portfolios that end up winning in the long run.

1. Assume that FTX has 10b in outstanding liabilities. There is 1b of cash already recovered.

Optimistically, the 5b of liquid assets (including the original 1b of cash) plus the less-liquid assets sum to USD 5.5b. Pessimistically, the original billion stands, but the 5b is a fantasy, and liquidators manage to scrounge up another 1b, for a recovery of USD 2b.

So, call it 45¢ or 10¢ on the dollar, assuming 10¢ on the dollar goes to the trustee either way. Then the probability weights which cohere with the market price are: $$(10¢ * {6 \over 7}) + (45¢ * {1 \over 7}) = 15¢$$.

Note the many sources of possible error, here. The outstanding liabilities might be more than 10b. The probabilities might be wrong. The trustee might charge more than usual for such an exotic case. Etc.

2. If you pay 15¢ with a 50 / 50 shot of getting either 10 or 45¢ back, you either lose 5¢ or gain 30¢, and you compute the Kelly bet $$f*$$ with:

$f* = {p_{win} \over 5¢:15¢} - {p_{lose} \over 30¢:15¢} = 150\% - 25\% = 125\%$

Or at a 20¢ claim price with the same payouts:

$f* = {50\% \over 10¢:20¢} - {50\% \over 25¢:20¢} = 100\% - 40\% = 60\%$

Equivalently, a firm whose whole portfolio was on FTX might be justified in selling 40 to 20% (half to full-Kelly fractions) of their claims even if they’re undervalued at 20¢, which is a neat way to explain how market prices might be too low in the presence of involuntary, concentrated positions, even with total information symmetry.

And of course betting half-Kelly is recommended because “the Kelly strategy marks the boundary between aggressive and insane investing” and all that. Though in this situation one might be sincerely tempted to go out in the same manner that FTX lived, and YOLO it. There are certainly philosophical reasons to do so, in addition to the dreary pragmatic ones.

3. You could also optimize for other things, like how many of the simulated portfolios have a positive return after some number of bets, which yields different results.