# Risk Update: March 2019 – St. Petersburg Paradox.

A very good way to understand what it is that we do, or at least try to do, is to make an analogy to what, in the world of mathematics, is known as the St Petersburg Paradox.

In the financial world of other-people’s-money, skewed upside-only incentive structures and faux probability-based risk measures, the distortions in the casino’s willingness to offer the “game” can get pretty interesting. As a simple visualisation, let’s look at some actual payoffs in current market pricing. As a clean form of the “game” let’s use a Convexity swap (a package of a variance swap and a volatility swap, in opposite directions) where we can isolate the pure convexity between volatility and volatility squared (variance) on EUR FX. From the perspective of the casino operator, the guy offering the game, and looking at the risk based upon short term probability of outcomes (think Value at Risk), he might view “selling” this bet as below.

So for total maximum potential upside of €35,000 (about €140 per day), this casino staff is willing to let somebody play the game every day for one year. In coin toss terms, the casino has about a 50% chance of keeping the money as the player tosses a tails first go, and something in the 95 percentile (based on recent historical outcomes) of at worst breaking even. He bases his judgement on probability, and all but ignores expected return. Given that his regulator allows him to account for risk as though this is a reasonable measure of risk, it is quite possible that the casino will happily sell this game over and over at ever decreasing charges to the players. The probability of return is high, and the risk is an accounting function, not an accountability function.

So what does this game look like from the other side? What if we looked at it on more of an expected return perspective? We can expand the potential range of outcomes over the long term historical range of EUR volatility, and turn it around for how the “player” or buyer of the game might look at it.

You need to look quite closely now to see the potential “cost” that the player has fronted to the casino to play this game, and you can see that fairly quickly, should we start getting into a succession of heads in our coin tosses, that the infinite potential returns start to be realised. At this price, this looks like a pretty attractive “buy” to play a St Petersburg Paradox game. So the buyer of the game, has a 50% chance of losing €35,000, and about a 95% chance of never making money, but has somewhere in the 5% remaining tail some possibility of making €3,000,000 or even more.

In reality, of course, our game of euro convexity is even better than the coin-toss game. In the coin-toss game, each individual toss is independent from other tosses. Just because you have tossed four or five heads in a row, it doesn’t change the probability of the next toss. That is not necessarily the case in our game of convexity. There is most certainly an element of reflexivity, or maybe more accurately Self Organised Criticality, in the convexity game.

The fact that the casino staff, due to his faux risk measure, hasn’t backed his potential (indeed expected) losses with capital, almost certainly means once we go beyond two, three, or four of his standard deviations, his boss may start aggressively bidding up to buy us out of the game! This is precisely how crises materialise. Too many wannabe casino operators get too complacent in competing to sell infinite expected loss games at ever-lower upfront prices due to an overly short term, probability based, no personal accountability business model.

So what is the key to being a good buyer of St Petersburg Paradox games? Well for starters, try not to pay too much, but probably more important is to structure your playing so that you can play as many times for as long as possible. You need to extend the game into the long term world of expected return. Second, make sure the casino has enough capital to pay you your infinite winnings. In our world of convexity, we like to structure it so that the casino has to pay us after every coin toss, not just at the end of the game. In return, we are happy to post the €35,000 upfront, or in a margin account.