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.

The gist of the paradox is a game where a fair coin is tossed and the player wins two dollars if heads comes up, and the amount of winnings doubles if heads come up in successive tosses. At any point that tails comes up the game is over and the player gets his accumulated payout. Simple math shows that the player has a 50% chance of $2, 25% of $4, 12.5% of $8 and so on, thus an “expected return” of $1 for each toss indefinitely, so an infinite expected return. The paradox is generally presented as “What would you be willing to pay to play the game?” We might argue, given the functioning of (fiduciary driven) financial markets, the more relevant question comes the other way around: “What would somebody be willing to accept to offer the game?” This, in a nutshell, is what vol selling is all about. How little will somebody accept for an unbounded potential loss? As we constantly discuss, this is the different time driven perspective between probability and expected return. As many readers will have heard us discuss, which side of a 100:1 payout would you want to be on for one number on a Roulette Wheel, with 1:36 odds? The answer likely depends on how many spins of the wheel you are playing, and with whose capital you are gambling. If your relevant measuring period is one spin, and you are using somebody else’s capital, probability likely dominates your decision making and you are willing to risk the $100 with a very high probability of winning the $1. On the other hand, over a large number of spins, with added emphasis, if it is your own capital, the expected value will start to take on greater significance.

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.

Figure 1: Payout on short 1-year EUR/USD 100k vega convexity swap over 2 Std Dev of recent historical data series
Source: Bloomberg

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.

Figure 2: Payout on long 1-year EUR/USD 100k vega convexity swap over long term historical data series
Source: Bloomberg

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.

So again, in Paradox type numbers – What’s the probability of the buyer losing his max upfront payment? About 50% (coin comes up tails the first time). What is the probability of the buyer getting to breakeven? About 12.5% (heads comes up three times in a row). What’s the probability of the buyer making money? About 5% (it moves outside the 95 percentile range). What’s the probability of one number on a 36 number Roulette Wheel? About 2.77%. What’s the expected return on a St Petersburg Paradox game, or a 100:1 payout on a Roulette Wheel, or on a euro convexity swap with a sufficient number of plays? About infinite.

Read our Disclaimer by clicking here

One Comment

  1. […] We gave a real-world example of this sort of payoff way back in our March 2019 Update, comparing a Convexity Swap to a St. Petersburg Paradox game: https://convex-strategies.com/2019/05/06/risk-update-q1-2019/. […]

Comments are closed.