St. Petersburg Paradox, the Martingale, and Risk Management

Many financial practitioners have long been interested in gambling and games of chance, from Ed Thorpe's, a former hedge fund manager, seminal work on blackjack card counting Beat the Dealer, to the famous hedge fund titan poker tournament (Take 'Em to School Poker Tournament). The similarity is more than skin deep, with gambling methods such as the Kelly Criterion even making their way downtown to investment strategy and risk management. In this post, we're going to discuss the mathematics and implications for finance of two different gambling related topics: the St. Petersburg Paradox and the Martingale Strategy.

St. Petersburg Paradox

First proposed by Bernoulli, the St. Petersburg paradox is a simple game that illustrates how people's utility function differs from pure expected value theory. Given a fair coin, the coin is flipped until a heads does not appear and the player collects $2^n$ dollars where $n$ is the number of heads. We can calculate the expected value easily by summing the probabilities and payoffs:

\[E = \sum_{i=1}^\infty \frac{1}{2^i} 2^i = 1 + 1 + \cdots + 1 = \infty\]

Even though the expected value in infinity, most people are willing to pay relatively little to play such a game, since it's unlikely that the player will win a large sum, and much more likely to not win any money or only make a moderate amount. The long tail of large potential profit forms a geometric distribution, a discrete distribution which models the likelihood of $n$ number of events happening in sequence:

\[\text{P}(X = n) = q^{n-1}p\]

Where $q = 1 - p$. Since we're using a fair coin, this simply becomes:

\[\text{P}(X = n) = \left(\frac{1}{2}\right)^n\]

The cumulative distribution function (CDF) is simply the chance that this doesn't happen:

\[\text{P}(X \leq n) = 1 - \left(\frac{1}{2}\right)^n\]

In order to make this a little more concrete, let's look at a graph of the likelihood and payoffs:

Payoff vs # Heads

And a graph of the CDF:

Payoff vs # Heads

As we can see, the likelihood of receiving larger and larger payoffs becomes exponentially smaller, with 93.75% of the time the player getting 4 or less heads in a row, netting a maximum payoff of only \$16.

Now imagine that instead of only playing this game once, you could pay a certain amount and play over and over again until you lost all of your bankroll. How much would you be willing to pay now? Though probably not an infinite number of dollars, the value of the game has gone up significantly for you. Assuming you have an ample bankroll to weather the storm, over the long run you will profit handsomely: making small amounts occasionally interspersed with massive windfalls.

The Martingale

The Martingale Strategy is a popular betting strategy that has been used by centuries by gamblers. The idea is simple: every time you lose a bet, place a subsequent bet that is double the size so that you can make back your losses, plus one extra dollar. Assuming an infinite bankroll, the payoff will always be 1 dollar:

\[2^n - (2^{n-1} + 2^{n-2} \cdots + 2^0) = 1\]

$2^n$ represents your winnings, while the sum of $2^{n-1}$ to $2^0$ are your losses up to that point. For example, let's use the simple example of betting on a fair coin. We have 3 dollars, so we can make a maximum of two bets of \$1 and \$2. The expected value looks like:

\[\text{E} = 0.5 (\$1) + 0.5 (0.5 (\$1) + 0.5 (-\$3)) = 0\]

Even though the expected value is 0, the chance of losing \$3 is only 25%, while the chance of winning \$1 is 75%. In the general case, we need $2^n-1$ dollars to place $n$ bets, where the chance of winning $1 is:

\[\text{P}(\$1) = 1 - \left(\frac{1}{2}\right)^n\]

And losing:

\[\text{P}(-\$(2^n-1)) = \left(\frac{1}{2}\right)^n\]

Let's look at a graph of our chance of winning as a function of our bankroll:

Martingale

Much like the St. Petersburg Paradox, the Martingale exhibits a long-tail, except while the former has a long-tail of a windfall, the Martingale has a long-tail of absolute ruin. In general, if you want a $\left(\frac{1}{2}\right)^n$ chance of ruin, then you need $2^n-1$ dollars. So for example, to push your chance of losing it all to around one in a million, you would need about a million dollars to make that happen; and with such a large capital base, a one dollar profit in the best case isn't particularly appealing.

The Martingale at first glance looks like a reasonable strategy, because the risk of ruin is pushed towards a very long, sometimes unobservable, tail – which people have a hard time intuitively reasoning about. The low probability risk of losing it all is not correctly taken into account, much in the same way that the low probability of a massive windfall isn't taken into account for the intuitive fair value of the St. Petersburg Paradox.

Application to Investing

Imagine you were a manager of a trading desk, and one of your traders was offering the St. Petersburg Paradox to clients for a hefty fee while simultaneously Martingale betting on some derivative. You are unaware of what he's actually doing, but you can see the return stream from his trades. He's only been working a couple months and yet, he's minting money every day, though occasionally suffers some moderate to large losses. Based on his record, you might think that he's doing really well and that while his returns are moderately volatile, he generates a lot of profit, so that his risk-adjusted returns are pretty good.

But you would be wrong, the trader is taking on massive levels of risk. But from your perspective of only looking at his return streams, this risk is non-existent, simply because the disastrous long-tail outcome has never been observed. When a distribution of returns exhibits a lot of excess kurtosis and skew, one needs to think very hard about how to appropriately manage risk. Popular risk models such as Value at Risk (VaR) that estimate the worst 95% or 99% outcome become insufficient when the return distribution deviates significantly from that of a log-normal distribution. If you used a 99% VaR model to measure the risk of your trader's strategies, it would fail to capture almost all of it, leaving you and your firm unknowingly exposed to existential extinction events.

Conclusion

While it's rare that any return stream is perfectly modeled by a log-normal distribution (most returns exhibit excess kurtosis and left-side skew), it's a good approximation for some, and completely unfit for others. Strategies such as selling volatility/variance insurance or selling far out-of-the-money calls or puts share many properties with the example of the aforementioned trader. It might look like superior risk-adjusted returns are being generated, but in fact this risk is just concentrated into a long-tail and thus unobservable until disaster strikes. This is often likened to "picking up pennies in front of a steam-roller."

Thank for reading, I hope you enjoyed this post! It's a little different than usual, but a recent Matt Levine article about a Canadian pension fund blow-up (from taking long-tail risk) got me thinking about the similarities between St. Petersburg Paradox and the Martingale Strategy and long-tail risky strategies such as selling variance insurance. No code for this post, as only a couple of trivial graphs were made.