Lessons from the St. Petersburg Paradox
The St. Petersburg paradox provides valuable insights for options trading and risk management.
The Game
Consider a casino game where:
Player starts with $2 and a fair coin
On heads: Player doubles their money
On tails: Game ends, player takes home winnings
The Paradox
What would be a fair, rational entry fee for this game? Blaise Pascal, the father of modern probability, proposed that for games with random payouts, the most critical statistic is the expected value. This concept is supported by the Law of Large Numbers, which can be demonstrated through Monte Carlo simulations. According to the Efficient Market Hypothesis, the fair price for such a game should be its expected value. However, this leads to a paradox.
Expected Value Calculation
The expected value of this game is infinite: E=∑n=1∞2n⋅12n=∑n=1∞1=∞E=∑n=1∞2n⋅2n1=∑n=1∞1=∞This suggests that a rational player should be willing to pay any finite amount to play the game, which is counterintuitive.
Resolving the Paradox
The St. Petersburg paradox is often viewed as a problem in the philosophy of rational choices. To resolve it, we invoke the concept of "utility":
Logarithmic Utility: A rational price for the game would be on the order of log₂(total house capital), even though logarithmic growth is slow.
Diminishing Marginal Utility: For most individuals and companies with cash constraints, winning $1 million rather than nothing is more impactful than the difference between winning $1 billion and $1.001 billion. Money received only turns into utility according to a concave function.
Applications to Options Trading
Beyond Average P/L: While average profit/loss is an important trade metric, the St. Petersburg paradox demonstrates that investors need to consider additional features.
Long-Term Survival: The Law of Large Numbers promises convergence, but it's crucial for small trading operations to trade strategically to stay in the game long-term and avoid account blow-ups.
Key Metrics for Survival: To ensure longevity in trading, focus on:
Probability of Profit (PoP)
Conditional Value at Risk (CVaR) at 5%
The Greeks (Delta, Gamma, Theta, Vega)
Trading Strategy: Trade small positions and trade very often to leverage the law of large numbers while managing risk.
Conclusion
The St. Petersburg paradox highlights the importance of considering utility, risk management, and long-term survival in trading strategies. By focusing on a broader set of metrics beyond just expected value, traders can develop more robust and sustainable approaches to options trading.