Reasoning About Variance in a Trading Game

Two bets can share an expected value and still differ enormously in risk, and a trading desk has to size for the difference.

Reasoning about variance in a trading game means comparing two bets or strategies with the same expected value and explaining which one a trading desk should actually prefer, based on how the spread of outcomes interacts with position limits, capital, and the number of times the bet gets repeated. Interviewers ask these questions specifically because expected value alone cannot distinguish between a strategy a desk can safely run and one that will blow through a risk limit before it pays off.

Consider two games offered side by side. Game one pays $10 with certainty. Game two pays $20 with probability 0.5 and $0 with probability 0.5, giving an expected value of 0.5 times 20, which is also $10. Both games have the same expected value, but game two has a standard deviation of $10, while game one has a standard deviation of zero.

Asked to choose between them for a single play, most interviewers accept either answer if it is justified correctly, since the choice depends on the size of the stake relative to the bankroll and on whether the game is repeated. What they are testing is whether you can name the standard deviation of each game specifically, rather than saying one is riskier without quantifying by how much.

Playing game two once carries $10 of standard deviation around a $10 expected value, a ratio of 1 between the two. Playing it 100 times independently, the expected total becomes $1,000, and because variance is additive across independent plays while standard deviation grows only with the square root of the number of plays, the standard deviation of the total grows to 10 times the single-play standard deviation, or $100. The ratio of standard deviation to expected value has fallen from 1 to 0.1, meaning the repeated version of the same bet is proportionally far less risky than a single play of it.

This is the mathematical reason a trading desk can profitably run a strategy with a modest edge on any single trade, provided the strategy is repeated often enough and the trades are close to independent of each other. A single trade with a small edge looks close to a coin flip. A thousand such trades look close to a certainty, which is the entire justification for high frequency, high repetition trading models used by firms such as Jane Street.