The dice part was fine, I can compute doubles probability in my sleep.
Start by clearly defining the random variables and payout structures for each game, then compute expected value and variance using probability theory. Compare each bet's expected value to its cost to identify positive EV bets, and consider risk (variance) in your assessment.
Pro tip: In trading interviews, always discuss the risk-adjusted return (e.g., Sharpe ratio) and not just expected value, as Optiver values understanding of risk management.
Clearly state the rules, possible outcomes, probabilities, and payout for each game (dice, coin, card).
For each bet, calculate EV = Σ (probability × payout) and subtract the cost of the bet if applicable.
Calculate variance = Σ probability × (payout - EV)^2 for each bet to measure risk.
Compare each bet's EV to zero (or to the cost) to determine if it's positive expected value.
Consider variance, risk-adjusted returns, and potential for ruin, especially if multiple bets are available.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
This is where I felt the most out of my depth.
Model the problem as a linear program where you maximize the minimum payoff across all outcomes subject to the constraint that the total allocation equals 1 (or a fixed budget). If the optimal minimum payoff is greater than 0, a risk-free arbitrage exists; otherwise, it does not. Then extract the allocation from the LP solution.
Pro tip: In practice, always check for negative-cost arbitrage first by solving the dual LP or using the no-arbitrage condition: a risk-free arbitrage exists if and only if there is no set of nonnegative state prices that price all bets consistently. This can be faster and more insightful.
Let x_i be the fraction of budget allocated to bet i. Enumerate all possible outcomes (e.g., combinations of game results) and compute the payoff of each bet in each outcome.
Set up a linear program: maximize t subject to sum(x_i) = 1, x_i >= 0, and for each outcome j, sum_i x_i * payoff_{i,j} >= t. This finds the maximum guaranteed minimum payoff.
Solve the LP using a standard solver. If the optimal t > 0, a risk-free arbitrage exists. If t = 0, no arbitrage exists (but a nonnegative payoff with some positive is possible only if t=0 and some outcomes have positive payoff).
If t > 0, the optimal x_i values give the allocation that guarantees at least t in every outcome. If t = 0, check if there is an allocation with nonnegative payoffs and at least one strictly positive; this is a weaker condition and may require a secondary LP.
Verify the solution by computing payoffs for each outcome. Discuss real-world factors like transaction costs, liquidity, and the assumption of divisibility.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Kelly criterion I know well enough to talk about confidently.
Start by defining the objective as maximizing long-term growth of the bankroll, then introduce the Kelly criterion as the theoretically optimal solution. Compare full Kelly, fractional Kelly, and fixed-fraction betting in terms of growth rate, risk of ruin, and practical considerations like estimation error and variance. Conclude with a recommendation that balances growth and risk, such as half-Kelly, and justify it based on the edge and variance.
Pro tip: Emphasize that in practice, edges are estimated with error, so full Kelly is overly aggressive; fractional Kelly provides a safety margin and better risk-adjusted returns. Mention that Optiver values robust decision-making under uncertainty, so highlighting the trade-off between theoretical optimality and practical robustness will impress.
State that the goal is to maximize the expected logarithm of wealth over T rounds, assuming independent bets with known edge and variance. Clarify that at least one positive-EV bet is available each round.
Explain that full Kelly maximizes expected log wealth and is optimal for long-run growth. Provide the formula for optimal fraction f* = edge / variance (or odds-based for binary bets).
Discuss that full Kelly maximizes growth but has high volatility and risk of ruin if edge is overestimated. Fractional Kelly (e.g., half-Kelly) reduces volatility and drawdowns at the cost of slightly lower growth. Fixed-fraction ignores edge and variance, leading to suboptimal growth or excessive risk.
Recommend a fractional Kelly approach (e.g., half-Kelly) because it balances growth and risk, is robust to estimation errors, and is commonly used in practice. Mention that the fraction should depend on confidence in the edge estimate and risk tolerance.
Note that in real trading, edges and variances are estimated and may change over time, so dynamic rebalancing and risk limits are important. Also mention that fixed-fraction can be appropriate if the edge is small or uncertain, but it is not growth-optimal.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
CLT application is straightforward if you have the mean and variance of a single round's profit, which I had estimated earlier.
First, clearly define the bet-sizing strategy and the stochastic process for per-round profit, then use the Central Limit Theorem to approximate the cumulative profit after T rounds as normal. Compute the mean and variance of the per-round profit, then calculate the probability that the sum exceeds zero using the normal CDF. Discuss how simplifying assumptions (e.g., independence, identical distribution) enable fast mental math while acknowledging potential inaccuracies.
Pro tip: Emphasize that in trading interviews, the ability to quickly derive a reasonable approximation and sanity-check it with extreme cases (e.g., T=1, T=large) is often more valued than a precise but slow calculation. Mention that you would validate the normal approximation with a quick simulation if time permits.
State the exact rule for bet size (e.g., fixed fraction of bankroll, Kelly criterion) and derive the mean and variance of the profit for a single round. Assume each round's outcome is independent and identically distributed.
Using linearity of expectation and the variance sum formula for independent variables, calculate the expected value and variance of the total profit after T rounds. For a fixed bet size, these scale linearly with T.
For large T, approximate the cumulative profit as normally distributed with the computed mean and variance. Justify the use of CLT by noting that T is typically large and the per-round profits are independent and bounded.
Standardize the threshold (0) using the normal approximation: P(Profit > 0) ≈ 1 - Φ((0 - μ_T)/σ_T), where Φ is the standard normal CDF. If μ_T > 0, this probability increases with T; if μ_T < 0, it decreases.
Explain that the normal approximation is fast and often sufficient for mental estimation, but it ignores skewness and kurtosis, which may matter for small T or extreme bets. Mention that simulation or exact calculation (e.g., binomial for binary outcomes) can improve accuracy at the cost of speed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.