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2385When a Control Variate Can Make Things Worse 15Why can a badly chosen control coefficient increase variance instead of reducing it?数学困难derivation未尝试面试订阅2386Infer Regime Probability From Conditional Means 16A payoff has conditional mean 5 in a calm regime and -2 in a stress regime. The unconditional mean is 2.2. What calm-regime probability is implied?数学简单数值题未尝试免费2387Why Importance Sampling Stays Unbiased 17Why does reweighting by the likelihood ratio keep an importance-sampling estimator unbiased for the original expectation?数学中等derivation未尝试免费2388Why Rare Events Cause Huge Monte Carlo Noise 18Why can a rare-event payoff have an unstable Monte Carlo estimate even when most simulated paths look harmless?数学中等derivation未尝试免费2389Why Regime Averaging Can Beat Crude Mixing 19Why can separately estimating conditional means by regime and then averaging them outperform one crude pooled Monte Carlo run?数学困难derivation未尝试面试订阅2390Why Antithetics May Fail on a Highly Nonmonotone Payoff 20Why can antithetic pairing fail to reduce variance for a payoff that oscillates strongly across the state space?数学困难derivation未尝试面试订阅239195% Half-Width From Sample Volatility and Path Count 21A Monte Carlo estimator has sample standard deviation 5 across n=400 paths. Using the normal approximation, what is the 95% half-width?数学简单数值题未尝试免费2392Required Paths for a Target Standard Error 22A crude estimator has standard deviation 6. How many paths are needed to bring the standard error down to 0.3?数学简单数值题未尝试免费2393RMSE of an Unbiased Monte Carlo Estimate 23An unbiased estimator has standard deviation 4 over one path and uses n=64 independent paths. What is the RMSE of the sample mean?数学中等数值题未尝试免费2394Confidence Interval From Mean and Standard Error 24A Monte Carlo estimate is 12.0 with standard error 0.4. Using the normal approximation, what 95% confidence interval do you report?数学中等数值题未尝试免费2395Why Bias Still Matters Alongside Variance 25Why is a low-variance Monte Carlo estimator still problematic if it is biased?数学困难derivation未尝试面试订阅5641Monte Carlo Price From Sample Payoffs 1A risk-neutral Monte Carlo run produced discounted-at-maturity payoffs [12.0, 4.0, 0.0, 8.0, 16.0] for an option. If the continuously compounded rate is 0.03 and maturity is 1, what is the time-0 Monte Carlo price estimate?数理金融简单数值题未尝试面试订阅5642Monte Carlo Price From Sample Payoffs 2A risk-neutral Monte Carlo run produced discounted-at-maturity payoffs [5.0, 0.0, 11.0, 3.0, 7.0] for an option. If the continuously compounded rate is 0.04 and maturity is 0.5, what is the time-0 Monte Carlo price estimate?数理金融简单数值题未尝试面试订阅5654Asian Path Payoff 4A simulated path for an arithmetic-average Asian call is [95, 92, 90, 97] with strike 94. What payoff does this single path contribute to the Monte Carlo estimator?数理金融中等数值题未尝试面试订阅