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4691Variance Shock Half-Life 1A mean-reverting stochastic-vol model has mean-reversion speed kappa = 1.5. What is the half-life of a variance shock?数理金融简单数值题未尝试面试订阅4692Remaining Variance-Shock Fraction After Nine MonthsA mean-reverting stochastic-vol model has variance drift d v t = kappa(theta-v t)dt + ... with kappa = 1.2. What fraction of an initial variance shock v 0-theta is expected to remain after 0.75 years?数理金融简单数值题未尝试面试订阅4693Time Until Only 20% of a Variance Shock RemainsA mean-reverting stochastic-vol model has kappa = 2. After how many years is only 20% of the initial variance shock expected to remain?数理金融简单数值题未尝试面试订阅4694Expected One-Year Forward VarianceIn a mean-reverting stochastic-vol model, current variance is v0 = 0.09, long-run mean is theta = 0.04, and mean-reversion speed is kappa = 1.5. What is E[v 1]?数理金融简单数值题未尝试面试订阅4695Expected Variance Drop Over Six MonthsIn a mean-reverting stochastic-vol model, variance starts at v0 = 0.16, long-run mean is theta = 0.09, and kappa = 1. What is the expected drop v0 - E[v 0.5 ] after half a year?数理金融简单数值题未尝试面试订阅4696Fair Average Variance Over Horizon 6In a mean-reverting stochastic-vol model, variance starts at v0 = 0.04, long-run mean is theta = 0.09, mean-reversion speed is kappa = 1.5, and horizon is T = 1. What expected average variance E[(1/T)∫ 0 T v s ds] and annualized volatility sqrt of that average variance does the model imply?数理金融中等数值题未尝试面试订阅4697Long-Run Mean Implied by an Average-Variance TargetSuppose expected average variance over one year is E[(1/T)∫ 0 T v s ds] = 0.097927 in a mean-reverting stochastic-vol model with current variance v0 = 0.12, kappa = 1, and T = 1. What long-run mean theta is implied?数理金融中等数值题未尝试面试订阅4698Annualized Volatility from Expected Forward VarianceCurrent variance is v0 = 0.04, long-run mean is theta = 0.09, mean-reversion speed is kappa = 2, and horizon is T = 0.5. What expected forward variance E[v T] and annualized volatility sqrt(E[v T]) does the model imply?数理金融中等数值题未尝试面试订阅4699Starting Variance Implied by an Average-Variance TargetA stochastic-vol model has long-run mean theta = 0.05, mean-reversion speed kappa = 1, and horizon T = 1. If expected average variance over that year is 0.081606, what starting variance v0 is implied?数理金融中等数值题未尝试面试订阅4700Mean-Reversion Speed Implied by a One-Year Forward Variance TargetA stochastic-vol model has current variance v0 = 0.16 and long-run mean theta = 0.04. If the expected variance one year ahead is E[v 1] = 0.10, what mean-reversion speed kappa is implied?数理金融中等数值题未尝试面试订阅4701Stochastic Vol Scenario 11Why can stochastic volatility explain a persistent equity-index downside skew more naturally than a constant-volatility Black-Scholes model?数理金融中等essay未尝试面试订阅4702Stochastic Vol Scenario 12What does vol-of-vol control economically in a stochastic-vol model?数理金融中等essay未尝试面试订阅4703Stochastic Vol Scenario 13Why is mean reversion in variance such a central ingredient of practical stochastic-vol models?数理金融中等essay未尝试面试订阅4704Stochastic Vol Scenario 14What dynamic feature can stochastic volatility generate that deterministic local-vol surfaces often struggle to reproduce?数理金融中等essay未尝试面试订阅4705Stochastic Vol Diagnostic 15Before choosing a stochastic-vol model, what should you ask first about the empirical smile behavior you want to capture?数理金融中等essay未尝试面试订阅4706Stochastic Vol Diagnostic 16If mean-reversion speed kappa rises while other parameters stay fixed, what happens to the persistence of a volatility shock?数理金融中等essay未尝试面试订阅4707Stochastic Vol Diagnostic 17If the long-run variance mean theta increases, what usually happens to long-dated variance expectations?数理金融中等essay未尝试面试订阅4708Stochastic Vol Diagnostic 18If spot-vol correlation becomes more negative, what usually happens to downside skew?数理金融中等essay未尝试面试订阅4709Stochastic Vol Diagnostic 19If vol-of-vol increases materially, what usually happens to uncertainty about future variance?数理金融中等essay未尝试面试订阅4710Stochastic Vol Diagnostic 20Why do stochastic-vol parameter differences often show up more clearly in longer-dated options than in very short-dated ones?数理金融中等essay未尝试面试订阅