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4671Regime Time Fraction From RMS Local Vol 6Along one realized path, local volatility equals 0.22 for a fraction f of the year and 0.3 for the rest. If the path-consistent RMS local volatility is observed to be 0.2709, what fraction f of the year was spent in the first regime?数理金融中等数值题未尝试面试订阅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4716Admissible Higher-Strike Call Interval 1A call with strike 80 trades at 25. Ignoring discounting and using only no-arbitrage monotonicity and slope bounds across strikes, what admissible price interval does that imply for the call with higher strike 90?数理金融简单数值题未尝试面试订阅4718Admissible Higher-Strike Call Interval 3A call with strike 100 trades at 10.5. Ignoring discounting and using only no-arbitrage monotonicity and slope bounds across strikes, what admissible price interval does that imply for the call with higher strike 110?数理金融简单数值题未尝试面试订阅4721Butterfly Repair Or Ceiling 6For equally spaced strikes 90, 100, 110, call prices at the wings are C(90)=18.2 and C(110)=8.1. What is the largest arbitrage-free value the middle call C(100) can take under butterfly convexity?数理金融中等数值题未尝试面试订阅4722Butterfly Repair Or Ceiling 7For equally spaced strikes 80, 100, 120, call prices at the wings are C(80)=26 and C(120)=7. What is the largest arbitrage-free value the middle call C(100) can take under butterfly convexity?数理金融中等数值题未尝试面试订阅4723Butterfly Repair Or Ceiling 8For equally spaced strikes 95, 100, 105, calls trade at 14, 12.5, and 10. By how much must the middle call price be reduced to remove the butterfly arbitrage?数理金融中等数值题未尝试面试订阅4724Butterfly Repair Or Ceiling 9For equally spaced strikes 85, 95, 105, calls trade at 20, 14.8, and 10.5. How much could the middle call price rise before butterfly convexity is first violated?数理金融中等数值题未尝试面试订阅4725Butterfly Repair Or Ceiling 10For equally spaced strikes 100, 110, 120, calls trade at 11.5, 8.4, and 5.8. How much could the middle call price rise before butterfly convexity is first violated?数理金融中等数值题未尝试面试订阅4726Calendar Repair Size 11At one strike, the shorter-maturity call with T=0.5 trades at 8.2 and the longer-maturity call with T=1 trades at 9.4. What minimum upward adjustment to the longer-maturity quote would restore basic calendar monotonicity?数理金融中等数值题未尝试面试订阅4727Calendar Repair Size 12At one strike, the shorter-maturity call with T=1 trades at 11 and the longer-maturity call with T=2 trades at 12.8. What minimum upward adjustment to the longer-maturity quote would restore basic calendar monotonicity?数理金融中等数值题未尝试面试订阅