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2805Squared Distance to the Center of the SquareA point is chosen uniformly in the unit square. What is the expected squared distance from the point to the center (1/2,1/2)?脑筋急转弯中等derivation未尝试面试订阅2808Variance of a Coordinate in the DiskA point is chosen uniformly from the unit disk. Let X be its x-coordinate. Compute Var (X).脑筋急转弯中等derivation未尝试面试订阅2812Expected Squared Distance in the Unit SquareTwo points are chosen independently and uniformly in the unit square. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅2828Mean and Variance After ThinningUnder the thinning setup above, derive E[Y] and Var (Y) in terms of E[X] and Var (X).概率中等derivation未尝试面试订阅2869Chebyshev for a Monte Carlo MeanAn unbiased Monte Carlo estimator averages n=100 i.i.d. draws with variance 9. Use Chebyshev's inequality to bound the probability that the sample mean deviates from its target by at least 0.5.概率简单derivation未尝试面试订阅2932Conditional Variance One Step AheadIn a branching process, let the offspring distribution have variance 2. Express \[ Var (Z n+1 \mid Z n) \] in terms of Z n.概率简单derivation未尝试面试订阅2933Variance Recursion across GenerationsLet a branching process have offspring mean m and variance 2. Show that \[ Var (Z n+1 )=m 2 Var (Z n)+ 2 E[Z n]. \]概率中等derivation未尝试面试订阅3976Asian Call versus Vanilla CallTwo arithmetic-average Asian calls share strike 100 and terminal price 100. Path A is [100, 120, 90, 100]; path B is [100, 95, 100, 100]. Which payoff is larger, and by how much?金融与交易中等derivation未尝试面试订阅3977Sensitivity to a One-Day SpikeAn arithmetic-average Asian call with strike 53 sees fixings [50, 52, 54, 60]. Under standard equal weights its payoff is based on the simple average. If the final fixing is instead double-counted, by how much does the payoff change?金融与交易中等derivation未尝试面试订阅3978Average-Price versus Average-Strike FocusAn arithmetic-average Asian call with strike 81 has already fixed [80, 84, 79], with one fixing left. What final fixing puts the option exactly at breakeven?金融与交易中等derivation未尝试面试订阅3979More Fixings and SmoothingAn average-strike Asian call has already fixed [90, 92, 94], with the terminal price still unknown and also included in the average. At what terminal price does the payoff become exactly zero?金融与交易中等derivation未尝试面试订阅3980Why Asians Are Popular in Commodity HedgingAn arithmetic-average Asian put with strike 98 has fixings [100, 97, 95, 96]. If the lowest fixing 95 is later corrected to 99, by how much does the payoff change?金融与交易中等derivation未尝试面试订阅3986Why Averaging Changes the Option So MuchWhy is an average-price Asian option usually cheaper than the matching vanilla option?金融与交易中等essay未尝试面试订阅3987Why a Late Averaging Window Mimics VanillaWhy does moving the averaging window closer to maturity make an Asian behave more like a vanilla option?金融与交易中等essay未尝试面试订阅3988Why Asians Can Hedge Real Exposures BetterWhy can arithmetic-average and geometric-average Asians price differently even on the same underlying and dates?金融与交易中等essay未尝试面试订阅3990How to Sanity-Check an Asian Option AnswerWhen might a desk prefer an average-strike Asian to an average-price Asian?金融与交易中等essay未尝试面试订阅4031Residual Hedge Variance 1Spot volatility is 20% and correlation to the hedging futures is 0.8. Under the minimum-variance hedge, what residual spot volatility remains?金融与交易中等derivation未尝试面试订阅4033Residual Hedge Variance 3An exposure has unhedged volatility 25%. After applying the minimum-variance hedge, the residual volatility is 15%. What correlation between spot and futures does this imply?金融与交易中等derivation未尝试面试订阅4035Residual Hedge Variance 5Spot volatility is 18%. What minimum absolute correlation |rho| is needed so that the minimum-variance hedge leaves residual volatility no more than 9%?金融与交易中等derivation未尝试面试订阅4866Infer Antithetic Correlation From Variance Reduction 1A Monte Carlo desk averages each payoff with its antithetic partner. The variance of the antithetic average is observed to be 35% of the crude single-path variance. If the paired payoffs have equal variance, what correlation rho between the two payoffs is implied?数理金融简单数值题未尝试面试订阅