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4923Infer Missing Tail Observation 8Sorted empirical losses are [2, 2, 4, 5, 7, 10, 13, x]. Using alpha=0.75 and the same conventions, if empirical ES is 15, what is x?数理金融中等数值题未尝试面试订阅4924Infer Missing Tail Observation 9Sorted empirical losses are [1, 1, 2, 4, 4, 7, 9, x]. Using alpha=0.875 and the same conventions, if empirical ES is 11.5, what is x?数理金融中等数值题未尝试面试订阅4926Infer One-Day VaR From Multi-Day Scaling 11Under square-root-of-time Gaussian scaling, a 10-day VaR is 7.589466. What 1-day VaR does that imply?数理金融中等数值题未尝试面试订阅4928Infer Pareto Tail Index From ES To VaR Ratio 13For a Pareto tail, suppose ES/VaR = alpha/(alpha-1). If VaR is 8 and ES is 12, what tail index alpha is implied?数理金融中等数值题未尝试面试订阅4929Infer Pareto Tail Index From ES To VaR Ratio 14For a Pareto tail with ES/VaR = alpha/(alpha-1), if VaR is 5.5 and ES is 9.166667, what alpha is implied?数理金融中等数值题未尝试面试订阅4931Infer Portfolio Covariance Loading From Component VaR 16For a linear Gaussian portfolio, component VaR satisfies component i = w i * z alpha * (Sigma w) i / sigma p. If z alpha=1.645, sigma p=0.2, w i=0.6, and the reported component VaR is 0.11844, what covariance loading (Sigma w) i is implied?数理金融困难数值题未尝试面试订阅4932Infer Portfolio Covariance Loading From Component VaR 17For a linear Gaussian portfolio, z alpha=2.326, sigma p=0.3, w i=0.35, and the reported component VaR is 0.111260333. What covariance loading (Sigma w) i is implied?数理金融困难数值题未尝试面试订阅4933Infer Portfolio Covariance Loading From Component VaR 18For a linear Gaussian portfolio, z alpha=1.96, sigma p=0.25, w i=0.5, and the component VaR is 0.1176. What covariance loading (Sigma w) i is implied?数理金融困难数值题未尝试面试订阅4936Tail Severity Interpretation 21Two desks report the same 97.5% VaR, but one desk has a much larger 97.5% ES. What does that tell you about the shape of its tail losses beyond the VaR cutoff?数理金融困难essay未尝试面试订阅4937Backtesting Intuition 22Why is ES harder to backtest directly than VaR in day-to-day risk control?数理金融困难essay未尝试面试订阅4938Scaling Failure Intuition 23Why can square-root-of-time VaR scaling fail badly during volatility clustering?数理金融困难essay未尝试面试订阅4939Tail Preference Intuition 24Why is ES typically preferred to VaR when the main concern is tail severity rather than breach frequency?数理金融困难essay未尝试面试订阅4940Allocation Intuition 25Why is a single firm-wide VaR or ES number not enough for desk incentives unless it is broken into component contributions?数理金融困难essay未尝试面试订阅5541Black-Scholes Call 1Under Black-Scholes with spot 100, strike 100, risk-free rate 0.03, dividend yield 0, volatility 0.2, and maturity 1, what is the European call price?数理金融中等数值题未尝试面试订阅5542Black-Scholes Call 2Under Black-Scholes with spot 95, strike 100, risk-free rate 0.04, dividend yield 0.01, volatility 0.25, and maturity 0.5, what is the European call price?数理金融中等数值题未尝试面试订阅5543Black-Scholes Call 3Under Black-Scholes with spot 120, strike 110, risk-free rate 0.02, dividend yield 0, volatility 0.18, and maturity 1.5, what is the European call price?数理金融中等数值题未尝试面试订阅5546Black-Scholes Put 1Under Black-Scholes with spot 100, strike 100, risk-free rate 0.03, dividend yield 0, volatility 0.2, and maturity 1, what is the European put price?数理金融中等数值题未尝试面试订阅5547Black-Scholes Put 2Under Black-Scholes with spot 95, strike 90, risk-free rate 0.04, dividend yield 0.02, volatility 0.22, and maturity 0.5, what is the European put price?数理金融中等数值题未尝试面试订阅5548Black-Scholes Put 3Under Black-Scholes with spot 120, strike 130, risk-free rate 0.02, dividend yield 0, volatility 0.18, and maturity 1.5, what is the European put price?数理金融中等数值题未尝试面试订阅5551Forward Form And Exercise Probability 1A stock has spot 100, strike 100, rate 0.03, dividend yield 0.01, volatility 0.2, and maturity 1. Under Black-Scholes, what are the forward price F 0,T and the risk-neutral probability that the call finishes in the money?数理金融中等数值题未尝试面试订阅