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2030Recovering Scenario Weights From a Concave Impact Average 10Let V take values 0 and 3, with probabilities p and 1-p. If E[sqrt(1+V)] = 5/4, determine p and then compute sqrt(1+E[V]).数学中等derivation未尝试面试订阅2032Barrier Score Gap for a Two-State Utilization Model 12Let u(x)=-ln(1-x) on x<1. Suppose U equals 0 with probability 1/2 and 3/4 with probability 1/2. Compute E[u(U)] and u(E[U]).数学简单数值题未尝试免费2033Certainty-Equivalent Return From Expected Log Growth 13A strategy produces one-period returns of 0% or 60%, each with probability 1/2 on one dollar of wealth. Compute E[ln(1+R)] and the certainty-equivalent constant return r ce satisfying ln(1+r ce)=E[ln(1+R)].数学中等derivation未尝试面试订阅2036Quadratic Jensen Gap From Mean and Variance 16Let phi(x)=x 2. If a random variable X has E[X]=2 and Var(X)=5, what is E[phi(X)] - phi(E[X])?数学简单数值题未尝试免费2040Three-Scenario Square-Root Impact Gap 20Suppose V takes values 0, 3, and 8 with equal probability. Compute E[sqrt(1+V)] and sqrt(1+E[V]).数学困难数值题未尝试面试订阅2041Probability-Implied Reciprocal Buffer Score 21Leverage L takes values 1 and 4. If E[L] = 2.2, what probability p is on L=1, and what is E[1/(1+L)]?数学简单derivation未尝试免费2045Comparing Two Equal-Mean Schedules Under an Alternative Surcharge 25A utilization surcharge is c(q)=1/(2-q) on q<2. Schedule A is deterministic with Q=1. Schedule B uses Q=1/2 or 3/2 with probability 1/2 each. Compute E[c(Q)] for Schedule B and c(E[Q]) for the shared mean.数学困难数值题未尝试面试订阅2071Price a Trinomial Claim After Market Completion 1A one-period stock is 100 today and ends at 120, 100, or 80. The risk-free rate is 0. A quoted up-state digital that pays 1 only in the up state trades at 0.2, which completes the market. What unique no-arbitrage price does this imply for the claim paying 5, 1, and 0 in the up, middle, and down states?数理金融中等数值题未尝试面试订阅2081Worst Shortfall of a Simple Hedge 11A non-traded payoff pays 4, 1, and 6 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash -8 and Delta = 0.1 shares of stock. What is the worst-case shortfall of that hedge across the three states?数理金融简单数值题未尝试免费2082Worst Shortfall of a Simple Hedge 12A non-traded payoff pays 3, 5, and 1 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash 7 and Delta = -0.05 shares of stock. What is the worst-case shortfall of that hedge across the three states?数理金融简单数值题未尝试免费2094Incomplete-Market Pricing Intuition 24Why can one extra state-contingent quote complete the market in a trinomial model even if the stock and bond alone cannot?数理金融困难essay未尝试面试订阅2095Incomplete-Market Pricing Intuition 25Why do indifference prices depend on risk aversion while no-arbitrage intervals do not?数理金融困难essay未尝试面试订阅2096Infer Event-Window Length From a Fair Variance Strike 1A 252-day variance swap assumes instantaneous volatility 0.4 during an event window of unknown length d trading days and volatility 0.2 during the remaining days. If the annualized fair volatility strike is 0.3, approximately how many trading days does the event window last?数理金融简单数值题未尝试免费2101Recover the Missing Log Return From a Variance Fixing 6A variance-swap fixing uses RV = (252/n) * sum r i 2 with n=4. Three logged returns have already been recorded: [0.01, -0.02, 0.015]. What absolute missing return |r 4| would make the realized variance equal 0.04725?数理金融简单数值题未尝试免费2104Recover the Missing Log Return From a Variance Fixing 9A variance-swap fixing uses RV = (252/n) * sum r i 2 with n=4. Three logged returns have already been recorded: [0.012, 0.011, -0.009]. What absolute missing return |r 4| would make the realized variance equal 0.024885?数理金融简单数值题未尝试免费2105Recover the Missing Log Return From a Variance Fixing 10A variance-swap fixing uses RV = (252/n) * sum r i 2 with n=4. Three logged returns have already been recorded: [0.015, -0.005, -0.012]. What absolute missing return |r 4| would make the realized variance equal 0.045234?数理金融简单数值题未尝试免费2106Infer Remaining Flat Volatility From a Live Variance Mark 11A variance swap has observed 30 trading days out of 90. Realized variance so far is 0.035, and the current fair full-life variance strike is 0.038333. If the remaining window is assumed to carry one flat volatility level, what annualized volatility does that imply for the remaining life?数理金融中等数值题未尝试面试订阅2111Infer Stress Probability From a Variance Forecast 16A one-year variance forecast says annualized volatility will be 0.15 in a calm regime and 0.35 in a stress regime. If the fair variance-swap volatility strike is 0.24, what risk-neutral probability of the stress regime is implied by the variance forecast?数理金融中等数值题未尝试面试订阅2119Variance-Swap Modeling Intuition 24Why can a variance swap mark-to-market move even if the headline implied-vol surface looks unchanged at first glance?数理金融困难essay未尝试面试订阅2171Infer the Structural Default Put 1In a Merton-style structural view, firm assets are worth 120, the present value of promised debt is 97, and the observed equity value is 28.5. What risky debt value and what implied default put value on the assets are consistent with this decomposition?数理金融中等数值题未尝试面试订阅