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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2076Affine Replication Check in an Incomplete Market 6The stock is 100 today and ends at 120, 100, or 80 next period. Consider a claim paying 20, 10, and 0 in those three states. Can it be replicated exactly using only the stock and cash? If yes, give the hedge. If not, identify the replication obstruction.数理金融中等derivation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2086Recover the Completing Digital Quote 16A one-period trinomial stock ends at 120, 100, or 80 with zero interest. An up-state digital paying 1 only in the up state completes the market and trades at an unknown price q. A claim paying 5, 1, and 0 in the three states is observed to trade at 1.8. What q is implied?数理金融中等数值题未尝试面试订阅2091Incomplete-Market Pricing Intuition 21Why does a unique no-arbitrage price disappear as soon as the trinomial market has more states than traded securities?数理金融困难essay未尝试面试订阅2092Incomplete-Market Pricing Intuition 22Why does the superhedge naturally sit at the top of an incomplete-market price interval?数理金融困难essay未尝试面试订阅2093Incomplete-Market Pricing Intuition 23Why can a minimum-variance hedge still fail to pin down a unique no-arbitrage price?数理金融困难essay未尝试面试订阅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2116Variance-Swap Surface Intuition 21Two one-year equity option surfaces have the same ATM implied volatility, but Surface B has much more expensive downside puts than Surface A. Why can Surface B still imply a meaningfully higher fair variance-swap strike?数理金融困难essay未尝试面试订阅2117Variance-Swap Sampling Intuition 22A quarter has only two very large overnight gap moves and otherwise tiny close-to-close returns. Why can realized variance still end up far above what a smooth diffusion intuition would suggest?数理金融困难essay未尝试面试订阅2118Variance-Swap Modeling Intuition 23Why can simple-return and log-return fixing conventions differ materially once moves become large?数理金融困难essay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