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5650Monte Carlo Standard Error 5A Monte Carlo run has sample mean terminal payoff 9.8 and sample standard deviation 5 from n=1024 paths. With rate 0.025 and maturity 1.25, what are the time-0 price estimate, its standard error, and the approximate 95% confidence interval?数理金融中等数值题未尝试面试订阅5654Asian Path Payoff 4A simulated path for an arithmetic-average Asian call is [95, 92, 90, 97] with strike 94. What payoff does this single path contribute to the Monte Carlo estimator?数理金融中等数值题未尝试面试订阅5656Required Path Count 1A pricing engine estimates that the discounted payoff standard deviation is about 6. How many Monte Carlo paths are needed to drive the standard error down to at most 0.15?数理金融中等数值题未尝试面试订阅5657Required Path Count 2A pricing engine estimates that the discounted payoff standard deviation is about 3.5. How many Monte Carlo paths are needed to drive the standard error down to at most 0.08?数理金融中等数值题未尝试面试订阅5661Why Monte Carlo Fits Path DependenceWhy is Monte Carlo often a natural choice for pricing path-dependent derivatives?数理金融中等essay未尝试面试订阅5662Why Monte Carlo Converges SlowlyWhy do practitioners say plain Monte Carlo converges slowly even though it is conceptually simple?数理金融中等essay未尝试面试订阅5663Why Discounting Still Matters In SimulationWhy is it incorrect to average terminal payoffs from a risk-neutral simulation and stop there without discounting?数理金融中等essay未尝试面试订阅5664Why Monte Carlo Beats Trees In High DimensionWhy can Monte Carlo become more attractive than lattice methods as the number of risk factors grows?数理金融中等essay未尝试面试订阅5665Why Monte Carlo And Model Risk InteractWhy does a small Monte Carlo standard error not guarantee that the option price is actually reliable?数理金融中等essay未尝试面试订阅5877Risk-Neutral Probability From Tree FactorsA one-step binomial tree has up factor u=1.15, down factor d=0.88, continuously compounded rate r=0.05, and Δt=0.5. Compute the risk-neutral probability of an up move.数理金融简单数值题未尝试免费5878CRR Up/Down Factors From VolatilityIn a Cox-Ross-Rubinstein tree the volatility is σ=0.25 per year and each step is Δt=0.25 years. Using u=e σ√Δt and d=1/u, what is the up factor u (to four decimals)?数理金融简单数值题未尝试免费5879Replicating Delta On A One-Step TreeA stock at 50 moves in one step to 58 or 44. A European call struck at 52 is written on it. What is the replicating delta (shares per option) over this step?数理金融简单数值题未尝试免费5880Two-Step European PutOn a two-step binomial tree, spot=100, strike=100, u=1.1, d=0.9, r=0.05, Δt=1. Price the European put at time 0.数理金融中等数值题未尝试免费5881American Put Early Exercise On Two StepsPrice an American put with strike 100 on a two-step tree: spot=100, u=1.2, d=0.8, r=0.03, Δt=1. Give the time-0 value and state whether early exercise occurs at the first down node.数理金融困难数值题未尝试面试订阅5882Completing A Trinomial Probability SetA one-step trinomial tree has multipliers u=1.2, m=1, d=0.8. The middle probability is fixed at p m=0.6, r=0.04, Δt=1. Find the up-move probability p u that makes the discounted underlying a martingale (so p u+p m+p d=1 and E[S 1]=S 0 e rΔt ).数理金融困难数值题未尝试面试订阅5883One-Step Binomial Call With Dividend YieldA one-step binomial tree has spot=100, strike=100, u=1.1, d=0.9, rate r=0.05, continuous dividend yield δ=0.02, Δt=1. Using the dividend-adjusted risk-neutral probability, price the European call.数理金融中等数值题未尝试免费5884Real-World Versus Risk-Neutral Probability On A TreeOn the same binomial tree, an analyst estimates a real-world up probability of 0.65 from historical data, while the risk-neutral up probability is 0.52. Which probability should be used to price a derivative by discounted expectation, and what governs the gap between the two?数理金融中等essay未尝试免费5885Tree Versus Black-Scholes ConvergenceA one-step CRR binomial tree prices an at-the-money one-year European call at 9.95, while the Black-Scholes value with the same spot, strike, rate and volatility is 8.43. By how much does the coarse tree overprice the option, and what single change to the tree would most directly shrink this error?数理金融中等数值题未尝试免费5886Two-Step European Call Via Terminal WeightsOn a two-step recombining tree with spot=64, strike=70, u=1.25, d=0.8, r=0, Δt=1, price the European call by weighting the three terminal payoffs with the binomial probabilities q 2, 2q(1-q), (1-q) 2.数理金融中等数值题未尝试免费