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3376Halving h in the Trapezoid RuleA smooth integrand is approximated with the trapezoid rule. If the current error at step size h is about 0.08, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3378Halving h in Simpson's RuleA smooth integrand is approximated with the Simpson rule. If the current error at step size h is about 0.016, what is the rough error after halving the step size?数学中等derivation未尝试面试订阅3381Degree of Exactness of Two-Point GaussFor two-point Gauss-Legendre quadrature on [-1,1], what is the highest polynomial degree integrated exactly?数学中等derivation未尝试面试订阅3382Two-Point Gauss on x^2+1Using two-point Gauss-Legendre on [-1,1], approximate \int -1 1 (x 2+1)\,dx.数学中等derivation未尝试面试订阅3383Two-Point Gauss on x^4Using two-point Gauss-Legendre on [-1,1], what approximation do you get for \int -1 1 x 4\,dx?数学中等derivation未尝试面试订阅3386Why Trapezoid Overestimates a Convex FunctionWhy does the trapezoid rule typically overestimate the integral of a convex function on one panel?数学中等essay未尝试面试订阅3387Why Midpoint Often Beats Trapezoid on Smooth ProblemsWhy can the midpoint rule outperform the trapezoid rule on smooth integrands even though both are second-order?数学中等essay未尝试面试订阅3388When Simpson's Rule Is Especially AttractiveWhen is Simpson's rule especially attractive compared with midpoint or trapezoid?数学中等essay未尝试面试订阅3389Why Oscillatory Integrals Need CareWhy can standard coarse-grid quadrature be unreliable on highly oscillatory integrals?数学中等essay未尝试面试订阅3390Why Adaptive Refinement Helps Near KinksWhy is adaptive refinement often better than a uniform fine grid when the integrand has a kink or localized sharp feature?数学中等essay未尝试面试订阅4596Martingale Decomposition Arithmetic 6In the martingale decomposition of a call, the discounted share-side term is 38.2 and the call value is 6.8. What discounted strike-side term must be subtracted?数理金融中等数值题未尝试面试订阅4597Martingale Decomposition Arithmetic 7A call price is 5.4 and the discounted strike-side term K e -rT N(d2) is 24.2. What discounted share-side term S0 e -qT N(d1) is implied?数理金融中等数值题未尝试面试订阅4598Martingale Decomposition Arithmetic 8The prepaid forward is 48.5, the call price is 7.3, and the discounted strike is 50.1. What put price follows from put-call parity?数理金融中等数值题未尝试面试订阅4599Martingale Decomposition Arithmetic 9In a call decomposition, the discounted asset-side tail expectation is 44.1 and the discounted strike-side tail expectation is 35.7. What call value follows?数理金融中等数值题未尝试面试订阅4600Martingale Decomposition Arithmetic 10A call is worth 6.2, a put is worth 4.7, and the discounted strike is 51.5. What prepaid forward price is implied by put-call parity?数理金融中等数值题未尝试面试订阅5887Fair Variance Strike From a Discrete Option StripA one-year variance swap is replicated by a strip of OTM options. Using the Carr-Madan weighting w i = (ΔK / K i 2), the discount-adjusted strip values give sum i w i * price i = 0.0180 (in variance units before the 2/T scaling), and the linear forward-correction term contributes an additional 0.0020. With T = 1, the fair variance is K var = (2/T) * (strip + forward term). What is the fair annualized volatility strike (decimal)?数理金融中等数值题未尝试面试订阅5923Full-Information Uniform StoppingYou observe up to three independent draws from the Uniform(0,1) distribution, one at a time, and after each you may stop and collect the value just seen or discard it and continue (no recall of discarded values). If you reach the third draw you must take it. Knowing the distribution exactly, what stopping policy maximizes the expected value collected, and what is that expected value?概率简单数值题未尝试免费5924House-Selling Stationary ThresholdOffers arrive sequentially and independently, each Uniform(0,1). After each offer you either accept it (and stop) or reject it forever (no recall) and wait for the next, paying a fixed search cost c per rejected offer. There is no deadline. For c = 0.02, find the optimal stationary acceptance threshold and the expected net payoff under it.概率中等数值题未尝试免费5925Prophet vs Gambler, Two PrizesTwo prizes arrive in sequence, each an independent Uniform(0,1) value, revealed one at a time; you must accept one immediately when shown (no recall, and if you reject the first you must take the second). A prophet who sees both in advance collects E[max(X1,X2)]. Compute the prophet's expected reward, the best the online gambler can guarantee in expectation, and the ratio of gambler to prophet.概率简单数值题未尝试免费5929Pay-Per-Look With RecallYou may sample as many independent Uniform(0,1) values as you like, paying a cost c per sample. Recall is allowed, so at any time you may stop and collect the maximum value seen so far. With c = 0.125, find the optimal stopping rule (the reservation level r above which you stop) and the expected net payoff (max value collected minus total sampling cost).概率中等数值题未尝试免费