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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5909Bold Play to a Quadrupling GoalYou have \2 and want to reach \8 on an even-money game you win with probability p=0.4. You use bold play: each round you stake the most that keeps you from overshooting the goal, i.e. \min( current wealth ,\ 8- current wealth ). What is the probability bold play reaches the \8 goal?概率中等数值题未尝试免费5914Red-and-Black Bold Play from Three-QuartersIn red-and-black you bet on an even-money outcome that comes up with probability p=0.4, scaling all amounts so the goal is \1. You currently hold \0.75 and use bold play: stake \min( current ,\ 1- current ) each round, trying to reach \1 before reaching \0. What is the probability bold play reaches the goal?概率中等数值题未尝试面试订阅5915Timid Versus Bold to QuadrupleStarting with \1 you want to reach \4 on an even-money game you win with probability p=0.4, quitting when you reach \4 or go broke. Compute the probability of reaching \4 under (A) timid play, betting \1 each round, and (B) bold play, staking \min( current ,\ 4- current ). Which strategy gives the higher chance of reaching the goal?概率困难数值题未尝试面试订阅5916Most You Would Pay for a Perfect TestA product launch pays +30 if the market is receptive and -12 if it is not; receptivity has prior probability 3 10 . You may instead shelve the product for 0. A consultant offers a perfectly accurate test that reveals the true market state before you decide. What is the most you should be willing to pay for this test?概率中等derivation未尝试免费5922Three-Candidate Best-ChoiceThree candidates of distinct, unknown qualities arrive in uniformly random order. After each interview you learn only the candidate's rank relative to those seen so far, and you must immediately and irrevocably hire or reject. You want to maximize the probability of hiring the single best candidate. What is the optimal policy and the resulting probability of success?概率简单数值题未尝试免费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5926Settling For Top TwoFour items arrive in uniformly random order; you observe only relative ranks and must accept one irrevocably (if you reach the last item you take it). Unlike the classic problem, you win if the item you accept is either the best OR the second-best overall. Find the policy that maximizes your winning probability and that maximum probability.概率困难数值题未尝试面试订阅5927Minimize the Expected RankThree items arrive in random order; after each you learn its rank relative to those seen so far and must accept or reject irrevocably (the last is forced). Instead of trying to get the single best, you want to minimize the expected absolute rank of the item you finally accept (rank 1 = best, rank 3 = worst). Find the optimal policy and the minimum achievable expected rank.概率中等数值题未尝试免费5928Two Hiring AttemptsFour candidates arrive in random order with only relative ranks observable. You are allowed to make TWO irrevocable picks during the stream (you point at a candidate, that candidate is chosen, then later you may point at one more); you win if AT LEAST ONE of your two chosen candidates is the overall best. Choices are made online with no recall. What is the optimal strategy and the probability of winning?概率困难数值题未尝试面试订阅5931Discounted Offer StoppingEach period an offer arrives, iid Uniform(0,1). If you accept an offer of value x at period t, you receive beta t * x, where beta = 0.9 is a per-period discount factor (so waiting shrinks the value of any future acceptance). No recall, infinite horizon. Find the optimal stationary acceptance threshold and the expected discounted payoff from the start.概率中等数值题未尝试免费5932Catch the Last SuccessFive deals appear in sequence. Independently, each deal turns out to be 'live' with probability 0.2 (and 'dead' otherwise); you learn live/dead immediately when the deal appears and must accept or pass irrevocably with no recall. You win precisely if you accept the LAST live deal of the five. Using the odds-algorithm logic for such problems (sum the odds r i = p i/(1-p i) from the end until the running total first reaches 1, and start accepting any live deal from that index onward), find the optimal stopping index and your probability of winning.概率困难数值题未尝试面试订阅5933The Postdoc Problem (Pick Second-Best)Four applicants of distinct unknown qualities arrive in uniformly random order; after each you learn only its rank relative to those seen so far and must irrevocably accept or reject (if you reach the last you must take it). A famous twist: you win only if the applicant you accept is the SECOND-best of all four (the very best is taken by a rival institution, so picking the best is a loss). Find the optimal policy and the maximum probability of landing exactly the second-best.概率困难数值题未尝试面试订阅5934Sultan's Dowry with Five SuitorsA sultan offers a vizier the hand of one of 5 daughters, presented one at a time in uniformly random order. After meeting a daughter, the vizier learns only how she ranks (by dowry) relative to those already seen and must immediately and irrevocably accept or send her away; if he reaches the last he must take her. He wins only by choosing the single highest dowry. Among all 'skip the first r, then take the first daughter who beats every earlier one' rules, find the optimal r and the exact probability of winning.概率中等数值题未尝试免费5935Hiring With an Interview CostThree candidates of distinct unknown qualities arrive in uniformly random order; each candidate must be interviewed (in order) before you can judge their relative rank, and conducting each interview costs 0.05 utility. After interviewing a candidate you immediately and irrevocably hire or reject (if you reach the third you must hire). You receive payoff 1 if you hire the overall best candidate and 0 otherwise, minus the total interview cost incurred. Among the threshold rules 'reject the first r interviewed, then hire the first later record', find the optimal r and the resulting expected net payoff.概率中等数值题未尝试免费5936Sell Before the Deadline in a Falling MarketYou must sell an asset within 3 periods. In period t one offer arrives, drawn uniformly on [0, 1 - 0.2*(t-1)] (a declining market: the range is [0,1] in period 1, [0,0.8] in period 2, [0,0.6] in period 3). You accept and stop, or reject forever (no recall). If you reach period 3 you must accept that offer. Knowing these distributions, find the optimal acceptance thresholds and the expected sale price under the optimal policy.概率中等数值题未尝试免费5938Bird in the Hand vs Discounted WaitYou face two periods. In period 1 a reward X1 ~ Uniform(0,1) is offered; accept it now to receive X1, or wait. If you wait, in period 2 you must accept X2 ~ Uniform(0,1), but a reward received in period 2 is worth only a fraction beta = 0.8 of its face value (discounting). No recall. Find the optimal period-1 acceptance threshold and the expected payoff of the optimal policy.概率简单数值题未尝试免费