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2293LSM Implementation Judgment 13Why are tree benchmarks still useful even when the production engine is LSM Monte Carlo?数理金融困难essay未尝试面试订阅2294LSM Implementation Judgment 14Why does time-step placement matter in LSM beyond just making the simulation mesh 'finer'?数理金融困难essay未尝试面试订阅2295LSM Implementation Judgment 15Why is 'more paths' not a complete answer if the continuation specification itself is misshapen?数理金融困难essay未尝试面试订阅2721Unit and Domino Tilings of a Nine-Cell StripA strip of length 9 is tiled using monomers of length 1 and dominoes of length 2. Derive the generating function for the number a n of tilings, and compute a 9.脑筋急转弯简单数值题未尝试面试订阅2722Tiling With 1-, 2-, and 4-Step BlocksLet a n be the number of ways to tile a strip of length n using blocks of lengths 1, 2, and 4. Write the generating function A(x) and compute a 10.脑筋急转弯中等derivation未尝试面试订阅2725Coefficient in a Mixed Pole Generating FunctionFind the coefficient of x 8 in 1/((1-x) 2(1-x 3)).脑筋急转弯中等derivation未尝试面试订阅2727Binary Strings With Ones Congruent to One Mod FourHow many binary strings of length 9 have a number of ones congruent to 1 modulo 4?脑筋急转弯简单数值题未尝试面试订阅2730Submultisets With Size Divisible by ThreeFrom the multiset a,a,a,b,b,c,c,c,c , you may choose any submultiset. How many choices have total size divisible by 3?脑筋急转弯困难derivation未尝试面试订阅2732Weak Compositions With Uniform Upper BoundsHow many weak compositions of 15 into 5 parts have every part at most 4? Solve it from the generating function ((1-x 5)/(1-x)) 5.脑筋急转弯中等derivation未尝试面试订阅2735Compositions Into Ones and TwosLet c n be the number of ordered compositions of n using only parts 1 and 2. Use a generating function to show c n = F n+1 , and then compute c 10.脑筋急转弯简单数值题未尝试面试订阅2737Inventory Paths With Holds InsertedA trader executes 4 buys, 4 sells, and 2 holds. Buys and sells each change inventory by plus or minus 1, holds leave inventory unchanged. The trader starts flat, ends flat, and inventory may never become negative at any intermediate time. How many execution sequences are possible?脑筋急转弯困难derivation未尝试面试订阅2738Flat Inventory Paths With Five Buys and Five SellsA flat-start inventory process consists of exactly 5 buys and 5 sells, each changing inventory by plus or minus 1. Inventory may never go negative. How many feasible paths are there?脑筋急转弯中等derivation未尝试面试订阅2739Catalan Generating Function for Balanced Trade SkeletonsLet C n count buy/sell skeletons of length 2n that start and end at inventory 0 and never go negative. Write the generating function C(x)=sum n>=0 C n x n, derive the quadratic equation it satisfies, and compute C 4.脑筋急转弯困难derivation未尝试面试订阅2741Bid Ceiling With a Fixed Outside OptionA trader values a data license at v. In a sealed first-price auction against one rival, winning with bid b occurs with probability p(b). If skipping the auction preserves an outside payoff of c, what inequality must b satisfy for participating with bid b to weakly dominate skipping?脑筋急转弯中等derivation未尝试面试订阅2742When a Rebate Makes Aggressive Bidding RationalIn a first-price auction, a bidder of value v receives a fixed rebate r if she loses, and nothing if she skips. If bidding b wins with probability p(b), derive the expected payoff of bidding b and compare it with skipping.脑筋急转弯中等derivation未尝试面试订阅2743Scaled Uniform Values and Win ProbabilityThree bidders have i.i.d. private values from Uniform (20,80). In a first-price auction, find the symmetric equilibrium bid of a bidder with value 56. Also compute that bidder's probability of winning under equilibrium play.脑筋急转弯中等derivation未尝试面试订阅2744First-Price Equilibrium Under a Quadratic CDFTwo bidders have i.i.d. private values on [0,1] with CDF F(v)=v 2. In a first-price auction, derive the symmetric equilibrium bid function and compute the bid for value 0.6.脑筋急转弯中等derivation未尝试面试订阅2745Reverse Auction With Two SuppliersA buyer runs a sealed-bid procurement auction with two risk-neutral suppliers. Each supplier's cost is i.i.d. Uniform (0,1), and the lowest bid wins and is paid its own bid. Derive the symmetric equilibrium bid function s(c), and compute the expected profit of a supplier with cost 0.2.脑筋急转弯中等derivation未尝试面试订阅2746Truthful Bidding Under a Capacity PenaltyIn a second-price auction for one extra inventory slot, paying above your true value v can backfire because if you win while your realized usage is low, you incur a deterministic penalty k. If the penalty is independent of the auction price and only applies when you win, what is the effective value that should replace v in the usual Vickrey truth-telling argument?脑筋急转弯困难derivation未尝试面试订阅2747All-Pay Contest Break-Even EffortTwo equally skilled quants enter an all-pay contest for a prize worth V. A contestant who spends effort e while the rival spends e 0 wins with probability e/(e+e 0), and everyone pays their own effort regardless of outcome. Holding e 0 fixed, write the payoff of effort e and solve the equation that defines break-even effort.脑筋急转弯简单derivation未尝试面试订阅