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5772Level Perpetuity PVA perpetuity pays 7 at the end of every year forever, with the first payment one year from now. The discount rate is 0.035. What is its present value?金融与交易简单数值题未尝试面试订阅5789Skew To Recover From A ShortYou start short q0 = -150 shares; lot size is 50. Under a symmetric quote, bid-fill probability is 0.30 and ask-fill probability is 0.30. Under a bid-skew policy designed to buy back, bid-fill probability rises to 0.50 while ask-fill probability falls to 0.20. Compute the expected ending inventory under each policy, and state which leaves you closer to flat.金融与交易中等数值题未尝试免费5823Optimal Skew After A FillAfter a round-1 buy fill leaves you long 1 unit, you choose a round-2 quote skew s (in cents) that shifts both quotes down to encourage a sell. Selling probability is 0.3 + 0.1*s and expected edge per sell is 0.05 - 0.01*s, for s between 0 and 5. Round-2 expected edge is (selling probability)*(edge). Which integer s in 0,1,2,3,4,5 maximizes round-2 expected edge?金融与交易困难数值题未尝试面试订阅5826Expected Wait Until First FillYou repost the same quote every round; each round it fills independently with probability 0.2. Rounds continue indefinitely until the first fill. What is the expected number of rounds until (and including) the first fill?金融与交易简单数值题未尝试免费5830A dealer has just observed a buy order and updated the posteA dealer has just observed a buy order and updated the posterior probability that value is high to 0.70 (value high = 102, low = 98). A second buy order then arrives. In this model informed traders buy when value is high with probability 0.9 and when value is low with probability 0.1 (noise component already folded in). Treating the post-first-trade posterior as the new prior, what is the posterior P(high) after the second buy, and the new fair value?金融与交易中等数值题未尝试面试订阅5831Over a short window a dealer observes 7 buys and 3 sells. EaOver a short window a dealer observes 7 buys and 3 sells. Each trade is independently a buy with probability p, where p = 0.5 in the no-news state and p = 0.7 in the good-news state. The two states are equally likely a priori. Using the order imbalance as the signal, what is the posterior probability that the market is in the good-news state?金融与交易困难数值题未尝试面试订阅5832A dealer's clientele is 20% informed and 80% noise on the cuA dealer's clientele is 20% informed and 80% noise on the current trade. Informed traders, when present, always pick the profitable side; noise traders are random. The dealer has just been adversely picked off (filled on the wrong side). Going into the NEXT incoming order, the dealer believes nothing about the population has changed. What is the unconditional probability that the next order comes from an informed trader, and why is this the right number to price against rather than 0%?金融与交易中等数值题未尝试面试订阅5834Conditional on being filled on your ask, you know the trade Conditional on being filled on your ask, you know the trade was either from a noise trader (probability 0.65 given a fill) or an informed trader (probability 0.35 given a fill). If noise, the value is unchanged at your mid of 100. If informed, the value has moved against you to 100 + m for a buy, where the post-trade fair value among informed-driven fills is 103. What is the expected post-fill fair value of the asset, i.e. E[V | you got lifted]?金融与交易中等数值题未尝试面试订阅5846Name the Strategy from Its PayoffA position's expiry profit is flat and negative for low stock prices, slopes upward through a single break-even, then becomes flat and positive (capped) for high stock prices, with a kink at each of two strikes. No stock is held. Name the simplest two-leg option strategy that produces this profile and state its directional bias.金融与交易中等数值题未尝试免费5849Identify the Volatility StructureA position's expiry profit is large and positive far below a price L, declines linearly to a single flat negative minimum across a middle range, then rises linearly again and becomes large and positive far above a price U>L. No stock is held and the minimum loss is bounded. Name the simplest option strategy with this profile and the view it expresses.金融与交易中等数值题未尝试免费5874Sign Of Hedged P&L When Realized Beats ImpliedYou buy an option and delta-hedge it continuously to expiry. If realized volatility ends up higher than the implied volatility you paid, what is the sign of your hedged P&L, and which Greek explains why?数理金融简单essay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5942Collecting a Full Set of SixA vending machine dispenses one of 6 distinct toy types, each equally likely and independent across purchases. What is the expected number of purchases needed until you own at least one of every type?概率中等数值题未尝试免费5943Variance of Completing a Four-Sticker SetStickers come in 4 equally likely types, independent across packs. Let T be the number of packs you open until you have collected all 4 types. Compute Var (T).概率困难数值题未尝试面试订阅5944Two Unequal PrizesEach cereal box contains one prize: a common prize with probability 2/3 and a rare prize with probability 1/3, independent across boxes. What is the expected number of boxes you must open to have collected both prizes?概率中等数值题未尝试免费5945Two of EverythingA machine outputs one of 3 equally likely tokens per play, independent across plays. What is the expected number of plays until you hold at least TWO copies of each of the 3 token types?概率困难数值题未尝试面试订阅5946Halfway CollectionTrading cards come in 10 equally likely types, independent across packs. What is the expected number of packs needed until you own 5 distinct types (any 5, not a specific set)?概率中等数值题未尝试免费5947Two Stickers to GoYou are collecting 6 equally likely sticker types (independent packs). You currently hold exactly 4 distinct types. What is the expected number of ADDITIONAL packs needed to complete the set of 6?概率简单数值题未尝试免费5951Three Prizes, Unequal OddsA claw machine yields prize 1 with probability 1/2, prize 2 with probability 1/3, and prize 3 with probability 1/6, independently each play. What is the expected number of plays to collect all three prizes?概率困难数值题未尝试面试订阅