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5824Learning Your Fill RateYour unknown per-round fill probability is either 0.2 (prior 0.5) or 0.5 (prior 0.5). You quote 3 independent rounds and observe exactly 2 fills. What is the posterior probability that the true fill probability is 0.5?金融与交易中等数值题未尝试面试订阅5825Reprice After An Informative FillYour round-1 bid fills. A fill is more likely when an informed seller is present, so after a fill you assign probability 0.6 to fair value having dropped by 0.05 (and 0.4 to no change). In round 2 you re-quote and earn raw edge 0.04 if filled, with fill probability 0.5, but you also still hold the round-1 unit which loses the fair-value drop. What is the expected round-2 PnL including the revaluation of the held unit?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5827A dealer faces order flow where a fraction 0.4 of traders arA dealer faces order flow where a fraction 0.4 of traders are informed (always trade in the correct direction) and 0.6 are noise traders who buy or sell with equal probability. The true value is equally likely high or low, so informed traders buy half the time and sell half the time. You observe a buy order. What is the probability that the trader who just bought is informed?金融与交易中等数值题未尝试面试订阅5828True value is either 50 (low) or 60 (high), equally likely. True value is either 50 (low) or 60 (high), equally likely. A fraction 0.25 of order flow is informed and trades in the correct direction; the remaining 0.75 is noise that buys or sells 50/50. A dealer is hit on the bid (a sell order arrives) and gets filled. What is the dealer's expected loss per share relative to true value, i.e. the expected adverse move E[bid - V | sell], if the bid is quoted at the prior mean 55?金融与交易困难数值题未尝试面试订阅5829In a market the per-trade adverse-selection cost to a dealerIn a market the per-trade adverse-selection cost to a dealer is alpha*delta, where alpha is the informed fraction and delta is the value gap. Noise traders are willing to pay at most a half-spread of 0.30 before they stop trading entirely. The value gap is delta = 1.0. Above what informed fraction alpha does the market break down (no spread can both cover adverse selection and retain noise traders)?金融与交易困难数值题未尝试面试订阅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5833A dealer quotes a two-sided market and earns the half-spreadA dealer quotes a two-sided market and earns the half-spread s on every fill. A fraction alpha = 0.15 of fills are informed and cost the dealer the full value gap delta = 0.8 (the dealer is on the wrong side for the entire move). The remaining fills are noise and the dealer keeps s. The dealer wants expected profit per trade of at least 0.02. What is the minimum half-spread s the dealer must charge?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5835Two venues route flow to your quotes. Venue A flow is 5% infTwo venues route flow to your quotes. Venue A flow is 5% informed; Venue B flow is 40% informed. Both have the same value gap delta = 0.5, and informed flow costs you delta per fill while you earn half-spread s = 0.10 on all fills. You can only post on one venue. Compute expected profit per fill on each venue, and explain in one line which venue you choose and the adverse-selection reason.金融与交易中等数值题未尝试面试订阅5836A dealer quotes a single ask price A for a stock whose valueA dealer quotes a single ask price A for a stock whose value V is uniformly distributed on [40, 60]. An informed counterparty buys only when V > A (the quote is too cheap). Conditional on getting filled at ask A, what is the expected true value of the stock, and what does this imply about the loss the dealer makes on filled trades if A = 50?金融与交易困难数值题未尝试面试订阅5837Roll Implied Spread From AutocovarianceUnder Roll's model, transaction price changes have a first-order serial autocovariance of -0.0009 (in price-squared units). Estimate the implied effective spread.金融与交易中等数值题未尝试面试订阅5838Round-Trip Half-Spread CostQuotes are 80.00 bid / 80.10 ask and stay constant. A trader buys 500 shares at the ask, then later sells the same 500 shares at the bid. What is the total dollar cost of crossing the spread on the round trip, and what is it as a multiple of the half-spread per share?金融与交易简单数值题未尝试面试订阅5839Relative Spread In Basis PointsStock A trades around 20.00 with a quoted spread of 0.04; stock B trades around 200.00 with a quoted spread of 0.30. Express each quoted spread in basis points of its midpoint, and state which name is more expensive to cross on a relative basis.金融与交易简单数值题未尝试面试订阅5840Long Strangle Break-EvensA long strangle buys a put with strike 95 for premium 2 and a call with strike 105 for premium 3, where 105>95. What are the lower and upper break-even prices at expiry, and the total distance between them?金融与交易简单数值题未尝试免费5841Bear Put Spread EconomicsYou buy a put with strike 110 for premium 7 and sell a put with strike 100 for premium 3, where 110>100. What are the net debit, the break-even stock price, and the maximum profit at expiry?金融与交易中等数值题未尝试免费5842Long Call Butterfly ProfileA long call butterfly buys one 90 call, sells two 100 calls, and buys one 110 call for a net debit of 2 (strikes equally spaced). What is the maximum profit, the stock price at which it occurs, and the maximum loss?金融与交易中等数值题未尝试免费5843Long Call Break-Even and Loss FloorYou buy a single call with strike 50 for premium 4. What is the break-even stock price at expiry, the maximum possible loss, and the profit if the stock ends at 61?金融与交易简单数值题未尝试免费