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5817Expected Edge Under Cancel RiskPer posted quote you normally capture 0.030 of net edge. On 20 percent of posts a price move makes your quote stale before you trade; you cancel successfully on 70 percent of those stale cases (avoiding the trade entirely), but on the remaining stale cases you are picked off and lose 0.080 per share instead of earning the edge. The other 80 percent of posts fill normally at the full edge. What is your expected net edge per posted quote?金融与交易困难数值题未尝试面试订阅5818Quote Both Sides Or One SideQuoting only the ask: fill probability 0.40, net edge 0.025 per share. If you also add a bid, the ask economics are unchanged, but the bid fills with probability 0.30 at a net edge of -0.020 per share (its flow is toxic given your view). Each side is 100 shares. Which is better, quoting one side or both, and what is the higher expected PnL?金融与交易中等数值题未尝试免费5819Edge Per Unit Of RiskA quote earns an expected net edge of 0.018 per fill, and the PnL of each fill has a standard deviation of 0.12. What is the edge-per-unit-of-risk (expected edge divided by standard deviation) for this quote? Give the answer to three decimals.金融与交易简单数值题未尝试免费5820Expected Daily PnL From Volume And Fill RateThe product trades 2,000,000 shares per day. You are at the touch on enough of that flow that you fill 4 percent of daily volume passively, capturing a net edge of 0.0015 per share after all costs. What is your expected daily PnL?金融与交易中等数值题未尝试免费5821Expected Fills Over Many RoundsYou post the same two-sided quote in each of 8 independent rounds. In every round the quote fills with probability 0.35, and there is no inventory limit so fills do not interact. What is the expected number of fills over the 8 rounds?金融与交易简单数值题未尝试免费5822Cumulative PnL With Inventory CarryoverOver 3 rounds you quote to buy 1 unit each round when filled. Round 1 fills at price 100, round 2 fills at price 101, round 3 does not fill. After round 3 the unit is marked at fair value 102. Each unit also costs 0.10 in carry per round it is held to the end of round 3. What is total expected PnL, counting markup minus total carry?金融与交易中等数值题未尝试面试订阅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