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5810Expected Profit Over N QuotesOver a session you post 5000 quotes. Each quote has a fill probability of 0.08, and each fill earns a net edge of 0.25. What is your expected total PnL for the session?金融与交易简单数值题未尝试免费5812Probability-Weighted PnL Of A Skewed QuoteLong inventory, you skew quotes to sell. The ask fills with probability 0.45 earning net edge 0.020 per share; the bid fills with probability 0.10 but is toxic, earning net edge -0.030 per share. Each side quotes 100 shares. Assuming at most one side fills, what is the expected PnL of this skewed quote?金融与交易困难数值题未尝试面试订阅5813Blended EV Across Maker And Taker FillsOf your 1000 fills, 70 percent are passive maker fills earning 0.05 net edge plus a 0.001 rebate per share, and 30 percent are aggressive taker fills earning 0.05 net edge minus a 0.003 fee per share. Every fill is 100 shares. What is your expected total PnL across all fills?金融与交易困难数值题未尝试面试订阅5814Maker Rebate Versus Taker Fee TradeoffPosting passively, you earn a rebate of 0.002 per share but capture only 0.006 of edge with fill probability 0.30. Crossing aggressively, you pay a taker fee of 0.003 per share, capture 0.011 of edge, and fill with certainty. For a 100-share order, which approach has higher expected PnL?金融与交易中等数值题未尝试免费5815EV Across A Fill-Size DistributionWhen your bid is hit, the executed size follows a distribution: 50 shares with probability 0.5, 200 shares with probability 0.3, and 400 shares with probability 0.2. You earn a net edge of 0.02 per share regardless of size. Given that you are filled, what is the expected PnL of the fill?金融与交易中等数值题未尝试免费5816EV With Queue Position And Partial FillYour 300-share quote reaches the front of the queue with probability 0.6. If it does, it fully fills (300 shares); otherwise the queue clears only partway and you fill just 100 shares. Net edge is 0.015 per share in both cases. What is the expected PnL of the quote, accounting for queue position?金融与交易中等数值题未尝试免费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