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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5844Long Put Maximum GainYou buy a single put with strike 40 for premium 3. What is the break-even stock price at expiry and the maximum possible profit on the position?金融与交易简单数值题未尝试免费5848Short Straddle Profit and RiskYou sell (write) a straddle at strike 100, collecting a call premium of 6 and a put premium of 7. What is the maximum profit, the two break-even prices, and the profit if the stock ends at 118?金融与交易中等数值题未尝试免费5850Alternating Geometric SumEvaluate the infinite series sum (n=0) inf 5 * (-2/3) n.数学简单数值题未尝试免费5853Radius of Convergence via Ratio TestDetermine the radius of convergence of the power series sum (n=1) inf (3 n / n) x n.数学中等数值题未尝试免费5855p-Series Convergence ThresholdFor which of the following exponents p does the series sum (n=1) inf 1/n p converge: p = 1/2, p = 1, p = 3/2, p = 2? List all values for which it converges.数学简单数值题未尝试免费5858Integrating Factor with Variable CoefficientSolve the first-order linear ODE x'(t)+\dfrac x(t) t =t for t>0 with x(1)=2, and evaluate x(2).数学中等derivation未尝试面试订阅5859Separable Nonlinear ODE and Blow-UpSolve the separable ODE y'(x)=x\,y(x) 2 with y(0)=1. At what value of x>0 does the solution blow up?数学中等derivation未尝试面试订阅5860Overdamped Second-Order ODE with Distinct Real RootsSolve y''-5y'+6y=0 with y(0)=1 and y'(0)=0.数学中等derivation未尝试面试订阅5861Risk-Neutral Probability From A Quoted CallIn a one-period binomial model the stock has S0=100 and goes to Su=130 or Sd=90, with risk-free rate 0. A call struck at K=100 trades at price 12. Back out the implied risk-neutral probability of the up state from this option quote.数理金融简单数值题未尝试面试订阅5862Arrow-Debreu Prices From Option QuotesA stock has three future states with prices 120, 100, and 80; the risk-free rate is 0. Calls struck at 80 trade at 28 and calls struck at 100 trade at 8. Using the digital/butterfly decomposition, find the Arrow-Debreu price of the single highest state (the state where the stock ends at 120).数理金融困难数值题未尝试面试订阅