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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5847Risk Reversal Payoff RegionsWith no stock position, you sell a put with strike 90 for premium 5 and buy a call with strike 110 for premium 5, where 110>90. What is the net premium, and what is the position's profit at expiry if the stock ends at (a) 80 and (b) 120?金融与交易中等数值题未尝试免费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5868Theta Versus Gamma On A Hedged DayYou are long a delta-hedged option with gamma 0.04 on a stock at 100, priced at 20% implied vol. Over one trading day (1/252 year) the stock moves 1.5 points. Using gamma P&L = 0.5·gamma·(dS) 2 and theta P&L = -0.5·gamma·S 2·sigma impl 2·dt, what is the net P&L for the day, to four decimals?数理金融中等数值题未尝试免费5869Hedged P&L From The Variance GapApproximate the total P&L of a long delta-hedged option by 0.5·gamma·S 2·(sigma real 2 - sigma impl 2)·T. With gamma 0.05, S = 50, realized vol 0.30, implied vol 0.22, and T = 20/252 years, what is the approximate P&L, to four decimals?数理金融困难数值题未尝试面试订阅5871Vega P&L From A Vol RepricingYour position has total vega of 20 dollars per one full volatility point (i.e. per 1.00 change in sigma). Overnight the implied vol used to mark the book rises from 22% to 27% with spot unchanged. Ignoring all other Greeks, what is the mark-to-market vega P&L in dollars, to two decimals?数理金融简单数值题未尝试免费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5876The Daily Move That Breaks Even On GammaFor a delta-hedged long option, the gamma gain 0.5·gamma·dS 2 exactly offsets the daily theta 0.5·gamma·S 2·sigma impl 2·dt when the absolute daily move dS equals the implied break-even move. For S = 100, implied vol 18%, and dt = 1/252, what is that break-even daily move in points, to four decimals?数理金融困难数值题未尝试面试订阅5877Risk-Neutral Probability From Tree FactorsA one-step binomial tree has up factor u=1.15, down factor d=0.88, continuously compounded rate r=0.05, and Δt=0.5. Compute the risk-neutral probability of an up move.数理金融简单数值题未尝试免费5878CRR Up/Down Factors From VolatilityIn a Cox-Ross-Rubinstein tree the volatility is σ=0.25 per year and each step is Δt=0.25 years. Using u=e σ√Δt and d=1/u, what is the up factor u (to four decimals)?数理金融简单数值题未尝试免费5879Replicating Delta On A One-Step TreeA stock at 50 moves in one step to 58 or 44. A European call struck at 52 is written on it. What is the replicating delta (shares per option) over this step?数理金融简单数值题未尝试免费5882Completing A Trinomial Probability SetA one-step trinomial tree has multipliers u=1.2, m=1, d=0.8. The middle probability is fixed at p m=0.6, r=0.04, Δt=1. Find the up-move probability p u that makes the discounted underlying a martingale (so p u+p m+p d=1 and E[S 1]=S 0 e rΔt ).数理金融困难数值题未尝试面试订阅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5886Two-Step European Call Via Terminal WeightsOn a two-step recombining tree with spot=64, strike=70, u=1.25, d=0.8, r=0, Δt=1, price the European call by weighting the three terminal payoffs with the binomial probabilities q 2, 2q(1-q), (1-q) 2.数理金融中等数值题未尝试免费5891Who Owns the Class PriorTwo teams ship classifiers trained on a balanced 50/50 dataset, but the live population is 90% class 0. Team A used Gaussian discriminant analysis; Team B used logistic regression. Which model explicitly contains an estimate of the class prior P(y), and explain why that distinction makes one team's fix to the prevalence mismatch cleaner than the other's.机器学习中等essay未尝试面试订阅