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2028Log Carry Score Is Concave 8Let psi(x)=ln(1+x) on x>-1. Compare E[psi(X)] and psi(E[X]).数学中等derivation未尝试面试订阅2037Why Jensen Matters for Nonlinear Risk Transforms 17Why is it dangerous to plug an average state into a nonlinear convex risk transform and treat that as the average transformed risk?数学中等essay未尝试面试订阅2071Price a Trinomial Claim After Market Completion 1A one-period stock is 100 today and ends at 120, 100, or 80. The risk-free rate is 0. A quoted up-state digital that pays 1 only in the up state trades at 0.2, which completes the market. What unique no-arbitrage price does this imply for the claim paying 5, 1, and 0 in the up, middle, and down states?数理金融中等数值题未尝试面试订阅2076Affine Replication Check in an Incomplete Market 6The stock is 100 today and ends at 120, 100, or 80 next period. Consider a claim paying 20, 10, and 0 in those three states. Can it be replicated exactly using only the stock and cash? If yes, give the hedge. If not, identify the replication obstruction.数理金融中等derivation未尝试面试订阅2086Recover the Completing Digital Quote 16A one-period trinomial stock ends at 120, 100, or 80 with zero interest. An up-state digital paying 1 only in the up state completes the market and trades at an unknown price q. A claim paying 5, 1, and 0 in the three states is observed to trade at 1.8. What q is implied?数理金融中等数值题未尝试面试订阅2091Incomplete-Market Pricing Intuition 21Why does a unique no-arbitrage price disappear as soon as the trinomial market has more states than traded securities?数理金融困难essay未尝试面试订阅2092Incomplete-Market Pricing Intuition 22Why does the superhedge naturally sit at the top of an incomplete-market price interval?数理金融困难essay未尝试面试订阅2093Incomplete-Market Pricing Intuition 23Why can a minimum-variance hedge still fail to pin down a unique no-arbitrage price?数理金融困难essay未尝试面试订阅2106Infer Remaining Flat Volatility From a Live Variance Mark 11A variance swap has observed 30 trading days out of 90. Realized variance so far is 0.035, and the current fair full-life variance strike is 0.038333. If the remaining window is assumed to carry one flat volatility level, what annualized volatility does that imply for the remaining life?数理金融中等数值题未尝试面试订阅2118Variance-Swap Modeling Intuition 23Why can simple-return and log-return fixing conventions differ materially once moves become large?数理金融困难essay未尝试面试订阅2119Variance-Swap Modeling Intuition 24Why can a variance swap mark-to-market move even if the headline implied-vol surface looks unchanged at first glance?数理金融困难essay未尝试面试订阅2120Variance-Swap Modeling Intuition 25Why does a capped variance swap usually trade cheaper than an uncapped one with the same strike?数理金融困难essay未尝试面试订阅2121Caplet Quote Calibration 1A caplet desk prices under the natural forward measure using PV = P(0,T i+1 )*delta*B, where B is the forward-measure Black factor. A quoted caplet PV is 0.001671, the discount factor is 0.955, and the accrual is 0.25. What Black factor B is implied?数理金融中等数值题未尝试面试订阅2122Caplet Quote Calibration 2A caplet PV under its natural forward measure satisfies PV = P(0,T i+1 )*delta*B. The desk sees PV = 0.00212, accrual delta = 0.5, and Black factor B = 0.004709. What discount factor P(0,T i+1 ) is implied?数理金融中等数值题未尝试面试订阅2123Caplet Quote Calibration 3A caplet PV under its natural forward measure satisfies PV = P(0,T i+1 )*delta*B. The desk sees PV = 0.001323, discount factor P(0,T i+1 ) = 0.98, and Black factor B = 0.0054. What accrual fraction delta is implied?数理金融中等数值题未尝试面试订阅2125Caplet Quote Calibration 5A caplet PV under its natural forward measure satisfies PV = P(0,T i+1 )*delta*B. The desk sees PV = 0.004095, accrual delta = 1, and Black factor B = 0.0045. What discount factor P(0,T i+1 ) is implied?数理金融中等数值题未尝试面试订阅2126LMM Desk Intuition 6A caplet trader only needs one forward rate as underlying. Which numeraire is the natural first choice and why?数理金融中等essay未尝试面试订阅2127LMM Desk Intuition 7Why does each forward rate in an LMM naturally come with its own forward measure rather than one universal driftless measure for all forwards at once?数理金融中等essay未尝试面试订阅2128LMM Desk Intuition 8Why are caplet volatilities treated as primitive inputs in a market model rather than derived indirectly from a short-rate state?数理金融中等essay未尝试面试订阅2129LMM Desk Intuition 9Why is the forward rate itself often a better state variable for a caplet desk than a discount-bond price ratio written in abstract form?数理金融中等essay未尝试面试订阅