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4471State-Price Density Solve 1A stock and bond trade in a one-period two-state model with S0=100, Su=120, Sd=80, and gross risk-free return 1. Solve for the state-price densities λu and λd.数理金融中等数值题未尝试面试订阅4472State-Price Density Solve 2A stock and bond trade in a one-period two-state model with S0=50, Su=62, Sd=42, and gross risk-free return 1.02. Solve for the state-price densities λu and λd.数理金融中等数值题未尝试面试订阅4476Convex-Hull No-Arbitrage Test 1A one-period stock has S0=100, future states 118, 94, and 76, and risk-free rate 0. Use the discounted-price convex-hull test to decide whether an arbitrage-free pricing measure can exist.数理金融中等数值题未尝试面试订阅4478Convex-Hull No-Arbitrage Test 3A one-period stock has S0=80, future states 86, 82, and 70, and risk-free rate 0.02. Use the discounted-price convex-hull test to decide whether an arbitrage-free pricing measure can exist.数理金融中等数值题未尝试面试订阅4481Why Positivity MattersWhy is the positivity of state prices central to the First Fundamental Theorem intuition?数理金融中等essay未尝试面试订阅4482Existence Versus UniquenessWhy does the First Fundamental Theorem care about existence of an arbitrage-free pricing measure rather than uniqueness?数理金融中等essay未尝试面试订阅4483Why Discounting AppearsWhy does the discounted price process appear naturally in the theorem's finite-state reasoning?数理金融中等essay未尝试面试订阅4484No-Arbitrage Is Weaker Than CompletenessWhy can a market be arbitrage-free and still leave some claims without a unique price?数理金融中等essay未尝试面试订阅4485First DiagnosticIn a simple discrete model, what is the first thing you should check before talking about arbitrage-free pricing measures?数理金融中等essay未尝试面试订阅4491Equivalent Measure Family 1In a three-state market, one family of equivalent martingale measures consistent with traded prices is q(lambda) = (0.2 + 1lambda, 0.5 - 2lambda, 0.3 + 1lambda). For what lambda range are all three state probabilities strictly positive, and is the martingale measure unique?数理金融简单数值题未尝试面试订阅4492Free Dimension of EMM Family in a 4-State Rank-3 MarketA finite-state market has 4 states and traded payoff span rank 3. Assuming no arbitrage and strict positivity are otherwise satisfied, how many free parameters remain in the equivalent martingale measure family, and is the martingale measure unique?数理金融简单数值题未尝试面试订阅4502How Many Candidate Claims Are Automatically Uniquely Priced? 1In a 4-state market, the traded span is generated by payoffs B0, B1, and B2. Three candidate claims are A = 2 B0 - B1, C = B1 + B2, and D = [1,0,0,0]. How many of A, C, and D are automatically guaranteed a unique arbitrage-free price from the traded market alone?数理金融中等数值题未尝试面试订阅4503How Many Candidate Claims Are Automatically Uniquely Priced? 2In a 5-state market, traded payoffs span H1, H2, and H3. Candidate claims are U = H1 + 2 H3, V = H2 - H1, and W = [0,1,0,0,0]. How many of U, V, and W are guaranteed uniquely priced just from attainability?数理金融中等数值题未尝试面试订阅4504How Many Candidate Claims Are Automatically Uniquely Priced? 3A market's traded span contains F0 and F1 but not the full state space. Candidate claims are M = 3 F0 - 0.5 F1, N = [1,0,1,0], and P = [0,1,0,1]. How many of M, N, and P are automatically guaranteed unique prices?数理金融中等数值题未尝试面试订阅4505How Many Candidate Claims Are Automatically Uniquely Priced? 4In a 4-state market, traded payoffs span G0, G1, and G2. Candidate claims are A = G0 + G1, B = 2 G2 - G1, C = G0 - G2, and D = [0,0,0,1]. How many of A, B, C, and D are guaranteed uniquely priced from the given information?数理金融中等数值题未尝试面试订阅4506Why Counting Securities Is Not Enough for CompletenessA candidate says a 4-state market with 4 traded securities must be complete. Why is that claim too quick?数理金融中等essay未尝试面试订阅4507Why a Redundant Asset Does Not Collapse a Pricing IntervalIn an incomplete market, why does listing one more asset fail to eliminate price intervals if that asset's payoff lies in the old span?数理金融中等essay未尝试面试订阅4508Why an Incomplete Market Can Still Uniquely Price One ClaimWhy can a market be globally incomplete yet still assign a unique arbitrage-free price to one particular claim?数理金融中等essay未尝试面试订阅4509Why Strict Positivity Matters When Talking About Equivalent MeasuresWhy is strict positivity of state prices or implied probabilities part of the theorem's language rather than a cosmetic side condition?数理金融中等essay未尝试面试订阅4510Why Uniqueness of the Pricing Measure and Completeness Tell the Same Geometry StoryAt interview level, why should you think of 'unique martingale measure' and 'complete market' as two views of the same linear-algebra fact?数理金融中等essay未尝试面试订阅