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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4486Before Solving ProbabilitiesBefore solving for a risk-neutral probability, what inequality should you inspect first in a one-period model?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4496Span-Restoring Asset Choice 6In a three-state market, the currently traded payoffs are cash [1,1,1] and stock [80,100,120]. Which candidate extra payoff would actually make the market complete: A=[1,1,1], B=[0,1,0], or C=[2,2.5,3]?数理金融中等数值题未尝试面试订阅4497How Many New Independent Payoffs Are Needed in a 5-State Rank-2 Market?A market has 5 states and current traded-payoff rank 2. How many additional independent payoff directions are needed to make the market complete?数理金融中等数值题未尝试面试订阅4499One New Direction Added to a 7-State Rank-5 MarketA market has 7 states and current traded-payoff rank 5. A newly listed asset is known to add one genuinely independent payoff direction. After adding it, how many more independent payoff directions are still needed to make the market complete?数理金融中等数值题未尝试面试订阅4501Claim Price Interval From State-Price Family 11A family of state-price vectors is pi(lambda) = (0.25+lambda, 0.50-2lambda, 0.20+lambda) with lambda constrained to [0, 0.05]. For the claim payoff (0,1,3), what arbitrage-free price interval does the family imply?数理金融中等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