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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4511Why Price Intervals Come from Missing DirectionsWhy do arbitrage-free price intervals typically appear when the traded span misses some state-space directions?数理金融中等essay未尝试面试订阅4512What to Challenge First When Someone Says 'Three Securities, Three States, So Complete'A PM says 'there are three securities and three states, so the market is complete.' What is the first structural check you should push back with?数理金融中等essay未尝试面试订阅4513Why Attainability Comes Before Probability LanguageIf a nontraded claim seems to have several candidate prices, what should you verify before arguing about which probability measure is 'correct'?数理金融中等essay未尝试面试订阅4514Why the Same Market Can Have One Price for Replicable Claims and Many for OthersHow can one and the same market produce exact single prices for some claims and only intervals for others without contradiction?数理金融中等essay未尝试面试订阅4515Why an Extra Asset Must Add a New Direction, Not Just a New StoryWhen someone claims a newly listed security 'should help complete the market,' what makes that statement true or false?数理金融中等essay未尝试面试订阅