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2033Certainty-Equivalent Return From Expected Log Growth 13A strategy produces one-period returns of 0% or 60%, each with probability 1/2 on one dollar of wealth. Compute E[ln(1+R)] and the certainty-equivalent constant return r ce satisfying ln(1+r ce)=E[ln(1+R)].数学中等derivation未尝试面试订阅2043Log Carry Gap From Two Scenarios 23The desk wants to see the exact concave Jensen gap, not just the inequality direction. A desk scores carry through psi(x)=ln(1+x). Suppose X takes values 0 and 3 with probabilities 1/2 and 1/2. Compute E[psi(X)] and psi(E[X]).数学中等数值题未尝试面试订阅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2081Worst Shortfall of a Simple Hedge 11A non-traded payoff pays 4, 1, and 6 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash -8 and Delta = 0.1 shares of stock. What is the worst-case shortfall of that hedge across the three states?数理金融简单数值题未尝试免费2082Worst Shortfall of a Simple Hedge 12A non-traded payoff pays 3, 5, and 1 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash 7 and Delta = -0.05 shares of stock. What is the worst-case shortfall of that hedge across the three states?数理金融简单数值题未尝试免费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2094Incomplete-Market Pricing Intuition 24Why can one extra state-contingent quote complete the market in a trinomial model even if the stock and bond alone cannot?数理金融困难essay未尝试面试订阅2095Incomplete-Market Pricing Intuition 25Why do indifference prices depend on risk aversion while no-arbitrage intervals do not?数理金融困难essay未尝试面试订阅2246State-Mixture Copula Calibration 1A desk uses a two-state one-factor copula toy model: conditional on a calm systemic state a name defaults with probability 0.40%, and conditional on a stress state it defaults with probability 6.40%. The unconditional one-year default probability is 1.60%. What stress-state probability is implied?数理金融中等数值题未尝试面试订阅2247State-Mixture Copula Calibration 2In a two-state conditional-independence model, a name has calm-state default probability 0.60% and stress-state weight 20.00%. If the unconditional one-year default probability is 2.40%, what stress-state default probability is implied?数理金融中等数值题未尝试面试订阅2248State-Mixture Copula Calibration 3A name's one-year default probability is modeled as a mixture of a calm state and a stress state. The stress-state probability is 25.00%, the stress-state default probability is 5.50%, and the unconditional default probability is 1.93%. What calm-state default probability is implied?数理金融中等数值题未尝试面试订阅2251Basket Dependence Recovery 1Two names have one-year default probabilities 2.00% and 3.00%. A desk prices a two-name first-to-default basket using a horizon first-to-default probability of 4.40%. What joint default probability over the horizon is implied?数理金融简单数值题未尝试免费2252Basket Dependence Recovery 2A two-name basket has first-to-default probability 3.10%. Name A has default probability 1.80%, and the desk's dependence model implies joint default probability 0.40%. What default probability for name B is implied?数理金融简单数值题未尝试免费2253Basket Dependence Recovery 3Two names have one-year default probabilities 1.20% and 2.50%. The desk's copula calibration implies joint default probability 0.30%. What first-to-default probability does that imply?数理金融简单数值题未尝试免费2256Copula Desk Intuition 1A junior trader says 'conditional independence means no dependence.' Why is that wrong in a one-factor copula?数理金融困难essay未尝试面试订阅2257Copula Desk Intuition 2Why can a senior tranche be more sensitive to stronger dependence even when the equity tranche barely moves?数理金融困难essay未尝试面试订阅