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5884Real-World Versus Risk-Neutral Probability On A TreeOn the same binomial tree, an analyst estimates a real-world up probability of 0.65 from historical data, while the risk-neutral up probability is 0.52. Which probability should be used to price a derivative by discounted expectation, and what governs the gap between the two?数理金融中等essay未尝试免费5885Tree Versus Black-Scholes ConvergenceA one-step CRR binomial tree prices an at-the-money one-year European call at 9.95, while the Black-Scholes value with the same spot, strike, rate and volatility is 8.43. By how much does the coarse tree overprice the option, and what single change to the tree would most directly shrink this error?数理金融中等数值题未尝试免费5886Two-Step European Call Via Terminal WeightsOn a two-step recombining tree with spot=64, strike=70, u=1.25, d=0.8, r=0, Δt=1, price the European call by weighting the three terminal payoffs with the binomial probabilities q 2, 2q(1-q), (1-q) 2.数理金融中等数值题未尝试免费5887Fair Variance Strike From a Discrete Option StripA one-year variance swap is replicated by a strip of OTM options. Using the Carr-Madan weighting w i = (ΔK / K i 2), the discount-adjusted strip values give sum i w i * price i = 0.0180 (in variance units before the 2/T scaling), and the linear forward-correction term contributes an additional 0.0020. With T = 1, the fair variance is K var = (2/T) * (strip + forward term). What is the fair annualized volatility strike (decimal)?数理金融中等数值题未尝试面试订阅5888Variance-Notional Swap SettlementA variance swap is quoted with a variance notional of 5,000 per variance point (where a variance point is one unit of 100*sigma 2, i.e. payoff = VarNotional * (10000*sigma realized 2 - 10000*K vol 2)). The volatility strike is K vol = 0.20 and realized annualized volatility over the life is 0.25. What is the payoff to the long (decimal/number)?数理金融简单数值题未尝试免费5889Vega Notional From Variance NotionalA trader wants a variance swap that behaves locally like a vega notional of 40,000 (per vol point) at the current volatility strike of K vol = 0.25. Using the standard linearization that near the strike the variance-notional payoff has vega notional N vega = 2 * K vol * N var (with vol points and variance both in decimal-consistent units), what variance notional N var should be set (number)?数理金融简单数值题未尝试免费5890Jump Contribution to Realized VarianceOver a 252-day window, 251 days each have a squared log-return of 0.0001, and a single jump day has a log-return of -0.10. Realized variance is annualized as RV = (252/252) * sum r i 2 (i.e. RV = sum of squared daily log-returns, since there are 252 observations). What annualized realized variance results, and what would it have been without the jump day (give both as decimals)?数理金融中等数值题未尝试面试订阅5891Who Owns the Class PriorTwo teams ship classifiers trained on a balanced 50/50 dataset, but the live population is 90% class 0. Team A used Gaussian discriminant analysis; Team B used logistic regression. Which model explicitly contains an estimate of the class prior P(y), and explain why that distinction makes one team's fix to the prevalence mismatch cleaner than the other's.机器学习中等essay未尝试面试订阅5892Posterior from a Generative Gaussian ModelA generative classifier models one feature as Gaussian within each class with equal variance: x|Y=0 ~ N(0,1), x|Y=1 ~ N(2,1), and class prior P(Y=1)=0.5. Using Bayes' rule to convert this generative description into the discriminative posterior, compute P(Y=1|x=1.5).机器学习中等数值题未尝试面试订阅5893Deriving the Even-Money Kelly FractionYou repeatedly bet a fraction f of your current wealth on an even-money wager that wins with probability p>\tfrac12 (you gain the staked amount on a win, lose it on a loss). By maximizing the expected logarithm of your wealth multiplier over one round, derive the growth-optimal fraction f *.概率简单derivation未尝试免费5894Kelly Fraction at General Net OddsA favorable bet pays net odds b to 1: staking an amount, you gain b times the stake with probability p and lose the stake with probability 1-p. Betting a fraction f of wealth each round, derive the growth-optimal fraction f * in terms of b and p.概率简单derivation未尝试免费5895Maximum Growth Rate of a Kelly BettorAn even-money coin wins with probability p=0.6. You bet the growth-optimal (Kelly) fraction every round. Compute the resulting maximum expected log-growth rate per round, and express it in closed form in terms of p.概率中等数值题未尝试免费5896Why Half-Kelly Keeps Three-Quarters of the GrowthFor a small-edge repeated bet the expected log-growth is well approximated by the quadratic G(f)\approx f-\tfrac12 2 f 2, where and 2 are the per-round mean and variance of the bet's return. Using this approximation, find the optimal fraction f * and show what fraction of the maximal growth G(f *) is retained by betting half-Kelly, f=f */2.概率中等derivation未尝试面试订阅5897Overbetting to Twice KellyUnder the small-edge approximation G(f)\approx f-\tfrac12 2 f 2 for the expected log-growth of a repeated bet, the growth-optimal fraction is f *= / 2. At what (nonzero) betting fraction does the expected log-growth fall back to zero, and what does this say about the symmetry of growth around f *?概率中等数值题未尝试面试订阅5898Continuous Kelly for Normal ReturnsEach round you allocate a fraction f of wealth to a position whose one-period return R is approximately normal with small mean >0 and variance 2 (with 2\ll 2), so post-round wealth is multiplied by 1+fR. Using a second-order expansion of the log, derive the growth-optimal fraction f *.概率中等derivation未尝试面试订阅5899Betting Kelly on the Wrong ProbabilityAn even-money coin truly wins with probability p=0.55, but you overestimate it as p=0.65 and bet the Kelly fraction implied by your estimate. What is your actual long-run expected log-growth rate per round? Compare it to the growth you would have earned betting the correct Kelly fraction, and state what the sign of your actual growth implies.概率困难数值题未尝试面试订阅5900Higher Expected Return, Lower Compounded GrowthAn even-money coin wins with probability 0.6. Trader A always stakes the fraction f A=0.10 of wealth; Trader B always stakes f B=0.40. (i) Whose stake has the higher one-round expected (arithmetic) profit? (ii) Whose wealth compounds faster over many rounds? Explain the apparent conflict.概率中等数值题未尝试免费5901Expected Rounds to Double a Kelly BankrollA gambler bets the Kelly fraction on an even-money coin with win probability p=0.6 every round, so log-wealth is a random walk with positive drift. Let G be the per-round expected log-growth (the maximal Kelly growth rate). Using an optional-stopping argument on a suitable martingale, estimate the expected number of rounds until wealth first doubles. You may ignore overshoot past the doubling level.概率困难数值题未尝试面试订阅5902Kelly Sizing with an Unknown Win ProbabilityA coin's win probability is unknown, with prior \sim Beta (2,2). You observe 7 wins and 3 losses in calibration trials, then must place one even-money bet on the next flip, choosing a fraction f of wealth to maximize the expected log-wealth after that bet. What fraction should you bet, and why is the posterior mean (rather than, say, the posterior mode) the right quantity to plug into the Kelly formula?概率中等数值题未尝试面试订阅5903Capping the Single-Bet DrawdownYou bet a fraction f of wealth on an even-money coin with win probability p=0.65, but a risk rule forbids any single losing bet from cutting your wealth by more than 20\%. What fraction should you bet, and for which win probabilities p does this drawdown rule actually constrain you below the Kelly fraction?概率简单数值题未尝试免费