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5872Realized Vol From A Six-Day PathOver six days the daily returns were [0.010, -0.015, 0.022, -0.006, 0.013, -0.009]. Using realized vol = sqrt(252 × average(r 2)), what annualized realized volatility results (four decimals), and was it above or below a quoted 20% implied?数理金融中等数值题未尝试免费5873Straddle Premium Versus Expected MoveA stock at 120 has 28% annualized implied vol over the next 30 trading days. Under a lognormal-approximated normal model, the expected absolute move is S·sigma·sqrt(T)·sqrt(2/pi). What is that expected absolute move, to four decimals (T = 30/252)?数理金融中等数值题未尝试免费5874Sign Of Hedged P&L When Realized Beats ImpliedYou buy an option and delta-hedge it continuously to expiry. If realized volatility ends up higher than the implied volatility you paid, what is the sign of your hedged P&L, and which Greek explains why?数理金融简单essay未尝试免费5875Annualizing A Daily MoveA trader observes that the options market is pricing a 1.2% one-day one-sigma move. Using 252 trading days and sigma ann = daily move·sqrt(252), what annualized implied volatility does this imply, to four decimals?数理金融简单数值题未尝试免费5876The Daily Move That Breaks Even On GammaFor a delta-hedged long option, the gamma gain 0.5·gamma·dS 2 exactly offsets the daily theta 0.5·gamma·S 2·sigma impl 2·dt when the absolute daily move dS equals the implied break-even move. For S = 100, implied vol 18%, and dt = 1/252, what is that break-even daily move in points, to four decimals?数理金融困难数值题未尝试面试订阅5877Risk-Neutral Probability From Tree FactorsA one-step binomial tree has up factor u=1.15, down factor d=0.88, continuously compounded rate r=0.05, and Δt=0.5. Compute the risk-neutral probability of an up move.数理金融简单数值题未尝试免费5878CRR Up/Down Factors From VolatilityIn a Cox-Ross-Rubinstein tree the volatility is σ=0.25 per year and each step is Δt=0.25 years. Using u=e σ√Δt and d=1/u, what is the up factor u (to four decimals)?数理金融简单数值题未尝试免费5879Replicating Delta On A One-Step TreeA stock at 50 moves in one step to 58 or 44. A European call struck at 52 is written on it. What is the replicating delta (shares per option) over this step?数理金融简单数值题未尝试免费5880Two-Step European PutOn a two-step binomial tree, spot=100, strike=100, u=1.1, d=0.9, r=0.05, Δt=1. Price the European put at time 0.数理金融中等数值题未尝试免费5881American Put Early Exercise On Two StepsPrice an American put with strike 100 on a two-step tree: spot=100, u=1.2, d=0.8, r=0.03, Δt=1. Give the time-0 value and state whether early exercise occurs at the first down node.数理金融困难数值题未尝试面试订阅5882Completing A Trinomial Probability SetA one-step trinomial tree has multipliers u=1.2, m=1, d=0.8. The middle probability is fixed at p m=0.6, r=0.04, Δt=1. Find the up-move probability p u that makes the discounted underlying a martingale (so p u+p m+p d=1 and E[S 1]=S 0 e rΔt ).数理金融困难数值题未尝试面试订阅5883One-Step Binomial Call With Dividend YieldA one-step binomial tree has spot=100, strike=100, u=1.1, d=0.9, rate r=0.05, continuous dividend yield δ=0.02, Δt=1. Using the dividend-adjusted risk-neutral probability, price the European call.数理金融中等数值题未尝试免费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