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5563Why Dividend Yield Lowers CallsWhy does a higher continuous dividend yield lower a European call price in Black-Scholes?数理金融中等essay未尝试面试订阅5564Why Time Value Can Rise Even OTMWhy can an out-of-the-money option still gain value as maturity extends, even if intrinsic value is zero today?数理金融中等essay未尝试面试订阅5565Why Forward Moneyness Organizes PricesWhy is forward moneyness often a cleaner way to compare options than spot moneyness when rates or dividends matter?数理金融中等essay未尝试面试订阅5616One-Step Binomial Call 1In a one-step binomial tree, spot is 100, strike is 105, up factor is 1.1, down factor is 0.92, the continuously compounded rate is 0.04, and Δt=1. What is the European call price?数理金融中等数值题未尝试面试订阅5617One-Step Binomial Call 2In a one-step binomial tree, spot is 80, strike is 75, up factor is 1.12, down factor is 0.9, the continuously compounded rate is 0.03, and Δt=0.5. What is the European call price?数理金融中等数值题未尝试面试订阅5631One-Step Trinomial Call 1A one-step trinomial tree uses spot 100, strike 102, multipliers u=1.12, m=1, d=0.9, risk-neutral probabilities (0.25, 0.5, 0.25), rate 0.03, and Δt=1. What is the European call price?数理金融中等数值题未尝试面试订阅5632One-Step Trinomial Call 2A one-step trinomial tree uses spot 80, strike 75, multipliers u=1.1, m=1, d=0.92, risk-neutral probabilities (0.3, 0.45, 0.25), rate 0.04, and Δt=0.5. What is the European call price?数理金融中等数值题未尝试面试订阅5861Risk-Neutral Probability From A Quoted CallIn a one-period binomial model the stock has S0=100 and goes to Su=130 or Sd=90, with risk-free rate 0. A call struck at K=100 trades at price 12. Back out the implied risk-neutral probability of the up state from this option quote.数理金融简单数值题未尝试面试订阅5862Arrow-Debreu Prices From Option QuotesA stock has three future states with prices 120, 100, and 80; the risk-free rate is 0. Calls struck at 80 trade at 28 and calls struck at 100 trade at 8. Using the digital/butterfly decomposition, find the Arrow-Debreu price of the single highest state (the state where the stock ends at 120).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