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5845Zero-Cost-ish Collar BoundsYou own stock at 100, buy a put with strike 90 for premium 4, and sell a call with strike 115 for premium 3. What is the net premium paid, the maximum profit, and the maximum loss of the collar at expiry?金融与交易中等数值题未尝试免费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5863Spot The Arbitrage In A Price SetIn a one-period two-state world (states U and D) with risk-free rate 0, asset A pays 2,1 and trades at 1.4, asset B pays 1,3 and trades at 1.7. A bond paying 1,1 trades at 1. Do these three prices admit a strictly positive state-price vector, or is there an arbitrage? Report the state prices if they exist, otherwise state 'arbitrage'.数理金融中等数值题未尝试面试订阅5864Cheapest Dominating Portfolio CostCalls with strikes 80 and 100 trade at 22 and 8. You want a static portfolio of these two calls whose payoff dominates that of a strike-90 call in every terminal state. What is the minimum cost of such a dominating (super-replicating) portfolio?数理金融简单数值题未尝试面试订阅5865Digital Upper Bound from Call SpreadA cash-or-nothing digital call pays 1 if S T > 100 and 0 otherwise. Calls with strikes 95 and 100 trade at 9 and 6. Using a static call-spread super-hedge, what is the tightest model-free upper bound on the digital's price?数理金融中等数值题未尝试面试订阅5866Digital Lower Bound from Call SpreadA cash-or-nothing digital call pays 1 if S T > 100 and 0 otherwise. Calls with strikes 100 and 105 trade at 6 and 4. Using a static call-spread sub-hedge, what is the strongest model-free lower bound on the digital's price?数理金融中等数值题未尝试面试订阅5867Daily Break-Even Move From Implied VolA stock trades at 80 with 32% annualized implied volatility. To roughly the nearest cent, what one-day one-sigma move (in price points) does that implied volatility price in, using 252 trading days?数理金融简单数值题未尝试免费5870How Many Daily Moves To Break EvenA stock at 100 has 25% annualized implied vol, so its one-day one-sigma move is about 1.5749 points. An at-the-money straddle costs 5 points. Treating break-even as the cumulative absolute move equalling the premium, how many one-day one-sigma moves does the premium correspond to, to four decimals?数理金融中等数值题未尝试免费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