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5614Why A Directional Winner Can Still Lose On VolHow can an option buyer be directionally right on the stock yet still lose money because implied exceeded realized?数理金融中等essay未尝试面试订阅5615Why Selling Rich Vol Is Not Free MoneyIf implied volatility is usually above realized, why is systematically shorting options still risky?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5621Tree Hedge Ratio 1At a one-step node, the underlying can move from 100 to 110 or 90. The option value in those two states is 12 and 2. What delta and cash position replicate the option over this step?数理金融中等数值题未尝试面试订阅5624Tree Hedge Ratio 4At a one-step node, the underlying can move from 95 to 109.25 or 83.6. The option value in those two states is 9.5 and 1.5. What delta and cash position replicate the option over this step?数理金融中等数值题未尝试面试订阅5626Two-Step Binomial Call 1Use a two-step binomial tree with spot 100, strike 100, up factor 1.1, down factor 0.9, rate 0.03, and step size Δt=0.5. What is the European call price at time 0?数理金融中等数值题未尝试面试订阅5628Two-Step Binomial Call 3Use a two-step binomial tree with spot 120, strike 115, up factor 1.08, down factor 0.94, rate 0.02, and step size Δt=1. What is the European call price at time 0?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5636Why Trees Handle Early Exercise WellWhy are tree methods often more natural than Black-Scholes closed form for pricing American options?数理金融中等essay未尝试面试订阅5637Why Risk-Neutral Probability Is Not A ForecastWhy should the binomial-tree risk-neutral probability not be interpreted as your actual forecast of an up move?数理金融中等essay未尝试面试订阅5638Why More Steps HelpWhy does increasing the number of steps in a recombining tree often improve pricing accuracy?数理金融中等essay未尝试面试订阅5639Why Trees Beat Monte Carlo For Early ExerciseWhy can a simple lattice be more useful than plain Monte Carlo when exercise timing matters?数理金融中等essay未尝试面试订阅5640Why Recombining Structure MattersWhy is a recombining lattice computationally attractive compared with a tree that branches without recombining?数理金融中等essay未尝试面试订阅5641Monte Carlo Price From Sample Payoffs 1A risk-neutral Monte Carlo run produced discounted-at-maturity payoffs [12.0, 4.0, 0.0, 8.0, 16.0] for an option. If the continuously compounded rate is 0.03 and maturity is 1, what is the time-0 Monte Carlo price estimate?数理金融简单数值题未尝试面试订阅5642Monte Carlo Price From Sample Payoffs 2A risk-neutral Monte Carlo run produced discounted-at-maturity payoffs [5.0, 0.0, 11.0, 3.0, 7.0] for an option. If the continuously compounded rate is 0.04 and maturity is 0.5, what is the time-0 Monte Carlo price estimate?数理金融简单数值题未尝试面试订阅5646Monte Carlo Standard Error 1A Monte Carlo run has sample mean terminal payoff 10.5 and sample standard deviation 4.2 from n=400 paths. With rate 0.03 and maturity 1, what are the time-0 price estimate, its standard error, and the approximate 95% confidence interval?数理金融中等数值题未尝试面试订阅5647Monte Carlo Standard Error 2A Monte Carlo run has sample mean terminal payoff 6.2 and sample standard deviation 2.8 from n=625 paths. With rate 0.04 and maturity 0.5, what are the time-0 price estimate, its standard error, and the approximate 95% confidence interval?数理金融中等数值题未尝试面试订阅5648Monte Carlo Standard Error 3A Monte Carlo run has sample mean terminal payoff 14 and sample standard deviation 7.5 from n=900 paths. With rate 0.02 and maturity 1.5, what are the time-0 price estimate, its standard error, and the approximate 95% confidence interval?数理金融中等数值题未尝试面试订阅