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中文题目A 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
打开 →Determine the radius of convergence of the power series sum_(n=1)^inf (3^n / n) x^n.
打开 →Near a simple root r with f'(r) != 0, what local rate does Newton's method typically enjoy?
打开 →For which of the following exponents p does the series sum_(n=1)^inf 1/n^p converge: p = 1/2, p = 1, p = 3/2, p = 2? List all values for which it converges.
打开 →A trader is long spot and short one futures contract. Initially spot is 100 and futures is 104, so the cash basis is -4. Later the basis changes by 1.5. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →A trader is long spot and short one futures contract. Initially spot is 80 and futures is 76, so the cash basis is 4. Later the basis changes by -2. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →A trader is long spot and short one futures contract. Initially spot is 52 and futures is 52.8, so the cash basis is -0.8. Later the basis changes by 0.6. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →A trader is long spot and short one futures contract. Initially spot is 120 and futures is 123, so the cash basis is -3. Later the basis changes by 2. Ignoring financing, what is the trade's marked-to-market PnL change?
打开 →optimization · gradient-descent · line-search · convergence · iterative-methods · newton-method · quasi-newton · bfgs
打开 →Let $X_n \sim \text{Binomial}(n, \lambda/n)$ for fixed $\lambda > 0$ and $n > \lambda$. (a) Write down the characteristic function $\varphi_{X_n}(t) = E[e^{itX_n}]$ in closed form. (b) Show that $\lim_{n \to \infty} \varphi_{X_n}(t) = e^{\lambda(e^{it} - 1)}$ for every $t \in \
打开 →某私募的策略经理把过去 12 个月的日 P&L 平均值定为 0.06%,准备据此外推年度回报。这种"样本均值即真均值"的隐含假设到底有多牢靠?——回答它需要两条极限定理: 大数定律 (law of large numbers, LLN)说"公式 足够大时样本均值确实贴近真均值"; 中心极限定理 (central limit theorem, ...
打开 →If the risk-free rate rises while the state payoffs stay the same, which piece of the one-step replicating portfolio is directly affected even before the hedge ratio changes?
打开 →A second-order finite-difference scheme uses Richardson extrapolation E_R = (4E_fine - E_coarse)/3. If E_R=10.6 and E_fine=10.4, what coarse-grid estimate E_coarse is implied?
打开 →A second-order finite-difference scheme uses Richardson extrapolation E_R = (4E_fine - E_coarse)/3. If E_R=2.8 and E_coarse=2.4, what fine-grid estimate E_fine is implied?
打开 →Let $N_\lambda\sim\mathrm{Poisson}(\lambda)$. Show that \[ \frac{N_\lambda-\lambda}{\sqrt{\lambda}}\Rightarrow N(0,1) \quad\text{as }\lambda\to\infty \] by working directly with characteristic functions.
打开 →Let $X_1,X_2,\dots$ be i.i.d. with $P(X_i=1)=P(X_i=-1)=1/2$. Show, using characteristic functions, that \[ \frac{X_1+\cdots+X_n}{\sqrt n}\Rightarrow N(0,1). \]
打开 →Let $X_n\sim\mathrm{Binomial}(n,\lambda/n)$ with fixed $\lambda>0$. Use characteristic functions to show that $X_n\Rightarrow \mathrm{Poisson}(\lambda)$.
打开 →Why does making the binomial time step smaller make the replication route feel closer to continuous Black-Scholes hedging?
打开 →Why does adding more binomial steps usually make backward induction more informative about dynamic hedging rather than less?
打开 →Why does the cash basis $S-F$ become negative in contango markets?
打开 →Why does the cash basis $S-F$ become positive in backwardation?
打开 →Why can a cash-futures hedge still leave risk even if the futures notional matches the cash position perfectly?
打开 →Why does basis often shrink as a futures contract approaches expiry?
打开 →Why is basis worth monitoring as its own state variable rather than only watching spot and futures separately?
打开 →周五下午两点，你在上海某私募的因子研究组里收到一张 12,000 × 600 的设计矩阵——600 个候选 alpha 因子在沪深300 成分股上 18 个月日频的横截面暴露。组合经理希望你下班前给一组系数，明早接入回测。你写下普通最小二乘（ordinary least squares, OLS）的闭式解 beta = np.linalg.solve(X.T...
打开 →深圳某私募的多因子研究员手头有 60 个交易日的沪深300 成分股横截面收益,外加一份「因子动物园」(factor zoo)清单:动量、价值、质量、低波,再加上 70 多个另类与基本面因子,合计 公式 个候选预测变量、公式 个观测——一个典型的 公式 病态设计矩阵。她直接套用上一模块的普通最小二乘(ordinary least squares, OLS),解...
打开 →周一开盘后 15 分钟，沪深300 ETF 期权（300ETF options on SSE）的隐含波动率（implied volatility, IV）整体上抬了 3 个 vol。你在一家私募的做市账户上挂着一组 50ETF 与 300ETF 近月平值 call，定价模型需要把每张合约的市场报价反推成 IV。上一节用梯度下降跑过同样的题：在某些深度虚值（o...
打开 →钩子：当一次完整梯度要四个小时 某上海百亿私募的研究员准备把一套基于沪深300 成分股的多因子神经网络 α 信号搬上生产。训练集是过去 5 年的日频面板：约 180 万行样本 × 300 只成分股 × 80 个特征。前两课的工具一一被排除——海森矩阵（Hessian matrix, 公式）装不进显存，L BFGS 一次方向计算也要把整批数据过一遍。退到最朴素...
打开 →Why is a simple root f'(r) != 0 the standard clean setting for fast Newton convergence?
打开 →Why do exact gradient descent convergence and the normal-equation solution agree for OLS?
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