课程平稳性与 ARMA 模型 · 时间序列分析
AR、MA 与 ARMA 过程
周一开盘前，某沪深300 量化私募的研究员把昨天打捞回来的 1500 个日内对数收益样本（log returns）丢进 R，画了一张样本 ACF：lag 1 大约 0.18，lag 2 大约 0.05，再往后几乎全部落进 Bartlett 带里。她想问的是：这条「拖尾」曲线像不像一阶自回归（autoregressive, AR）模型该有的样子？如果是 AR，...
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中文题目课程平稳性与 ARMA 模型 · 时间序列分析
ARMA 模型的识别、估计与预测
周一早盘，某私募的时间序列研究员把过去 200 个交易日的对冲组合超额收益丢进 statsmodels。她想确认这条曲线是不是一个干净的 ARMA 过程——若是，残差就是一组白噪声，可以挂上下一阶段的 GARCH；若不是，她得回去重做特征工程。问题是：用 AR(1)、MA(1)、ARMA(1, 1) 还是 ARMA(2, 1)？拟合完之后怎么知道这一支模型确...
打开 →题目1836 · 统计
AR(1) Forecast Error Variance 1
For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.6 and Var(e_t) = 1, what is the h = 3 step forecast error variance?
打开 →题目1838 · 统计
AR(1) Forecast Error Variance 3
For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.5 and Var(e_t) = 2.25, what is the h = 4 step forecast error variance?
打开 →题目1821 · 统计
AR(1) Multi-Step Forecast 1
A signal follows X_t = 0 + 0.6 X_(t-1) + e_t with Var(e_t) = 2 and current value X_t = 10. What is the h = 3 step forecast E[X_(t+3) | X_t]?
打开 →题目1822 · 统计
AR(1) Multi-Step Forecast 2
A signal follows X_t = 4 + 0.7 X_(t-1) + e_t with Var(e_t) = 1.5 and current value X_t = 8. What is the h = 2 step forecast E[X_(t+2) | X_t]?
打开 →题目1823 · 统计
AR(1) Multi-Step Forecast 3
A signal follows X_t = -1 + 0.8 X_(t-1) + e_t with Var(e_t) = 1 and current value X_t = 3. What is the h = 4 step forecast E[X_(t+4) | X_t]?
打开 →题目5181 · 金融与交易
Arbitrage Direction 1
Spot is 100, maturity is 1 years, and the funding rate is 0.03. The quoted forward price is 104. Assuming no income and no frictions, which arbitrage direction is indicated, and what is the mispricing per unit relative to fair value?
打开 →题目5182 · 金融与交易
Arbitrage Direction 2
Spot is 90, maturity is 0.5 years, and the funding rate is 0.04. The quoted forward price is 91. Assuming no income and no frictions, which arbitrage direction is indicated, and what is the mispricing per unit relative to fair value?
打开 →课程波动率与机制转换模型 · 时间序列分析
ARCH 与 GARCH 模型
某私募(private fund)的风控会上，研究员甩出沪深300 日收益的实证表：日内收益序列本身的自相关系数 公式 在滞后 公式 时几乎全部落在 公式 的 Bartlett 带内；可一旦把同一条序列 平方 再画一次 ACF，从滞后 1 到滞后 60 全是正值、缓慢衰减。再算样本峰度：5.8——远大于正态分布(Gaussian distributi...
打开 →题目6026 · 统计
ARCH(1) as the Beta-Zero Special Case
A GARCH(1,1) reduces to ARCH(1) when $\beta=0$: $h_t=\omega+\alpha r_{t-1}^2$. With $\omega=0.7$ and $\alpha=0.3$, compute the unconditional variance $\bar h$ as a decimal.
打开 →题目1443 · 数学
Arcsine Cubic Remainder
Compute lim_{x->0} [arcsin x - x] / x^3.
打开 →题目3262 · 数学
Area of a Parallelogram from a Unit Square
The map $x=u+v,\ y=u-v$ sends the unit square $0\le u,v\le 1$ to a parallelogram. What is its area?
打开 →题目3265 · 数学
Area of a Polar Wedge
Use the polar Jacobian to compute the area of the region $0\le r\le 2,\ 0\le \theta\le \pi/3$.
打开 →题目3261 · 数学
Area of an Ellipse from the Unit Disk
The map $x=2u,\ y=3v$ sends the unit disk $u^2+v^2\le 1$ to an ellipse in the $(x,y)$-plane. What is the area of that ellipse?
打开 →题目3264 · 数学
Area Scaling Under a Triangular Map
The map $x=u,\ y=u+2v$ sends the unit square to a parallelogram. Compute the image area.
打开 →题目3966 · 金融与交易
Arithmetic-Average Call I
An arithmetic-average Asian call uses four equally weighted fixings. The first three fixings are [100, 102, 98], the strike is 100, and the desk wants the final payoff to be 4. What fourth fixing is required?
打开 →题目3968 · 金融与交易
Arithmetic-Average Call II
A weighted arithmetic-average Asian call uses weights [1, 1, 1, 2] on four fixings. The first three fixings are [50, 52, 51], the strike is 53, and the trader wants payoff 1. What final fixing is required?
打开 →题目3970 · 金融与交易
Arithmetic-Average Call III
An average-strike Asian call uses four equally weighted fixings including the terminal price. The first three fixings are [90, 94, 95]. What terminal price makes the payoff exactly 5?
打开 →题目3967 · 金融与交易
Arithmetic-Average Put I
An arithmetic-average Asian put uses four equally weighted fixings. The first three fixings are [80, 78, 82], the strike is 81, and the desk wants the final payoff to be 2.5. What fourth fixing is required?
打开 →题目3969 · 金融与交易
Arithmetic-Average Put II
A weighted arithmetic-average Asian put uses weights [1, 1, 1, 2]. The first three fixings are [40, 39, 42], the strike is 41, and the target payoff is 0.5. What final fixing is required?
打开 →题目1844 · 统计
ARMA Identification or Simplification 4
You observe the diagnostic statement: AIC prefers ARMA(2,1) but BIC prefers ARMA(1,1). What is the correct modeling conclusion?
打开 →题目5862 · 数理金融
Arrow-Debreu Prices From Option Quotes
A 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
打开 →题目1841 · 统计
ARMA Identification or Simplification 1
You observe the diagnostic statement: ACF tails off geometrically, PACF cuts after lag 1. What is the correct modeling conclusion?
打开 →题目1842 · 统计
ARMA Identification or Simplification 2
You observe the diagnostic statement: ACF cuts after lag 1, PACF tails off. What is the correct modeling conclusion?
打开 →题目1843 · 统计
ARMA Identification or Simplification 3
You observe the diagnostic statement: Both ACF and PACF tail off. What is the correct modeling conclusion?
打开 →题目1845 · 统计
ARMA Identification or Simplification 5
You observe the diagnostic statement: (1-0.5L) X_t = (1-0.5L) e_t. What is the correct modeling conclusion?
打开 →题目5857 · 数学
Arithmetico-Geometric Series
Evaluate the infinite series sum_(n=1)^inf n * (1/2)^n.
打开 →课程平稳性与 ARMA 模型 · 时间序列分析
非平稳性、单位根与 ARIMA 模型
周三下午两点半,你在上海某私募(private fund)的统计套利(statistical arbitrage)团队碰到一桩争执。两只沪深300 成份股里的地产龙头,对数价格(log price)序列各自带强烈趋势,放在同一张图上几乎平行。新来的研究员把两条序列直接做 OLS,跑出 公式、系数 公式 值 18,准备开仓。资深 PM 一句话拦下他:「先做单位...
打开 →题目5827 · 金融与交易
A dealer faces order flow where a fraction 0.4 of traders ar
A dealer faces order flow where a fraction 0.4 of traders are informed (always trade in the correct direction) and 0.6 are noise traders who buy or sell with equal probability. The true value is equally likely high or low, so informed traders buy half the time and sell half the t
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