题目6043 · 随机过程
Autocorrelation of a Stationary OU Process at a Lag
A stationary OU process has mean-reversion speed kappa = 0.7. What is the autocorrelation between X_t and X_{t+2}?
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中文题目题目2672 · 机器学习
Autocorrelation-Corrected Sample Size
A monthly feature is observed for 60 months and behaves roughly like an AR(1) series with lag-1 autocorrelation $\rho=0.6$. Using the heuristic $n_\text{eff}\approx n(1-\rho)/(1+\rho)$, what is the effective sample size?
打开 →题目680 · 脑筋急转弯
Automaton Visit Repeat 5
A deterministic signal automaton has 81 possible internal states. Starting from any state, after how many visited states must some state have appeared at least twice?
打开 →题目5837 · 金融与交易
Roll Implied Spread From Autocovariance
Under Roll's model, transaction price changes have a first-order serial autocovariance of -0.0009 (in price-squared units). Estimate the implied effective spread.
打开 →题目6038 · 统计
Sign and Size of Lag-1 Autocorrelation
A stationary spread obeys X_(t+1) = -0.4 X_t + epsilon_(t+1) with iid zero-mean shocks. What is the lag-1 autocorrelation of X_t, and what does its sign say about period-to-period dynamics?
打开 →题目1768 · 统计
A Lagged Variable Is Not Automatically a Valid Instrument
Someone proposes using yesterday's order-flow imbalance as an instrument for today's imbalance in a return-impact regression. Why is this not automatically a valid instrument in financial data?
打开 →题目4502 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 1
In a 4-state market, the traded span is generated by payoffs B0, B1, and B2. Three candidate claims are A = 2 B0 - B1, C = B1 + B2, and D = [1,0,0,0]. How many of A, C, and D are automatically guaranteed a unique arbitrage-free price from the traded market alone?
打开 →题目4504 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 3
A market's traded span contains F0 and F1 but not the full state space. Candidate claims are M = 3 F0 - 0.5 F1, N = [1,0,1,0], and P = [0,1,0,1]. How many of M, N, and P are automatically guaranteed unique prices?
打开 →题目2419 · 机器学习
Why Low Bias Is Not Automatically Desirable
Why is 'lower bias' not automatically a sufficient argument for preferring one model over another?
打开 →题目4503 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 2
In a 5-state market, traded payoffs span H1, H2, and H3. Candidate claims are U = H1 + 2 H3, V = H2 - H1, and W = [0,1,0,0,0]. How many of U, V, and W are guaranteed uniquely priced just from attainability?
打开 →题目4505 · 数理金融
How Many Candidate Claims Are Automatically Uniquely Priced? 4
In a 4-state market, traded payoffs span G0, G1, and G2. Candidate claims are A = G0 + G1, B = 2 G2 - G1, C = G0 - G2, and D = [0,0,0,1]. How many of A, B, C, and D are guaranteed uniquely priced from the given information?
打开 →题目2861 · 概率
Why the Difference of Two Copies Is Automatically Symmetric
Let $X$ and $Y$ be i.i.d. with characteristic function $\phi(u)$. Show that the characteristic function of $D=X-Y$ is $|\phi(u)|^2$, and conclude that $D$ is symmetric about $0$.
打开 →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，...
打开 →题目6040 · 统计
Two-Period Variance Ratio of an AR(1)
Returns are generated by a stationary AR(1) with autoregressive coefficient 0.5. The Lo-MacKinlay variance ratio at lag 2 is VR(2) = Var(r_t + r_(t+1)) / (2 Var(r_t)). Compute VR(2) and state whether it signals momentum or mean reversion.
打开 →模块2.3.1 · 数学与统计能力 · 时间序列分析
平稳性与 ARMA 模型
time-series · stationarity · autocorrelation · acf · pacf · white-noise · random-walk · ar
打开 →模块3.4.2 · 编程 · C++ 与低延迟
模板与现代 C++
cpp · cpp17 · templates · function-template · class-template · type-deduction · auto · structured-bindings
打开 →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 →课程模板与现代 C++ · C++ 与低延迟
Lambda 表达式、函数工具与 constexpr
国内某私募衍生品桌的研究主管要在不动 Monte Carlo 引擎的前提下，对同一笔 CSI 300 ETF（510300.SH）4.30 行权价的欧式 call 跑三种 payoff——call、put、二元 digital。C++98 时代的答案是一棵 PayoffBase 指针继承树；C++11 之后的答案变成一行 lambda：把它直接传进 pric...
打开 →课程Git 与代码质量 · 量化开发的软件工程
代码质量自动化：格式化、静态检查、测试与 pre-commit
周三下午两点。一家 A 股 私募 的资深同事打开你的 MR，标题是 feat(risk): 添加 沪深300 因子 z 列至业绩归因 。改了 12 个文件。二十秒之内审查线上铺满了「文件末尾多一个空行」「这个 import 没用到」「第 47 行行尾有空白」「import 没排序」之类的评论。你能感到审查时间正在漏走——这些评论没一条是关于你因子逻辑对不对的...
打开 →课程高级衍生品 · 衍生品
利率、信用与结构化衍生品
Hook（开场场景）. 某资管公司多策略组合的固收风险经理，在月末复盘时盯着账上三笔头寸：（A）规模 5 亿元的 5 年期 FR007 利率互换（IRS），付固收浮，固定端 2.45%；（B）一笔参考某城投平台的 CRMW 1 亿元名义；（C）一只挂钩中证500 的 18 个月雪球结构化产品，由头部券商收益凭证渠道发出，规模 3 亿元，敲入线 75%、月...
打开 →课程神经网络 · 机器学习理论
反向传播与自动微分
反向传播与自动微分 Hook：四分钟一步的梯度 你刚加入一家以沪深300 alpha 为主力的私募（private fund），上手第一件事是把上一课那张 5 层、宽度 128 的多层感知机（multi layer perceptron, MLP）跑通——目标是用一个标准的 Barra 因子模型（factor model）的截面特征去拟合 公式，本质上是在学...
打开 →课程平稳性与 ARMA 模型 · 时间序列分析
平稳性与自相关函数
某私募(private fund)交易日下午四点,你的 PM 把过去 500 个交易日的策略净值推过来,问:这条曲线的均值真的稳定吗?波动率有没有结构性变化?只看一条路径,凭什么相信估出来的均值与自相关有意义?这是时间序列分析(time series analysis)的元问题。横截面统计里你有 公式 个独立同分布(i.i.d.)样本,推断建立在「重复抽样」...
打开 →课程模板与现代 C++ · C++ 与低延迟
智能指针与移动语义
国内某私募 CSI 300 ETF 期权桌的资深 C++ engineer 在审一份六年前写就的策略库——它要进 live engine。他贴在每一份源码上的 PR review 评论只有一行：「这里裸 new ——改成 std::make unique 。」这份库是 C++03 风格写的， delete 散布在异常处理路径上，等一个错位的 throw 就足...
打开 →课程模板与现代 C++ · C++ 与低延迟
模板与泛型编程
国内某私募的 C++ 研究桌周一例会：新入职的研究员上线了 mean double 、 mean float 、 mean long double 三份函数——同一个九行的均值计算被复制了三次，只是浮点精度不同。Senior C++ engineer 的 review 意见只有一句话：「这里要写成模板。」周五新人交回的版本里，三份代码合成了一个 templa...
打开 →课程模板与现代 C++ · C++ 与低延迟
现代 STL 与词汇类型
国内某私募 CSI 300 ETF 期权桌的风险分析师在翻夜间对账日志：四十笔 510300 期权报价的隐含波动率（IV）显示为整齐的 1.0 。这不是市场信号，而是上一代 IV 求解器在「未收敛」时使用的 sentinel value。当下游的偏斜模型把 1.0 一起平均进去，报告的偏斜被肉眼可见地拖偏，早会因此浪费了三十分钟去追一个根本不存在的数字。修复...
打开 →题目268 · 概率
ABCA in a Four-Symbol Stream
Each symbol of an iid stream is chosen uniformly from $\{A,B,C,D\}$. Compute the expected waiting time until $ABCA$ first appears.
打开 →题目274 · 概率
Either 123 or 321 on a Fair Die
A fair six-sided die is rolled repeatedly. Let $T$ be the first time that either $1,2,3$ or $3,2,1$ appears as a consecutive length-3 block. Compute $E[T]$.
打开 →题目264 · 概率
Expected Time Until ABC or CBA
A symbol stream is iid and uniform on $\{A,B,C\}$. What is the expected number of symbols until either $ABC$ or $CBA$ first appears?
打开 →题目271 · 概率
Finish 1-2-3 After Already Rolling 1,2
A fair six-sided die is rolled repeatedly. You already know that the current observed suffix is exactly $1,2$. Starting from here, what is the expected additional number of rolls until $1,2,3$ first appears?
打开 →题目269 · 概率
Finish ABCA After Current Suffix ABC
An iid stream over $\{A,B,C,D\}$ is uniform. Suppose the current observed suffix is exactly $ABC$. From this point onward, what is the expected additional number of symbols until $ABCA$ first appears?
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