常用公式与定理 / Formulas and Theorems

本页结构

核心概念

条件概率、全概率公式与贝叶斯公式 Conditional probability, total probability and Bayes rule
期望线性性、无意识统计学家法则、全期望与全方差 Linearity of expectation, LOTUS, total expectation and total variance
协方差、相关系数与常见分布之间的联系 Covariance, correlation and links between standard distributions

学习顺序

把文字题拆成事件、条件变量和目标量。 Translate word problems into events, conditioning variables and target quantities.
存在隐含状态影响收益时，用全期望分层处理。 Use total expectation when a hidden state drives the payoff.
在拆分概率或期望前先确认独立性是否成立。 Check independence before factorizing probabilities or expectations.

A deep understanding of core statistical principles is crucial for modeling financial markets, pricing derivatives, and managing risk.

深入理解核心统计原理对于金融市场建模、衍生品定价和风险管理至关重要。

These laws govern how probabilities are calculated and updated, forming the basis for statistical inference and decision-making under uncertainty.

这些定律规定了概率的计算和更新方式，构成了不确定性下统计推断和决策的基础。

Conditional Probability, Bayes' Theorem, and Law of Total Probability

条件概率、贝叶斯定理和全概率定律

Consider events $A_1, \dots, A_n$ which form a partition of the sample space (i.e., they are mutually exclusive and collectively exhaustive) and an event $B$ .

考虑形成样本空间分区的事件 $A_1, \dots, A_n$ （即，它们是互斥的且共同详尽的）和事件 $B$ 。

Concept	Formula	Description
Conditional Probability	$\mathbb{P}(A \mid B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$	The probability of event $A$ occurring given that event $B$ has already occurred.
Law of Total Probability	$\mathbb{P}(B) = \sum_{i=1}^n \mathbb{P}(B \cap A_i) = \sum_{i=1}^n \mathbb{P}(B \mid A_i)\mathbb{P}(A_i)$	Used to find the marginal probability of an event $B$ when the sample space is partitioned.
Bayes' Theorem	$\mathbb{P}(A_1 \mid B) = \frac{\mathbb{P}(B \mid A_1)\mathbb{P}(A_1)}{\mathbb{P}(B)}$	Relates the posterior probability $\mathbb{P}(A_1 \mid B)$ to the prior $\mathbb{P}(A_1)$ and the likelihood $\mathbb{P}(B \mid A_1)$ . Relevance: Crucial for updating beliefs as new data arrives.

概念	公式	描述
条件概率	$\mathbb{P}(A \mid B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)}$	假设事件 $B$ 已经发生，则事件 $A$ 发生的概率。
总概率定律	$\mathbb{P}(B) = \sum_{i=1}^n \mathbb{P}(B \cap A_i) = \sum_{i=1}^n \mathbb{P}(B \mid A_i)\mathbb{P}(A_i)$	用于求样本空间分区时事件 $B$ 的边际概率。
贝叶斯定理	$\mathbb{P}(A_1 \mid B) = \frac{\mathbb{P}(B \mid A_1)\mathbb{P}(A_1)}{\mathbb{P}(B)}$	将后验概率 $\mathbb{P}(A_1 \mid B)$ 与先验 $\mathbb{P}(A_1)$ 和似然 $\mathbb{P}(B \mid A_1)$ 相关。相关性：对于新数据到来时更新信念至关重要。

Moments describe the shape and location of a probability distribution. Understanding their properties is key to manipulating random variables in models.

矩描述了概率分布的形状和位置。了解它们的属性是操纵模型中随机变量的关键。

Law of the Unconscious Statistician (LOTUS)

无意识统计学家定律（LOTUS）

The expected value of a function of a random variable $g(X)$ can be calculated without first finding the distribution of $Y=g(X)$ .

可以计算随机变量 $g(X)$ 的函数的期望值，而无需首先找到 $Y=g(X)$ 的分布。

Law of Total Expectation and Variance

总期望和方差定律

These laws are essential for models where one random variable depends on another (e.g., a two-stage process or a mixture model).

这些定律对于一个随机变量依赖于另一个随机变量的模型（例如两阶段过程或混合模型）至关重要。

Concept	Formula	Description
Total Expectation	$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$	The overall expected value of $X$ is the expected value of the conditional expectation of $X$ given $Y$ .
Total Variance	$\mathrm{Var}(X) = \mathrm{Var}(\mathbb{E}[X \mid Y]) + \mathbb{E}[\mathrm{Var}(X \mid Y)]$	The total variance is the sum of the variance of the conditional mean (between-group variance) and the mean of the conditional variance (within-group variance).

概念	公式	描述
总期望	$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]]$	$X$ 的总体期望值是给定 $Y$ 的情况下 $X$ 的条件期望的期望值。
总方差	$\mathrm{Var}(X) = \mathrm{Var}(\mathbb{E}[X \mid Y]) + \mathbb{E}[\mathrm{Var}(X \mid Y)]$	总方差是条件均值的方差（组间方差）与条件方差的均值（组内方差）之和。

Intuitively, the Law of Total Expectation says that if we "average over all averages" of $X$ obtained by some information about $Y$ , we obtain the true average. Similarly, the Law of Total Variance says that the true variance comes from two sources: between samples (the first term) and within samples (the second term).

直观地说，总期望定律说，如果我们对通过有关 $Y$ 的一些信息获得的 $X$ 进行“平均”，我们就得到了真实的平均值。同样，总方差定律指出，真正的方差有两个来源：样本之间（第一项）和样本内（第二项）。

Covariance and Correlation

协方差和相关性

These measure the linear relationship between two random variables $X$ and $Y$ .

这些测量两个随机变量 $X$ 和 $Y$ 之间的线性关系。

Key Properties of Variance and Covariance:

方差和协方差的关键属性：

$\text{Var}(aX + b) = a^2\text{Var}(X)$
$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y)$
If $X$ and $Y$ are independent, $\text{Cov}(X, Y) = 0$ , and $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ . Note: The converse is not always true (uncorrelated does not imply independent).

$\text{Var}(aX + b) = a^2\text{Var}(X)$
$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y)$
如果 $X$ 和 $Y$ 独立，则 $\text{Cov}(X, Y) = 0$ 和 $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ 。 注意： 相反的情况并不总是正确（不相关并不意味着独立）。

Common Relationships Between Distributions

分布之间的常见关系

Relationship	Formula	Relevance
Sum of Bernoullis	$X_1, \dots, X_n \sim \text{Bernoulli}(p) \text{ IID} \implies \sum_{i=1}^n X_i \sim \text{Binom}(n, p)$	Foundation of the Binomial Option Pricing Model.
Sum of Poissons	$X_i \sim \text{Poisson}(\lambda_i) \text{ independent} \implies \sum_{i=1}^n X_i \sim \text{Poisson}\left(\sum_{i=1}^n \lambda_i\right)$	Used in modeling cumulative event counts (e.g., defaults) over time.
Sum of Normals	$X_i \sim N(\mu_i, \sigma_i^2) \text{ independent} \implies \sum_{i=1}^n X_i \sim N\left(\sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma_i^2\right)$	Fundamental for portfolio theory and risk aggregation.

关系	公式	关联
伯努利变量之和	$X_1, \dots, X_n \sim \text{Bernoulli}(p) \text{ IID} \implies \sum_{i=1}^n X_i \sim \text{Binom}(n, p)$	二项式期权定价模型的基础。
泊松变量之和	$X_i \sim \text{Poisson}(\lambda_i) \text{ independent} \implies \sum_{i=1}^n X_i \sim \text{Poisson}\left(\sum_{i=1}^n \lambda_i\right)$	用于对随时间变化的累积事件计数（例如违约）进行建模。
正态变量之和	$X_i \sim N(\mu_i, \sigma_i^2) \text{ independent} \implies \sum_{i=1}^n X_i \sim N\left(\sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma_i^2\right)$	投资组合理论和风险聚合的基础。

These theorems provide the theoretical justification for many statistical and financial models, particularly those involving large samples or long time horizons.

这些定理为许多统计和金融模型提供了理论依据，特别是那些涉及大样本或长时间范围的模型。

Central Limit Theorem (CLT)

中心极限定理 (CLT)

Let $X_1, X_2, \dots, X_n$ be a sequence of i.i.d. random variables with mean $\mu$ and finite variance $\sigma^2$ . As $n \to \infty$ , the distribution of the standardized sample mean approaches the standard normal distribution:

令 $X_1, X_2, \dots, X_n$ 为独立同分布的序列。均值为 $\mu$ 和有限方差 $\sigma^2$ 的随机变量。作为 $n \to \infty$ ，标准化样本均值的分布接近标准正态分布：

Relevance: Justifies the use of the Normal distribution to model asset returns, as returns are the sum of many small, independent price changes. It also underpins statistical inference (e.g., confidence intervals, hypothesis testing).

相关性： 证明使用正态分布来模拟资产回报是合理的，因为回报是许多小的、独立的价格变化的总和。它还支持统计推断（例如置信区间、假设检验）。

Law of Large Numbers (LLN)

大数定律 (LLN)

The LLN states that as the number of trials increases, the average of the results obtained from a large number of independent and identically distributed random variables converges to the expected value.

LLN 指出，随着试验次数的增加，从大量独立且同分布的随机变量获得的结果的平均值会收敛到期望值。

Relevance: Guarantees that Monte Carlo simulations will converge to the true expected value as the number of simulations increases.

相关性： 保证随着模拟次数的增加，蒙特卡罗模拟将收敛到真实的期望值。

Markov's and Chebyshev's Inequalities

马尔可夫和切比雪夫不等式

These inequalities provide bounds on the probability that a random variable deviates from its mean, even when the full distribution is unknown.

这些不等式为随机变量偏离其均值的概率提供了界限，即使完整分布未知。

These formulas are indispensable for derivative pricing and continuous-time modeling.

这些公式对于衍生品定价和连续时间建模是不可或缺的。

Ito's Lemma

伊藤引理

Ito's Lemma is the fundamental rule of differentiation for stochastic processes, particularly those involving Brownian motion (Wiener process). It is the stochastic equivalent of the chain rule in standard calculus.

伊藤引理是随机过程微分的基本规则，特别是涉及布朗运动（维纳过程）的随机过程。它是标准微积分中链式法则的随机等价物。

For a function $G(t, X_t)$ where $X_t$ follows the Ito process $dX_t = \mu(X_t, t) dt + \sigma(X_t, t) dW_t$ , the differential $dG$ is:

对于函数 $G(t, X_t)$ ，其中 $X_t$ 遵循 Ito 过程 $dX_t = \mu(X_t, t) dt + \sigma(X_t, t) dW_t$ ，微分 $dG$ 为：

Relevance: Used to derive the Black-Scholes Partial Differential Equation (PDE) and to find the process followed by a function of an asset price (e.g., the log-price).

相关性： 用于推导 Black-Scholes 偏微分方程 (PDE) 并查找资产价格函数（例如对数价格）所遵循的过程。

Geometric Brownian Motion (GBM)

几何布朗运动 (GBM)

GBM is the most common model for asset prices $S_t$ in continuous time, assuming log-returns are normally distributed.

GBM 是连续时间内资产价格 $S_t$ 最常见的模型，假设对数收益呈正态分布。

$\mu$ : Drift (expected return)
$\sigma$ : Volatility
$dW_t$ : Wiener process (Brownian motion)

$\mu$ : Drift (expected return)
$\sigma$ : Volatility
$dW_t$ : Wiener process (Brownian motion)

The solution for $S_t$ is Lognormal: $S_t = S_0 \exp\left( \left(\mu - \frac{1}{2}\sigma^2\right) t + \sigma W_t \right)$ .

$S_t$ 的解是对数正态： $S_t = S_0 \exp\left( \left(\mu - \frac{1}{2}\sigma^2\right) t + \sigma W_t \right)$ 。

Black-Scholes-Merton (BSM) Formula (European Call Option)

Black-Scholes-Merton (BSM) 公式（欧式看涨期权）

The BSM formula provides a closed-form solution for the price of a European call option $C$ :

BSM 公式为欧式看涨期权 $C$ 的价格提供了封闭式解：

where:

其中：

$S$ : Current stock price
$K$ : Strike price
$r$ : Risk-free interest rate
$T-t$ : Time to maturity
$\sigma$ : Volatility of the stock return
$N(\cdot)$ : Cumulative distribution function of the standard normal distribution

$S$ : Current stock price
$K$ : Strike price
$r$ : Risk-free interest rate
$T-t$ : Time to maturity
$\sigma$ : Volatility of the stock return
$N(\cdot)$ : Cumulative distribution function of the standard normal distribution

Risk-Neutral Valuation

风险中性估值

The First Fundamental Theorem of Asset Pricing states that in a market with no arbitrage, there exists at least one risk-neutral measure $\mathbb{Q}$ under which the price of any derivative $V$ is the discounted expected value of its payoff, $V_T$ , under this measure.

资产定价第一基本定理指出，在没有套利的市场中，至少存在一个风险中性测度 $\mathbb{Q}$ ，其中任何衍生品 $V$ 的价格都是其收益 $V_T$ 在此测度下的贴现预期值。

Relevance: This is the core principle of modern derivative pricing. The BSM formula is derived by applying this principle to the GBM process under the risk-neutral measure. The key change is that the drift $\mu$ of the asset price process is replaced by the risk-free rate $r$ .

**相关性：**这是现代衍生品定价的核心原则。 BSM公式是将该原理应用于风险中性测度下的GBM过程而推导出来的。关键的变化是资产价格过程的漂移 $\mu$ 被无风险利率 $r$ 取代。

补充讲解

按信息分层

Condition on information

贝叶斯、全概率和全期望本质上都是信息记账工具。先写清什么已观测、什么隐藏、目标量需要如何更新。

Bayes, total probability, and total expectation are all bookkeeping tools for information. Write down what is observed, what is hidden, and what quantity must be updated.

方差有两个来源

Variance has two sources

全方差公式把组间不确定性和组内不确定性分开。它常用于混合模型、市场状态切换收益和条件风险估计。

The law of total variance separates uncertainty between groups from uncertainty inside each group. This is useful for mixture models, regime-switching returns, and conditional risk estimates.

不相关不等于独立

Uncorrelated is not independent

很多概率化简需要独立性，而不仅是不相关。面试中明确说出这个条件，可以避免错误拆分概率或期望。

Many probability shortcuts require independence, not just zero covariance. In quant interviews, explicitly naming this condition prevents invalid factorization of probabilities or expectations.

常用公式与定理