📚 Mathematical Modeling: Asset Models and Investment Strategy Analysis | 数学建模：资产模型与投资策略分析
Mathematical modeling plays a central role in modern finance, allowing investors to quantify risk, price assets, and construct optimal portfolios. This article explores core asset models and investment strategy analysis from a mathematical perspective.
数学建模在现代金融中扮演着核心角色，它使投资者能够量化风险、对资产定价并构建最优投资组合。本文从数学视角探讨核心资产模型与投资策略分析。
1. The Role of Mathematical Models in Finance | 数学模型在金融中的作用
Financial mathematics transforms economic intuition into precise equations. A model is a simplified representation of reality, built from assumptions and data, used to predict or explain asset behavior.
金融数学将经济直觉转化为精确的方程。模型是现实的简化表示，基于假设和数据构建，用于预测或解释资产行为。
The key steps of mathematical modeling include: (1) identifying the problem, (2) making assumptions, (3) defining variables and parameters, (4) constructing equations, (5) solving or simulating, and (6) validating the model.
数学建模的关键步骤包括：（1）识别问题；（2）做出假设；（3）定义变量与参数；（4）构建方程；（5）求解或模拟；（6）验证模型。
- Models require a balance between realism and tractability. 模型需要在现实性与可处理性之间取得平衡。
- A good model should be testable and adaptable to new data. 好的模型应可检验并能适应新数据。
2. Asset Returns and Probability Distributions | 资产收益率与概率分布
Let the price of an asset at time t be S_t. The simple return over one period is R_t = (S_t – S_{t-1}) / S_{t-1}, while the continuously compounded (log) return is r_t = ln(S_t / S_{t-1}).
设资产在时间 t 的价格为 S_t。单期简单收益率为 R_t = (S_t – S_{t-1}) / S_{t-1}，而连续复利（对数）收益率为 r_t = ln(S_t / S_{t-1})。
Log returns are preferred in modeling because they are time-additive and can be approximated by a normal distribution under the random walk hypothesis.
对数收益率在建模中更受青睐，因为它具有时间可加性，且在随机游走假设下可近似为正态分布。
r_t = ln(S_t) − ln(S_{t−1}), R_t ≈ r_t + ½ r_t²
The expected return and variance are the two fundamental statistics. For a portfolio with weights w_i, the expected return is a weighted average, but the variance includes covariances between assets.
预期收益率和方差是两个基本统计量。对于权重为 w_i 的投资组合，预期收益率是加权平均值，但方差包含资产之间的协方差。
3. Mean-Variance Portfolio Optimization | 均值-方差投资组合优化
Harry Markowitz introduced the mean-variance framework in 1952. The investor chooses portfolio weights w = (w₁, w₂, …, w_N) to minimize variance for a given expected return, or to maximize expected return for a given variance.
哈里·马科维茨于1952年提出均值-方差框架。投资者选择权重向量 w = (w₁, w₂, …, w_N)，在给定预期收益下最小化方差，或在给定方差下最大化预期收益。
The portfolio expected return is E(R_p) = Σᵢ wᵢ E(Rᵢ), and the variance is σ_p² = Σᵢ Σⱼ wᵢ wⱼ Cov(Rᵢ, Rⱼ).
投资组合预期收益为 E(R_p) = Σᵢ wᵢ E(Rᵢ)，方差为 σ_p² = Σᵢ Σⱼ wᵢ wⱼ Cov(Rᵢ, Rⱼ)。
min σ_p² = wᵀ Σ w subject to wᵀ μ = μ_p, wᵀ 1 = 1
Here Σ is the covariance matrix, μ is the vector of expected returns, and 1 is a vector of ones.
这里 Σ 是协方差矩阵，μ 是预期收益向量，1 是元素全为1的向量。
4. Efficient Frontier and Diversification | 有效前沿与分散化
The set of all optimal portfolios forms the efficient frontier. Any portfolio on the frontier offers the highest expected return for a given level of risk, or the lowest risk for a given return.
所有最优投资组合构成有效前沿。前沿上的任意组合在给定风险水平下提供最高预期收益，或在给定收益下提供最低风险。
Diversification reduces unsystematic risk. If assets are not perfectly correlated, the portfolio variance is less than the weighted sum of individual variances.
分散化降低非系统性风险。若资产之间并非完全相关，组合方差将小于各资产方差的加权和。
- Correlation coefficient ρ_ij = Cov(Rᵢ, Rⱼ) / (σᵢ σⱼ) lies between −1 and 1. 相关系数 ρ_ij = Cov(Rᵢ, Rⱼ) / (σᵢ σⱼ) 介于 −1 和 1 之间。
- The lower the correlation, the greater the benefit of diversification. 相关性越低，分散化收益越大。
5. Capital Asset Pricing Model (CAPM) | 资本资产定价模型
The CAPM, developed by Sharpe, Lintner, and Mossin, links an asset’s expected return to its systematic risk measured by beta.
资本资产定价模型（CAPM）由夏普、林特纳和莫森提出，将资产的预期收益与其由贝塔衡量的系统性风险联系起来。
E(Rᵢ) = R_f + βᵢ [E(R_m) − R_f]
where R_f is the risk-free rate, E(R_m) is the expected market return, and βᵢ = Cov(Rᵢ, R_m) / Var(R_m).
其中 R_f 是无风险利率，E(R_m) 是市场预期收益，βᵢ = Cov(Rᵢ, R_m) / Var(R_m)。
The market risk premium E(R_m) − R_f is the reward for bearing one unit of systematic risk. The security market line plots expected return against beta.
市场风险溢价 E(R_m) − R_f 是承担一单位系统性风险的回报。证券市场线以预期收益为纵轴、贝塔为横轴。
6. Beta and Systematic vs. Unsystematic Risk | 贝塔与系统性/非系统性风险
Total risk = systematic risk + unsystematic risk. Systematic risk affects all assets (e.g., inflation, interest rates). Unsystematic risk is specific to a company or industry.
总风险 = 系统性风险 + 非系统性风险。系统性风险影响所有资产（如通胀、利率），非系统性风险是公司或行业特有的风险。
Beta measures the sensitivity of an asset’s returns to market movements. β > 1 implies aggressive, β < 1 implies defensive.
贝塔衡量资产收益对市场波动的敏感度。β > 1 表示进取型，β < 1 表示防御型。
In the CAPM, only systematic risk is priced because unsystematic risk can be diversified away. A portfolio with many assets tends toward β equal to the market beta.
在CAPM中，只有系统性风险被定价，因为非系统性风险可以通过分散化消除。包含大量资产的投资组合，其贝塔趋近于市场贝塔。
7. Multi-Factor Models | 多因子模型
To improve on the CAPM, multi-factor models introduce additional risk factors. The Fama-French three-factor model adds size (SMB) and value (HML) factors.
为了改进CAPM，多因子模型引入了额外风险因子。法玛-弗伦奇三因子模型增加了规模因子（SMB）和价值因子（HML）。
E(Rᵢ) − R_f = βᵢ (E(R_m) − R_f) + sᵢ SMB + hᵢ HML
SMB (Small Minus Big) is the return difference between small and large-cap stocks. HML (High Minus Low) is the return difference between high book-to-market and low book-to-market stocks.
SMB（小减大）是小盘股与大盘股之间的收益差。HML（高减低）是高账面市值比与低账面市值比股票之间的收益差。
Factor models are estimated using linear regression, with factor loadings indicating exposure to each risk source.
因子模型通过线性回归估计，因子载荷表示对各风险源的暴露程度。
8. Monte Carlo Simulation for Asset Prices | 资产价格的蒙特卡洛模拟
In practice, closed-form solutions are often unavailable. Monte Carlo simulation generates many random price paths using the geometric Brownian motion model:
在实践中，常常没有解析解。蒙特卡洛模拟利用几何布朗运动模型生成大量随机价格路径：
dS_t = μ S_t dt + σ S_t dW_t, S_{t+Δt} = S_t exp[(μ − ½σ²)Δt + σ√Δt Z]
where Z is a standard normal random variable. This discretization is known as the Euler-Maruyama method.
其中 Z 是标准正态随机变量。该离散化方法称为欧拉-丸山法。
- Simulate thousands of paths to estimate the distribution of future wealth. 模拟数千条路径以估计未来财富的分布。
- Monte Carlo methods are flexible for options, portfolios, and risk metrics. 蒙特卡洛方法适用于期权、投资组合和风险度量。
9. Risk Measures: VaR and CVaR | 风险度量：VaR 和 CVaR
Value at Risk (VaR) is the maximum loss over a target horizon at a given confidence level. For a portfolio return R_p, the α-level VaR is the negative of the α-quantile.
风险价值（VaR）是在给定置信水平下、目标期限内的最大损失。对于组合收益 R_p，α 水平 VaR 是 α 分位数的负值。
P(R_p ≤ −VaR_α) = 1 − α
Conditional VaR (CVaR), also called Expected Shortfall, is the expected loss given that the loss exceeds VaR.
条件风险价值（CVaR），也称预期亏损，是在损失超过VaR的条件下损失的期望值。
CVaR is a coherent risk measure because it satisfies subadditivity, whereas VaR does not. This makes CVaR preferable for portfolio optimization.
CVaR 是一致性风险度量，因为它满足次可加性，而VaR不满足。这使得CVaR更适合用于投资组合优化。
10. Performance Evaluation and Investment Strategy | 绩效评估与投资策略
To compare risk-adjusted performance, the Sharpe ratio measures excess return per unit of total volatility:
为了比较风险调整后绩效，夏普比率衡量单位总波动所带来的超额收益：
Sharpe = (E(R_p) − R_f) / σ_p
The Treynor ratio uses beta instead of sigma, and Jensen’s alpha is the intercept of the regression of portfolio returns on market returns.
特雷诺比率使用贝塔代替标准差，詹森阿尔法是组合收益对市场收益回归的截距项。
Investment strategies can be classified as passive (indexing) or active (seeking alpha). The efficient market hypothesis implies that active management may not consistently beat the market after fees.
投资策略可分为被动型（指数化）和主动型（寻求阿尔法）。有效市场假说表明，主动管理在扣除费用后可能无法持续跑赢市场。
11. Model Limitations and Risk | 模型的局限性与模型风险
All models rely on assumptions. The normal distribution assumption underestimates tail risk because financial returns often exhibit fat tails and skewness.
所有模型都依赖于假设。正态分布假设低估了尾部风险，因为金融收益常呈现厚尾和偏态。
Parameter estimation error is significant. Small changes in expected returns or covariances can lead to very different optimal portfolios, a phenomenon called “estimation risk.”
参数估计误差影响显著。预期收益或协方差的微小变化会导致最优组合截然不同，这称为”估计风险”。
Model risk refers to losses arising from using an incorrect or misspecified model. Stress testing and robust optimization techniques help mitigate this risk.
模型风险是指因使用错误或错误设定的模型而导致的损失。压力测试和稳健优化技术有助于降低这种风险。
12. Conclusion | 结论
Mathematical modeling provides a systematic framework for understanding asset behavior and designing investment strategies. From Markowitz’s efficient frontier to multi-factor models and simulation methods, the analysis enables investors to make data-driven decisions under uncertainty.
数学建模为理解资产行为和设计投资策略提供了系统性框架。从马科维茨的有效前沿到多因子模型和模拟方法，这些分析使投资者能够在不确定性下做出数据驱动的决策。
However, models are simplifications. A prudent investor must combine quantitative tools with qualitative judgment, and always be aware of the limits of every model.
然而，模型是简化。审慎的投资者必须将量化工具与定性判断相结合，并始终意识到每个模型的局限性。
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