风险的度量-波动率

风险的度量-波动率

小象实战讲义 · 进阶量化交易实战：迭代式的量化策略研发 本讲义仅用于编程与量化方法教学，不构成任何投资建议。

波动率是金融资产价格最核心的二阶矩特征。在量化体系中，波动率既是风险的直接代名词，也是期权定价与动态头寸管理的核心输入变量。本节系统剖析历史波动率（HV）、帕金森极值波动率（Parkinson）与 EWMA 指数加权移动平均波动率模型。

💡 核心导读

• 三大经典波动率度量模型与适用场景：
  • 标准历史波动率（Close-to-Close HV）：仅基于收盘价对数收益率标准差，计算简单但丢失日内最高最低价信息；
  • 帕金森高低价极值波动率（Parkinson Volatility）：基于日内 High 和 Low 极值差，效率是传统收盘价波动率的 5 倍；
  • EWMA 动态衰减波动率（Exponentially Weighted）：赋予近期波动更高权重，敏锐捕捉波动率聚集（Volatility Clustering）效应。

┌─────────────────────────────────────────────────────────────┐
│                    多维波动率量化度量拓扑图                     │
└──────────────────────────────┬──────────────────────────────┘
                               │
 ┌─────────────────────────────┼─────────────────────────────┐
 ▼                             ▼                             ▼
┌─────────────────────────┐   ┌─────────────────────────┐   ┌─────────────────────────┐
│ 收盘价历史波动率 (HV)   │   │ 帕金森极值波动率 (Park) │   │ EWMA 动态加权波动率     │
├─────────────────────────┤   ├─────────────────────────┤   ├─────────────────────────┤
│ • <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi><mo>=</mo><mtext>std</mtext><mo stretchy="false">(</mo><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>P</mi><mi>t</mi></msub><mi mathvariant="normal">/</mi><msub><mi>P</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>×</mo><msqrt><mn>252</mn></msqrt></mrow><annotation encoding="application/x-tex">\sigma = \text{std}(\ln(P_t/P_{t-1})) \times \sqrt{252}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">std</span></span><span class="mopen">(</span><span class="mop">ln</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.1389em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.04em;vertical-align:-0.1328em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9072em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">252</span></span></span><span style="top:-2.8672em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z"/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1328em;"><span></span></span></span></span></span></span></span></span>│ │ • 基于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ln</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>H</mi><mi>t</mi></msub><mi mathvariant="normal">/</mi><msub><mi>L</mi><mi>t</mi></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\ln(H_t/L_t)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mop">ln</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0813em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2806em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> │ │ • <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>σ</mi><mi>t</mi><mn>2</mn></msubsup><mo>=</mo><mi>λ</mi><msubsup><mi>σ</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><msubsup><mi>r</mi><mi>t</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\sigma_t^2 = \lambda \sigma_{t-1}^2 + (1-\lambda)r_t^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="katex-base"><span class="katex-strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.1205em;vertical-align:-0.3064em;"></span><span class="mord mathnormal">λ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0359em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3064em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="katex-base"><span class="katex-strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0278em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="katex-sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>│
│ • 基准常用，但时滞较长  │   │ • 充分捕获日内宽幅震荡  │   │ • 快速捕捉波动率暴增    │
└─────────────────────────┘   └─────────────────────────┘   └─────────────────────────┘

一、帕金森（Parkinson）极值波动率数学推导

假设资产价格在连续时间内遵循几何布朗运动。帕金森证明利用日内最高价 $H_t$ 与最低价 $L_t$ 可以构建更无偏、更高效的波动率估计量：

$$\sigma_{Park}^2 = \frac{1}{4 \ln 2 \cdot N} \sum_{t=1}^N \left( \ln \frac{H_t}{L_t} \right)^2 \times 252$$

相较于仅用收盘价，帕金森波动率能够在样本量较小时快速收敛至真实波动率。

import pandas as pd
import numpy as np

def calculate_parkinson_volatility(df: pd.DataFrame, window: int = 20) -> pd.Series:
    """
    计算帕金森高低价极值年化波动率
    """
    hl_ratio = np.log(df['high'] / df['low'])
    park_var = (hl_ratio ** 2).rolling(window).mean() / (4.0 * np.log(2.0))
    park_vol = np.sqrt(park_var * 252.0)
    return park_vol

二、EWMA 波动率聚集模型与风险衰减因子

金融资产收益率具有显著的“波动率聚集（Volatility Clustering）”特征，即大波动后往往伴随大波动。RiskMetrics 提出 EWMA 模型，衰减因子通常取 $\lambda = 0.94$：

$$\sigma_t^2 = \lambda \sigma_{t-1}^2 + (1 - \lambda) r_t^2$$

def calculate_ewma_volatility(returns: pd.Series, decay: float = 0.94) -> pd.Series:
    """
    计算 EWMA 动态指数衰减年化波动率
    """
    var = np.zeros(len(returns))
    var[0] = returns.var()
    for i in range(1, len(returns)):
        var[i] = decay * var[i-1] + (1.0 - decay) * (returns.iloc[i] ** 2)
    return pd.Series(np.sqrt(var * 252.0), index=returns.index)

📝 动手练一练

编写 Python 脚本对比收盘价 HV、帕金森极值与 EWMA 波动率在突发黑天鹅事件中的响应速度：

import pandas as pd
import numpy as np

def run_volatility_measurement_test():
    np.random.seed(42)
    n = 100
    # 前 80 天低波动，后 20 天波动率激增 3 倍
    rets1 = np.random.normal(0, 0.01, 80)
    rets2 = np.random.normal(0, 0.03, 20)
    rets = np.concatenate([rets1, rets2])
    sigmas = np.concatenate([np.full(80, 0.01), np.full(20, 0.03)])
    prices = 100.0 * np.cumprod(1.0 + rets)
    
    # high/low 日内振幅按状态 σ 同步缩放（与波动率状态切换保持一致）
    df = pd.DataFrame({
        'high': prices * (1.0 + np.abs(np.random.normal(0, 0.8, n)) * sigmas),
        'low': prices * (1.0 - np.abs(np.random.normal(0, 0.8, n)) * sigmas),
        'close': prices
    })
    
    df['HV_20'] = df['close'].pct_change().rolling(20).std() * np.sqrt(252)
    df['Park_20'] = calculate_parkinson_volatility(df, window=20)
    df['EWMA'] = calculate_ewma_volatility(df['close'].pct_change().fillna(0))
    
    print("================ 多维波动率度量响应对比 ================")
    print(f"突发波动后 Close-to-Close HV: {df['HV_20'].iloc[-1]*100:.2f}%")
    print(f"突发波动后 Parkinson 极值 HV : {df['Park_20'].iloc[-1]*100:.2f}%")
    print(f"突发波动后 EWMA 动态波动率   : {df['EWMA'].iloc[-1]*100:.2f}%")
    assert df['EWMA'].iloc[-1] > df['EWMA'].iloc[70], "EWMA 波动率未能灵敏响应突发风险！"
    print("✅ 波动率量化度量与模型对比测试 PASS")
    print("======================================================")

if __name__ == '__main__':
    run_volatility_measurement_test()

运行结果输出：

================ 多维波动率度量响应对比 ================
突发波动后 Close-to-Close HV: 32.90%
突发波动后 Parkinson 极值 HV : 35.69%
突发波动后 EWMA 动态波动率   : 27.51%
✅ 波动率量化度量与模型对比测试 PASS
======================================================

本章小结

1. 根据场景选择波动率模型：长期资产配置选用标准历史波动率，日内与高频风控优先选用帕金森极值与 EWMA。
2. 捕捉波动率聚集现象：EWMA 模型通过指数衰减权重实现对突发市场剧震的秒级敏锐捕捉。
3. 波动率是仓位规模的核心输入：基于波动率倒数进行动态头寸分配（Volatility Targeting），平滑组合净值波动。

📋 行动清单 (Action Checklist)

• 运行波动率对比代码，调整 EWMA 衰减因子 $\lambda$（0.90 vs 0.97）观察响应敏感度；
• 编写 Garman-Klass 波动率模型（结合开盘、收盘与高低价）；
• 预习下一节，掌握止损止盈模块与动态跟踪止损带的设计。

— 小象教研组