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Quant Corner 数量分析 加中金融
权益类、大宗商品类 Equities/Commodities
Stocks’ uniqueness lies in their structure, as they are, in fact, a call option
股票的独特性在于它们的结构,因为它们实际上是标的公司资
on the underlying company’s asset (Merton, 1974). Due to that nature,
产的看涨期权(Merton,1974)。由于这种性质,股票受所谓 stocks are driven, by what is called “the leverage effect”. The leverage
的“杠杆效应”驱动。杠杆效应现象是由 Fischer Black 在 1976 effect phenomenon was developed by Fischer Black (1976), and
年提出的。它假设随着资产价格的下跌,公司变得更加杠杆化 assumes that as asset prices decline, companies become mechanically
(其债务/权益增加),这使得它们的违约概率更高,因此股 more leveraged (their debt/equity rises), which makes their default
价回报率与波动率之间呈负相关关系。 probability higher, hence negative correlation between stock price
return and volatility.
为了适应这种动态的股票市场,从业者转向恒定弹性方差
To accommodate this dynamic equity market practitioners turn to CEV
(CEV)模型。最初的模型是由 John Cox 于 1975 年开发的,在
(Constant Elasticity Variance) model. While the original model was
Linetsky和 Mendoza 2009年的工作之后得到了广泛的应用。该模 developed by John Cox in 1975, it became widely used after Linetsky
型显示了各种应用程序和附加功能,以适应各种动态变化,包 and Mendoza 2009 work that showed various applications and
括跳跃和不连续性等,并具有为多种资产定价、估计信用利差 additions to accommodate various dynamics (including jumps and
的能力。 discontinuity), and ability to price wide range of assets (including credit
spreads).
除了随机波动率模型外,还有局部波动率模型(Derman 等,
Other to stochastic volatility models there are local volatility models
Kani 1994,Dupire 1994)。局部波动率模型不是通过参数使动
(Derman et. Kani 1994, Dupire 1994). Local volatility model, instead of
态模型符合模型预期,而是从具有不同行权价和到期日的设定
using parameters to fit the dynamic to model, extract the volatility from
交易期权中提取波动率,然后从观察到的/流动性中计算隐含 set traded options with different strikes and maturities, after calculating
波动率后,采用插值法以达到平滑的波动率曲面。期权链流动 the implied (observed/liquid) volatilities, an interpolation is being
性越强,波动性就越能反映出合理的隐含波动率。 applied to allow smooth volatility surface. The more liquid our option
chain will be, the more our volatility will reflect the fair implied volatility.
前景展望—粗糙的波动率
Looking forward into the future — Rough Volatility
随着波动率建模的发展,研究人员和从业人员已经开始对观察 As volatility modeling evolved researchers and practitioners have
到的波动率和波动率动态行为提出一些问题。 started raising some questions with regards to the behavior of realized
volatility, and volatility dynamic.
主要问题是,研究人员会问自己:在波动率分析中,几何布朗
运动的假设是否成立。要了解为什么这个过程是他们研究的主 Mainly, researchers asked themselves whether the assumption of GBM
题,我们可能需要了解几何布朗运动的关键假设。 holds water in volatility analysis. To understand why this process was
the subject of their research we probably need to understand what the
几何布朗运动假设该过程没有记忆,即每个步骤中的噪声都与 key assumption of GBM is…
先前的步骤无关.即马尔可夫过程。因此,该噪声(或波动性) Geometric Brownian motion assumes that the process has no memory,
具有零自相关性。如果该假设成立,那么我们的过程将与时间 in a sense that the noise in each step is independent of the previous
的平方根成比例增长 steps (AKA — Markovian process), so theoretically the noise (or
volatility) has zero autocorrelation. If that assumption holds, our
= + √∆ process grows proportional to the square root of time
如我们所见,等式的第一部分是过程的漂移部分,而第二部分 = + √∆
是噪声/挥发性部分。
As we can see, the 1st part of the equation is the drift part of the
因此我们需要问:“波动过程中是否有记忆?” ,换句话说, process, while the 2nd part is the noise/volatility part.
如果波动率增量呈正/负相关,该怎么办? 几何布朗运动无法
So we need to ask ourselves “ what if volatility process does have
模型时间序列行为的这一特征,因为它假设该过程的内存为零。 memory?” (or in other words, what if the volatility increments are
因此我们需要转向另一种布朗运动—fBM,分数布朗运动。 positively/negatively correlated). That feature of timeseries behavior
cannot be modeled by GBM, as it assumes zero memory of the
fBM 是布朗运动的概括,但是它的行为是独特的,因为它允许
process, so we need to turn to another type of Brownian motion —
增量彼此独立。为了说明增量的自相关,我们转向赫斯特指数 fBM (or fractional Brownian motion)
(Mandelbrot 和 Van Ness,1968 年)。该指数基本上是(0,1)
范围内的数字,它描述了过程的自相关(均值回复)程度。 fBM is a generalization of Brownian motion, but it’s unique in its
behavior, as it allows increments to be not independent of each other.
让我们探索 H 的不同状态: To account for the autocorrelation of the increments we turn to the
Hurst Exponent (Mandelbrot & Van Ness in 1968). This exponent
H = 0.5-增量互不相关。这是布朗运动的特例 basically is a number in (0,1) range that describes the degree of
autocorrelation (mean-reversion) of the process.
H <0.5-增量呈负相关。时间序列的粗糙边缘。动态是均值回复。
Let’s explore the different states of H:
H> 0.5-增量呈正相关。时间序列的边缘更平滑。动态呈现趋势。
H = 0.5 — increments are uncorrelated of each other. A special case
of Brownian motion
H <0.5 — increments are negatively correlated. rougher edges of the
time series. dynamic is mean reverting
H>0.5 — increments are positively correlated. smoother edges of the
time series. dynamic is trending.
CCFA JOURNAL OF FINANCE June 2021
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