Forward Volatility And Event Jump Volatility
When we sell options over such an event, it can be helpful to calculate the jump in the underlying price that is being implied by the option prices. To find the implied jump, we compare the ATM implied volatilities of the front-month and second-month options. The assumption is that just before the event, most of the difference will due to the uncertainty because of the announcement. There may also be other reasons for the difference, but we hope the event is the dominant one.
First note that if the front-month implied volatility is lower than that of the second month, the event is not being projected to cause any movement. If the front month is higher than the second month, we need to first calculate the forward volatility, σ12, the volatility being implied from the first expiration at T1 until the second expiration at T2. This is
\[ \sigma_{12} = \sqrt{\frac{\sigma^2_2T_2 - \sigma_1^2T_1}{T_2-T_1}} \]
- \(\sigma_1\) is the implied volatility of the front month
- \(\sigma_2\) is the implied volatility of the second month
The volatility attributed to the event, \(\sigma_{E}\), is the difference of the front-month volatility and the forward volatility. \[ \sigma_{\rm E} = \sqrt{T_1(\sigma_1^2 - \sigma_{12}^2)} = \sqrt{(\sigma_1^2 - \sigma_{12}^2)\frac{T_1T_2}{T_2-T_1}} \] Now, we find that the expected absolute return (the jump) is \[ E(|R|) = \sqrt{\frac{2}{\pi}}\sigma_{\rm E} \] So we can find the expected jump from the front two implied volatilities and compare this to our estimate of the actual move in the underlying.
附录
日波动率
日波动率 = 收益率的平方求均值后开方
标准差定义为 \[ s = \sqrt{\frac{1}{N} \sum^{N}_{t=1}(R_t)^2} \] 假设样本足够大，日波动率\(\sigma =s\)
日收益率
每日绝对收益率(去除方向)的均值表现 \[ E(|R_t|) = \frac{1}{N}\sum^{N}_{t=1}|R_t| \] 前者是统计学的表达方式，很抽象，无法与现实的标的涨跌或者日收益率直接关联起来，后者则可以直接观察，日常的标的涨跌就能简单地换算成心中(期望)的绝对收益率。至于将日收益率与日波动率关联起来的公式 \[ E(|R_t|) = \sqrt{\frac{2}{\pi}}\sigma \] 书中并没有提供相关的推导，我也困惑了一段时间，受到另一位知乎作者的启发，终于搞清楚了其中关联起来的数学逻辑，有了深刻的理解才能帮助更好地去运用它，这里也做分享。
正态分布
若随机变量\(X\)服从一个位置参数为\(\mu\)、尺度参数为\(\sigma\)的正态分布，记为： \[ X \sim N(\mu, \sigma^2) \] 则其概率密度函数为 \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}} \] 正态分布的数学期望值或期望值\(\mu\)等于位置参数，决定了分布的位置；其方差\(\sigma^2\)的开平方或标准差\(\sigma\)等于尺度参数，决定了分布的幅度。
根据delta-hedging操作中日终不留敞口的逻辑，假设均值\(\mu=0\)，正态分布简化为 \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{x^2}{2 \sigma^2}} \]
转换推导
基础假设对数收益率符合正态分布，并且均值为0，这样绝对收益率的均值等于每项收益率乘以其分布概率，具体推导如下：
\[\begin{equation} \begin{aligned} E(|R_t|) &= \int_{-\infty}^{+\infty} \left|x\right| \times \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{x^2}{2 \sigma^2}} {\rm d}x\\ &=\frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^{+\infty} \left|x\right|e^{-\frac{x^2}{2 \sigma^2}}\\ &=\frac{1}{\sigma\sqrt{2\pi}} \times 2\sigma^2\\ &=\sqrt{\frac{2}{\pi}}\sigma \end{aligned} \end{equation}\]
以上\(\sigma\)为日波动率，转化为年化波动率，于是 \[ E(|R_t|) = \frac{1}{\sqrt{252}} \sqrt{\frac{2}{\pi}} \sigma_{\rm 年化} \approx \frac{\sigma_{\rm 年化}}{19.90} \]
\[ E(|R_t|) = \frac{1}{\sqrt{243}} \sqrt{\frac{2}{\pi}} \sigma_{\rm 年化} \approx \frac{\sigma_{\rm 年化}}{19.54} \]
定积分
\[ \int_{-\infty}^{+\infty} \left|x\right|e^{-\frac{x^2}{2 \sigma^2}} = \begin{cases} 2 \sigma^{2} & \text{for}\: \frac{\pi}{2} > 2 \left|{\arg{\left (\sigma \right )}}\right| \\\int_{-\infty}^{\infty} e^{- \frac{x^{2}}{2 \sigma^{2}}} \left|{x}\right|\, dx & \text{otherwise} \end{cases} \]
1 | from sympy import Symbol, exp, pi, oo, Integral |