標的資產的波動率,槓桿與衍生工具的波動率之間的關係


0

如果我想降低投資組合的風險,那麼瑣碎的事情就是從較高的波動性變為較低的波動率以獲得更好的夏普比率。它已經列出了股票的波動率,但是未列出期權類型"牛證"的波動率。我不認為它會像槓桿乘以潛在資產波動那樣簡單。但是考慮到基礎資產的波動性和槓桿率,那麼我應該能夠找到衍生產品的槓桿率("牛證")。

例如,如果槓桿是5倍,則有證書https://www.morganstanley.com/ied/etp-server/webapp/svc/document/finalTermsheetVersion?isin=GB00BG5W8D15&version=1

但是,由於槓桿作用是5倍,衍生產品的波動率是標的資產波動率的5倍,這可能是過分簡單的估計。

標的股票的波動今天在不同的信息來源中列出,並且該數字也有所不同,可能是因為不同的度量方法使用了不同的時間窗口。

在這種情況下,是否有一個公式可以使用,假設我已經獲得了基礎資產的波動性,並且獲得了該資產類別中衍生產品的槓桿作用?

2

The key variable is indeed the derivative's elasticity $\Omega$ (aka leverage, Lambda). It is defined by $$\Omega=\frac{\frac{\partial V}{V}}{\frac{\partial S}{S}}=\frac{\partial V}{\partial S}\frac{S}{V}=\Delta\frac{S}{V}.$$

Here, $V$ corresponds to the value of the derivative and $S$ to its underlying.

This number measures how much riskier the derivative is compared to its underlying. Intuitively, it tells you by how much the derivative price changes in percent if the price of the underlying asset changes by one percent (ceteris paribus).

Importantly, volatility and market beta (in a CAPM sense) are linear in the elasticity. That is, $\sigma_V=|\Omega|\cdot\sigma_S$ and $\beta_V=\Omega\cdot \beta_S$. You need the absolute value for to ensure that the volatility is positive (puts, for example, have a negative elasticity). These equations are derived, for example, in Cox and Rubinstein (1985).

For example, a call option has an elasticity of greater than one (easy to see in a Black Scholes world: $C=S\Delta-Ke^{-rT}N(d_2)\leq S\Delta\implies1\leq\frac{S\Delta}{C}=\Omega_C$). Thus, a call option is riskier than the underlying asset. On the other hand, a put option (acting as an insurance) has a negative elasticity and thus has a lower systematic risk than its underlying.