# 標的資產的波動率，槓桿與衍生工具的波動率之間的關係

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.