The relationship between volatility of underlying asset, leverage and the volatility of the derivative

Question

If I want to lower the risk of the portfolio then the trivial thing to do is change from higher volatility to lower for a better Sharpe ratio. It already lists the volatility for the stocks but the volatility for option type "bull certificate" is not listed. I don't expect it to be as simple as the leverage multiplied with the underlying asset volatility. But given the volatility of the underlying asset and the leverage then should I be able to find the leverage of the derivate ("bull certificate").

For example with 5 times leverage bullish there is a certificate https://www.morganstanley.com/ied/etp-server/webapp/svc/document/finalTermsheetVersion?isin=GB00BG5W8D15&version=1

But it is probably a gross oversimplication that the volatility of the derivate is 5 times the volatility of the underlying asset just because the leverage is 5 times.

The volatility of the underlying stock is today listed in different information sources, and that number is also different possibly because the different measures have used different time windows.

Is there a formula to use in this case, assuming that I am given the volatility of the underlying asset and given the leverage of the derivate in this asset class?

Answer 1

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.