How to create a volatile market, by combining less volatile markets?

Question

This might be against the law of gravity, but I'll give a try 🙂

Is there a way to combine two financial products $p_1$ and $p_2$, into a single product $p_c$ that is more volatile than its components?

Mathematically, if the daily returns of the original products are:

$$r_1\sim \mathcal{N}(\mu,\,\sigma^{2})$$ $$r_2\sim \mathcal{N}(\mu,\,\sigma^{2})$$

Can I build a financial product whose daily returns are:

$$r_c\sim \mathcal{N}(\mu_c,\,\sigma_c^{2})$$ $$\sigma_c > \sigma$$

Careful:

• I am not looking to increase volatility of absolute returns. (e.g. returns in USD). That is easy, just use leverage. But with leverage come proportionally higher transaction costs. So you don't have an extra edge in trading.

• I am looking to increase volatility of relative returns (e.g. percentage returns). I want to obtain higher volatility, at stable transaction costs. That would be an extra hedge in trading.

Answer 1

Throw in correlation as the additional variable.

Similarly, volofvol could be another candidate to play with.

A spread option could have a larger volatility than either of the two underliers.

Answer 2

Let $X_1$ and $X_2$ be your two assets and $C$ your financial product. For now we only assume products which are a linear combination of $X_1$ and $X_2$ with no shorting allowed, hence: $$\begin{align} & C=\alpha X_1 + (1-\alpha)X_2 \\ & 0\leq \alpha \leq 1 \end{align}$$ Letting $\rho$ be the correlation between returns $r_1$ and $r_2$, we have: $$\begin{align} \sigma_c^2&=\alpha^2\sigma^2+(1-\alpha)^2\sigma^2+2\alpha(1-\alpha)\sigma^2\rho \\ &=(1-2\alpha+2\alpha^2)\sigma^2+2\alpha(1-\alpha)\sigma^2\rho \end{align}$$ You are asking under which conditions: $$\sigma_c^2>\sigma^2$$ Namely: $$(1-2\alpha+2\alpha^2)+2\alpha(1-\alpha)\rho>1$$ Rearranging and letting $\theta=2(1-\rho)\geq0$: $$\theta\alpha^2-\theta\alpha+1>1$$ Namely: $$\theta\alpha(\alpha-1)>0$$ but there is no value of $\alpha$ for which this inequality is enforced given our initial constraint $0\leq \alpha\leq1$. On the other hand, if you allow short-selling, namely $\alpha$ can take values lower than $0$ or higher than $1$, so some sort of leverage (where leverage in one asset is financed by shorting the other one), you observe that you manage to increase the volatility of the product $\sigma_c$ with respect to the underlying volatility $\sigma$.