# Beta vs. Implied Volatility statistical arbitrage using options

Let two underlyings, $S_{1}$ and $S_{2}$, are correlated and $\beta$ is the slope of their returns linear regression, that is, it says how much $S_{1}$ co-variates with $S_{2}$ variance.

For instance, let

$$\beta=\dfrac{\sigma_{S_{1}S_{2}}}{\sigma^{2}_{S_{2}}}=0.83$$

that is, when $S_{2}$ raises by $1\%$ $S_{1}$ goes up by $0.83\%$; in this example we can assume to know the true value of $\beta$, then no estimation error is present.

Now consider two Call options: the former, $c_{1}$, is written on $S_{1}$ and the latter, $c_{2}$, is written on $S_{2}$, and they have both the same moneyness (e.g. 102%).

According to BMS formula, the implied volatility, $v_{1}$, extrapolated from $c_{1}$ is greater than the one, $v_{2}$, extrapolated from $c_{2}$.

For instance, let

$$v_{1}-v_{2}=6\%$$

on annual basis.

$S_{1}$ and $S_{2}$ are strongly correlated and their linear regression $R^{2}$ is above $0.8\approx0.9$.

What about buying $S_{2}$ Gamma selling $S_{1}$ Gamma in order to get a zero cost position but having sold (bought) an implied volatility greater (smaller) than realized volatility?

• Actually I am trading this strategy. While the gross performance is more or less lined up with the expected one (that is, strong correlation and $\beta$ are persistent over time), net performance are very sensitive to transaction costs. $\beta$ are estimated with Gaussian Kalman filter, $R^{2}$ are extrapolated from the state-space fitting on the time series. – Lisa Ann Jul 3 '13 at 8:50