# 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?

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This is definitely a valid (and possibly viable) strategy.

I think that your constraint of zero costs is a red herring and serves no useful purpose beside forcing you to take lopsided bets in the direction of the cheaper option. I would try instead to build a portfolio that has zero vega (hedged against overall moves in market-wide implied volatility) and zero delta (accomplished by overlaying a stock position). If possible may be able to combine additional options in your portfolio (i.e., more than two) to shield yourself against gamma exposure.

I am quite sure that other players in the market are doing this, so my guess is that this strategy, if well-diversified, could make good money before costs, but would be likely a non-trivial task to make it profitable after costs (as with most things in finance).

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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

You could sell a high realized volatility against a low implied all day and bust out in a month. Doing this as a inter-stock spread isn't going to make it much of a better trade. If you want to take advantage of realized vol vs. implied vol you need a model that describes the relationship between the two.

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Could you provide any example of such a model? – Lisa Ann Jul 3 '13 at 6:40