You're probably familiar with implied volatility, which is a calculation derived from option prices. In theory it reflects the market's expectations for future volatility.

Implied correlation, however, is something different. It's the market's expectation for the *correlation* of a stock against the S&P 500 ETF SPY – not how volatile it will be..And we can calculate this statistic from a new set of options available on Apple (NASDAQ:AAPL).

In a previous post I explained how options on NASDAQ's new Alpha Indexes track the relative performance of a stock vs. the overall market – providing a way for investors to isolate the outperformance of stock against the SPY ETF.

For example, the Alpha Index options on Apple trade under the symbol AVSPY. Whether the market or the stock rises or falls, the AVSPY index will go up if Apple outperforms SPY and go down if the stock underperforms.

## Volatility between two assets: Expectations of correlation

Given that these options are for an index that represents the *relationship* between two assets, what should the volatility of these options be? Turns out It depends a lot on correlation.

Think of it this way. If Philadelphia gets a lot of rain this year, nearby Baltimore probably will too – and vice versa. You could say that the relationship between rain in Philadelphia and the rain in Baltimore wouldn't be all that volatile even if rainfall rates themselves are volatile. Why? Because the rainfall rates are likely to be highly correlated.

But what about the relationship between rain in Philadelphia and Seattle? They're not likely to be highly correlated, thus the relationship between these two cities in terms of rainfall could be very volatile.

## Defining Alpha Index volatility

With a normal option, volatility is one of the inputs you use to calculate the fair value of the option. Since we are not certain what volatility will actually be, the market price of the option defines an implied volatility.

But the volatility of an Alpha option is a lot more complicated. These Alpha Indexes were designed based on research by Professors Jacob Sagi and Robert Whaley at the Owen School of Management at Vanderbilt University.

According to their research paper (available in the Education section at NASDAQ's Alpha Index site), because there are two lognormal distributions to consider, the volatility of an Alpha Index option has to account for two volatilities and correlation between the stock and the market.

** **I modified the formula to eliminate the Greek characters, which makes the formula easier to understand. (Note: Other forms of outperformance indexes may have different equations)

## Implied correlation: Looking forward not back

So here's something interesting. What if we were to solve this equation for the correlation? If we know the price of the Alpha Index options, and the expected volatility for AAPL and SPY, then the price of an AVSPY implies the level of correlation we can expect between AAPL and SPY

In my view, this is an exciting development. Right now all we can measure is past correlation, but implied correlation is by definition *forward* looking. For example, here's a look at the historical correlation between AAPL and SPY.

I looked back over a three-month basis, but that's the problem with historical correlation. Everyone will use a different lookback period. Implied correlation, however, is determined by Alpha Index option prices.

So compare that historical correlation to this chart that shows *implied* correlation based on AVSPY options since they started trading on April 18 (shortly before Apple's last earnings release).

In my previous article on the topic I mentioned that AVSPY options would usually be expected to have a volatility somewhere between AAPL and SPY's. And so far, they've traded that way as you can see here.

Note that AVSPY options cost a lot more compared to AAPL options back in April than they do now. My interpretation? AAPL was not expected to be highly correlated with the market with news pending.

I think most of us expect that AAPL will have a reasonable level of correlation to the market, but what if it doesn't – especially as an earnings release looms?

Well then we can expect higher implied volatilities, and thus higher prices, for these AVSPY options. Here's a look at the theoretical volatility for an AVSPY option assuming the market has a volatility of 15% and AAPL has volatility of 25%.

Note that when the correlation is somewhere between 0.3 and 0.8, (the yellow area of the chart) AVSPY's volatility should be between SPY and AAPL. However, if correlation gets really high, AVSPY options get a lot less expensive. If the correlation is very low, then AVSPY options will get very expensive on a relative basis.

As I mentioned in my previous article, there's not a lot of liquidity in these options yet, but if they do become popular, I can see ways to use this information to take advantage of these options.

You could trade around correlation levels – buying AVSPY options when they're relatively less expensive and selling them when they're pricey. Note that last time an earnings report loomed, AVSPY options were almost as expensive as AAPL's, but then trended down.

Perhaps upcoming earnings announcements are a good time to sell AVSPY options – either as a hedge or directional bet.

Or if your investment strategy is based on Apple's outperformance or underperformance with the market over the long term, then these options can help you hedge your risk. Or you can simply use these options to speculate on AAPL's relative performance against SPY on a short-term basis.

But even if you never trade options, knowing what the market expects in terms of correlation on a stock like Apple could help you make more informed decisions.

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