This article is a refinement of the theory backing the previous one (Can the Spread of the VIX Over a GARCH Model Predict Hedge Fund Returns). I was trying to clarify the logical steps to permit the regression to be meaningful --- i.e. to establish the causality of the link between the risk premium acquired by writing S&P 500 options and the profits made by traders.

We assert that hedge funds, by selling shares in the profitability of a trading strategy, are essentially writing option contracts which must be hedged by executing their trading strategy and the income they receive from their clients is the risk premium embedded in the spread between the option selling price and it's fair value (which is the value realized by the dynamic hedging strategy).

The question is how is that risk premium valued and what is the theoretical link to the VIX-GARCH spread? I will introduce an additional hypothesis, which is a more concrete argument than that made previously.

Each trading strategy is different, but there are commonalities in the general changes in the price of risk. So we can model the risk premium for any given strategy very much as we model the returns of common stocks. We represent it as a linear combination of an idiosyncratic risk premium and a systematic risk premium, with a "beta" to the systematic premium. Additionally, we will assume that the risk premium beta is likely to be positive and significantly different from zero.

In this framework, every strategies premium income is correlated with each other, just as every stock's returns are correlated with each others. It's also quite straightforward to see that the profitability of a put writing strategy is explicitly dependent on the spread between VIX and an empirically accurate model of the actual volatility of the S&P 500, which we model with a simple GARCH(1,1) variance process. Therefore, we expect the risk premium income of a hedge fund strategy to be positively correlated with the risk premium income of put writing, which the "risk beta" to be empirically established. This the the theoretical construct we need to make our regression a reasonable operation.

However, using this framework we can now reason that some particular strategies might have a stronger exposure to the systematic component of risk premia than others. So it makes sense to look at regressions between the hedge fund sector index returns and the VIX-GARCH spread. I will present the results of these regressions, as I do them, in future posts.