# Serial Autocorrelation And Derivatives

The figure shows the 252-day rolling SAC for the Dow Jones Industrial index. It is clear from this that there has been a longstanding trend since the late 1970s going from extremely positive SAC to the current extremely negative SAC. (I imagine you’ve all noticed this lately!) Positive SAC is rather like trend following, negative SAC is rather like profit taking. (I use ‘rather like’ because technically speaking trending is, in s.d.e. terms, the function of the growth or dt term, whereas SAC is in the random term.) The current level has been seen before, in the early thirties, mid 1960s and late 1980s. (Note that what I have plotted here is a very simplistic SAC measure, being just a moving window and therefore with all the well-known faults. The analysis could be improved upon dramatically, but the consequences would not change.)

As far as pricing and hedging of derivatives is concerned there are three main points of interest (as I say, mentioned in very, very few quant books!).

1) The definition of ‘volatility’ is subtly different when there is SAC. The sequence +1, -1, +1, -1, +1, has perfect negative SAC and a volatility of zero! (The difference between volatility with and without SAC is a factor of SQR(1 – rho^2), where rho is the SAC coefficient.

2) If we can hedge continuously then we don’t care about the probability of the stock rising or falling and so we don’t really care about SAC! (A fun consequence of this is that options paying off SAC always have zero value theoretically.)

3) In practice, however, hedging must be done discretely. And this is where non-zero SAC becomes important. If you expect that a stock will oscillate up and down wildly from one day to the next, like the above +1, -1, +1, -1, example, then what you should do depends on whether you are long or short gamma. If gamma is positive then you trade to capture the extremes if you can. Whereas if you are short gamma then you can wait, because the stock will return to its current level and you will have gained time value. Of course this is very simplistic, and for short gamma positions requires nerves of steel!

That’s just a brief introduction to a much-ignored topic. I hope it will inspire some discussion!

P