What is Serial Autocorrelation and Does it Have a Role in Derivatives?

Short answer

Serial autocorrelation (SAC) is a temporal correlation between a time series and itself, meaning that a move in, say, a stock price one day is not independent of the stock move on a previous day. Usually in quantitative finance we assume that there is no such memory, that's what Markov means. We can measure, and model, such serial autocorrelation with different 'lags.' We can look at the SAC with a one-day lag, this would be the correlation between moves one day and the day before, or with a two-day lag, that would be the correlation between moves and moves two days previously, etc.

Example

Figure 2.12 shows the 252-day rolling SAC, with a lag of one day, 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 that many people instinctively felt this!)

Long answer

Very, very few people have published on the subject of serial autocorrelation and derivatives pricing and hedging. Being a specialist in doing things that are important rather than doing what everyone else does, I am obviously one of those few!

Positive SAC is rather like trend following, negative SAC is rather like profit taking. (I use 'rather like' because technically speaking trending is, in stochastic differential equation 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

Figure 2.12: The 252-day rolling SAC, with a lag of one day, for the Dow Jones Industrial index.

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 yj — p2, where p 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!

References and Further Reading

Bouchaud, J-P, Potters, M & Cornalba, L 2002 Option pricing and hedging with temporal correlations. International Journal of Theoretical and Applied Finance 5 1-14

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

What is Dispersion Trading?

Short answer

Dispersion trading is a strategy involving the selling of options on an index against buying a basket of options on individual stocks. Such a strategy is a play on the behaviour of correlations during normal markets and during large market moves. If the individual assets returns are widely dispersed then there may be little movement in the index, but a large movement in the individual assets. This would result in a large payoff on the individual asset options but little to payback on the short index option.

Example

You have bought straddles on constituents of the SP500 index, and you have sold a straddle on the index itself. On most days you don't make much of a profit or loss on this position, gains/losses on the equities balance losses/gains on the index. But one day half of your equities rise dramatically, and one half fall, with there being little resulting move in the index. On this day you make money on the equity options from the gammas, and also make money on the short index option because of time decay. That was a day on which the individual stocks were nicely dispersed.

Long answer

The volatility on an index, ai, can be approximated by

where there are N constituent stocks, with volatilities a, weight Wi by value and correlations pj. (I say 'approximate' because technically we are dealing with a sum of lognormals which is not lognormal, but this approximation is fine.)

If you know the implied volatilities for the individual stocks and for the index option then you can back out an implied correlation, amounting to an 'average' across all stocks:

Dispersion trading can be interpreted as a view on this implied correlation versus one's own forecast of where this correlation ought to be, perhaps based on historical analysis.

The competing effects in a dispersion trade are

• gamma profits versus time decay on each of the long equity options

• gamma losses versus time decay (the latter a source of profit) on the short index options

• the amount of correlation across the individual equities.

In the example above we had half of the equities increasing in value, and half decreasing. If they each moved more than their respective implied volatilities would suggest then each would make a profit. For each stock this profit would depend on the option s gamma and the implied volatility, and would be parabolic in the stock move. The index would hardly move and the profit there would also be related to the index option's gamma. Such a scenario would amount to there being an average correlation of zero and the index volatility being very small.

But if all stocks were to move in the same direction the profit from the individual stock options would be the same but this profit would be swamped by the gamma loss on the index options. This corresponds to a correlation of one across all stocks and a large index volatility.

Why might dispersion trading be successful?

• Dynamics of markets are more complex than can be captured by the simplistic concept of correlation.

• Index options might be expensive because of large demand, therefore good to sell.

• You can choose to buy options on equities that are predisposed to a high degree of dispersion. For example, focus on stocks which move dramatically in different directions during times of stress. This may be because they are in different sectors, or because they compete with one another, or because there may be merger possibilities.

• Not all of the index constituents need to be bought. You can choose to buy the cheaper equity options in terms of volatility.

Why might dispersion trading be unsuccessful?

• It is too detailed a strategy to cope with large numbers of contracts with bid-offer spreads.

• You should delta hedge the positions which could be costly.

• You must be careful of downside during market crashes.

References and Further Reading

Grace, D & Van der Klink, R 2005 Dispersion Trading Project. Technical Report, Ecole Polytechnique Federale de Lausanne

Found a mistake? Please highlight the word and press Shift + Enter