Mean Reversion, Momentum, As Well As Volatility Term Construction

Everybody know that volatility depends on the measuring frequency: the criterion divergence of 5-minute returns is dissimilar from that of daily returns. To endure precise, if z is the log price, as well as then volatility, sampled at intervals of τ, is 

volatility(τ)=√(Var(z(t)-z(t-τ)))

where Var agency taking the variance over many sample times. If the prices actually follow a geometric random walk, as well as then Var(τ)≡Var((z(t)-z(t-τ)) ∝ τ, as well as the volatility just scales amongst the foursquare beginning of the sampling interval. This is why if nosotros mensurate daily returns, nosotros require to multiply the daily volatility by √252 to obtain the annualized volatility.

Traders too know that prices hit non actually follow a geometric random walk. If prices are hateful reverting, nosotros volition notice that they hit non wander away from their initial value every bit fast every bit a random walk. If prices are trending, they wander away faster. In general, nosotros tin terminate write

Var(τ)  ∝ τ^(2H)

where H is called the "Hurst exponent", as well as it is equal to 0.5 for a truthful geometric random walk, but volition endure less than 0.5 for hateful reverting prices, as well as greater than 0.5 for trending prices.

If nosotros annualize the volatility of a mean-reverting cost series, it volition terminate upwardly having a lower annualized volatility than that of a geometric random walk, fifty-fifty if both accept precisely the same volatility measured at, say, 5-min bars. The reverse is truthful for a trending cost series.  For example, if nosotros attempt this on AUDCAD, an plainly mean-reverting fourth dimension series, nosotros volition larn H=0.43.

All of the higher upwardly are well-known to many traders, as well as are inwards fact discussed inwards my book. But what is to a greater extent than interesting is that the Hurst exponent itself tin terminate alter at some fourth dimension scale, as well as this alter sometimes signals a shift from a hateful reversion to a momentum regime, or vice versa. To come across this, let's plot volatility (or to a greater extent than conveniently, variance) every bit a exercise of τ. This is oft called the term construction of (realized) volatility. 

Start amongst the familiar SPY. nosotros tin terminate compute the intraday returns using midprices from 1 minutes to 2^10 minutes ( 17 hrs), as well as plot the log(Var(τ)) against log(τ). The fit, shown below,  is excellent. (Click figure to enlarge). The slope, divided past times 2, is the Hurst exponent, which turns out to be 0.494±0.003, which is really slightly mean-reverting.

But if nosotros hit the same for daily returns of SPY, for intervals of 1 twenty-four hours upwardly to 2^8 (=256) days, nosotros notice that H is now 0.469±0.007, which is significantly mean reverting. 

Conclusion: hateful reversion strategies on SPY should run amend interday than intraday.

We tin terminate hit the same analysis for USO (the WTI unsmooth fossil oil futures ETF). The intraday H is 0.515±0.001, indicating pregnant trending behavior. The daily H is 0.56±0.02, fifty-fifty to a greater extent than significantly trending. So momentum strategies should run for unsmooth fossil oil futures at whatsoever reasonable fourth dimension scales.

Let's plough at nowadays to GLD, the golden ETF. Intraday H=0.505±0.002, which is slightly trending. But daily H=0.469±0.007: significantly hateful reverting! Momentum strategies on golden may run intraday, but hateful reversion strategies for certain run amend over multiple days. Where does the transition occur? We tin terminate examine the term construction closely:

We tin terminate come across that at or thence 16-32 days, the volatilities depart from immediately business extrapolated from intraday frequencies. That's where nosotros should switch from momentum to hateful reversion strategies.

One side banking concern complaint of interest: when nosotros compute the variance of returns over periods that straddle ii trading days as well as plot them every bit exercise of log(τ), should τ include the hours when the marketplace was closed? It turns out that the answer is yes, but non completely.  In fellowship to hit the nautical chart higher upwardly where the daily variances initially autumn on the same immediately business every bit the intraday variances, nosotros accept to count 1 trading twenty-four hours every bit equivalent to 10 trading hours. Not 6.5 (for the the States equities/ETF markets), as well as non 24. The precise position out of equivalent trading hours, of course, varies across dissimilar instruments.

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Industry Update

• Nick over at mintegration.eu discusses the novel intraday databases at Quandl as well as Kerf.
• Factorwave.com (Euan Sinclair's creation) started a novel forum: slack.factorwave.com. It has some really active as well as in-depth discussions of many trading as well as investing topics.
• Prof. Matthew Lyle at Kellogg School of Management has a novel newspaper out that relates fundamentals to variance guide chances premiums: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2696183.

Upcoming Workshops

• April 20-21: Mean Reversion Strategies. 

There are a lot to a greater extent than to hateful reversion strategies than just pairs trading. Find out how to thrive inwards the electrical flow depression volatility environs favorable to this type of strategies.