By Sergei Belov, Ernest Chan, Nahid Jetha, and Akshay Nautiyal ABSTRACT We applied Corrective AI (Chan, 2022) to a trading model that takes advantage of the intraday seasonality of forex returns. Breedon and Ranaldo (2012) observed that foreign currencies depreciate vs. the US dollar during their local working hours and appreciate during the local working hours of the US dollar. We first backtested the results of Breedon and Ranaldo on recent EURUSD data from September 2021 to January 2023 and then applied Corrective AI to this trading strategy to achieve a significant increase in performance. Breedon and Ranaldo (2012) described a trading strategy that shorted EURUSD during European working hours (3 AM ET to 9 AM ET, where ET denotes the local time in New York, accounting for daylight savings) and bought EURUSD during US working hours (11 AM ET to 3 PM ET). The rationale is that large-scale institutional buying of the US dollar takes place during European working hours to pa

Everybody know that volatility depends

on the measurement frequency: the standard deviation of 5-minute returns is

different from that of daily returns. To be precise, if z is the log price,

then volatility, sampled at intervals of τ, is

volatility(τ)=√(Var(z(t)-z(t-τ)))

where Var means taking the variance over many sample times. If the prices really

follow a geometric random walk, then Var(τ)≡Var((z(t)-z(t-τ)) ∝ τ, and the volatility simply scales with the square root of

the sampling interval. This is why if we measure daily returns, we need to

multiply the daily volatility by √252 to obtain the annualized volatility.

Traders also know that prices do not really follow a geometric random walk. If

prices are mean reverting, we will find that they do not wander away from their

initial value as fast as a random walk. If prices are trending, they wander

away

*faster*. In general,

we can write

Var(τ) ∝ τ^(2H)

where H is called the "Hurst exponent", and it is equal to 0.5 for a

true geometric random walk, but will be less than 0.5 for mean reverting

prices, and greater than 0.5 for trending prices.

If we annualize the volatility of a mean-reverting price series, it will end up

having a lower annualized volatility than that of a geometric random walk, even

if both have exactly the same volatility measured at, say, 5-min bars. The

opposite is true for a trending price series. For example, if we try this

on AUDCAD, an obviously mean-reverting time series, we will get H=0.43.

All of the above are well-known to many traders, and are in fact discussed in

my book. But what is more interesting is

that the Hurst exponent itself can change at some time scale, and this change sometimes signals a shift from a mean reversion to a momentum regime, or vice

versa. To see this, let's plot volatility (or more conveniently, variance) as a

function of τ. This is often called the term structure of (realized)

volatility.

Start with the familiar SPY. we can compute the intraday returns using

midprices from 1 minutes to 2^10 minutes (~17 hrs), and plot the

log(Var(τ)) against log(τ). The fit, shown below, is excellent. (Click

figure to enlarge). The slope, divided by 2, is the Hurst exponent, which turns

out to be 0.494±0.003, which is very slightly mean-reverting.

But if we do the same for daily returns of

SPY, for intervals of 1 day up to 2^8 (=256) days, we find that H is

now 0.469±0.007, which is

*significantly*mean reverting.

Conclusion: mean reversion

strategies on SPY should work better interday than intraday.

We can do the same analysis for USO (the WTI crude oil futures ETF). The

intraday H is 0.515±0.001, indicating significant trending behavior. The

daily H is 0.56±0.02, even more significantly trending. So momentum

strategies should work for crude oil futures at any reasonable time scales.

Let's

turn now to GLD, the gold ETF. Intraday H=0.505±0.002, which is slightly

trending. But daily H=0.469±0.007: significantly mean reverting! Momentum

strategies on gold may work intraday, but mean reversion strategies certainly

work better over multiple days. Where does the transition occur? We can examine

the term structure closely:

We can see that at around 16-32 days, the volatilities depart from

straight line extrapolated from intraday frequencies. That's where we should switch from momentum to mean reversion strategies.

One side note of interest: when we compute

the variance of returns over periods that straddle two trading days and plot them as function of log(τ), should τ

include the hours when the market was closed? It turns out that the answer is

yes, but not completely. In order to produce the chart above where the

daily variances initially fall on the same straight line as the intraday

variances, we have to count 1 trading day as equivalent to 10 trading hours.

Not 6.5 (for the US equities/ETF markets), and not 24. The precise number of equivalent trading hours, of course, varies across different instruments.

===

**Industry Update**

- Nick over

at mintegration.eu discusses the new

intraday databases at Quandl and Kerf. - Factorwave.com

(Euan Sinclair's creation) started a new forum: slack.factorwave.com.

It has some very active and in-depth discussions of many trading and

investing topics. - Prof. Matthew

Lyle at Kellogg School of Management has a new paper out that relates fundamentals to variance risk

premiums: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2696183.

**Upcoming Workshops**

- April 20-21:

Mean Reversion Strategies.

There are a lot more to mean reversion strategies than just pairs

trading. Find out how to thrive in the current low volatility environment favorable to

this type of strategies.

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