## Is EUR/USD mean reverting?

Nassim Taleb in his book “Dynamic Hedging” provides an empirical rule to validate if the market has mean reversion tendency. It states that if volatility lowers if the longer time frame is used, then the mean reversion takes place.

### Variation ratio method

Variance ratio is computed by dividing the variance of returns estimated from longer intervals by the variance of returns estimated from shorter intervals, (for the same measurement period), and then normalizing this value to one by dividing it by the ratio of the longer interval to the shorter interval. A variance ratio that is greater than one suggests that the returns series is positively serially correlated or that the shorter interval returns trend within the duration of the longer interval. A variance ratio that is less than one suggests that the return series is negatively serially correlated or that the shorter interval returns tend toward mean reversion within the duration of the longer interval.

Symbol | Minute vol | Hour vol | 4 Hours vol | Day vol |
---|---|---|---|---|

EUR/USD | 12.28% | 11.71% | 12.02% | 12.14% |

USD/CHF | 12.87% | 11.64% | 11.61% | 10.75% |

EUR/CHF | 10.25% | 9.06% | 8.86% | 9.23% |

GBP/USD | 11.87% | 11.03% | 11.11% | 10.08% |

EUR/GBP | 11.03% | 9.54% | 9.80% | 9.52% |

So, we might conclude that USD/CHF and GBP/USD has some mean-reverting property during the day, others are mean-reverting only in hour time frame at best.

### Parkinson number

The Parkinson number, or High Low Range Volatility, developed by the physicist, Michael Parkinson, in 1980 aims to estimate the volatility of returns for geometric brownian motion using the high and low in any particular period. The Parkinson number will be called P:

We have and the close-to-close registered high and the registered low respectively in a time frame .

For currencies in 2010 Parkinsons numbers are below. They are annualized by .

Symbol | Minute P | Hour P | 4 Hours P | Day P |
---|---|---|---|---|

EUR/USD | 8.72% | 11.36% | 11.63% | 11.72% |

USD/CHF | 9.05% | 11.56% | 11.72% | 11.30% |

EUR/CHF | 7.04% | 8.95% | 8.99% | 8.96% |

GBP/USD | 8.52% | 10.89% | 10.99% | 10.80% |

EUR/GBP | 7.93% | 9.46% | 9.63% | 9.83% |

### Parkinson-to-volatility ratio

For Geometric Brownian Motion (GBM) Parkinson-to-volatility ratio should be:

For general trading strategies the market maker edge is strongest in case where the ratio is higher than 1.67. It is otherwise better to follow a trend.

Deviation from this value shows how much the process deviates from GBM. The table below confirms that USD/CHF doesn’t follow GBM. It looks quite convincing to follow a trend in USD/CHF and EUR/USD.

Month | Daily P | Daily vol | P/vol |
---|---|---|---|

1 | 0.5616% | 0.5151% | 1.0901 |

2 | 0.5858% | 0.5695% | 1.0287 |

3 | 0.5699% | 0.5249% | 1.0856 |

4 | 0.5343% | 0.5327% | 1.0030 |

5 | 0.7439% | 0.6426% | 1.1576 |

6 | 0.6677% | 0.4863% | 1.3730 |

7 | 0.6591% | 0.6454% | 1.0212 |

8 | 0.6715% | 0.6748% | 0.9951 |

9 | 0.6226% | 0.6223% | 1.0004 |

10 | 0.6658% | 0.6149% | 1.0827 |

11 | 0.6474% | 0.6312% | 1.0256 |

12 | 0.7060% | 0.6652% | 1.0613 |

Month | Daily P | Daily vol | P/vol |
---|---|---|---|

1 | 0.5081% | 0.4997% | 1.0168 |

2 | 0.6125% | 0.6356% | 0.9636 |

3 | 0.5419% | 0.5382% | 1.0069 |

4 | 0.5388% | 0.6079% | 0.8864 |

5 | 0.9768% | 1.0170% | 0.9604 |

6 | 0.6844% | 0.5936% | 1.1529 |

7 | 0.6858% | 0.5488% | 1.2497 |

8 | 0.6075% | 0.6783% | 0.8956 |

9 | 0.6099% | 0.6579% | 0.9270 |

10 | 0.6826% | 0.6818% | 1.0012 |

11 | 0.7392% | 0.7753% | 0.9534 |

12 | 0.6473% | 0.6461% | 1.0019 |

#### Readings

Dynamic Hedging: Managing Vanilla and Exotic Options by Nassim Nicholas Taleb

Thank you…the variance ratio method is crystal. How do you use the P or GK number (as Taleb says)to “derive the max and min in any given day” and “understand the distribution of stop losses”

KKJanuary 10, 2013 at 07:43

Honestly I tried to find some other bits of info to find how to do that but I failed. There is a closed form formula for geometric brownian motion but it is hard to find a reliable source. So I resorted to Monte Carlo methods, i.e. generate 1 mln of GBM paths and get estimations.

fxpaulJanuary 15, 2013 at 00:50