# FxPaul

Math in finance or vice versa

## 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.

Annualized volatilities of year 2010
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:
$P = \sqrt{\frac{1}{n}\frac{1}{4\log 2} \sum_{i=1}^n {\left[\log \frac{S^H_i}{S^L_i}\right]^2} }$
We have $S^H_i$ and $S^L_i$ the close-to-close registered high and the registered low respectively in a time frame $i$.

For currencies in 2010 Parkinsons numbers are below. They are annualized by $\sqrt{N}$.

Parkinson numbers of year 2010
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:
$\frac{P}{\sigma} = 1.67$

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.

USD/CHF Parkinson-to-volatility ratio
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
EUR/USD Parkinson-to-volatility ratio
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

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

Written by fxpaul

June 16, 2011 at 20:52