Estimate range from gArch model

[momop]

I am posting this with some hesitation because this may be a noob question. Hopefully forum members won't downvote. I have been playing with forex data trying to build a daily trend follow scalper. I came across gArch and it gives me a volatility estimate - but I don't know if there's a way to use that information to predict the trading day's range (Hi-Low). In my backtest, I have 1600 trading days and the strategy trades on 25% of days and if there's a way to estimate the range from volatility with a confidence interval, it would help select trend days better. Simply using EMA doesn't give good results.



Obviously, I am not a quant - so would greatly appreciate pointers.

[contango_and_cash]

Assuming you are f_garch(close)?



If you are, then you have the garch vol for close. Theoretically i suppose you could do it on open, high, low etc.



That would give you some "range".

[zach]

volatility in the forex spot market is goofy. i spent a lot of time trying to model it and got very frustrated. forex is unique in that you can see massive moves (if i remember right i saw a 10-sigma move on the eurodollar when i was trading it, which would've been around 2010-11) in very brief spans of time. a general rule of thumb in forex is on the majors you see around 100 pips of movement within a 24 hour period. GARCH is a good starting point, perhaps a blended GARCH as suggested by the above poster would be helpful. I spent a good amount of time on autocorrelation studies (Hurst exponent) with decent results as well.

[aickley]

That's a question that as been popping up at work so I will write this down.



What is quite well researched is the inverse problem, namely estimating vol from range (assuming the underlying follows a BM). By using OHLC one can construct much more precise estimators compared to the usual variance estimator from closing prices. This literature starts with Parkinson (1980) and a brief summary can be found in this note of Santander written by Colin Bennet or in his book.



Getting back to your question, assuming the underlying price follows a BM without drift its range can be easily calculated from the volatility (Feller (1951)). The mean and the variance of a range up to time T are

[img]/User%20Files/5020/Latex-Equation-12529.gif[/img]

while the density is given by

[img]/User%20Files/5020/Latex-Equation-12530.gif[/img]

where L(x) is the cdf of a Kolmogorov distribution. So given your daily vol estimate you can get the estimate for a range, but personally I don't think it will be any accurate.



One thing to keep in mind is that range is by definition based on extremes so it is probably quite difficult to forecast it exactly. A good experiment is to plug in your typical daily vol in the formulas above (in place of \sigma \sqrt{T}) to see how much information about range do you get when the model and the parameters are known perfectly.



I would guess that a more promising way might be to go to the intraday scale and check how range behaves there as well as how it scales when you increase the sampling period. Speaking of the dynamics, there will be definitely a seasonal component as well as quite a bit of autocorrelation. One can attempt forecasting tomorrow's range given the today's observations. Probably it's a good idea to do that at different sampling frequencies and then combine the results. The place to start is probably Martens, van Dijk (2007).



Feller (1951) https://projecteuclid.org/euclid.aoms/1177729589

Martens, van Dijk (2007) http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.487.4123

[aickley]

^^^ There were issues with posting links

[momop]

Thanks all for the response. Helpful pointers that I will investigate and report back.



@contango_and_cash: I was using High-Low returns for fGARCH estimation and predict tomorrow's volatility. Then I used the predicted volatility to estimate the actual range by linear fitting the previous 30 day's predicted vs actual_range values. This was no better than simply using a EMA on the daily ranges.



@zach: I will look into Hurst exponent. Are there any minimum number of data points over which Hurst estimate would work better ?



@aickley: I will spend more time to digest your ideas Brief look into Parkinson, I might have accidentally used the Hi-Low in a similar way - but will delve deeper. Thanks!

[aickley]

BTW, it worth mentioning the scaling law of range divided by the standard deviation (as a function of window size) is related to the Hurst exponent (R/S analysis). The theory is for stationary processes so one must be cautious.