Kalman Filters

[BlueHorseshoe 10]

There doesn't appear to be a thread anywhere on TL discussing Kalman Filters - obviously there should be!

 

Here's a link to an Ernie Chan post giving a description of the potential utility for Kalman Filters in linear regression models, and a more general post from another blog:

 

http://epchan.blogspot.co.uk/2011/04/many-facets-of-linear-regression.html

 

http://intelligenttradingtech.blogspot.co.uk/2010/05/kalman-filter-for-financial-time-series.html

 

Does anyone have any knowledge, experience, or thoughts on how to apply Kalman Filters?

 

 

Thanks

 

BlueHorseshoe

[Obsidian 10]

Hi BHS,

that looks interesting, I will have a look at it during the weekend. though I highly doubt that I have enough mathematical background

[SIUYA 11]

Off the top of the head - it will give similar results to exponential moving avgs, and as with all of these things, the mean that is either being reverted to, or exploding away from is the key issue in that it moves as well......

 

Might make for a good volatility filter maybe.....but I guess it all depends on what you are planning to do with and while I am not mathematically up to it.....my guess is that it will be much like your conclusion that a simple MA is still a great and simple way to measure trend.

(I hope I am wrong )

[BlueHorseshoe 10]

Off the top of the head - it will give similar results to exponential moving avgs, and as with all of these things, the mean that is either being reverted to, or exploding away from is the key issue in that it moves as well......

 

Hi SIUYA,

 

One of the attractions from my reading was that the KF calculation is recursive, and I'm always looking for ways to remove (easily curve-fitted) inputs such as MA length.

 

Unfortunately, upon further reading it would seem that the Kalman Filter also has an input. Rather than controlling a lookback length as with an MA, the input adjusts the sensitivity to 'noise'. So the benefit that I imagined existed in this respect isn't there.

 

Might make for a good volatility filter maybe.....but I guess it all depends on what you are planning to do with and while I am not mathematically up to it.....my guess is that it will be much like your conclusion that a simple MA is still a great and simple way to measure trend.

(I hope I am wrong )

 

My enquiries are for a slightly different reason . . . I'm fed up with hard stops and I'm looking for other ways to manage risk. As you know, I considered options. I'm now looking at the possibility of incorporating elements of stat arb, and Kalman Filters are one of the two common procedures used in calculating an instrument's beta. But that's something for another thread.

 

I have been able to piece together a Kalman Filter from the formulas and information I've found online - I'll share the code for this on this thread shortly.

 

I guess I was hoping that this thread might elicit input from someone with experience of pairs trading . . .

 

Cheers,

 

BlueHorseshoe

[SIUYA 11]

Unfortunately, upon further reading it would seem that the Kalman Filter also has an input. Rather than controlling a lookback length as with an MA, the input adjusts the sensitivity to 'noise'. So the benefit that I imagined existed in this respect isn't there.

 

 

I thought about trying to use something like a range bar that modified it self based on some volatility element.....something similar maybe. (beyond my pay grade) and use a visual best fit model.

maybe you just have to continually optimise on the last X days and modify using that?

 

However, isnt the point about using a ATR or similar filter the same in terms of then modifying quantities traded? Or are you looking to go where the stops are adjusted but the qty stays the same?

 

I guess I was hoping that this thread might elicit input from someone with experience of pairs trading . . .

 

pairs trading.....double the risk with half the profit.

[BlueHorseshoe 10]

Here is an attempt at EasyLanguage code for a Kalman Filter, based on what I have been able to find from wikipedia etc:

 

  
Inputs:
G(0.0001);

Variables:
X(0),
Y(0),
K(0);

If Currentbar>1 then begin
X=(K+((C-K)*(Squareroot(G*2))));
Y=(Y+(G*(C-K)));
K=X+Y;
End;

Plot1(K);  

 

If there's anyone who has knowledge of EL and Kalman Filters who can check this over then that would be greatly appreciated.

 

Cheers,

 

BlueHorseshoe

[onesmith 10]

Do you have the exact specification in a succinct format? Compare it to this formula which uses a more intuitive logic flow to return the same values as the formula you posted.

 

input: g(.0001);
var: k(c), dif(0), avg(0), vel(0), sqrt2g(squareroot(2*g));

if currentbar>1 then begin
   dif=c-k;
   avg=k+sqrt2g*dif;
   vel=vel+g*dif;
   k=avg+vel;
   plot1(k,"k");
end;

[BlueHorseshoe 10]

Do you have the exact specification in a succinct format? Compare it to this formula which uses a more intuitive logic flow to return the same values as the formula you posted.

 

input: g(.0001);
var: k(c), dif(0), avg(0), vel(0), sqrt2g(squareroot(2*g));

if currentbar>1 then begin
   dif=c-k;
   avg=k+sqrt2g*dif;
   vel=vel+g*dif;
   k=avg+vel;
   plot1(k,"k");
end;

 

Hi Onesmith,

 

Thanks for your reply.

 

Unfortunately the difficulties of finding any kind of consistency of mathematical notation between different resources, given my limited mathematical capabilities, means that I have do not have a succinct specification from which I am working. Hence I am not totally confident that my formula is based upon a correct interpretation.

 

Would there be any advantage to your arrangement of the formula in terms of processing etc?

 

Regards,

 

BlueHorseshoe

[onesmith 10]

The primary advantage is readability facilates understanding how it works. It's efficiencies such as declaring squareroot(2*g) only become significant if the concept is viable and you have a way to exploit it.

[BlueHorseshoe 10]

The primary advantage is readability facilates understanding how it works. It's efficiencies such as declaring squareroot(2*g) only become significant if the concept is viable and you have a way to exploit it.

 

The concept that I am exploring is the use of a cointegrated pair as a risk management tactic within an existing directional strategy. In other words, entries will be derived from the behaviour of a single instrument that has alpha, rather than from the spread between the two instruments as is normal in stat arb.

 

I have no way of knowing whether this will work until I am able to program it in at least an approximate way, but all the building blocks I require (cointegration, covariance, beta . . .) are completely new to me and understanding them is quite a chore.

 

Incidentally, I am led to understand that Kalman Filters are sometimes used in place of expected value or arithmetic mean within the covariance calculation we were discussing in another thread . . . hence my whole questioning in that thread may be completely moot.

 

It all takes so bloody long . . .

 

BlueHorseshoe

[Nicolas32 10]

BlueHorseshoe,

 

Your code here-above is supposed to calculate what?

The "optimal" ratio between the two price series?

Something different?

 

Kalman filter is a methodology to calculate "adaptative" things. There is not one unique formula. It depends on what we try to do.

 

Let's A and B be the 2 instruments.

 

In pairs trading, typically:

- cointegration is checked (Dickey-Fuller or other) on the long term (typically > 1 year).

- hedge ratio (let's call it gamma) is calculated on the in-sample data by linear regression

- then, on out-of-sample data, we enter "long A short B with appropriate position sizing" each time the spread A-gamma*B departs too much from its mean.

 

For this "basic" approach, Kalman filter is not really useful.

 

It may become useful if you want to calculate a shorter-term gamma, in order to have a more "dynamic" and short-term spread.

 

Then, we may consider these 2 equations

{ B[t] (observed) = gamma[t] (to be assessed by KF) * A[t] (observed) + noise (unknown)

{ gamma[t] = gamma[t-1] + noise (unknown)

 

We have the 2 typical equations (state equation and measurement equation) on which we can the KF methodology.

 

There is a research paper on intraday pairs trading which implements the above KF approach (as well as other methodologies) to assess a short-term gamma:

Dunis and al

Statistical Arbitrage and High-Frequency Data with an Application to Eurostoxx 50 Equities

March 2010

http://www.ljmu.ac.uk/Images_Everyone/Jozef_1st(1).pdf

 

What is above is only my understanding so... may be wrong!

 

Nicolas

 

 

 

 

 

Nicolas

[BlueHorseshoe 10]

BlueHorseshoe,

 

Your code here-above is supposed to calculate what?

The "optimal" ratio between the two price series?

Something different?

 

Kalman filter is a methodology to calculate "adaptative" things. There is not one unique formula. It depends on what we try to do.

 

Let's A and B be the 2 instruments.

 

In pairs trading, typically:

- cointegration is checked (Dickey-Fuller or other) on the long term (typically > 1 year).

- hedge ratio (let's call it gamma) is calculated on the in-sample data by linear regression

- then, on out-of-sample data, we enter "long A short B with appropriate position sizing" each time the spread A-gamma*B departs too much from its mean.

 

For this "basic" approach, Kalman filter is not really useful.

 

It may become useful if you want to calculate a shorter-term gamma, in order to have a more "dynamic" and short-term spread.

 

Then, we may consider these 2 equations

{ B[t] (observed) = gamma[t] (to be assessed by KF) * A[t] (observed) + noise (unknown)

{ gamma[t] = gamma[t-1] + noise (unknown)

 

We have the 2 typical equations (state equation and measurement equation) on which we can the KF methodology.

 

There is a research paper on intraday pairs trading which implements the above KF approach (as well as other methodologies) to assess a short-term gamma:

Dunis and al

Statistical Arbitrage and High-Frequency Data with an Application to Eurostoxx 50 Equities

March 2010

http://www.ljmu.ac.uk/Images_Everyone/Jozef_1st(1).pdf

 

What is above is only my understanding so... may be wrong!

 

Nicolas

 

 

 

 

 

Nicolas

 

Hi Nicolas,

 

I only just checked back on this thread and saw your response - very helpful, thanks!

 

BlueHorseshoe

[BlueHorseshoe 10]

Please note that the EL code given in the thread above is a square root formulation intended to increase the stability of the calculation. The change of decimal place in the gain factor is necessary for this purpose, but I have also found that it has the added advantage of increasing the granularity when the gain ratio is made to be dependent upon another value. Also, please note that the gain is an input in this code, and it is not therefore truly recursive. You'll find the formulation for calculating optimal gain on the wikipedia page for KF.

 

BlueHorseshoe