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Assume we have two instruments A and B. Also time is increasing from 1 to n. Let's say that A1 is price of instrument A at time 1. Let's assume that A and B are highly correlated instruments. Then we can try to compute TruePrice of stock B from stock A.

Then if Bn is more enough than TruePrice we sell B, and if Bn is less enough than TruePrice we buy B. I think that would be so-called statistical arbitrage.

I would prefer not to discuss pronse and cons of this scheme in general. Cause it proved to work cause I'm using it for a long time and still profitable.

What I really want to discuss is how can we compute TruePrice having what we have. Let me describe two algorithms:

  1. TruePrice = Bn * ( (A1 + A2 + .... + An) / (B1 + B2 + .. + Bn) )
  2. TruePrice = Bn / n * (A1 / B1 + A2 / B2 + .... An / Bn)

Also please note that Ai and Bi are measurements of price. This could be median low close or anything else of the certain interval (i'm using median now). The last items An and Bn are life i.e. changing while trading to reflect current situation.

The question is what are prons and cons of each of these (1 and 2) algorithms? Probably you can suggest something else?

Now I'm using 1 but I'm not satisfied with it. When instrument A grow (so An increase, other Ai are fixed as they in "past") TruePrice is not changing enough. And you can see from formula that TruePrice of stock B depends a lot on Bn what is actually current B price. So TruePrice of stock B depends on stock B too much what is not good. I think TruePrice of stock B should more depends on correlated An, but it doesn't.

I'm not sure if 2 would solve this problem.

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1) I assume you're using beta and not correlation. 2) This is not statistical arbitrage. 3) The best way to compare two strategies is to back-test them over a long time period. 4) There needs to be hedging or some sort of neutralization when trading security B to prevent an inadvertent directional bet; usually this is done by taking the opposite direction on A. 5) It generally helps to take into account transaction costs when deciding a good value to trade at. –  chrisaycock Jun 30 '12 at 1:55
    
@chrisaycock I want to know the difference between 1 and 2 from mathematical point of view. What are prons and cons of these calculations? What behavior should I expect from TruePrice when these algorithms are used, or they are pretty similar? –  javapowered Jun 30 '12 at 17:26
    
@chrisaycock - I don't necessarily agree with OP's strategy, but based on the Wikipedia link: isn't the OP indeed relying on a "statistical mispricing" of an instrument, betting on its move toward an "expected value"? Do you say it is not statistical arbitrage by definition, or due to the lack of rigor in his claims? (I ask because I don't know; not to challenge.) –  Andrew Cheong Apr 25 '13 at 0:25
    
@acheong87 Stat arb involves a basket of assets. The trader determines an idealized portfolio based on an alpha model (a score of desirability for each asset) and a risk model (a list of risk-factor exposures for each asset). The OP could be describing pairs trading, though he doesn't indicate whether he takes the opposing position of security A. Inside the Black Box explains these concepts. –  chrisaycock Apr 25 '13 at 2:04
    
@chrisaycock - Ah, I see. Thanks for the explanation. –  Andrew Cheong Apr 25 '13 at 15:10
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1 Answer 1

up vote 2 down vote accepted

This is probably not a good question for Q.SE—despite its application in your strategy, you're (perhaps unaware) asking a general math question: the difference between the ratio of means and the mean of ratios. There are numerous results online that distinguish the difference, but I assume the terminology wasn't apparent to you at the time. I've come across some elementary books and sites that use one or the other though, seemingly arbitrarily and without thought, e.g. when smoothing the stochastic; so maybe it'll be useful to comment on it:

Your first formula, $$B_n\frac{\sum{A_i}}{\sum{B_i}} = B_n\frac{\frac{1}{n}\sum{A_i}}{\frac{1}{n}\sum{B_i}}$$ is the ratio of means.

Your second formula, $$B_n\frac{1}{n}\sum{\frac{A_i}{B_i}}$$ meanwhile, is the mean of ratios.

It's not really about advantage/disadvantage, but what you seek from an analytical perspective.

The first formula is not sensitive to how the series compare per point in time. For example, both $A'_i$, a wildly fluctuating series whose average is $\mu'$, and $A''_i$, a constant series that is always $\mu'$, would yield the same result.

The second formula, however, is sensitive to each point in time, and averages the various measurements of comparison, over time.

I'm skeptical about your method, to say the least, but as you insist, I'll leave that discussion out. Given your claims, though, I imagine the second formula, the mean of ratios, is what you're after, (1) because you appear to be calculating whether one instrument is an "overestimator" or "underestimator" of the other, and I imagine you'd like to calculate the average of those comparisons (estimations) at each point in time, not the comparison (estimation) of two averages across all points in time, and (2) because the mean of ratios is not sensitive to absolute price levels (only relative price levels per point in time), e.g. if the price jumped up 200% in the middle of your series and you were using the ratio of means, differences in the latter half of the series would dominate in effect over the former half (unless you're using log-normalized values).

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thanks, currently i've switched to "mean of ratios" and it seems it's much better. –  javapowered May 21 '13 at 5:58
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