Log normal returns of financial instruments and lack of...

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jasonz
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Log normal returns of financial instruments and lack of...

Post by jasonz »

(...lack of symmetry of profitability of long vs short entries)

Hi all, I'm pleased to be a new member of the boards. So by way of saying hello I will give you some background on myself.

I guess I've been interested in trading systems since around 1993 when I worked for a Reuters subsidiary in Sydney developing a piece of software called "Reuters Terminal Graphics", and got to know our consultant futures trader, read his back issues of his trading magazines, and become generally fascinated in my early 20s.

Since then, I've gone on to study financial mathematics and work in investment banking software development hand in hand with traders in areas such as interest rate, equity and credit derivatives.

Whenever I've explored mechanically traded systems and seen their performance statistics in my copy of tradestation, one thing that always bothered me was lack of symmetry of the profitability of long versus short entries. My financial mathematics study of the last few years struck me yesterday, and it gave me an idea about it that I wanted to share with you.

Now these biases seem to show up across all sorts of systems and timeframes.

Now it occurred to me that this would be expected if log-returns are assumed to be (roughly) normally distributed, as opposed to the simple returns being normally distributed.

Another consideration in regards this is the treatment of back adjusted contracts. I have for many years thought of back-adjustment through addition as wholely satisfactory. It occurs to me however that one loses the "return" (as a % basis) by simple arithmetic adjustment of prior contracts.

All this is bubbling in my head as I am considering, given my background, the use of options or log returns for measurement of volatility, and attempting to determine whether certain variances in smile possibly could be leading indicators of direction, or imply future volatility in its own right.

My last thought on these matters is whether daily calibration of stochastic volatility models such as GARCH could possibly be used to create predictions of volatility.

Volatility, as I describe here, would be a unit measurement for describing where either stops should be placed or entries be placed, or for weighting investments, and further providing information regarding re-weighting of contracts, etc. It is also useful, of course, for measuring relative value particularly in selling options straddles which seem to be profitable systematic approaches ( www.oxeye.co.uk )

These are just some ideas I am interested to explore. I am wondering whether anyone else has considered any of these thoughts.

The specific issue of log returns, there appears to be a "jensen's inequality" around the mean which would explain how a trend caught 50% of the way up would be of greater value than a trend caught 50% of the way down - which would be entirely explanatory of the reason why long entries generate greater profits than short entries.

Say that the mean price of a stock as 50. It would oscillate by going up exp(25%), then retrace to exp(-25%) below. If you only ever catch the 2nd part of the move, on the downward part the monetary value of each downward movement is always diminishing. Conversely the upwards movement is always implicitly expanding in value. This would be entirely expected in financial theory. (As imperfect as it is with kurtotic, skewed distributions with stochastic volatility.)

This was the specific point I really wanted to raise for discussion as I don't think anyone on this message board is making mention of the weaknesses that appear to be there in dealing in non-log-returns, and whether in fact we may create far better resolution of trading system behaviour and profitability by doing so.

Maybe instead we should be looking to take what we can from modern financial mathematics and build statistical systems that measure return, excursion, etc on a log return basis, at least internally. The question is how one models slippage and commission.

We seem to get there in a round-about fashion in many of the traditional statistics of systematic trading system building, but I have an intuition that things would be far clearer if we moved to logarithmic measurement. If one looks at "r-multiples", even there log-r-multiples would be more statistically interesting, as the impact of the r-multiple loss is non-linear on a portfolio.

That is to say, if one r-multiple is 10%, and two r-multiples are 20%, then a loss of 10% needs a (100/90-1)% improvement, but a loss of 20% needs a (100/80-1)% recovery.

If r-multiples were statistics upon log-adjusted returns, then one would know they needed a 1R% recovery or 2R% recovery.

excetera. I just provide this for discussion, I hope these might stimulate some ideas or discussion.

I really found the sluggo's postings I've been reading this weekend on random entry very interesting, thanks for all the good posts that I've read on here. I hope this can add something, and that I haven't lost everyone! :)

kind regards
jasonz
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Joined: Sat Apr 14, 2007 10:37 am

Post by jasonz »

A simpler way of expressing it is to say that one leverages up as an asset price increases, and one leverages down as it decreases.

Implicitly this is due to traversal being approximately a log-normally distributed process, instead of a normally distributed process, however to talk of it in simpler terms of leverage make it simpler.

It is also the case that a stop loss of a short position has implicitly a greater risk than a stop loss of a long position under this assumption of log normal distribution of returns.

These reasons give a mathematical insight as to why shorts are less profitable than longs. Another aspect of it is that a long price movement is unconstrained in dollar terms, whereas a short trend is constrained to approaching 0. This raises the spectre that short trading systems are in fact intrinsically different to long trading systems - that they are not one and the same at all, that using the same parameters may be suboptimal particularly if the stops are a large percentage of the total contract valuation.

Lastly there will be inflation effects, as well as skewness of price movements (short trends being supposedly more violent in stock markets for instance), which may account for long versus short profit differences, however, there is a simpler issue regarding the inequality of exp(price_increase)-1 > 1-exp (price_decrease) if price_increase == price_decrease.

I would appreciate any comments on these thoughts, they have just been rattling around my head this weekend...
jankiraly
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Location: San Diego

Post by jankiraly »

Try to avoid the mental constraint of only thinking about one system trading one stock. Real hedge funds / CTAs / professional traders have multiple systems trading large portfolios of stocks (or futures, or forex pairs, or options, or ...), all at once.

These traders have many simultaneous positions. The simplistic analysis of Geometric Average Trade and Terminal Wealth Relative and all the rest of the Ralph Vince / optimal f songbook, which begins with the one-system-one-stock assumption that trades are neatly arranged single-file in sequential order, is woefully inadequate. It simply doesn't apply.

Luckily, we no longer have to plug trade outcomes into crude mathematical simplifications like optimal-f, or "expectancy", or "The System Quality Number". We now have accurate software trading simulators which correctly handle the intricacies of multiple simultaneous systems having simultaneous positions in numerous different instruments. Instead of building approximations, we can just simulate the full and complete situation. It's easier, it's more accurate, and (today) it's fast as blazes.
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