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Friday, August 17, 2018

Statistical Mechanics of Algorithmic Day Trading SP500, Part I : Probability Distribution of The Daily Return

      “Number rules the universe.”― Pythagoras


SP500 market is characterized by both deep liquidity and high daily volumes. The popular instruments for trading SP500 index are SPY, SPX and Emini (/ES) futures.   At current prices of SP500 a daily move in Emini futures can easily exceed $1000 while, for example, ThinkOrSwim trading platform requires only ~ $6000 as the initial margin. Consequently, a SP500 daytrader is attracted by the idea to get rich fast like a moth to a flame; more than 90% of retail traders end up losing money. It is reasonable to ask thenWhy do people trade at all?- and is it possible to make money by day trading SP500. The answer to the first question is not an easy one as it involves many facets of our life. I believe that to a large degree it is related to being in a poor financial situation.

To answer the second question I used the statistics of one regular trading session of SP500 index. The probability distribution of the daily return of SP500 was presented in the first part of this publication. It was shown that day trading of SP500 is similar to a coin toss with even payout and with a small favor for betting from the long side. In the second part, a generic methodology of algorithmic day trading for a retail investor will be derived based on the calculated statistics. 


PART I : Introduction

Price Patterns and/or Technical Indicators (TA) are usually proposed as a way to describe the price behavior of SP500 or any other market for that matter.  TA is an attractive framework for day trading because a human been is naturally a pattern recognition system. Many retail traders look for the Holy Grail in the realm of TA with a great persistence but nearly always end up with a mediocre result at best. No wonder that FOMO-FUD dichotomy is a typical state of a TA trader. These emotions - Fear of Missing Out (FOMO), Fear of Uncertainty and Doubt (FUD) can be avoided altogether when our expectation of a trade result is expressed in numbers. 

Probability Distribution of the daily return of SP500 index.

SP500 historical data were taken from finance.yahoo.com. Since Jan. 3, 1950 till Jul. 13,2018 there were 17,254 trading days. To obtain the probability distribution (the probability density of a daily return, p(r)), the histogram of the daily returns in % was calculated and normalized by dividing over 17,254. Figures 1 to 3 show the result. Figure 1 shows the major part of the distribution. Figures 2 and 3 show rare events - tails or black & white swans - that SP500 experienced since 1950.

Figure 1.

The maximum of p(r) is around 0.2% which reflects SP500's tendency to be in a  bull market.

 Figure 2.



Figure 3.


Probability P(r).

The integral of p(r) from minus infinity to r, P(r), gives the probability of having a daily return less than r. P(r) is shown in Figure 4.


In particular, the chance of a negative daily return is given by P(0) which is 0.47. Accordingly, the chance of a positive daily return is 0.53.
The integral of r*p(r) from r1  to r2 divided by ( P(r2)-P(r1)), R(r1,r2), gives the  daily return averaged over [r1,r2].
In particular, the average negative return is given by R(-inf.,0) which is -0.65%. The positive return is given by R(0,inf.) which is  0.64%.
 R(-inf.,inf.)=0.033% is the expected average return for day trading from the long side.

 PART I : Conclusion & Discussion

Day trading of SP500 can be viewed as a daily toss of the unfair coin -0.47 tails, 0.53 heads - with the even payout, 0.6%. This imaginary coin is slightly in favor of trading from the long side.  As of today, SP500 is at $2800 which translates into $46 of the expected daily win for trading Emini futures from the long side.  Bid-Ask differential and the cost of trading for a retail trader amounts to $31 per one Emini futures roundtrip; 250 trading days *$(46-31)/($2800*50)=2.7% per year before tax. Thus the expected return, 2.7%, is well below the rate of $ devaluation, 4.5%. On the other note, a black swan can ruin a multiyear profitable streak. 

In the second part,  a generic algorithm for day trading will be derived based on the calculated probability distribution. A reader will find out that the return of the proposed algorithmic trading is expected to beat $ dollar devaluation by a significant margin while profiting from the swan events.

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