Probability Distribution of The Daily Return

      “Number rules the universe.”― Pythagoras

 Introduction

A reader will find out that Statistical Mechanics of Day Trading doesn't use Price Patterns and Technical Analysis as a way to describe the price behavior. A human being in a natural environment is rewarded for pattern recognition skill as it helps to discover the relationship between cause and effect. Hence the psychological expectation that a price pattern (cause) - price action (effect) relationship does exist. This expectation is a trap, many traders look for the Holy Grail in the realm of TA with a great persistence but nearly always end up in fear and distress. These emotions - Fear of Missing Out (FOMO), Fear of Uncertainty and Destruction (FUD) -  can be avoided altogether when instead of the psychological expectation trading is based on the mathematical expectation. The probability distribution of the daily return of SP500 is presented in this part of Statistical Mechanics of Day Trading.  It was shown that day trading of SP500 is similar to a coin toss which slightly favors betting from the long side. 

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 17,362. 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. 

Highly liquid SP500 market behaves very much in an agreement with the efficient market hypothesis. Highly liquid SP500 market behaves very much in an agreement with the efficient market hypothesis. 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 next part, a generic methodology of algorithmic day trading for a retail investor 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.