Take Profit & White Swans

INTRODUCTION

The probability distribution of SP500 daily return was calculated and posted in Part I. The calculations showed that the Efficient Market Hypothesis is a good approximation, i.e. most of the time the daily return of SPX index is a variable with close to zero expectation which equates day trading to gambling against a house. The house has the statistical advantage due to the trading fees. In Part II, it was shown that both for longs and shorts a positive expectation can be achieved using  "cut your losses early and let your winners run" approach. 

On algorithmic level this approach was formulated as follows:

1. Every trading day: if no position open SPX position at the close of the trading session;
2. Next day: if return < StopLoss than close the position.


When the only parameter in this algorithm (StopLoss) is optimized the algorithm is expected to outperform the total return of SP500 (see Part II). Here we modified the algorithm by introducing the TakeProfit variable.



ALGORITHM

1. Every trading day: if no position open SPX position at the close of the trading session;
2. Next day: if ( return < StopLoss )  { close the position}
                     else{
                             if ( return > TakeProfit )  {close the position}
                     }
Based on the results in Part II  the StopLoss variable wass set as follows:

 StopLoss = 1/4*StandardDeviation,

The TakeProfit variable was varied using the following formula:

 TakeProfit = x*StandardDeviation.

The calculated return averaged over years 2007 to 2017  is shown below.

Keep in mind that historical Standard Deviation of SPX index is about 1.25%.  The result means that taking profit before 1.25% is expected to decrease the long-term return of the algorithm almost 4 times for shorts and 2 times for longs. Moreover, taking profit before 3 SD  still significantly decreases the performance.

BLACK & WHITE SWANS

Let us simplify Swan definition to a session close with an abnormal daily return. For example, the normal distribution with the mean and the standard deviation of SPX index predicts 24 days (since 1950) with the daily return 3 SD (3.75% ) or more. There were 102 days when SP500 printed 3.75% or more. For a 10x leverage day trader, a 3 SD move against the position is a Black Swan by any means. The Black Swan side of the distribution is taken care of by "cut your losses early" part of the algorithm. At the same time "let your winners run" part accumulates White Swan events providing that the Take Profit parameter is large enough.

CONCLUSIONS

In algorithmic day trading, the take profit parameter has to be 3 or more standard deviations.

Taking profit early - less than 1 SD - results in day trading with a negative expectation.