Applied stochastic models in business and industry. Stock timing using genetic algorithms, страница 6

^*  t_l the beginning date of a learning period,

^*  d_l the duration of this learning period,

^*  d_m the longest period entering the technical trading rules (for example, moving averages need 50 days before the time of computing), and ^* d_t is the length of the test period.

The variable t_l then has to verify:

d_m5t_l5T  d_t  d_l

In fact, we cannot start the learning period before d_m because we need at least d_m trading days to calculate some of the indicators; on the other hand, if we start after T  d_t  d_l; then the test period will end after T; which is not possible.

Adjustment of endowments. We consider here a specific expert e^k obtained for security k; at date t_l; it is assumed that e^k is endowed with a sure wealth w^k₀ and n^k₀ stocks. Let p₀^k be the initial opening price of the stock; the total wealth of the expert at the beginning of the process is then

W₀k ¼ wk₀ þ nk₀p₀k

At the end of the first date, the closing price P₁^k is observed; as mentioned earlier, three decisions are possible, buying, selling or doing nothing. Assume for example that the signal is ‘sell’. If the expert sells all his stocks, he then possesses only cash. Consequently, he cannot continue to sell on the next day because he has already sold all his portfolio. The problem is symmetric if the signal is ‘buy’.

We arbitrarily decided to allow a fixed percentage of adjustment (50% in the empirical test); that is to say that the expert can only buy an amount equal to w^k₀=2; rounded to obtain an entire number of stocks. In the same manner, only half of the portfolio of stocks can be sold when a selling signal appears. Obviously, the parameter defining the adjustable part of the portfolio can be fixed by the user.

Consider, for example, a buying order at the end of date 0. The trade will be executed at the opening price of the next day, that is p₁^k: If we denote by c the rate of transaction costs, then the number of stocks to be bought is equal to

                                                                                  wk         

                                                Dnk₁ ¼ Ent Pkð1 0þ cÞ ¼ nk1  nk0

0

The evolution of wealth is then

wk1 ¼ wk0  nk1  nk0p1kð1 þ cÞ

nk₁ ¼ nk₀ þ Dnk₁

It is worth noticing that we take into account the real market conditions in defining the number of stocks to be bought by considering P₀^k; which is observed at the end of the first day, while the variation of wealth is calculated with respect to the actual transaction price p₁^k; that is the next opening price:

When posting an order at the end of a period t; one cannot know the future wealth W_tþ^k ₁; because the transaction price is not known with certainty. It means that our strategy, allowing only a part of the portfolio to be traded, is prudent and ensures that the expert will possess enough cash to trade.

Measurement of performance. The learning period extends from t_l to t_l þ d_l; the benchmark we considered is an optimal expert, in this time frame, who is assumed to know the entire sequence of prices. He then always takes the right decision and obtains an optimal wealth at the end of the period. Obviously, the experts cannot attain such a performance but can be ranked according to the difference of final wealths. This benchmark is also used to fix the stopping criteria of the genetic algorithm. There are two ways to stop it;

Table I. Dates of the experiments.