УДК 519.246

В работе рассматривается проблема оптимального размещения ресурсов в вероятностных средах с некоторым ростом. В такой среде задачей трейдера является получение максимального прироста капитала в результате реинвестиций. Управление в такой среде можно сравнить с игрой, в которой трейдер располагает априорной информацией о множестве возможных в будущем исходов в совокупности с их достоверностями и полезностями, и на основе этой информации он делает ставки на выбранное подмножество благоприятных для него исходов. Основной проблемой в такой игре является поиск алгоритма ставок, обеспечивающего асимметрию шансов трейдера, т.е., положительное математическое ожидание полезности ставки. Второй проблемой является определение количества ресурсов, выделенных трейдеру на каждую ставку.

Бұл жұмыста ықтималдылық ортада қорды біршама өсетіндей етіп оптималды орналастыру мәселесі қарастырылады. Осындай ортадағы трейдердің мақсаты рет қайта қаржыланғанда капиталдың максималды өсуін алу. Осындай ортадағы басқаруды, трейдер келешекте мүмкін болатын нәтижелер жиынының нақтылығы мен тиімділігі туралы априорлы мәліметті білетін ойынға теңеуге болады. Және осы мәліметке сүйене отырып, ол өзіне тиімді болатын жиынға қояды. Бұл ойындағы негізгі мәселе, трейдердің мүмкіншіліктерін асимметриялығын қамтамасыз ететін, яғни қойылу пайдалылығы математикалық күтімінің оң болатындығын қамтамасыз ететін, қойылу алгоритмін іздеу. Екінші мәселе – трейдердің әрбір қойылуға бөлінетін қордың мөлшерін анықтау.

ABSTRACT

The problem of the optimal placing of resources in the likelihood environment is considered in that work. In that environment the issue of the trader is to receive the maximum gain to the capital as a result of N reinvestments. The management in that environment could be compared with the game, where the trader owns aprioristic information about the set of possible future outcomes in aggregate with its reliability and utility. Based on that information he stakes on the chosen subset of favorable for him outcomes. The major problem of that game is a search of an algorithm of the stakes which provides an asymmetry of the trader chances, i.e. positive mathematical expectations of the stake utility. The second problem is a determination of the quantity of resources provided to the trader for each stake.

Keywords: optimal placing, trader, management, capital, algorithm of the stakes, mathematical expectations, maximum, risk, models

PROBLEM STATEMENT

Let’s consider a problem of optimum placing of resources in likelihood environments with some growth. An example of such environment is investment activity on a securities market in a case, when the saved up incomes together with a fixed capital of the trader (the economic subject actively working on a securities market) repeatedly reinvested, showing growth or reduction (for example, «effect of compound percent», «percent for percent»). A problem of the trader in such environment is reception of the maximum gain of the capital as a result of capital reinvestment under conditions: 1) the probability of ruin asymptoticly is close to 0, 2) the size and probability maximum capital “ subsidence” on “way” from 1st to N th transaction are limited, 3) the capital of the trader reaches sizes, not smaller set, for the fixed or smaller number of transactions.

The majority of social and economic environments are characterised by uncertainty. Uncertainty is expressed that the result of the applied management to the person exercising this administration (trader), at the moment of decision-making is known only with some reliability. Management in such environment can be compared to game in which the trader has the aprioristic information on set of possible outcomes in the future in aggregate of their credibility and utility, and on the basis of this information he stakes on the chosen subset of outcomes favorable for it. Prizes increase utility of the trader, and losses – reduce it.

The basic problem in such game is search of algorithm of the rates providing asymmetry of chances of the trader, i.e., a positive population mean of utility of the rate. In investment business such algorithm is called as mechanical trading system (MTS). MTS provides recognition potentially advantageous positions, statistically confirming asymmetry of chances. But we assume that MTS with a positive population mean already exists.

The second problem is definition of quantity of the resources, allocated to the trader on each rate. Let’s consider a case with two possible outcomes of game. Let on the winning rate the trader receives yield y, and on losing rate loss- I from the size of the rate. Then counting on unit of the enclosed capital as an outcome of one rate the trader can receive the final capital W_{T} (Terminal wealth)

Provided that the trader reinvesting on each k th rate a constant share (fixed fraction) f from the capital which has been saved up by it after k - 1previous rates, we will have outcomes of 1st, 2nd, …, N th rates

, … ,

(2)

Where on “way” from N rates k of attempts have appeared winning and N-k – losing. The capital share (1-f) remains in the form of liquidity or is put in risk free actives, providing reserves under the future losses. W_{T} (N, k)– branching process – «a binary tree» (Figure 1).

Figure 1. Binary tree

In this case the formula (2) shows that the capital of the trader changes not linearly, and exponential, namely has geometrical growth (a geometrical progression). Setting function of utility of the trader as the size or as rate of increase of the capital on “way” from N transactions (“way” – the scenario), we will note two circumstances:

1. The chosen criteria of utility are dynamic (time moment here – events of transactions), a decision-making problem – multiperiod, it means search of multistage strategy of placing of the capital. It distinguishes a problem of the trader from a problem of management of a portfolio which theory has been developed by Markowitz [1]. The approach of Markowitz – one-periodical.

2. The portfolio theory of Markowitz uses E - V criteria (Mean – Value – Covariance of Return) with arithmetic-mean growth and a risk measure as dispersion-covariance (volatility) growth process. However, volatility rather mediately expresses risk of the trader as which experts usually understand probable losses of the trading capital. And the problem of the trader assumes compound criterion of growth and a risk measure in the obvious form – as probabilities of the maximum losses of the capital for fixed time.

Let’s consider possible criteria of optimality in a problem of the trader. The most obvious criterion – maximization of the expected size of the final capital – result of N transactions (on all scenarios of length N):

(3)

In case of two possible financial results of each rate it is had binomial scenarios – chains of length N with k wins, and each rate can be considered within the limits of the scheme of Bernoulli with two outcomes y – with probability p and l with probability q=1-p. The binomial model of Coke– Ross – Rubinshtejn can serve in such statement by model of a stream of financial results of investment activity of the trader, [2, 3, 4] (CRR-model of scenarios of movement of the prices of a base active for an estimation of fair cost of an option for this active). Now we will generalise on a case of continuous time. For this purpose we use results for Brownian motion process. Within the scope of CRR-model we have:

(4)

From here for criterion function we will receive

.(5)

From criterion

.(6)

Receive extremum

.

As we consider MTS with asymmetric chances, then py+ql >0 and, as a rule, in investment practice IyI<1 and IlI<1 we will have f*< 0 and If*I>1. Such decision has no “investment” sense if not to consider possibility of loan of the capital and «short sales» (short selling). Within the limits of these restrictions f* [0.1].

Besides and therefore

And its derivative strictly increase on [0,1]

Therefore we receive an optimum on piece border f*=1 – in case of game with positive asymmetry of chances and f*=0 – in case of game with negative chances of a prize (i.e. optimum strategy is each time to put «all or anything»).

The second possible criterion in a problem of the trader has offered Feller [5]. It has considered criterion of minimisation of probability of casual full loss by the trader of the capital – «probabilities of crash» (“crash” occurs, if as a result of k th rate W_{T}(N,k)=0). Under the of Feller’s formula, the trader should minimise the size of each rate that inevitably attracts profit short-reception. Thereby and this criterion is unacceptable from the practical point of view. Besides, Feller has shown that if to stake each time the available capital “crash” is inevitable: P_{kpax}=1-p^{N} → 1 at the big scenarios.

There is a requirement for criterion which would do something an average in any sense between maximisation of growth of the final capital (guaranteeing thereby “crash” on Feller) and minimisation of probability of “crash” (minimising thereby expected growth). Such asymptotically optimum strategy has been offered by Kelly [6] in the work, executed by it in interests of corporation Bell Laboratories and the devoted problem of a signal transmission in noisy lines.

Let’s consider factor of a geometrical gain of the capital in the scenario of length N. As, W_{T}=W_{T}(N,0) then

. (7)

Then for multiplier a geometrical progression

(8)

And using convergence of frequency to probability in the scheme of Bernoulli [7], at the large N we will receive

. (9)

Kelly has suggested to choose optimum strategy of placing of a resource such that it maximises average expected geometrical growth h(f).

Conditions of an optimality of 1st order give

(10)

(11)

The formula (11) also gives the optimum size on Kelly reinvested capital shares. It is possible to notice that

, (12)

I.e. h(f) it is continuous and strictly monotonously decreases on (0,1). Besides, h(0)=py+ql>0 that according to a condition of asymmetry of chances of MTS. Therefore h(f) has a unique maximum on [0,1].

Search optimum on Kelly of strategy of management by the capital for mechanical trading system. Let the statistics of financial results (P&L) MTS of the trader (for transaction of MTS everyone oh profitablenesses are known, there is a sample of transactions) is known. Not always distribution P&L is distribution of Bernoulli. Meanwhile, statement of a problem of the trader demands binomial scenarios of trade.

Nevertheless, in trading practice there is one important case when distribution P&L can be close to distribution Bernoulli. It is a case of MTS with the fixed conditions of an exit from a position – so-called «profit protection» (Take Profit) and «loss fixing» (Stop Loss). In MTS practice usually sells positions when the profit or the loss have reached certain size (as a rule, this size is function of the size of an open position in case of Stop Loss and volatility function of market price in case of Take Profit). The sequence of results of rates of such MTS can be modeled by the scheme of Bernoulli approximately.

The deviation from the scheme of Bernoulli arises in that case when on MTS additional conditions are imposed, for example, compulsion of closing of all open positions in the end of trading session As both Take Profit, and Stop Loss usually get out within 80 – 90 percentage confidential intervals of fluctuations of market prices ,events End of Day Close («sale of a position at the price of last transaction») arise seldom, but frequently corresponding to them P&L are located far from (two) atoms Bernoulli MTS distributions. These events can be considered as extreme (extreme values) with values P&L lying in “tails” of distribution. It is possible to show that in many cases these “tails” are not “heavy”, i.e. frequencies corresponding to them are scornfully small in comparison with Bernoulli, and as a first approximation it is possible to use binomial model. For MTS stress-testing it is possible to approximate “tails” sedate functions or distributions of type of Pareto [7, 8] as it is accepted in the theory of extreme values. Then distribution P&L is represented in the form of a mix Bernoulli atoms and sedate “tails”.

REFERENCES

1. Markowitz H. Portfolio Selection // Journal of Finance. 1952. 7. No. 1. P. 152–164.

2. Cox J.C., Ross R.A., Rubinstein M. Option pricing: a simplified approach. // Journal of Financial Economics. 1979. V. 7, No 3.

3. Shiryaev A.N. Base of the stochastic financial mathematics. The facts. Models. Moscow. 1998. V. 1. 512 p.

4. Shiryaev A.N. Base of the stochastic financial mathematics. Theory. Moscow. 1998. V. 2. 544 p.

5. Feller W. An Introduction to Probability Theory and Its Applications. John Wiley, New York. 1966. V.1. 659 p.

6. Kelly J.L. A new interpretation of information rate // Bell System Technical Journal. 1956. V. 35. P. 89–95.

7. Orynbay M.S., Shakenov R.K., Shakenov K.K About the numerical decision of some problems of risk with distribution of type of Pareto. The international 11th interuniversity conference on mathematics and the mechanics, devoted to the 10 anniversary of the Euroasian National University of L.N. Gumilev. 25 – on May, 26th, Astana. Theses of reports. Astana, 2006. P. 152.

8. Dzhekel P. Application of method Monte Carlo in finance. Moscow, “Internet trading”. 2004. 156 p.